Table of Contents
Cover
Table of Contents
Series Page
Title Page
Copyright Page
List of Figures
List of Tables
Preface
Acknowledgments
Notations
1 Introduction
1.1 An Overview of the Development of Structural Reliability
Theory
1.2 Basic Concepts
1.3 Contents of this Book
References
2 Method of Uncertainty Analysis
2.1 Classification of Uncertainty
2.2 Probability Analysis Methods
2.3 Fuzzy Mathematical Analysis Method
2.4 Gray Theory Analysis Method
2.5 Relative Information Entropy Analysis Method
2.6 Artificial Intelligence Analysis Method
2.7 Example: Risk Evaluation of Construction with Temporary
Structure Formwork Support
References
3 Reliability Analysis Method
3.1 First-Order Second-Moment Method
3.2 Second-Order Second-Moment Method
3.3 Reliability Analysis of Random Variables Disobeying
Normal Distribution
3.4 Responding Surface Method
References
4 Numerical Simulation for Reliability
4.1 Monte-Carlo Method
4.2 Variance Reduction Techniques
4.3 Composite Important Sampling Method
4.4 Importance Sampling Method in V Space
4.5 SVM Importance Sampling Method
References
5 Reliability of Structural Systems
5.1 Failure Mode of Structural System
5.2 Calculation Methods for System Reliability
5.3 Example: Reliability of Offshore Fixed Platforms
5.4 Analysis on the Reliability of a Semi-Submersible Platform
System
References
6 Time-Dependent Structural Reliability
6.1 Time Integral Method
6.2 Discrete Method
6.3 Calculation of Time-Dependent Reliability
6.4 Structural Dynamic Analysis
6.5 Fatigue Analysis
References
7 Load Combination on Reliability Theory
7.1 Load Combination
7.2 Load Combination Factor
7.3 Calculation of Partial Coefficient of Structural Design
7.4 Determination of Load Combination Coefficient and
Design Expression
7.5 Example: Path Probability Model for the Durability of a
Concrete Structure
References
8 Application of Reliability Theory in Specifications
8.1 Requirements of Structural Design Codes
8.2 Expression of Structural Reliability in Design
Specifications
8.3 Example: Target Reliability and Calibration of Bridges
8.4 Reliability Analysis of Human Influence
References
Index
Also of Interest
End User License Agreement
List of Tables
Chapter 2
Table 2.1 Representation of membership function.
Table 2.2 Digital representation of gray scale.
Table 2.3 Indices and weights of a risk evaluation system for
fastener-type st...
Table 2.4 Rating scale table.
Table 2.5 Weighting of individual differences.
Table 2.6 Expert ratings.
Table 2.7 Correlation matrix for each index in the D-layer;
relative weight an...
Table 2.8 Relative weighting and correlation coefficient matrix
of materials a...
Table 2.9 Relative weight and correlation matrix of index.
Chapter 3
Table 3.1 Relationship between reliability index and failure
probability Pf.
Table 3.2 Comparison of results.
Table 3.3 Probabilistic characteristics of the random variables
in Example 3.
Table 3.4 LS-SVM learning results.
Table 3.5 Probabilistic characteristics of random variables in
Example 4.
Table 3.6 Effect of sample numbers on calculated results.
Chapter 4
Table 4.1 Simulation results of σ1 versus σ when G(X)=3.0 - x.
Table 4.2 Results of different sampling simulation methods.
Table 4.3 Relationship between area ratio of sampling ellipse
δ0 and k0.
Chapter 5
Table 5.1 Soil parameters.
Table 5.2 Uncertainty of soil parameters.
Table 5.3 Understanding the soil calculation model.
Table 5.4 Bearing capacity of a single pile.
Table 5.5 Bearing capacity under different supporting
boundary conditions.
Table 5.6 Statistical results for shear capacity and simulation
of the structu...
Table 5.7 Failure probability obtained by different reliability
calculation me...
Table 5.8 Wave parameters for a 100-year-return period.
Table 5.9 Data on sectional force and bending moment of each
working condition...
Table 5.10 Data for limit state parameters in each working
condition.
Table 5.11 Calculated variable distribution types.
Table 5.12 Stochastic models of calculated variables.
Table 5.13 Reliability index and failure probability of a semisubmersible pla...
Table 5.14 Calculated values for sectional force of semisubmersible platform ...
Table 5.15 Resistance parameters for semi-submersible
structural joints.
Table 5.16 Reliability data for local nodes.
Table 5.17 Overall reliability of target platform.
Chapter 6
Table 6.1 Statistical standard deviation of high frequency
mooring force range...
Table 6.2 Statistical standard deviation of low frequency
mooring force range.
Table 6.3 Mooring force discovery series under different
effective wave height...
Table 6.4 Wave ocean state distribution.
Table 6.5 Low-frequency moving force cycles.
Table 6.6 False damage and false life of the main pipe joints.
Table 6.7 Design parameters of a submarine pipeline.
Table 6.8 Calculated case.
Table 6.9 Structural modal analysis results.
Table 6.10 Fatigue life and failure probability of a suspended
pipeline in dif...
Table 6.11 Fatigue life and failure probability of a pipeline
under different ...
Table 6.12 Fatigue life and failure probability of a pipeline at
different wav...
Table 6.13 Fatigue life and failure probability of pipeline at
different water...
Table 6.14 Fatigue life and failure probability of pipelines with
different di...
Table 6.15 Fatigue life and failure probability of pipeline at
different resid...
Table 6.16 Wave dispersion map of the South China Sea
(ΣP=100).
Table 6.17 Fatigue reliability analysis parameters in the S-N
curve method.
Table 6.18 Fatigue reliability analysis parameters for fracture
mechanics.
Table 6.19 Fatigue reliability index and failure probability of
key nodes base...
Table 6.20 Fatigue reliability index and failure probability of
key nodes base...
Table 6.21 S-N curves in different environments.
Table 6.22 Fatigue reliability index of key nodes at the No. 1
connection for ...
Chapter 7
Table 7.1 Different state combinations that cause crossing.
Table 7.2 Parameters of various random loads.
Table 7.3 Concentration of chloride ion on a concrete surface.
Table 7.4 Distribution of the standard value of compressive
strength of a conc...
Table 7.5 Calculation parameters and distribution types.
Table 7.6 Calculation parameters and distribution types.
Chapter 8
Table 8.1 Target reliability index of current building
structures in China.
Table 8.2 Factor for importance of structure γ0.
Table 8.3 Load adjustment coefficient of service life for
structural design γL...
Table 8.4 Signs of durability limit state of various structures [816]
.
Table 8.5 Annual target reliability and failure probability of
bearing capacit...
Table 8.6 Annual target reliability and failure probability of
the serviceabil...
Table 8.7 Calibration operating condition.
Table 8.8 Function distribution and parameters *.
Table 8.9 Load ratio.
Table 8.10 Recommended cost for reliability calibration.
Table 8.11 Statistical data for geometric parameter
uncertainty KA.
Table 8.12 Geometric size distribution of components without
the influence of ...
Table 8.13 Standard deviation of concrete strength.
Table 8.14 Estimation criterion for error coefficient EF.
Table 8.15 Human error rate and distribution parameters for
degree of influenc...
Table 8.16 Influence of different human errors on the buckling
strength of for...
Table 8.17 Occurrence of human error.
Table 8.18 Distribution of tightening torque on bolts in
different parts.
Table 8.19 Average value of skid resistance for fasteners
under different bolt...
Table 8.20 Comparison of failure probability.
List of Illustrations
Chapter 2
Figure 2.1 Three types of transfer function.
Figure 2.2 Diagram of two-layer BP neural netbook structures.
Figure 2.3 Diagram of support vectors.
Figure 2.4 Diagram of regression support vector machine.
Figure 2.5 Fuzzineation of score values.
Figure 2.6 Membership function of the evaluation grade.
Chapter 3
Figure 3.1 Diagram of structure failure probability.
Figure 3.2 Responding surface function.
Figure 3.3 Response surface method based on LS-SVM.
Figure 3.4 Number of FEM calculations.
Figure 3.5 Portal frame calculation diagram.
Figure 3.6 Calculation diagram for Example 4.
Chapter 4
Figure 4.1 Probability density function with truncated
distribution.
Figure 4.2 Approximate parabolic surface of V space.
Figure 4.3 Important sampling area of V space.
Figure 4.4 Relationship between principal curvature k and
sampling elliptic pa...
Figure 4.5 Influence of different confidence a on simulation
results.
Chapter 5
Figure 5.1 Load-path relationship.
Figure 5.2 Different strength-deformation (R-A) relations.
Figure 5.3 Fault tree.
Figure 5.4 Event tree of structure.
Figure 5.5 Failure diagram of structure.
Figure 5.6 Series system.
Figure 5.7 Two-dimensional failure region for reliability
problem of structura...
Figure 5.8 Two simple parallel systems.
Figure 5.9 Condition system.
Figure 5.10 The impact of correlation on system security
indications.
Figure 5.11 Simple experimental design of two variables.
Figure 5.12 Systematic enumeration process.
Figure 5.13 t ∼ z curve.
Figure 5.14 Q ∼ Z curve.
Figure 5.15 P-y curve of soil.
Figure 5.16 p-y curve of sandy soil.
Figure 5.17 Calculation model of pile.
Figure 5.18 Load-bearing capacity under axial compression.
Figure 5.19 Load-bearing capacity under axial tension.
Figure 5.20 Lateral bearing capacity with different pile top
constraints.
Figure 5.21 Deterministic analysis of computational structure
model.
Figure 5.22 Shear and bending bearing capacity and structural
placement diagra...
Figure 5.23 Statistical results and probability analysis of
bearing capacity.
Figure 5.24 3D FEM model of a semi-submersible platform.
Figure 5.25 Analysis of structural reliability of a semisubmersible platform.
Figure 5.26 Reliability evaluation procedure for a semisubmersible platform.
Chapter 6
Figure 6.1 Sample function of random process of load effect.
Figure 6.2 Sample function and failure time of safety limit
state process Z(t)...
Figure 6.3 Transcendence of random process vector X(t).
Figure 6.4 Sample function of nonstationary load effect and
resistance.
Figure 6.5 Sample function of load effect and resistance (when
resistance is c...
Figure 6.6 Typical risk function.
Figure 6.7 Variation trend of risk function in different
structural stage.
Figure 6.8 Sample functions of vector stochastic processes.
Figure 6.9 Sample function and spectral density of random
process.
Figure 6.10 Probability density function of Rayleigh
distribution.
Figure 6.11 Analysis on the relationship between input and
output spectral den...
Figure 6.12 Coordinate system of a single point mooring
offshore jacket platfo...
Figure 6.13 Structural model of a BZ28-1 SPM platform.
Figure 6.14 Prototype cross section of an oil pipeline.
Figure 6.15 Foree spectrum of pipeline nodes.
Figure 6.16 Power spectrum of pipeline midspan
displacement response.
Figure 6.17 Linear and nonlinear calculation of maximum
stress spectrum of mid...
Figure 6.18 Vibration displacement response spectrum of
pipeline at different ...
Figure 6.19 Vibration stress response spectrum of pipeline at
different water ...
Figure 6.20 Pipeline reliability index and peak stress
spectrum at different w...
Figure 6.21 Vibration displacement response spectra of
pipelines with differen...
Figure 6.22 Vibration stress response spectrum of pipelines
with different dia...
Figure 6.23 Reliability index and peak stress spectrum of
pipelines with diffe...
Figure 6.24 Vibration displacement response spectrum of
pipeline at different ...
Figure 6.25 Vibration stress response spectrum of pipeline at
different residu...
Figure 6.26 Pipeline reliability index and peak stress
spectrum at different r...
Figure 6.27 Fatigue reliability analysis process for deep-water
semi-submersib...
Figure 6.28 Fatigue reliability analysis for a deep-water semisubmersible pla...
Figure 6.29 Schematic diagram of the connection between the
platform column an...
Figure 6.30 Comparison of calculated results for fatigue
reliability index.
Chapter 7
Figure 7.1 The combination of random process.
Figure 7.2 Typical sample function of mixed rectangular
update stochastic proc...
Figure 7.3 Borges process combination.
Figure 7.4 TR combination diagram of three load
combinations.
Figure 7.5 Process combination of three rectangular wave.
Figure 7.6 Comparison of several combination rules.
Figure 7.7 Flow chart of specification method 1.
Figure 7.8 Flow chart of specification method 2.
Figure 7.9 Corrosion path model.
Figure 7.10 Corrosion multi-path model.
Figure 7.11 Carbonation diagram.
Figure 7.12 Curve of pH value and critical chloride
concentration.
Figure 7.13 Simulated flow diagram.
Figure 7.14 Bridge structural status.
Figure 7.15 Number of cracks in piers.
Figure 7.16 PDF of main rebars.
Figure 7.17 CPDF of main rebars.
Figure 7.18 PDF of corrosion-induced crack width.
Figure 7.19 CPDF of corrosion-induced crack width.
Figure 7.20 Time-dependent CPDF of main rebar.
Figure 7.21 Time-dependent CPDF of corrosion-induced crack
width.
Figure 7.22 Structural damage to the bridge.
Figure 7.23 PDF of chloride threshold value.
Figure 7.24 PDF of time to corrosion initiation of main rebars.
Figure 7.25 PDF of time to crack initiation of concrete.
Figure 7.26 PDF of corrosion ratio of main rebars.
Figure 7.27 CPDF of corrosion ratio of main rebars.
Figure 7.28 PDF of corrosion-induced crack width.
Figure 7.29 CPDF of corrosion-induced crack width.
Chapter 8
Figure 8.1 Limit state of structural design.
Figure 8.2 Wind speed span-time rate distribution curve of a
bridge.
Figure 8.3 Calibration process.
Figure 8.4 Human error event tree.
Figure 8.5 Block diagram of human error simulation program
for E3 and E7.
Figure 8.6 Block diagram of human error simulation programs
for E1(a), E1(b) a...
Figure 8.7 Human error simulation program block diagram for
E8 and E9.
Figure 8.8 Flow chart for structural system reliability
calculation in constru...
Figure 8.9 Influence of human error.
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Structural Reliability in Civil
Engineering
Wei-Liang Jin
Qian Ye
and
Yong Bai
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List of Figures
Figure 2.1 Three types of transfer function.
Figure 2.2 Diagram of two-layer BP neural netbook structures.
Figure 2.3 Diagram of support vectors.
Figure 2.4 Diagram of regression support vector machine.
Figure 2.5 Fuzzineation of score values.
Figure 2.6 Membership function of the evaluation grade.
Figure 3.1 Diagram of structure failure probability.
Figure 3.2 Responding surface function.
Figure 3.3 Response surface method based on LS-SVM.
Figure 3.4 Number of FEM calculations.
Figure 3.5 Portal frame calculation diagram.
Figure 3.6 Calculation diagram for Example 4.
Figure 4.1 Probability density function with truncated distribution.
Figure 4.2 Approximate parabolic surface of V space.
Figure 4.3 Important sampling area of V space.
Figure 4.4 Relationship between principal curvature k and
sampling elliptic parameters a, b and k (β=3, Δβ=1.0, δ0=0.8).
Figure 4.5 Influence of different confidence a on simulation results.
Figure 5.1 Load-path relationship.
Figure 5.2 Different strength-deformation (R-A) relations.
Figure 5.3 Fault tree.
Figure 5.4 Event tree of structure.
Figure 5.5 Failure diagram of structure.
Figure 5.6 Series system.
Figure 5.7 Two-dimensional failure region for reliability problem
of structural system.
Figure 5.8 Two simple parallel systems.
Figure 5.9 Condition system.
Figure 5.10 The impact of correlation on system security
indications.
Figure 5.11 Simple experimental design of two variables.
Figure 5.12 Systematic enumeration process.
Figure 5.13 t ∼ z curve.
Figure 5.14 Q ∼ Z curve.
Figure 5.15 P-y curve of soil.
Figure 5.16 p-y curve of sandy soil.
Figure 5.17 Calculation model of pile.
Figure 5.18 Load-bearing capacity under axial compression.
Figure 5.19 Load-bearing capacity under axial tension.
Figure 5.20 Lateral bearing capacity with different pile top
constraints.
Figure 5.21 Deterministic analysis of computational structure
model.
Figure 5.22 Shear and bending bearing capacity and structural
placement diagram.
Figure 5.23 Statistical results and probability analysis of bearing
capacity.
Figure 5.24 3D FEM model of a semi-submersible platform.
Figure 5.25 Analysis of structural reliability of a semisubmersible
platform.
Figure 5.26 Reliability evaluation procedure for a semisubmersible
platform.
Figure 6.1 Sample function of random process of load effect.
Figure 6.2 Sample function and failure time of safety limit state
process Z(t).
Figure 6.3 Transcendence of random process vector X(t).
Figure 6.4 Sample function of nonstationary load effect and
resistance.
Figure 6.5 Sample function of load effect and resistance (when
resistance is constant).
Figure 6.6 Typical risk function.
Figure 6.7 Variation trend of risk function in different structural
stage.
Figure 6.8 Sample functions of vector stochastic processes.
Figure 6.9 Sample function and spectral density of random
process.
Figure 6.10 Probability density function of Rayleigh distribution.
Figure 6.11 Analysis on the relationship between input and output
spectral density function of offshore platform structure.
Figure 6.12 Coordinate system of a single point mooring offshore
jacket platform.
Figure 6.13 Structural model of a BZ28-1 SPM platform.
Figure 6.14 Prototype cross section of an oil pipeline.
Figure 6.15 Foree spectrum of pipeline nodes.
Figure 6.16 Power spectrum of pipeline midspan displacement
response.
Figure 6.17 Linear and nonlinear calculation of maximum stress
spectrum of midspan section of suspended pipeline for various
cases.
Figure 6.18 Vibration displacement response spectrum of pipeline
at different water depths.
Figure 6.19 Vibration stress response spectrum of
Figure 6.20 Pipeline reliability index and peak stress spectrum at
different water depths.
Figure 6.21 Vibration displacement response spectra of pipelines
with different diameters.
Figure 6.22 Vibration stress response spectrum of pipelines with
different diameters.
Figure 6.23 Reliability index and peak stress spectrum of pipelines
with different outer diameters.
Figure 6.24 Vibration displacement response spectrum of pipeline
at different residual stresses.
Figure 6.25 Vibration stress response spectrum of pipeline at
different residual stresses.
Figure 6.26 Pipeline reliability index and peak stress spectrum at
different residual stresses.
Figure 6.27 Fatigue reliability analysis process for deep-water
semi-submersible platform structure.
Figure 6.28 Fatigue reliability analysis for a deep-water
semisubmersible platform.
Figure 6.29 Schematic diagram of the connection between the
platform column and the transverse brace.
Figure 6.30 Comparison of calculated results for fatigue reliability
index.
Figure 7.1 The combination of random process.
Figure 7.2 Typical sample function of mixed rectangular update
stochastic process with given probability density function.
Figure 7.3 Borges process combination.
Figure 7.4 TR combination diagram of three load combinations.
Figure 7.5 Process combination of three rectangular wave.
Figure 7.6 Comparison of several combination rules.
Figure 7.7 Flow chart of specification method 1.
Figure 7.8 Flow chart of specification method 2.
Figure 7.9 Corrosion path model.
Figure 7.10 Corrosion multi-path model.
Figure 7.11 Carbonation diagram.
Figure 7.12 Curve of pH value and critical chloride concentration.
Figure 7.13 Simulated flow diagram.
Figure 7.14 Bridge structural status.
Figure 7.15 Number of cracks in piers.
Figure 7.16 PDF of main rebars.
Figure 7.17 CPDF of main rebars.
Figure 7.18 PDF of corrosion-induced crack width.
Figure 7.19 CPDF of corrosion-induced crack width.
Figure 7.20 Time-dependent CPDF of main rebar.
Figure 7.21 Time-dependent CPDF of corrosion-induced crack
width.
Figure 7.22 Structural damage to the bridge.
Figure 7.23 PDF of chloride threshold value.
Figure 7.24 PDF of time to corrosion initiation of main rebars.
Figure 7.25 PDF of time to crack initiation of concrete.
Figure 7.26 PDF of corrosion ratio of main rebars.
Figure 7.27 CPDF of corrosion ratio of main rebars.
Figure 7.28 PDF of corrosion-induced crack width.
Figure 7.29 CPDF of corrosion-induced crack width.
Figure 8.1 Limit state of structural design.
Figure 8.2 Wind speed span-time rate distribution curve of a
bridge.
Figure 8.3 Calibration process.
Figure 8.4 Human error event tree.
Figure 8.5 Block diagram of human error simulation program for
E3 and E7.
Figure 8.6 Block diagram of human error simulation programs for
E1(a), E1(b) and E2.
Figure 8.7 Human error simulation program block diagram for E8
and E9.
Figure 8.8 Flow chart for structural system reliability calculation in
construction period under the influence of human errors.
Figure 8.9 Influence of human error.
List of Tables
Table 2.1 Representation of membership function.
Table 2.2 Digital representation of gray scale.
Table 2.3 Indices and weights of a risk evaluation system for
fastener-type steel pipe formwork support construction.
Table 2.3 Indices and weights of a risk evaluation system for
fastener-type steel pipe formwork support construction.
Table 2.4 Rating scale table.
Table 2.5 Weighting of individual differences.
Table 2.6 Expert ratings.
Table 2.7 Correlation matrix for each index in the D-layer; relative
weight and correlation coefficient matrix of fastener and pole
index.
Table 2.8 Relative weighting and correlation coefficient matrix of
materials and erection indices.
Table 2.9 Relative weight and correlation matrix of index.
Table 3.1 Relationship between reliability index and failure
probability Pf.
Table 3.2 Comparison of results.
Table 3.3 Probabilistic characteristics of the random variables in
Example 3.
Table 3.4 LS-SVM learning results.
Table 3.5 Probabilistic characteristics of random variables in
Example 4.
Table 3.6 Effect of sample numbers on calculated results.
Table 4.1 Simulation results of σ1 versus σ when G(X)=3.0 - x.
Table 4.2 Results of different sampling simulation methods.
Table 4.3 Relationship between area ratio of sampling ellipse δ0
and k0.
Table 5.1 Soil parameters.
Table 5.2 Uncertainty of soil parameters.
Table 5.3 Understanding the soil calculation model.
Table 5.4 Bearing capacity of a single pile.
Table 5.5 Bearing capacity under different supporting boundary
conditions.
Table 5.6 Statistical results for shear capacity and simulation of the
structure.
Table 5.7 Failure probability obtained by different reliability
calculation methods.
Table 5.8 Wave parameters for a 100-year-return period.
Table 5.9 Data on sectional force and bending moment of each
working condition.
Table 5.10 Data for limit state parameters in each working
condition.
Table 5.11 Calculated variable distribution types.
Table 5.12 Stochastic models of calculated variables.
Table 5.13 Reliability index and failure probability of a
semisubmersible platform.
Table 5.14 Calculated values for sectional force of semisubmersible
platform node.
Table 5.15 Resistance parameters for semi-submersible structural
joints.
Table 5.16 Reliability data for local nodes.
Table 5.17 Overall reliability of target platform.
Table 6.1 Statistical standard deviation of high frequency mooring
force range.
Table 6.2 Statistical standard deviation of low frequency mooring
force range.
Table 6.3 Mooring force discovery series under different effective
wave heights.
Table 6.3 Mooring force discovery series under different effective
wave heights.
Table 6.4 Wave ocean state distribution.
Table 6.5 Low-frequency moving force cycles.
Table 6.6 False damage and false life of the main pipe joints.
Table 6.7 Design parameters of a submarine pipeline.
Table 6.8 Calculated case.
Table 6.9 Structural modal analysis results.
Table 6.10 Fatigue life and failure probability of a suspended
pipeline in different cases.
Table 6.11 Fatigue life and failure probability of a pipeline under
different span lengths.
Table 6.12 Fatigue life and failure probability of a pipeline at
different wave heights.
Table 6.13 Fatigue life and failure probability of pipeline at
different water depths.
Table 6.14 Fatigue life and failure probability of pipelines with
different diameters.
Table 6.15 Fatigue life and failure probability of pipeline at
different residual stresses.
Table 6.16 Wave dispersion map of the South China Sea (ΣP=100).
Table 6.17 Fatigue reliability analysis parameters in the S-N curve
method.
Table 6.18 Fatigue reliability analysis parameters for fracture
mechanics.
Table 6.19 Fatigue reliability index and failure probability of key
nodes based on the S-N curve method.
Table 6.20 Fatigue reliability index and failure probability of key
nodes based on fracture mechanics.
Table 6.21 S-N curves in different environments.
Table 6.22 Fatigue reliability index of key nodes at the No. 1
connection for different S-N curves.
Table 7.1 Different state combinations that cause crossing.
Table 7.2 Parameters of various random loads.
Table 7.3 Concentration of chloride ion on a concrete surface.
Table 7.4 Distribution of the standard value of compressive
strength of a concrete cube.
Table 7.5 Calculation parameters and distribution types.
Table 7.6 Calculation parameters and distribution types.
Table 8.1 Target reliability index of current building structures in
China.
Table 8.2 Factor for importance of structure γ0.
Table 8.3 Load adjustment coefficient of service life for structural
design γL.
Table 8.4 Signs of durability limit state of various structures.
Table 8.5 Annual target reliability and failure probability of
bearing capacity limit state.
Table 8.6 Annual target reliability and failure probability of the
serviceability limit state.
Table 8.7 Calibration operating condition.
Table 8.8 Function distribution and parameters *.
Table 8.9 Load ratio.
Table 8.10 Recommended cost for reliability calibration.
Table 8.11 Statistical data for geometric parameter uncertainty KA.
Table 8.12 Geometric size distribution of components without the
influence of human factors.
Table 8.13 Standard deviation of concrete strength.
Table 8.14 Estimation criterion for error coefficient EF.
Table 8.15 Human error rate and distribution parameters for
degree of influence.
Table 8.16 Influence of different human errors on the buckling
strength of formwork support systems.
Table 8.17 Occurrence of human error.
Table 8.18 Distribution of tightening torque on bolts in different
parts.
Table 8.19 Occurrence of human error.
Table 8.20 Distribution of tightening torque on bolts in different
parts.
Table 8.21 Average value of skid resistance for fasteners under
different bolt tightening torques.
Table 8.22 Comparison of failure probability.
Preface
Engineering structural reliability refers to the ability of a structure to
complete predetermined functions within a specified time and under
specified conditions, while the degree of structural reliability is a
mathematical measure of reliability. According to the definition, the
reliability of engineering structures should include three aspects: the
first is the part of the structure itself, including structural resistance,
structural type, and structural reuse; the second is the external effects
that the structure is subjected to, including direct, indirect, and
combined effects on the structure; the third involves the basic methods
of structural reliability, including the calculation method of reliability,
analysis of system reliability, and calculation of dynamic reliability.
Therefore, the reliability of engineering structures mainly involves the
basic methods of reliability, which is also the main content of this book.
The theoretical research on structural reliability flourished in the
1970s with the transition of structural design codes from the allowable
stress design method to the probability-based limit state design
method, while the domestic research work was relatively synchronized
with the foreign research. However, in terms of basic theoretical
research on structural reliability, there is a significant gap between the
domestic research and the foreign research, which is basically modified
according to the foreign regulatory systems, which means it is in a
“running” stage compared to similar international research. With the
continuous deepening of understanding and research on structural
reliability theory in the domestic academic and engineering
communities, especially the great discussion on structural reliability in
the 1990s, it is necessary to consider both the theoretical system of
structural specifications based on reliability and the practical
functional requirements of structures in the application of engineering
structures. This is mainly reflected in the formulation of unified
standards for structural design reliability in the early 20th century.
Changing “the structural reliability” to “the degree of structural
reliability” is the biggest highlight of the unified standard formulation,
which means it is in the “parallel” stage with similar international
research. With the continuous progress of research on structural
reliability theory by Chinese scientific and technological workers, and
the deepening understanding of engineering structural reliability
issues by engineering technicians, the establishment of China’s
regulatory system and the application of engineering structural
reliability will be more perfect, and it is fully possible to achieve a
“leading” stage compared to similar international research. This is also
the purpose of writing this book.
This book consists of eight chapters, mainly introducing the
development overview and basic concepts of the basic theory of
reliability, uncertainty analysis methods, reliability calculation
methods, simulation methods of reliability, system reliability analysis,
time-varying structural reliability, load and load combination methods,
the application of reliability in specifications, and the application of
reliability theory in practical engineering.
This book can be used as a textbook and teaching reference for
graduate and senior undergraduate students majoring in civil
engineering, water conservancy, highway, railway, port, ship and ocean
engineering in higher education institutions. It can also be a
professional reference book for engineering technicians and scholars
engaged in research and design in the fields of civil and industrial
architecture, municipal facilities, bridges, roads (highways and
railways), port and ocean engineering.
Acknowledgments
I would like to express my gratitude to Professor Guofan ZHAO of
Dalian University of Technology in China for introducing me to the
research field of structural reliability theory and application. In the
future, he will continue to provide strong support and assistance in
researching the reliability of marine structures, the durability of
concrete structures, and other engineering structures, which I will
never forget.
Thank you to Professor Eberhard LUZ from Stuttgart University in
Germany for providing me with a relaxed and enjoyable working
environment during my Humboldt research work from the autumn of
1991 to 1993, which enabled me to conduct research on uncertainty
and numerical simulation of reliability in structural reliability.
Thank you to Professor Torgeir MOAN from Norwegian University of
Science and Technology during my Norwegian Research Council’s
research work from 1994 to 1995. His extensive knowledge and
working environment in marine engineering structures have enabled
me to find new breakthroughs in the theory and application of
structural reliability.
Thank you to colleagues from China National Offshore Oil Corporation
(CNOOC) and the Engineering Reliability Committee of the Chinese Civil
Engineering Society (CCES) for achieving reasonable application of
structural reliability in structural design specifications, effectively
promoting the development of structural reliability theory and
application.
Since 1996, when I officially joined Zhejiang University, I have opened a
research direction in structural reliability, established a new course
called “Structural Reliability”, and trained many doctoral and master’s
students. They all play important roles in their respective positions.
This book also reflects their research achievements in the field of
structural reliability. Here, I would like to express my heartfelt gratitude
to them through this book.
I would like to express my special gratitude to Dr. Qian YE and
Professor Yong BAI for their joint efforts and writing, which ultimately
led to the formation of this manuscript.
The work of this book has received strong support from projects such
as the National Natural Science Foundation of China (NSFC) and the
Ministry of Science and Technology (MOST) of China; thank you to the
teachers and graduate students of the research team on structural
reliability at Zhejiang University, as well as to friends from all walks of
life for their strong support and assistance in the publication of this
book.
Dr. Wei-Liang JIN
Qiushi Distinguished Professor
Zhejiang University, P.R. China
Notations
a
Current crack length in Fracture mechanics model
A
Deflection of structural systems; Experience adjustment
coefficient
aa
Aeff
The limit on crack length under certain functions after
bearing secondary cyclic loads within its designed service
life
Effective sample area
Alimit
Maximum deflection of structural system
a0
Initial crack length
Aq
Gross area of pile tip
As
Surface area of pile body
Awhole
Sampling area
B
b(X)
BQ
Proposition supported by new experimental results
Stress at any position in the structural system
Deviation coefficient of Q
BSC
Deviation coefficient of SC
C
CkX
Test constants in Fracture
Effect coefficient for converting load into effect
The specified limits for the structure or component body to
meet the requirements for normal use
Kurtosis coefficient
CL
Lift coefficient of wave force
CsX
Skewness coefficient
d
Truncated values in truncated distribution functions
D
Fatigue damage
Outer diameter of pile
Effects caused by dead load
Effects caused by the average value of dead load
Df
Structural damage area
dij
Fatigue damage due to wave, low or high frequency
combination stress Si under the sea case i and the wave
direction j
Safety region of stochastic process in the whole life of
structure
The displacement vector of all nodes in the element
DS
de
E
EF
Ei
Standard value effect of seismic loads
Error factor
Subjective uncertainty
ejk
Error term due to spatial averaging
Ek
Plastic failure of the first failure mode
f
f(X)
Surface friction force per unit area
Joint density function of variables X(=(x1, x2, …, xn))
fGray(z)
The built-in function of gray variable
fHi
Zero crossing rate of high-frequency mooring force
fi
Average zero crossing rate
Fi
ith failure mode
fk
Standard values of material properties
fLi
Zero crossing rate of low-frequency mooring force
fwi
Wave zero crossing rate
ft
Concrete tensile strength
Fij
ith failed component in the jth failure mode
Fmax X
Cumulative distribution function of X at maximum value
FMi(x)
FN(n)
Cumulative distribution function for maximum load effects
of various combinations
Cumulative distribution function in time integration method
fR()
Probability density function for the whole structure
fR(t)
Instantaneous probability density function of structural
time-varying resistance
fRi()
Probability density function of the strength of the i-th link
frsf (x)
Response surface function
Fs
Structural failure function
fS(t)
Instantaneous probability density function of time-varying
load effects
Probability density function with time-varying state
fX(x, t)
Probability density function of Xi at xi point
Conditional probability density function under given
condition X2|X1
g(•)
Functional function space composed of single limit state
function
G(•)
Gi
Functional function space composed of multiple limit state
functions
The importance of subjective uncertainty
Gmax
Maximum allowable stress of structural system
H(ω)
Frequency response function
h(x)
H(x)
Importance sampling probability density function for the
variable x
Shannon entropy
Hk
Characteristic wave height
hT(t)
Risk function
hN(n)
Risk function in time integration method
hV()
i
Importance sampling probability density function for the
variable v
Radius of gyration
I
J
Total error of commonly used
Jacobian matrix
K
Structural stiffness
Traditional model describing the fatigue life of components
or structures under constant stress amplitude
Lateral earth pressure coefficient
k
Initial modulus of soil
KA
Ka
The ratio of actual and standard values of geometric features
of structural components
Rankine active earth pressure coefficient
Klimit
Ultimate structural stiffness
K0
Coefficient of static earth pressure
l
Number of support vectors in SVM
L
Li
Effects caused by live load Unit length
Persistent live load
lij
Number of ith effective mode under jth condition
LN(n)
Reliability function in time integration method
Effect caused by the average distribution of live load at any
time point
Lr
Standard value effect of roof live load
Temporary live load
The effect caused by the average distribution of the
maximum service life of live load
m
Random variables in Traditional model describing the
fatigue life of components or structures under constant
stress amplitude
Test constants in fracture mechanics model
mE
Influence degree of human error
Mi
The magnitude of subjective uncertainty
Mj
Plastic resistance moment in the jth segment
n
N
Number of components in the ith failure mode in the failure
mode method
number of times a given load is applied in a time integration
method
Total number of structural failures/sampling simulations
N(s)
Relationship between material fatigue parameters
Nc
Dimensionless bearing capacity coefficient of cohesive soil
ni
Actual number of cycles under stress amplitude Si
Ni
Number of stress cycles at constant stress amplitude
nL
Number of basic time periods
nLi
Low frequency cycles
N0
Number of cycles that the structure must be able to
withstand to meet design requirements
Nq
Dimensionless bearing capacity coefficient of sandy soil
nWi
Number of cycles of waves
P*
Design verification points corresponding to the maximum
possible failure probability of the structure
P(A)
Subjective level of belief in proposition A
Degree of subjective negation of proposition A by humans
P(Ai)
Prior probability
P(AiB)
Revised posterior probability
p(xi)
Distribution probability of the ith discrete point of a variable
P(error) Probability of truncated failure modes
Pf
Structural failure probability
pf(t)
Instantaneous failure probability at a certain moment
pf(tL|r)
Conditional failure probability under given structural
resistance
pk(t)
Probability of an event occurring within a time interval
P0
Effective overburden pressure of soil at calculation point
Pp(s)
Probability distribution of random stress
Pr(F90)
Probability distribution value with 90% error rate
Pr(F10)
Probability distribution value with 10% error rate
ps
Peak factor for variable s in SRSS
PS
Probability reliability of structures
pX
Peak factor for variable X in SRSS
q
Unit pile end bearing capacity
Q
Dynamic pile end bearing capacity
External effects borne by the structure
Q1
Single parameter load system
Qf
Friction force of pile body
Given load
Qi
External load
Qp
Support force at pile end
R
Structural resistance
Standard value effect of rain load
R*
Coordinates of design verification points for resistance of
structural components
R(t)
R(·)
Structural time-varying resistance
Resistance function of structural components
RfL(τ)
Autocorrelation of transverse wave force
ri
The number of repetitions of variable loads during the
design reference period
Total number of time periods for each load Si(t) during the
design reference period
rk
Load increment
RK
Standard value of resistance of structural components
ratio of the ith structural component to the jth load effect
RXX (τ)
Autocorrelation of stationary stochastic process
Ru(τ)
Autocorrelation of water particle velocity
S
Effect of action
Standard value effect of snow load
SC
Resistance of offshore platform structural systems (ultimate
bearing capacity)
S(t)
Time-varying load effect
Stochastic process of comprehensive effect of nth kinds of
loads
S(ω)
Stress power spectrum
Extreme distribution of load effects within one year
Maximum load effect in the event
Corresponding load effect of the structure under a given
load
SfL(ω)
Spectral density of transverse wave force
Coordinates of design verification points for dead load
SGK
Dead load standard value
SHi
Hot spot stress amplitude of high-frequency tube nodes
under working conditions
Si
Constant stress amplitude
Si(t)
ith combined load with time-variance
Si(t)
Stochastic process of the kind of ith load effect Effective
mode of the ith structure
Si(t0)
Random variable at any time point for the type of ith load
effect
Structural failure in the ith effective mode
Large value distribution of load effect during the duration of
the ith load effect
SLi
SM
Low frequency hot spot stress amplitude under working
conditions
Maximum value of load combination
SM
Maximum comprehensive load
Maximum value of the jth load during the design reference
period
Smax(tL)
The maximum value of load effect within the service life of
the structure
Coordinates of design verification points for live loads
SQK
Live load standard value
Ss
Structural effective
The effect caused by the average distribution of the
maximum service life of snow load
Hotspot stress amplitude of working condition waves
SX(ω)
(mean square) Spectral density
Su
Shear strength of cohesive soil
Sη(ω)
Spectral density function of ocean waves
T
Return period
Tf
Fatigue life
Return period
tL
Structural service life
|Tu (ω)|
Transfer function of horizontal velocity of wave water
quality points
u*
Maximum likelihood points on failure surface in U-space.
u
u(t)
Displacement vector of any point within the element
Horizontal velocity of wave water quality points at the depth
of the pipeline axis
up
Observation points for the overall state of the established
v
Average rate of event occurrence
Crossing rate of vector stochastic process leaving security
region
vi
Occurrence rate
vi(u)
Crossing rate of stochastic process Xi(t)
vmi
Average arrival rate of mixed stochastic process pulse
Positive crossing zero frequency of stress processes
vp
Maximum value frequency
w
Parameters related to the strength of all loads acting on the
structure in plastic theory
W
Wind load/Effects caused by wind load, creep, shrinkage, or
temperature changes
Effect caused by the average distribution of the maximum
service life of wind load
x*
Maximum likelihood points on failure surface in X-space.
X
Random function X=X(x1, x2, …, xn)
Support vector in SVM
X(t)
Stationary stochastic process
X(t)
Stochastic process vector
x(t)
Value of X(t)
Maximum value during the time period τ2
Xa
The actual strength or performance of the structural
XD
The best estimation point for design points
xE
Parameter value when no one is wrong
XG
Internal force generated by the standard value of dead load
Internal force caused by live load standard value (i.e. load
effect standard value)
Xi
xm
Samples generated by the importance sampling function in
SVM
Parameter value in case of human error
Xm
Mean point
New average point
xt
Random variable
Xr
Performance that the structure needs to achieve within its
designed service life
y*
Maximum likelihood points on failure surface in Y-space.
y50
Displacement value corresponding to strain value ε50
z
Local displacement of piles
Z
Structural functional function
[Z]
Failure limits in physical synthesis method
Z(t)
Safety limit state process
Maximum value during the time period τ1
α
Confidence coefficient of sampling function
α1
Characteristic parameters of stress spectrum
α2
Characteristic parameters of stress spectrum
αc1
Average axial compressive strength of the concrete
Gamma Function of the concrete
αi
Sensitivity coefficient
αk
Standard value of geometric dimensions
β
Reliable indicators
βij
Reliability indicator of the jth component under the limit
state design expression at the time of the i-th component
βT
Target reliability indicators
Target reliability index of ith component
γD
Partial coefficient of effects caused by dead load
γG
Partial coefficient of permanent load
γL
Partial coefficient of effects caused by live load
γm
Model uncertainty parameters
γQ
Partial coefficient of live load
Partial coefficients for variable loads
γ0
Structural importance coefficient
γR
Partial coefficient of resistance for structural components
γT
γW
Partial coefficient of effects caused by uneven settlement,
creep, shrinkage, or temperature changes
Partial coefficient of effects caused by wind load
γu
Uncertainty parameters for ultimate strength calculation
δ
Dirac function
δ0
Effective sampling area ratio (the ratio of effective sampling
area to the entire sampling area)
δt
Any little time increment
Internal friction angle of sandy soil
Δ
Damage parameters in the linear damage accumulation rule
Δi
Relative deflection with Qi
ΔK
Variation amplitude of stress intensity factor in fracture
mechanics model
ΔKth
ΔS
Threshold value of change amplitude of stress intensity
factor
Applied stress amplitude
Δβ
Effective sampling area of sampling function
ε
relative error
ε50
Strain at 50% maximum stress in undisturbed soil
undrained test
η(t)
Wave height function
θj
Plastic turning angle of section at the jth point
κi
Principal curvature of the failed surface in ith axil
λj
Slenderness ratio
λBE
Average estimate of the impact of human error
λm
The mth order moment of stress spectrum
λUB
Maximum estimate of the impact of human error
μA(x)
Membership function of A, simply called as μA(x)
Average duration of the ith action
Reduction coefficient of concrete considering brittleness
Average compressive strength of concrete cubes
μsi
The average value of the distribution of individual load
effect sections
Moment of the maximum value distribution during the
design reference period
μSM
Average of the distribution of maximum combined effects
μst
The mean of the maximum value distribution during the
design reference period
μi
Average duration of stochastic process Xi(t)
μZ
Mean value of functional function variable Z
ξ
Gamma function
ρ
Ratio of live load to standard value of dead load
σst
The second moment of the maximum value distribution
during the design reference period
σZ
Mean square deviation of functional function variable Z
τi
Pulse duration
υX(r)
Wearing rate (average passing rate per unit time)
υi
Average occurrence rate of the ith effect
φ
φ[·]
Stability coefficient
Probability density function of standard Normal distribution
Φ[.]
Cumulation distribution function of standard Normal
distribution
ψ
Load combination value coefficient
ψC
Combination value coefficient of secondary variable
ψci
Combination value coefficient of the ith variable load
Γ
Stress spectrum width parameter
⊗(z)
Grey variables formed by changes in basic variables within a
fixed interval
π (xi /R) Subjectivity of parameter uncertainty xi/R
1
Introduction
Civil engineering is the general term for all sciences and technologies
involved in the construction of various land engineering facilities. It is a
technical discipline that studies engineering facility structures, as well
as rock, soil and the environment, and their interaction with
engineering facilities. Civil engineering is the cornerstone of national
economic development, the bearing structure of all industrial and civil
buildings, bridges and aqueducts built across rivers and lakes,
breakwaters built in oceans, as well as sea-crossing bridges and
offshore platforms. These constructions are primarily made of steel,
wood, masonry, concrete and reinforced concrete, and are collectively
referred to as engineering structures. They are designed to carry loads
consisting of equipment, people and vehicles, and to withstand wind,
rain, snow, sunshine, waves, currents, earth pressure and earthquakes.
Whether the engineering structure is safe or not has a direct bearing on
people’s property, lives, safety and health, as well as on the progress of
national modernization. Therefore, an engineering structure should be
able to perform a variety of designed functions during its service life,
without the need for excessive maintenance, while ensuring the safety,
serviceability and durability of the structure. These are the basic
concepts of engineering structural reliability [1-1][1-2][1-3].
There are some uncertainties in terms of structural design and use,
which will inevitably affect structural resistance and load effect to a
certain extent. In the early stages of structural design, people
sometimes evaluate the influence of uncertainty on an engineering
structure by means of a safety factor. This is also used as an evaluation
index for civil engineering, without taking the randomness of these
uncertainties into account. In fact, the relationship between safety
factor and structural reliability is not all that clear, since structures with
the same safety factor may have different levels of reliability. This
demonstrates that the safety factor alone cannot accurately reflect the
reliability of engineering structures.
Structural reliability is a subject concerning uncertainty research, and
is designed to ascertain the effects of uncertainties arising from the
entire life cycle of an engineering structure (including design,
construction, application and maintenance) on its safety, serviceability
and durability. With the development of computational science,
engineering structures are required to be increasingly precise and
intelligent, but in practice, the design and construction of engineering
structures remains an iterative process in the conventional sense, far
from being able to meet the needs of social development. If the
uncertainty of design parameters is not taken into account, the benefits
of accurate structural analysis will be overwhelmed by the use of safety
indices determined roughly from experience. Therefore, it is of great
significance to take parameter randomness into reasonable
consideration during engineering design. Structural engineering should
not only meet the pre-defined functional requirements, but also help to
save costs as far as possible. This requires paying attention to the
uncertainties existing in practical engineering, so that the structure can
be designed scientifically using a more rational and realistic method,
that is to say, a design method based on structural reliability [1-4][1-5][16][1-7][1-8][1-9][1-10][1-11][1-12][1-13]
.
Moreover, the reliability of in-service structures cannot be ignored
either. This is because there are also many uncertainties arising from
the construction and use of an engineering structure. These include
load uncertainty, environmental uncertainty, resistance uncertainty,
and effect uncertainty. Such uncertainties may bring about potential
safety hazards, which may in turn lead to structural failure, resulting in
a disastrous accident, causing great economic losses and endangering
people’s lives and safety [1-14][1-15][1-16][1-17][1-18][1-19][1-20][1-21][1-22].
Therefore, it is imperative to analyze and evaluate the reliability of all
engineering structures [1-23].
Compared with structural uncertainties, the uncertainties of loads and
load effect on structures are even more important. Generally speaking,
the environmental effects on engineering structures, such as wind load,
temperature action, seismic action and marine environmental impact,
are all reflected in a random fashion, while the environmental effects of
structural design are expressed as extreme values, quite different from
how the environment acts in nature. However, as far as structural
design is concerned, it is necessary to take its reliability into
consideration [1-24]. Therefore, the environmental effects on structures,
and their dynamic impact, constitute the scope of dynamic structural
reliability. This is extremely important for the reliability of an
engineering structure throughout its life cycle, requiring the highest
degree of attention [1-25].
1.1 An Overview of the Development of
Structural Reliability Theory
In the early 1920s, Mayer et al. [1-26] applied the probability theory and
mathematical statistics to the reliability analysis of engineering
structures. In 1947, Freudenthal [1-27] published an article titled “The
Safety of Structures”, marking the beginning of systematic research on
how to apply reliability to structural design. In the 1950s, the concept
of “reliability” had drawn widespread attention in the field of civil
engineering. In the 1970s, researchers focused more on how to use the
reliability method for structural design specifications. Later, some
countries introduced reliability into their national standards, ushering
in a period of practical application. Since the 1980s, research on the
reliability of engineering structures has risen to the system level. Today,
many countries now pay increasing attention to reliability research,
with it having extensive applications.
The research on structural reliability primarily focuses on the following
aspects: (i) basic theory of structural reliability and associated
calculation methods; (ii) issues related to the reliability of structural
systems; (iii) issues related to the reliability of structures under
dynamic loads; (iv) issues related to the reliability of a structure’s
fatigue load; (v) issues related to the reliability of geotechnical
engineering structures; and (vi) issues related to evaluation of the
reliability of in-service engineering structures.
1.1.1 Method of the Degree of Reliability Calculated
Structural reliability, which is theoretically based on the probability
theory, focuses on the determination of functions, the search for failure
modes, the calculation of failure probability, and the characteristic
statistics of random variables. Major calculation tools include the finite
element method (FEM), the boundary element method (BEM) and
network analysis techniques, while the main calculation methods
include numerical simulation, proximate calculation, optimization and
intelligent analysis.
(1) Fast integration methods
Fast integration methods include the first-order reliability method
(FORM), the second-order reliability method (SORM) and other higherorder reliability methods, of which FORM and SORM are most
commonly used for reliability analysis in engineering practice.
In 1974, Hasofer and Lind [1-28] expanded the limit state equation at the
checking point, and proposed an improved FORM, known as the
checking point method. On this basis, Rackwitz and Fiessler [1-29]
proposed a checking point method that could consider the actual
distribution of random variables. Later, the Rackwitz-Fiessler method
was adopted by the Joint Committee on Structural Safety (JCSS), and
named the “JC” method. Marked by the put-forward of the “JC” method,
FORM effectively reached maturity for use in reliability analysis in the
1980s, and there was little foreign research literature on FORM
published after the 1990s.
In 1996, Zhao Guofan et al. [1-30] associated relevant random variables
with generalized stochastic space on the basis of their practical analysis
method, further proposing a method for processing non-normal
variables and associated random variables. The calculation of the
partial derivative of variables in the limit state equation is an important
part of reliability calculation by FORM. Xu Jun et al. [1-31] proposed a
partial derivative calculation method for rational polynomial functions
with regard to highly nonlinear and complex limit state equations. The
Rosenblatt transformation is decomposed in reference [1-32], with the
relevant normally distributed variables transformed into irrelevant
normally distributed variables through mapping transformation, and
then into independent standard-normally distributed variables through
orthogonal transformation, deriving a more general and applicable
form of FORM.
For structural reliability analysis, Fiessler et al. [1-29] first proposed an
SORM considering the second-order surface effect that could be used
for second-order expansion at the design point in U space. Then,
Breitung [1-33] proposed an asymptotic formula considering the effects
of the principal curvature of the limit state surface at the checking point
for failure probability calculation using SORM. This formula is now
widely applied in reliability analysis. Although SORM already effectively
achieved maturity in the 1990s thanks to the efforts of academic
researchers at home and abroad, especially Fiessler, Rackwitz, Tvedt
and Breitung [1-29][1-34][1-35], even more progress was made in SORM
research in later years.
Considering the difficulty in calculating the Hessian matrix in SORM,
Der Kiureghian [1-36], Zhao Yan-Guang and Tetsuro Ono [1-37][1-38]
adopted empirical points for fitting the curves, obtaining a quadric
curve and revealing the principal curvature of the surface, thus further
improving SORM. Li Yungui [1-39] applied Laplace’s integral
approximation theory to the approximate calculation of reliability in
generalized stochastic space and orthogonal random space. This
method is applicable to the high nonlinearity of functions; the
asymptotic method remains the type of SORM that also considers the
use of the second partial derivative term in a nonlinear function; when
used to calculate the reliability index, this method shows good accuracy
by taking the partial derivative of the limit state equation and obtaining
Taylor’s series, but it is troublesome to solve some complex functions
that are difficult to take the derivative of.
In 1998, Der Kiureghian and Dakessian [1-40] proposed a reliability
calculation method for multi-design-point limit state equations.
Calculation accuracy and calculation complexity are often contradictory
to each other, making it necessary to choose calculation methods with
varying levels of complexity for different objects. In 1999, Zhao Yan-
gang and Tetsuro Ono [1-41] set conditions suitable for both FORM and
SORM.
Some researchers have explored higher-order reliability calculation
methods. Tichy [1-42] proposed a first-order third-moment reliability
calculation method; Zhao [1-43][1-44] proposed a higher-order moment
reliability analysis method by normal transformation of variable order
moments; Zhao et al. [1-45] proposed a fourth-moment analysis method
for reliability calculations.
In conclusion, although SORM is already well established, academic
researchers at home and abroad have improved it in different aspects,
enhancing its serviceability. Considering its low calculation complexity,
SORM shows good accuracy when used to solve linear and weakly
nonlinear limit state equations. However, because higher-order
methods have a more complex calculation process than SORM, and
since the larger the number of expansion times, the higher the order
and the higher the calculation complexity, there is no generally
recognized short-cut calculation method with strong serviceability.
What is worse, there is no significant improvement in accuracy,
meaning such methods are rarely adopted for reliability analysis in
engineering practice.
(2) Numerical simulation methods
The fast integration methods for reliability calculation, such as
FORM/SORM, are quite similar to each other in how they handle the
nonlinear and non-normally distributed variables of nonlinear limit
state equations. Errors become obvious if the limit state equations are
highly nonlinear. The Monte-Carlo method has been brought to the
forefront so that it can be used to obtain more accurate results for
structural reliability calculations.
The Monte-Carlo method is a numerical simulation method based on
the probability and statistics theory. This method avoids some of the
mathematical difficulties that arise from the process of structural
reliability analysis, making it unnecessary to consider the nonlinearity
of functions or the complexity of the limit state surface. It is highly
intuitive, accurate and applicable. However, the frequency of simulation
in the Monte-Carlo method is inversely proportional to the failure
probability, while the failure probability of structures is usually very
low. So, when the Monte-Carlo method is directly adopted for reliability
analysis, the sampling size must be very large in order to ensure
satisfactory accuracy. But owing to its poor efficiency, the Monte-Carlo
method is difficult to apply to the reliability analysis of actual
engineering structures. For this reason, it is very important to find a
way of improving the efficiency of the Monte-Carlo method. At present,
variance reduction techniques are used to improve simulation accuracy,
including antithetic sampling, conditional expectation variance
reduction, importance sampling, stratified sampling, controlled
variables and correlated sampling.
Importance sampling is one of the most widely used, effective and
popular variance reduction techniques for the calculation of structural
reliability. Commonly used importance sampling methods include
general importance sampling, updated importance sampling, adaptive
importance sampling and directional importance sampling [1-46].
The computing paradigm in the importance sampling method,
proposed by Melchers [1-47] is quite representative and often adopted
by other researchers. In this method, the probability density function of
an N-dimensional independent normal distribution is taken as the
sampling function, while the design point obtained by SORM is taken as
the center point, achieving good results.
Hohenbichler and Rackwitz [1-48] proposed an updated importance
sampling method, one based on the reliability index obtained by FORM
or SORM. The distance from the sampling point along β to the failure
surface in normal space is calculated to iteratively correct and update
the calculation results of failure probability. This method is suitable for
the actual form of the failure surface and has better adaptability to
nonlinear problems. Also, calculation accuracy is improved due to the
introduction of analytical methods. However, this method needs to
constantly solve the limit state equations, leading to high calculation
complexity when the limit state equations are hard to solve.
Bucher [1-49] proposed an adaptive iterative search strategy to select a
sampling center and sampling variance for importance sampling, and
built an adaptive importance sampling method. It requires a little
preliminary analysis, but the initial checking point and sampling
variance, which are difficult to determine, have great influence on the
convergence of the calculation. In 1999, Au and Beck [1-50] created an
importance sampling point with the aid of Markov chains, and
proposed a new adaptive sampling computation framework. With good
robustness, this method is unaffected by the level of failure probability
and the shape of the limit state surface.
Directional sampling is a method by which samples are taken randomly
along the direction of a radius vector in the spherical coordinate system
to analyze structural reliability. The directional importance sampling
method was first proposed by Au and Beck [1-51]. Later, Ditlevsen,
Melchers and Gluver [1-52] applied this method to reliability analysis.
The basic idea of directional importance sampling is as follows.
Considering that the distance from each sampling point to the origin of
coordinates in the N-dimensional independent standard normal space
obeys the N-DoF distribution, truncated probability density without the
β spherical domain can be taken as an importance sampling function to
calculate the failure probability.
Jin Wei-Liang [1-53][1-54] combined the importance sampling technique
with conditional expectation variance reduction, putting forward a
composite importance sampling method. Owing to the adoption of
conditional expectation variance reduction, the sampling points are
limited in the failure domain, thus raising the effectiveness of sampling,
while the importance sampling function of truncated distribution helps
to improve the efficiency of the sampling calculation. Through an
orthogonal transformation, the original random variables are switched
into another standard random space, with a V-space importance
sampling method established that fully considers the geometric
characteristics of the failure surface (e.g., maximum likelihood,
gradient, curvature, etc.).
Dong Cong et al. [1-55] solved the problem of seeking all design points in
the generalized multi-design point problem based on a generalized
genetic algorithm that he had proposed for global optimization of
nonlinear systems in an unconnected domain. By establishing a
recursive bound-tolerance algorithm, they solved the problem of
compression and combination for generalized multi-design points, and
advanced a theory of adaptive importance sampling based on the
generalized genetic algorithm.
Wu Bin, Ou Jinping et al. [1-56] analyzed the characteristics of structural
dynamical reliability, proposed an importance sampling method, and
put forward a method for selecting importance sampling functions and
a concrete expression for them with respect to white noise load.
When the Monte-Carlo method is used to analyze problems, a random
number should be generated first; samples then need to be taken
randomly according to the probability distribution of random variables.
Although the frequency of sampling can be brought under control
thanks to improvements in sampling technology, for a large and
complex structure, the calculation complexity remains very high,
making its use somewhat limited. The sample size should be minimized
in order to improve calculation efficiency. At present, a number theorybased method, such as computer sampling or Latin sampling, is often
used to generate a pseudo-random number in place of randomly
collected samples for reliability analysis.
The Latin hypercube sampling method is used directly for failure
estimation, but its efficiency is poor. It has little obvious advantage
compared with direct Monte-Carlo simulation. Olsson, Sandberg and
Dahlblom [1-57] combined Latin hypercube sampling with importance
sampling, putting forward an importance sampling method based on
Latin sampling, and then proposed a measure for reducing the
correlation among Latin sample points and a solution for orthogonal
space. Examples show that this method can effectively improve the
efficiency of numerical simulation.
Wu and Zhao [1-58] proposed a number theory-based method for solving
multi-dimensional normal distribution functions. This method can be
used to precisely calculate the failure probability of a structural system;
examples show that this method has enough accuracy.
In summary, numerical simulation is easy to perform and capable of
solving problems efficiently, but the effectiveness of direct analog
computation is very low, making it imperative to develop a high-
efficiency numerical simulation method. Researchers throughout the
world have done a lot of work to improve the performance of numerical
simulation, with a variety of new high-efficiency algorithms being
proposed. As can be expected, new high-performance numerical
simulation methods will surely emerge in large numbers with the
development of computer software and hardware technology.
(3) Responding surface method
During the reliability analysis of complex structures, structural
functions cannot normally be (or are not) expressed explicitly;
furthermore, only some discrete empirical points can be obtained by
numerical algorithms (e.g., FEM) or from experimental studies. If
Monte-Carlo simulation is used for reliability analysis, extensive
numerical analysis will have to be conducted. Under the circumstances,
the responding surface method is therefore a good choice.
Wong [1-59] first proposed a quadratic polynomial responding surface
method for reliability calculations, using quadratic polynomials
containing linear terms and cross terms to approximately fit the real
limit state of structures, and then locating the sampling center at the
average point. When there are many random variables, the number of
samples required for fitting will increase rapidly when adopting this
method. Moreover, it will be difficult to reflect the main characteristics
of the failure surface if the sampling center is located at the average
point. Later, Bucher et al. [1-60] improved this method, with the cross
terms in the quadratic polynomials ignored and the quadratic terms
retained. At the same time, the sampling center was approximately
located on the failure surface due to a linear interpolation between the
average point and the checking point obtained by fitting. When
Bucher’s method is adopted for calculation, only 2n+1 experimental
point is needed to uniquely determine 2n+1 unknown number.
However, when Bucher’s iterative strategy is used for calculation, the
convergence process may be unstable owing to obvious fluctuations in
the response surface during iteration. This is because the sampling
center is likely to come from extrapolation. Therefore, Rajashekhar [161]
gave different weights according to the distance between the
empirical point and the real limit state surface in a bid to maximize the
fitting precision of the limit state equations at the design points,
thereby improving the accuracy of failure probability calculations. Kim
[1-62]
projected the sampling point onto the linear response surface so
that it could be as close to the response surface as possible, enhancing
the convergence of the method. In recent years, Lee [1-63], Wong [1-64]
and Kaymaz [1-65] have also been committed to improving the nonlinear
fitting ability of the responding surface method, reducing its calculation
complexity and increasing its accuracy.
By learning the samples obtained from finite element calculation,
neural networks can precisely fit those highly nonlinear functional
relations which are difficult to accurately express by means of general
analytic expressions [1-66][1-67][1-68][1-69].
Tong and Zhao [1-70] combined the quadratic polynomial responding
surface method with the geometric method for reliability index
selection, accelerating the convergence speed of iteration. Gui and Kang
[1-71]
applied intelligent computing techniques, such as neural
networks, fuzzy mathematics and genetic algorithms, to the reliability
analysis to systematically study the intelligent calculation methods
essential for neural network response surface reconstruction, achieving
a good effect.
The Support Vector Machine (SVM) is a new AI technique that has many
advantages [1-72] over quadratic polynomials and neural network
function fitting. Yuan [1-73] carried out research into the application of a
least squares support vector machine (LS-SVM) to reliability
calculations, achieving good results.
In short, the responding surface method is an ideal method for
analyzing the reliability of large and complex structures in implicit limit
state equations. Traditional quadratic polynomial methods have limited
nonlinear fitting ability, meaning AI techniques, such as neural
networks and genetic algorithms, as well as fuzzy mathematics, can be
applied to reliability analysis. SVM is a new AI technique with excellent
small sample processing ability which has been developing rapidly in
recent years. The responding surface method has small samples,
meaning the SVM-based responding surface method has broad research
and application prospects.
1.1.2 Reliability Method of Structural Systems
The reliability analysis of actual engineering structures is usually not
just a matter of calculating the reliability of a failure mode, a
component or a surface, but involves calculating the reliability of
structural systems in multiple functions. The probability of system
reliability was worked out during the early stages of reliability
development. In 1969, Cornell [1-74] proposed a wide-bound formula for
the interval estimation method applied to system reliability calculation.
In 1979, Ditlevsen [1-75] found that for some examples, an excessively
large range of system failure probability was obtained by Cornell and
Moses’ method [1-76], and therefore proposed a narrow-bound formula
for system reliability calculations. Ang et al. [1-77] applied the idea of
fault tree analysis to the reliability of structural systems, and proposed
a point estimation method, i.e., the probabilistic network evaluation
technique (PNET) for system reliability analysis. With the
establishment of the above theory, an available solution can be
developed for system reliability analysis under the condition that the
main failure modes are known. But due to the difficulties in
mathematical and mechanical analysis, the research on system
reliability has developed very slowly. So far, the related achievements
have not yet been put into practice.
Numerical simulation is also a good method for calculating the
reliability of structural systems. Nie and Ellingwood [1-78] simulated the
reliability of structural systems using Fekete points generated in
advance; their findings show that compared with other homogeneous
point generation techniques (t-design, GLP), Fekete points are more
homogeneous and require fewer sampling points to replace random
dots for the analog computation of reliability. In this way, the frequency
of finite element analysis (FEA) can be effectively reduced when a finite
element program is used for structural calculation. But like other
pseudo-Monte-Carlo methods, this method has insufficient accuracy
when variable dimensions are high.
1.1.3 Load and Load Combination Method
In the late 1960s, Hasofer [1-79] proposed a combination of persistent
floor live load with temporary floor live load. Although this
combination rule was not applied to engineering practice, the idea of
this combination will continue to guide and promote the development
of the “Loads and Load Combination Method”. Der Kiureghian [1-80]
introduced FORM into the load combination method, proposing a firstorder second moment rule for load combinations. It is argued that the
first and second moments μst and σst of the maximum value distribution
of a single load effect in the design reference period [0, T] can be
expressed by the first second moments μs and σs distributed at any time
point.
(1.1)
(1.2)
The change of parameters p and q in the above equations, as well as
their value, is related to the type of load process and the average
frequency of load occurrence in [0, T]. It is very inconvenient to use the
parameters p and q in engineering practice, but this load combination
rule, which boasts a unique idea and a simple method, helps to promote
the innovation of theoretical research on random load combination.
In 1970, Turkstra [1-81] and Larrabee [1-82] proposed a load combination
rule (referred to as the TR rule) from a practical perspective. It is
argued that in a load combination, a certain load reaches its maximum
in the reference period, while the remaining loads occur at any given
time point. Therefore, when N variable loads are combined, they play a
dominant role in turn. The maximum value in [0, T] can be combined
with the value of the remaining N-1 loads at any time point to form N
combinations. The controlling value is taken as the approximate value
of the maximum load, as follows
(1.3)
where SMj represents the maximum value of the j-th load in [0, T], i.e.,
, which is the relative maximum value among the
values of the remaining N-1 loads at any time point.
In 1972, Ferry-Borges and Castanheta [1-83] proposed the isochronous
load combination model based on Turkstra’s idea. This then became the
basis for giving advice on load combinations in the Canadian
Construction Specifications. The expression of this model is shown in
Equation (1.4).
(1.4)
where S1(t), S2(t), …, SN(t) are the loads to be combined, while SM
represents the maximum value of the load combination.
In 1976, JCSS proposed another load combination rule (the JCSS
Combination Rule). It is assumed that the stationary binomial process
is uniformly selected as a probability model for the random load
process {Si(t), t ∈ [0, T]}, (i = 1, 2, ⋯, N). The total number of time
intervals for every Si(t) in [0, T] is flagged as ri, which is arranged in
order of magnitude of ri(r1 ≤ r2 ≤ ⋯ < rN). In the combination process,
the maximum value of a certain load, Si(t), in [0, T] is taken in turn.
Regarding the loads for which there are more than ri time intervals in
[0, T], the local maximum value at the previous time interval is taken,
while the relevant instantaneous value is taken for the remaining loads.
In this way, N combinations can be formed, with N relatively maximum
loads obtained. After this, a group of results that play a controlling role
can be taken as an approximation of SM.
Wen [1-11] came up with an idea similar to Hasofer’s rule, and further
studied how to combine class-a loads with class-b loads, with the load
combinations divided into two types:
1. When multiple class-b loads are involved, the combination can be
expressed as
(1.5)
where λi is the average occurrence rate of i-th action. λij = λiλj (μdi +
μdj), λijk = λiλjλk (μdi μdj + μdj μdk + μdk μdi),
,
where
Sj(t).
represents the overlap between two loads, Si(t) and
, where
represents the
overlap among three loads, Si(t), Sj(t) and Sk(t)
2. When there are both class-a and class-b loads involved in the
combination, the maximum of the combination in [0, T] can be
further calculated after combining the combined extrema of classb loads for the duration of the quadratic variation of class-a loads.
This can be expressed as follows:
(1.6)
where
.
Larrabee [1-82] studied the problem of load combinations involved in
various loading processes. The load combination rule proposed by Wen
[1-84]
is also of great theoretical importance and rigorously reasoned,
and boasts highly accurate results, but there only a few types of load
models were adopted. However, many parameters are involved in the
calculation, making it inconvenient to apply this rule in engineering
practice. Usually, it is used to verify the accuracy of other load
combination methods. Soares [1-85] discussed the combination of main
load effects in ship structures. Floris [1-86] used the stochastic analysis
method to study load combinations, and proposed a new practical
analysis method. Casciati and Colombi [1-87] discussed load
combinations and the related problem of fatigue reliability. Naess and
Royset [1-88] generalized Turkstra’s rule and applied it to related load
effect combinations. Gray and Melchers [1-89] studied the problem of
load combination by means of load space simulation, and Ellingwood [190]
studied the problem with respect to structures.
Wang and Padgett [1-91] corrected the load coefficient for reinforced
concrete bridges under the combined action of earthquakes and
erosion. Meimand and Schafer [1-92] investigated the effect of load
combinations on the reliability of cold-formed flexural steel members.
Al-Sibahy and Edwards [1-93] proposed a method for testing the
performance of new concrete block walls under the combined action of
axial loads and thermal exposure. Hmidan [1-94] investigated the
combined effect of sustained load and low temperature on the flexural
performance of damaged steel beams repaired with CFRP. Lantsoght [195]
built an extended strip model for slabs considering the combined
action of loads. Xu [1-96] built a method for simplifying the aseismic
design of high-rise buildings in a vertical composite framework system.
Feng and Dai [1-97] studied the modes of load combinations on highspeed railway bridges, and calculated the partial load factor in
accordance with the statistical parameters of high-speed railway bridge
loads and resistance by reference to the target reliability index
presented in the Code for Design on the Steel Structure of a Railway
Bridge. Chen [1-98] explored load combinations in structural design. By
introducing the isochronous load models widely used in other
applications together with the relevant combination theories, he
proposed theoretical formulas for the partial load factor and
combination coefficient, and explained how to apply these formulas
and theories within the load code. He also pointed out that, since load
combinations are under the control of the permanent load, structural
reliability can be adjusted automatically under a design condition
dominated by structural weight. This would occur based on the load
combination rule in the revised load code, without the need to
deliberately improve the safety factor for different specific occasions.
Dai [1-99] introduced and analyzed the possible common effect of load
combination value changes on structural design according to the
revised code for the design of building structures. They then gave
examples based on probability theory for comparison, and pointed out
problems that should be brought to the forefront and further explored.
Ou Jinping et al. [1-100] determined the probability distribution of
amplitude and random variables in any period of the random load
combination process in accordance with the principle of maximum
entropy. Then, by reference to the code design method, they discussed
the assurance rate of the load combination coefficient, and conducted
the Monte-Carlo test, verifying the accuracy of this load combination
method.
Gong and Zhao [1-101] proposed an analytic solution and a simplified
calculation formula for the probability distribution function with
respect to the combination of persistent variable load and temporary
variable load.
Su [1-102] introduced the provisions on loads and load combination in
the second part of the British code BS5400, providing load data for load
conversion in a bridge design made under BS code using homemade
software. He pointed out that, from the perspective of the total load
effect, the BS code focuses on safety while the Chinese code focuses on
cost-effectiveness.
Gu Ming et al. [1-103] established a joint probability distribution function
for the effect of wind loads on high-rise buildings in two orthogonal
directions based on high-frequency force balance experiments and the
Copula Frank function. The wind load combination coefficient was
calculated at a specific assurance rate.
Li [1-104] considered the correlation between wind-wave elements,
adopting Gumbel’s joint probability model to successfully combine
water flow with wind by means of wind currents. To study the effects of
wind load, wave load and environmental load, along with their
combinations on sea-crossing bridges, they combined wind load with
wave load by means of the combination model proposed by JCSS.
To sum up, Chinese and foreign researchers have focused on different
things while studying and applying random load combinations, which
remains a core part of reliability theory [1-104]. Foreign researchers have
concentrated on in-depth studies of the theory of random load
combinations based on the engineering reliability theory, and have
proposed different load combination rules. Chinese researchers have
focused on the application of the existing random load theory, and
applied this to random load combinations in the construction of roads
and bridges, high-speed railways, and civil buildings.
1.1.4 Engineering Applications
Thanks to its continuous development and improvement, the reliability
theory has laid a solid foundation for the safety evaluation of
engineering structures such as industrial and civil buildings, bridges
and marine structures. At present, the priority for the practical
application of reliability theory has been shifted from the reliability of a
single structural component (including time-independent reliability
and static reliability) to the reliability of a complex system. This
includes time-dependent reliability, life-cycle reliability and dynamic
reliability. Many outstanding academic researchers throughout the
world have carried out research into this form of reliability, achieving a
series of excellent results.
Representative foreign researchers include Frangopol, Moan and
Ellingwood. Based on the basic principle of reliability, Frangopol et al.
[1-105][1-106][1-107][1-108][1-109][1-110][1-111]
conducted an in-depth study
on the life-cycle reliability of concrete structures and marine structures
from the perspective of materials, corrosion, fatigue and repair. Moan et
al. [1-112][1-113] [1-114][1-115][1-116][1-117][1-118][1-119][1-120][1-121][1-122][1123]
conducted extensive research on the reliability analysis of typical
offshore structures such as fixed platforms, floating platforms, offshore
wind turbines and submarine pipelines, and calibrated the reliability
level of various offshore structures, greatly promoting the application
of reliability theory to offshore engineering. Ellingwood [1-61] [1-124][1-
125][1-126][1-127][1-128]
has also made outstanding contributions to the
development of reliability-based structural design theory, as well as
structural safety evaluation and risk analysis.
Chinese academic researchers started studying reliability in the 1950s.
In the 1960s, Zhao Guofan published a monograph titled “Engineering
Structural Reliability” [1-30], and published Structural Reliability Theory
[1-31]
in the early 21st century, reflecting China’s latest achievements in
reliability research. Gong Jinxin [1-4][1-30][1-31][1-32][1-129][1-130][1-131]
continues to carry out research into the reliability analysis method in
generalized random space considering inter-variable correlation, as
well as high-precision SORM, the fourth moment method, the system
reliability method and the time-dependent reliability method for
concrete structures. Wang Guangyuan and Lü Dagang et al. [1-6][1-132][1133][1-134][1-135][1-136][1-137]
have done a lot of research on the seismic
reliability of structures. Li Jie and Chen Jianbing et al. [1-25][1-138][1-139][1140][1-141][1-142][1-143][1-144][1-145][1-146]
conducted an in-depth study on
stochastic dynamics and the seismic reliability of large-scale complex
engineering networks, advancing a network connection reliability
analysis theory centered on the recursive decomposition of structural
functions. This enabled them to create a moment method system for
the functional reliability analysis of large-scale complex engineering
networks. Xu Jun et al. [1-147][1-148][1-149][1-150][1-151] conducted an indepth study on the efficient numerical methods of structural dynamic
reliability by combining stochastic structural dynamics with the
random finite element analysis of structures. Jin Wei-Liang et al. [1-152]
[1-153][1-154][1-155][1-156][1-157][1-158][1-159][1-160] [1-161][1-162][1-163][1-164][1165][1-166][1-167][1-168][1-169][1-170][1-171][1-172][1-173][1-174][1-175][1-176][1177] [1-178][1-179][1-180][1-181][1-182][1-183]
have done extensive research on
the strength and fatigue reliability of fixed offshore platforms, as well as
on the overall strength and fatigue reliability of floating platforms, the
fatigue reliability of submarine pipelines and the vibration comfort of
engineering structures.
Reliability theory is now being applied to numerous engineering
structures in various fields. With the continuous development of
computer technology and the accumulation of actual engineering data,
it is virtually guaranteed that the theory will continue to be built upon.
Reliability theory thus plays an increasingly vital role in engineering
design, evaluation and operation.
1.2 Basic Concepts
This section presents a basic description of the main concepts related
to the reliability theory, methodology and application.
1.2.1 Reliability and Degree of Reliability
During the life cycle of an engineering structure, it must be able to
function normally. The life cycle is a whole process, composed of
structural design, construction, operation, maintenance, repair,
reinforcement, transformation, demolition and reuse. There should be
special requirements for structural performance in these different
periods. Structural reliability refers to the ability of a structure to fulfill
its preset functions (safety, serviceability and durability) within the
prescribed time under the specified conditions. The above-mentioned
prescribed time is defined as the working life of a structure as given in
the Unified Standards for the Reliability Design of Engineering Structures,
while the specified conditions mean that the structure should meet the
requirements for normal design, construction, use and maintenance.
The preset functions refer to structural safety, serviceability and
durability. The requirements for these functions are as follows: (1) be
able to bear various types of pressure arising from construction and
use; (2) maintain good functional performance; (3) have enough
durability; (4) be able to maintain enough bearing capacity in case of
fire; and (5) be able to stay stable in case of an accident, such as
explosion, impact or artificial destruction. Therefore, structural
reliability is the generic term for structural safety, serviceability and
durability.
Structural reliability refers to the ability of a structure to fulfill its
preset functions within the prescribed time under the specified
conditions. Structural reliability is also the probabilistic measurement
of the reliability of a structure. Failure probability refers to the
probability that the preset functions fail to be exerted. The structural
reliability must be set with the cause and mode of structural failure in
mind (e.g., for a structure or structural component that collapses
suddenly without warning, its reliability should be higher than that of a
structure or structural component which can collapse before failing). It
must also consider the possible consequences of failure, such as the
human, material and financial resources needed to reduce the risk of
failure, as well as specific social and environmental conditions.
1.2.2 Uncertainty
Structural reliability is used to mathematically measure the effects of
structural uncertainty on structural safety, serviceability and durability
throughout a structure’s life cycle. Structural uncertainty refers to the
inability to accurately foresee what is going to occur to a structure or
structural component, and means that there is no inevitable
relationship between the conditions required for a possible occurrence
and its consequences.
Structural uncertainty, which exists throughout the life cycle of a
structure, manifests itself in the inability to precisely know the active
state of the structure, particularly the distribution of consequences and
losses. Structural uncertainty can be classified based on a structure’s
characteristics, manifestations, internal relations and attributes, such
as randomness, fuzziness and knowledge imperfection. It falls broadly
within the domain of objective and subjective uncertainties, physical,
statistical and model uncertainties, and parameter and system
uncertainties. Structural uncertainty analysis methods can be classified
into probability-based reliability calculations, fuzzy concept-based
reliability calculations, gray theory-based reliability calculations and
entropy theory-based reliability calculations, depending on the
uncertainty classification.
1.2.3 Random Variables, Random Functions and Random
Processes
Structural reliability and unreliability are uncertain events, brought
about by the uncertainty of relevant variables. In structural reliability
analysis, these variables are usually regarded as time-independent
random quantities, and include random variables random functions,
and time-dependent random processes.
Structural design and analysis are processes in which qualitative
analysis is combined with quantitative calculation. Quantitative
calculation refers to the mathematical and mechanical methods used to
calculate the variables involved. In reliability theory, the variables
directly used for design calculation are called basic random variables,
such as load, material strength, modulus of elasticity, and component
size. The probability characteristics of random variables can be
described in terms of their probability density function and probability
distribution function. The statistical features of random variables are
often used to reflect a certain probability characteristic of theirs. For
example, the average value (first moment) reflects the degree of
concentration of random variables, while the variance (second
moment) reflects the degree of dispersion of random variables.
For a two- or higher-dimensional random variable cluster, it can be
expressed by the random variable function X (x1, x2, …, xn) and by the
random function X.
The stochastic process X(t) is a random function based on time t, where
the value of X is a random variable at any time point. The value of X(t) is
determined by its probability density function fX(x, t). Of course, the
variable t can be replaced by any set of finite, or countable, infinite
values, such as the number of loads applied. Therefore, the stochastic
process X(t) can be divided into a continuous stochastic process and a
discrete stochastic process.
1.2.4 Functional Function and Limit State Equation
When the entire structure or a part of the structure exceeds a specific
state, a certain preset function will fail to be exerted. This specific state
is known as the limit state of the function. Therefore, when a structure
or component is in its limit state, a relational expression of various
relevant basic variables will be formed. This is called the limit state
equation.
The working state of the structural component within which it exerts
its preset functions can be described with the relationship between
action effect S (referring to the internal force and deformation caused
by load, such as axial force, bending moment, shear force, deflection or
crack width of the component) and structural resistance R (referring to
the ability of the component to resist the action effect, such as the
cross-sectional strength and stiffness of the component). This
expression is known as the structural function, and is represented by
Z=g(R, S).
1.2.5 Reliability Index and Failure Probability
The probability of having the preset functions exerted within the
prescribed time under the specified conditions is known as reliability
probability, while the probability of failing to have the preset functions
exerted under the above conditions is called failure probability. In
engineering practice, structural failure is the focus of concern, and
failure probability is mostly used to reflect structural reliability.
When the function Z = R − S, the probability of structural failure is
calculated according to the following formula:
(1.7)
(1.8)
Equations (1.7) and (1.8) offer formulas for failure probability given
known probability distributions of the structural function.
Generally, the distribution of Z depends on the probability distribution
of the random variables contained within it, and on the form of the
function. When the function contains n random variables, structural
failure probability is expressed as:
(1.9)
As can be seen, the classical integral expression is a high-dimensional
integral, and its dimensions are the same as the number of random
variables. When there are many random variables, it is very difficult to
perform calculations directly and apply it to engineering practice,
which is why the reliability index needs to be introduced.
It is assumed that Z obeys normal distribution and that its average
value is μZ while its standard deviation is σZ. Therefore, structural
failure probability can be given as follows
(1.10)
(1.11)
where
it called the reliability index, which has a one-to-one
correspondence with failure probability.
1.2.6 Member Reliability and System Reliability
The reliability probability of a single component is called component
reliability; the reliability probability of an overall structure is called
system reliability. An actual engineering structure is complex, and
ultimate structural failure depends on the overall behavior of a
structure. The research on the failure of the overall structural system is
an important aspect in reliability. Because the failure of the entire
structure is caused by the failure of a structural component, the failure
probability of the entire structure can be estimated according to the
failure probability of each structural component.
1.2.7 Time-Dependent Reliability and Time-Independent
Reliability
If the structural state remains unchanged throughout the service
period, reliability is referred to as time-independent reliability. In
engineering practice, some variables are random and time-dependent.
For example, the variable load acting on a structure changes from time
to time, and this structure will not be considered safe unless it is in a
secured state at every moment of its service life. Thus arises the
problem of time-dependent reliability. The problem of time-dependent
reliability can be first converted into a problem of time-independent
reliability by mathematical methods before being solved. Therefore, the
solution to the problem of time-independent reliability is the basis for
all reliability theory.
1.3 Contents of this Book
This book first introduces the related concepts and calculation methods
for structural reliability, and then analyzes the reliability of actual
engineering structures. The content is divided into the following parts:
This book introduces the basic theory and analysis methods of
structural reliability. The contents are as follows: An overview of the
basic theories and concepts of reliability, uncertainty analysis,
reliability calculations, numerical simulation of reliability, system
reliability analysis, reliability of time-varying structures, loads and load
combinations, and standard application of reliability.
Characterized by a combination of theory and practical engineering,
this book systematically describes the latest research achievements in
reliability theory to promote the development of reliability theory in
engineering practice. In this way, it provides a reference for improving
and revising the unified standard of structural reliability. The book is
characterized by a combination of theory and actual engineering,
meaning it can be used as a textbook and teaching reference book for
graduate students and senior undergraduates majoring in civil
engineering, water conservancy, highway engineering, railway
engineering, port engineering, shipbuilding and maritime engineering.
It is also a professional reference book for engineers, technicians and
academic researchers engaged in the research and design of civil and
industrial architecture, municipal facilities, bridges, roads (highways
and railways), ports and marine facilities.
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2
Method of Uncertainty Analysis
The problem of structural reliability is one of dealing with the uncertainty of things. If there is an inevitable
causal relationship between the conditions for the occurrence of an event and the results of such an
occurrence, then the event can be called a definite phenomenon. Likewise, if there is no inevitable causal
relationship between the conditions for the occurrence of an event and the results of the event, then such an
event is known as an indefinite phenomenon [2-1].
In fact, there are many indefinite phenomena in nature and engineering practice. In the analysis and
application of structural reliability, the reliability of a structure is affected by both subjective and objective
factors. For example, objective factors involved in engineering structure design include random variables
such as action, environmental effects, materials and geometric parameters. In the process of engineering
decision-making, people often make use of structural reliability analysis tools, thereby bringing about
subjective uncertainty. Artificial level of understanding should be taken into consideration, and it is also
necessary to clarify the impact of subjective uncertainty on reliability analysis and predictions. As a matter
of fact, the comprehensive effect of various uncertainties should be considered with respect to structural
reliability analysis, design, evaluation and prediction[2-2], especially when the reliability of an existing
structure is being assessed[2-3]. It should be noted that as people deepen their understanding of the
objective world, a variety of methods are constantly emerging to handle uncertainty, making it possible to
analyze the uncertainties of structural reliability in a more comprehensive way[2-4].
The uncertainty, in terms of its types, characteristics, forms and attributes, introduces uncertainty
classification methods, and sets forth methods for uncertainty analysis are classified in this chapter. These
methods include the probability analysis method, fuzzy mathematics method, gray theory method, relative
information entropy analysis method and artificial intelligence (AI) analysis method.
2.1 Classification of Uncertainty
In nature and engineering practice, there exist many indefinite phenomena. For example, during structural
design, it is impossible to know exactly the required size of a structure that can help effectively resist wave
load, wind load and snow load. Neither is it possible to know exactly whether the material characteristics of
such a structure are unique. In short, the uncertainty means that the occurrence or result of an event is
uncertain, or that the result cannot be foreseen before the occurrence of the event. This is mainly manifested
as follows: (1) the upper limit of various structures and loads, as well as the lower limit of material strength
are actually difficult to define; (2) even if there is such a natural limit, it is likely to be very uneconomical in
practical applications (extreme load); (3) the limit imposed by quality control and testing is not completely
effective, while a change may take place in the potential performance; and (4) even if there is an accepted
limit, its use is not always necessarily reasonable (maximum load, minimum resistance).
Due to the difference in research object and solution, the problems of structural uncertainty can be
expressed by the following classification methods.
2.1.1 Classification on Uncertainty Type
(1) Aleatory uncertainty
Aleatory uncertainty represents the intrinsic, inherent and fundamental uncertainty of basic variables. It
refers to the natural randomness of physical quantities and cannot be eliminated. Aleatory uncertainty is a
type of objective uncertainty, which is determined by internal factors and external conditions, such as the
uncertainty of load, material properties and geometric dimensions.
(2) Epistemic uncertainty
Epistemic uncertainty is a type of uncertainty arising from limited information and understanding. It can be
further divided into statistical uncertainty, model uncertainty and measurement uncertainty. The first is
caused by limited observations and depends on the sum of sample data and any existing knowledge; model
uncertainty is caused by the defects and idealization of the physical model; measurement uncertainty is
caused by the inaccuracy of the methods and tools used to evaluate a variable. Generally, knowledge
uncertainty can be reduced by collecting more information in a more careful manner, or by adopting a more
sophisticated model.
2.1.2 Classification on Uncertainty Characteristics
(1) Objective uncertainty
Objective uncertainty refers to all countable information from an event. It is random and can be described
mathematically, involving only variables.
(2) Subjective uncertainty
Subjective uncertainty refers to all uncountable information from an event, caused by human activities. It is
characterized by fuzziness, ignorance and incomplete knowledge. Often expressed by the Fuzzy theory, it
involves not only variables, but also system processes.
2.1.3 Classification on Form of Manifestation
(1) Random uncertainty
The concept of random uncertainty means that the result of an event is uncertain because the conditions for
its occurrence are out of control, but the result still has a clearly defined range of variation. Random
uncertainty (called randomness for short) boasts a certain category of items, and the concept of randomness
has a certain extension, but its connotation is uncertain. It can be expressed by a mathematical expression.
For example, variable uncertainty obeys normal distribution. In general, the reliability uncertainty analysis
method, which is a probability method, can be used.
(2) Fuzzy uncertainty
Fuzzy uncertainty (called fuzziness for short) is caused by the boundary uncertainty of variables,
characterized by “fuzziness”. Fuzzy things fall under an uncertain category. The concept of fuzziness has a
clear connotation, but its extension is unclear. It is usually described in fuzzy words. From a philosophical
point of view, fuzzy uncertainty is a higher form of random uncertainty. In general, the fuzzy random
structural reliability theory based on fuzzy set theory can be used.
(3) Incompleteness
Incompleteness is caused by a lack of knowledge. Its concept has a clear yet unclear connotation, i.e., “some
information is known, whereas some is unknown”. Normally, it can be described by means of the gray theory,
by the neural network theory and turbidity, and by the bifurcation theory. In general, reliability uncertainty
analysis methods, such as gray theory and information theory, can be adopted.
2.1.4 Classification on Uncertainty Attributes
(1) Parameter uncertainty[2-5]
Parameter uncertainty is caused by the uncertain knowledge of basic variables or events. As a fuzzy random
problem, it arises from the randomness, fuzziness and incompleteness of variables.
(2) System uncertainty [2-5]
System uncertainty is caused by the inadequacy of the theoretical model or failure probability. It is the rest
(in the sense of remainder) of parameter uncertainty. It is primarily caused by human activities and
incompleteness.
2.2 Probability Analysis Methods
2.2.1 Classical Probability Analysis Method
If the joint probability density function f(x) of random variables of a structure is known, then the failure
probability Pf of the structure can be expressed as:
(2.1)
where Df represents the structural failure zone. This kind of method is commonly used in the traditional
structural reliability method based on the probability characteristics of variables[2-6]. Subjective judgment is
completely ruled out because objective probability is adopted as part of classical probability theory and
reliability is calculated based strictly on assumed objective distribution. It is usually very difficult to
establish a total probability distribution function. In general, the first and second moments of distribution
parameters are used to approximately describe the uncertainty of basic variables.
2.2.2 Bayes Probability Method
Subjective judgment is expressed by the rule of objective probability. The new formula obtained from
conditional probability is as follows:
(2.2)
where P(Ai) represents prior probability; B can be regarded as a proposition supported by the new
experimental results. P(AiB) represents revised prior probability; in this way, it can not only combine
objective events together, but also combine objective probability with subjective judgment, greatly
broadening the application scope of the probability theory.
However, one of the premises of Bayes probability is:
(2.3)
When P(A) is used to express the degree of artificial subjective belief in proposition A, a conclusion can be
drawn, as follows
(2.4)
In other words, human cognition is only composed of certainty A and uncertainty , while there is no room
for “unknowns”. For example, P(A)=0.5 can mean “not knowing”, but there are often more than two results.
Therefore, when the Bayes method is applied to something which is unknown, a series of contradictions will
arise. This shows that the Bayes method is not an ideal tool for uncertainty inference.
2.3 Fuzzy Mathematical Analysis Method
2.3.1 Definition
Fuzzy mathematical analysis is primarily used to solve uncertainty problems with an inexact definition and
unclear boundaries. Zedeh[2-7] was the first person to introduce the concept of fuzzy subsets into the idea
that basic variables are not taken as deterministic values or stochastic variables, but rather represented by
discrete feature points. This means that the fuzzy subset A can be represented as
(2.5)
where μA(X) is the membership function of A; x ∈ U is a universe of discourse, 0 ≤ μA(X) ≤ 1, x → μA(X). The
above formula is called Zedeh notation, and is not a sum formula.
If μA(X) is a continuous function, then the Zedeh notation is
(2.6)
The rest of the fuzzy subset A is denoted as
respectively,
and
. The rule of fuzzy set is,
Union A ∪ B = C ⇔ for ∀ x ∈ U x ∈U, so
(2.7)
Intersection A ∩ B = D ⇔ For x∀∈U, so
(2.8)
The common representations of a membership function are as follows in Table 2.1:
In where, if [L] = μL, then
(2.9)
Table 2.1 Representation of membership function.
Word
Membership
0.0 0.1
0.2
0.3
0.4
0.5 0.6
0.7
0.8
0.9
1.0
Very Small (VS) 1.0 0.8464 0.4624 0.1024 0.0064 0.0
Small (S)
1.0 0.92
0.68
0.32
0.08
0.0
Medium (LS)
0.0 0.08
0.32
0.68
0.92
1.0 0.92
0.68
0.32
0.08
0.0
Large (L)
0.0 0.08
0.32
0.68
0.92
1.0
Very Large (VL)
0.0 0.2828 0.5657 0.8246 0.9592 1.0
2.3.2 Mode of Expression
The subjective uncertainty of a basic variable A can be expressed by the following steps.
Let subjective uncertainty be Ei. The measurement criteria are as follows: (i) Mi is the level of Ei; (ii) Gi is the
importance of Ei. So
(2.10)
For all subjective uncertainties, there is
(2.11)
There should be a fuzzy conditional relationship between M and A
(2.12)
Therefore, the fuzzy relationship between subjective uncertainty and basic variable A is as follows
(2.13)
This means the membership function derived from the importance of subjective uncertainty is as follows
(2.14)
Accordingly
(2.15)
For this method, the key is how to determine the membership function μA(X) of the basic variable A.
2.4 Gray Theory Analysis Method
The gray theory[2-8] can be used to study an uncertainty problem featuring a small sample size and
insufficient information. The gray system (G system) is a semi-open and semi-closed system for a certain
level of understanding, where some information is known while the rest is unknown, i.e., the information is
incomplete.
A system containing completely clear information is called a white system; a system containing unknown
information is called a black system; a system containing some clear information and some unclear
information is called a gray system. System information incompleteness is divided into four types: (1)
incomplete element information; (2) incomplete structural information; (3) incomplete boundary
information; and (4) incomplete information on operating behavior.
2.4.1 Basic Concept
The gray theory is characterized by information incompleteness and the non-uniqueness of results obtained.
Therefore, theories have been devised including the principle of information incompleteness and the
principle of process non-uniqueness. The solving process for non-uniqueness is a combination of qualitative
and quantitative analyses. By supplementing information, qualitative analysis can be used to determine one
or several satisfactory solutions. That is how the gray system is solved.
System information incompleteness is divided into four types: (1) incomplete element information; (2)
incomplete structural information; (3) incomplete boundary information; and (4) incomplete information
on operating behavior.
If the connotation and extension of an object are completely certain, it is white; if it has an uncertain
connotation and a certain extension, it is gray; if it has a certain connotation and an uncertain extension, it is
fuzzy; if both connotation and extension are uncertain, it is gray fuzzy.
For the basic variable z, its value changes at interval h, making it a gray variable, denoted by ⊗(z).
(2.16)
or expressed in discrete form, ⊗(z) = {hi, i ∈ I, hi, ∈ R}; different coefficients on ⊗(z) are expressed in
fGray(z), where fGray(z) is the built-in function of ⊗(z), or its whitening weight function [fGray(z) ∈ (0, 1)].
The whitening weight function has the following attributes:
1. fGray(z) ∈ (0, 1)
2. z represents gray scale, such as high, medium or low gray scale.
Expressed in discrete form,
2.4.2 Case Study
For the carbonized depth in concrete[2-9], this can be represented by a gray area, and a region can be set for
β.
(2.17)
where i = 0, 1, 2, ……, N, N +1. A is the factor of carbon velocity. B is a parameter of concrete carbonization, is
about 0.4 to 0.6. Given the concrete carbonization qualification index do, the failure probability of concrete
carbonation durability Pf = P(do ≤ d) and
. Table 2.2 shows the list of digital reperesentation
of gray scale.
Table 2.2 Digital representation of gray scale.
0
1
Worst Extremely
bad
2
3
Bad Relatively
bad
4
5
6
Relatively
good
Good Slightly
good
7
8
9
10
Not very
good
Very
good
Extremely
good
Best
Because the corresponding carbonized depth in concrete is a weighted interval number at time t, β also has
a weighted average. The specific calculation process is as follows:
1. Given time
The weight
are taken in proper order at the interval of
,
corresponding to each value is calculated and denoted as
. (Normally, N should not be too small).
2. The formula for
is as follows
(2.18)
where i = 0, 1, 2, ……, N, N +1.
3. It is assumed that both d0 and
are random variables which obey normal distribution. So
(2.19)
where i = 0, 1, 2, ……, N, N +1
4. A curve of
is drawn, and the weighted average
is calculated
(2.20)
2.5 Relative Information Entropy Analysis Method
According to the relative information entropy theory and the definition of parameter and system
uncertainty, the objective uncertainty of a basic variable X can be represented by the Shannon entropy[2-10]
(2.21)
where p(xi) represents the distribution probability of the ith discrete point of variable xi. If the subjectivity
π(xi/R) of parameter uncertainty comes mostly from that of the probability distribution of variable xi and
that of the variable distribution interval caused by R, then the subjectivity can be represented by a relative
information ambiguity function, while its relative information entropy[2-11] can be described by means of
Equation (2.22)
(2.22)
Considering that the limit state function z=g(X) is a function of basic variables and xi is a relatively fuzzy
variable with parameter uncertainty, let
be expressed as a function of relatively
−1
fuzzy variables, where g(x) has an inverse function, g (x). Therefore, z=g(X) is also a relatively fuzzy
variable. So,
(2.23)
The right end of the equation can be obtained from the combination rule Equation (2.23) of relative
information. Hence, the relative information entropy of the structure caused by parameter uncertainty can
be written as:
(2.24)
where superscript P represents parameter uncertainty and H(Z) is the Shannon entropy of probabilistic
reliability analysis. So
(2.25)
where Pf represents the probability of structural failure revealed by probabilistic reliability analysis.
The impact of system uncertainty is taken into consideration to build a relative information ambiguity
function for system uncertainty. Therefore, the total relative information ambiguity function of parameter
and system uncertainties can be obtained from Equation (2.26)
(2.26)
Similar to Equation (2.25), the relative information entropy of the structure is as follows
(2.27)
The Shannon entropy method is used to represent the impact of parameter uncertainty and system
uncertainty in order to compare with the results of probabilistic reliability analysis, i.e.,
(2.28)
or H(Z/R, R’), where
represents the equivalent structural failure probability, a
comprehensive index of structural reliability, containing both objective uncertainty and subjective
uncertainty; correspondingly, the structural safety index[2-12][2-13] is
(2.29)
2.6 Artificial Intelligence Analysis Method
2.6.1 Neural Networks
An ANN (Artificial Neural Network) is a theoretical mathematical model of the human brain and its
activities. Composed of a great many processing units properly interconnected, an ANN is a large-scale
nonlinear adaptive system[2-14]. Featuring excellent nonlinear mapping and associative memory capabilities,
ANNs are widely used for function fitting under various complex relations, and show both great flexibility
and good adaptability. ANNs helps to provide more accurate calculation results without the aid of explicit
mathematical and physical models. Moreover, they can boast good fault tolerance and adaptability. These
characteristics make ANNs suitable for the time-variant reliability evaluation of structures.
Among neural network algorithms, the most widely used is a multi-layered feed forward neural network
based on error back propagation, the BP (Back Propagation) neural network. The BP learning algorithm is
adopted in the BP neural network, which is composed of both forward propagation and back propagation.
Forward propagation means transmitting the input signal from the input layer to the output layer via the
hidden layer. If the signal is properly output from the output layer, the learning algorithm will end;
otherwise, back propagation will be enabled. Back propagation involves reversely calculating the error
signal (the difference between sample output and network output) along the original connection route, with
the gradient-descent algorithm used to adjust the weight and threshold of neurons in each layer to reduce
the 4error signal. BP is a widely used neural network characterized by high operability, low computational
complexity and low parallelism.
Structurally, the BP neural network consists of an input layer, an output layer and one or more hidden layers.
The neural network is a process of mapping from the input surface to the output surface. This type of
mapping is realized by means of the transfer function. There are two kinds of transfer functions commonly
used in the BP neural network: purelin and sigmoid. Sigmoid functions include the symmetric tansig
function and the asymmetric logsig function. See Figure 2.1.
If there are enough elements in the hidden layer, two layers can approximate any continuous function.
Figure 2.2 shows a typical BP neural network containing two hidden layers. P represents the input
parameter; IWi represents the set of weights in the i-th layer; F1 and F2 are transfer functions tansig and
purelin in the first and second hidden layers, respectively; a1 is the intermediate variable generated after the
input parameter is transmitted through the first hidden layer, and also the input parameter of the second
hidden layer; the output parameter a2 is obtained after a1 is transmitted through the second hidden layer.
Figure 2.1 Three types of transfer function.
Figure 2.2 Diagram of two-layer BP neural netbook structures.
The following is the specific process by which the BP responding surface method is used for structural
reliability analysis:
1. The number and statistical characteristics of random structural variables, and their corresponding
sampling points, are randomly generated.
2. A structural finite element model is used to calculate the structural response at the sampling point to
solve the structural limit state function. The function value and random variables constitute neural
network learning samples.
3. The BP neural network model is used to determine structural parameters of the neural network, with
learning samples applied to neural network training.
4. The weight and threshold of the trained neural network are substituted into the response function to
establish an explicit expression for the limit state function.
5. FORM is used to calculate the structural failure probability and complete the structural reliability
analysis.
2.6.2 Support Vector Machine
SVM (Support Vector Machine) is a general machine learning method based on the statistical learning
theory. Its idea originated from the support vector method proposed by Vapnik et al.[2-15] in 1963 to solve
the pattern recognition problem. SVM maps the original space of the problem to a higher-dimensional
feature space, where the classification problem can then be solved. When the classification problem is
solved, it is considered that only the vectors on the classification boundary play a role in classification; these
vectors are called support vectors. In Figure 2.3, a, b, c, d and e are support vectors. This is where the
method gets its name.
The principle of structural risk minimization is applied to SVM, enhancing the generalization ability of the
learning machine. This means that a small error is still bound to be generated even if a solution is obtained
from limited training samples. There is a strict theoretical basis for SVM, capable of solving practical
problems featuring a small sample size, nonlinearity, high number of dimensions and a local minimum.
Considering this, the implicit limit state equation can be reconstructed in a new manner for structural
reliability analysis.
For response surface reconstruction by SVM, it is essentially a process of building a regression support
vector machine (RSM)[2-16]. For SVM, regression is the same as pattern recognition, because it is also a
process of mapping the input vectors in the original space to a higher-dimensional feature space, where a
classification problem is taken into consideration. The mapping from the original space to a higherdimensional feature space is realized by using the inner product in the feature space described by the kernel
that meets the Mercer condition. Figure 2.4 shows the basic process of how RSM is implemented.
Figure 2.3 Diagram of support vectors.
Figure 2.4 Diagram of regression support vector machine.
(1) SVM response surface reconstruction
A quantitative method that makes a trade-off between approximation precision and approximate function
complexity for a given set of sample data is adopted for RSM. A set of feature subsets is selected from the
training set so that the linear division of the feature subsets are equivalent to the segmentation of the entire
dataset. For RSM in linear loss functions, the following optimization problem needs to be solved
(2.30)
where vector w and scalar b control the position of the optimal classification plane; (xi, yi) is a training point
set; C is a penalty function, reflecting the coordination between structural and empirical risk; ε is the error
insensitivity coefficient; ξ and ξ* are slack variables.
Under Kuhn-Trucker conditions, Equation (2.30) is transformed into the following dual problem:
(2.31)
where α* and α are Lagrange multipliers; l represents the number of support vectors.
Equation (2.31) is a convex set programming problem and has a unique solution. In other words, there is an
obvious theoretical basis for the determination of the network topology of SVM. This is also one of its
advantages over neural networks. The complexity of the process of solving Equation (2.31) is determined by
the number of samples with a nonzero weight, i.e., the support vectors. During the development of its
solving algorithms, SMO (Sequential Minimal Optimization) [2-17] was gradually developed on the basis of
the Chunking and Osuna algorithms. SMO can decompose a large-scale QP (Quadratic Programming)
problem into a series of smaller QP problems containing 2 Lagrange multipliers only, making it possible to
solve the primal problem by a semi-analytical method.
After Equation (2.31) is solved, the optimal regression function of RSM is as follows:
(2.32)
The training point set in Equation (2.32) is composed of the empirical points corresponding to the input and
the response obtained by the structural finite element method. It is the structural limit state equation
reconstructed by SVM. Different kernel functions can be used to obtain different response surface equations.
Commonly used kernel functions include linear kernel, polynomial kernel, RBF (radial basis function, or
Gaussian) kernel and two-layer neural network kernel. However, there is no satisfactory method for
choosing an appropriate kernel function, meaning they need to be selected based on experience.
(2) Deduction of partial derivative for SVM regression function
The FORM/SORM for reliability calculation requires the use of a partial derivative of the limit state equation.
A first-order partial derivative is required if FORM method is used for calculation. For the reconstruction of
the response surface Equation (2.32), let support vector
, and input vector
. In this way, the first-order partial derivative of the response surface equation
constructed by linear kernel, polynomial kernel, RBF kernel and two-layer neural network (Sigmoid) kernel
can be deduced. Examples are as follows:
(i) Linear kernel
Kernel function
(2.33)
Its response surface function
(2.34)
In which b is a constant, so
(2.35)
(ii) Polynomial kernel
Kernel function
(2.36)
Its response surface function
(2.37)
so
(2.38)
where d represents the order of the polynomials.
(iii) RBF kernel
Kernel function
(2.39)
Its response surface function
(2.40)
so
(2.41)
where γi is a constant, which determines the width of the function around the center point.
(iv) Sigmoid kernel
(2.42)
Its response surface function
(2.43)
so
(2.44)
Where v(X·Xi)is a linear kernel, but S[•] is function of v(X · Xi). c and b is a constant, respectively.
(3) Calculation steps for SVM-based responding surface method
The traditional responding surface method for structural reliability analysis consists of two processes: the
reconstruction of a local response surface near the expansion point and the search for the design point. For
the SVM-based responding surface method, SVM is used to reconstruct the response surface, while the
geometric reliability method is used to calculate the design point and structural reliability. The steps are as
follows:
i. Let the initial expansion center point be X(1)=(x1(1), x2(1), …, xn(1)), with the average point taken
normally;
ii. Use FEM or another method to calculate the function value at the current expansion point and each
subsequent expansion point: g(x1(k), x2(k), …, xn(k)) and g(x1(k), x2(k), …, xi(k)±fσi, …, xn(k)), where the
value range of f is generally set to 1∼3, and k represents the number of iteration rounds;
iii. Normalize the data of the above empirical points;
iv. Substitute normalized 2N +1 sample points into Equation (2.31) to solve the quadratic programming
problem, obtaining the value of a*, α and b, thus establishing SVM response surface Equation (2.32);
v. Use the geometric reliability calculation method to calculate the checking and reliability index β, with
the partial derivative deduced according to Equations (2.35) ∼(2.44);
vi. Check whether the reliability index |β(k)-β(k-1)| and checking point function value |g(xk)| meet the
given accuracy requirements. If the conditions are met, finish iteration to obtain the reliability index; if
the conditions are not met, calculate a new expansion point, and return to step 2 to restart iterative
computation.
2.7 Example: Risk Evaluation of Construction with Temporary
Structure Formwork Support
2.7.1 Basic Information of the Formwork Support Structure
A 10-storey beamless floor structure with a storey height of 3m, column grid size 6,000mm×6,000mm,
column size 550mm×550mm, and slab thickness of 200 mm; concrete strength of the concrete slab and
column: C30; reinforcement in positive and negative moment field: HPB235 steel and HRB335 steel. Area of
reinforcement at support: 1,214 mm2/m; area of reinforcement at midspan: 808 mm2/m. Fastener-type
steel pipe formwork is used for support. Steel pipe φ48 mm×3.5 mm; vertical rod spacing 750 mm; step
distance 1,700 mm. Section stiffness of the bracing system is 6.4×103 kN, determined after considering the
influence of wood keel on bracing stiffness and the depreciation effect of steel uprights.
In particular, the length is calculated together with the stable bearing capacity of the formwork support
system, according to the code.
(2.45)
Where, h is the step distance, a is the length of the rod extending from the top cross bar of the formwork. In
this example, a is 0. The maximum value calculated by the code is taken as the stable bearing capacity of the
formwork. When the stability coefficient is calculated using the Monte-Carlo method, if the length
calculation coefficient μj < 1.0, then μj is taken to be 1.
2.7.2 Establishment of Construction Risk Evaluation System
In order to evaluate the construction risk of fastener-type steel pipe formwork, it is necessary to first
establish a risk evaluation system. The construction safety of the fastener-type steel pipe formwork is
affected by a variety of factors, which can generally be divided into two categories, the first of which is onsite construction. Throughout the entire process, from scaffold design to dismantling, each link directly
affects formwork safety. The second category is safety management, which must be underlined throughout
the whole construction process, and which exerts a significant influence on formwork safety. Both of these
can be further divided into sub-indices. Scaffold collapse is the main type of safety accident that occurs, with
every link in the formwork erection process affecting safety. Design scheme, materials and erection,
inspection and acceptance, and loading and dismantling all fall under key influential indices, so the
construction indices are divided into these six aspects. Safety management and site construction are closely
related and interact with each other. Effective auxiliary safety management is entailed in the whole
formwork construction process. A multi-index and multi-level risk evaluation system[2-18] was duly
established based on the above analysis, and after soliciting advice from numerous design and construction
experts for fastener-type steel pipe formwork.
Table 2.3 Indices and weights of a risk evaluation system for fastener-type steel pipe formwork support
construction.
Level A
Index
Level B
Level C
Level D
Level E
Index (S/N) Key
Index
Key Score Index
Key Score Index
Score
(S/N)
(Weight) (S/N)
(Weight) (S/N)
(Weight)
Construction Safety
7.38
Safety
6.28(0.28)
risk of
management (0.45)
education
formwork
(1)
(1)
support
Personnel 8.18(0.36)
ability (2)
Safety
7.98(0.36)
protection
(3)
Onsite
9.08
Design
8.15(0.17)
construction (0.55)
scheme (4)
Inspection 6.88(0.15)
and
acceptance
(5)
Dismantling 5.86(0.12) Dismantling 6.73(0.58)
(6)
time (1)
Component 4.96(0.42)
dismantling
(2)
Materials
9.88(0.21) Steel pipe 8.93(0.48)
(7)
quality (3)
Load (8)
Fastener
9.86(0.52)
quality (4)
7.13(0.15) Dynamic
load (5)
6.21(0.44)
(6)Stacking 7.97(0.56)
load (6)
Erection (9) 9.48(0.20) Support (7) 5.05(0.15)
Bottom
6.14(0.18)
horizontal
tube (8)
Scissor
4.16(0.12)
support (9)
Vertical
rods (11)
Fastener
9.02(0.27) Position 6.07(0.43)
installation
selection
(10)
(1)
(2)Proper 8.06(0.57)
torque (2)
9.11(0.28) (3)Aspect (3)
8.14(0.28)
Step distance (4)
8.34(0.29)
Verticality (5)
6.00(0.21)
Connection mode (6)
6.36(0.22)
Key
Score
(Weight)
The evaluation system includes a total of five levels:
Level A: Overall objective of evaluation (formwork construction risk)
Level B: Evaluation index set (safety management and construction safety)
Level C: Subset of evaluation indices (safety education, personnel ability, safety protection, design
scheme, inspection and acceptance, dismantling, materials, loads and erection)
Level D: Subset of evaluation indices (dismantling time, component dismantling, steel pipe quality,
fastener quality, dynamic load, stacking load, support, bottom horizontal tube, scissor support, fastener
installation and vertical rods)
Level E: Bottom evaluation index set (position selection, proper torque, aspect, step distance, verticality
and connection mode)
Table 2.3 shows indices and weights of a risk evaluation system for fastener-type steel pipe formwork
support construction.
2.7.4 Index Weighting
In the evaluation system shown in Table 2.3, the importance of each index varies, making it necessary to
determine the weighting of each index. This index weighting can be determined by such methods as the
Analytic Hierarchy Process (AHP) [2-19], expert scoring method, or others. To be specific, an expert scoring
method is adopted to determine the appropriate weighting by means of a questionnaire, the respondents to
which are construction site managers. To reflect the influence of the respondents’ educational background
μ1, age μ2, working years μ3, and safety education μ4 on the weighting judgment, the questionnaire also
contains some questions regarding these aspects.
Table 2.4 Rating scale table.
Score Grade
10
9
8
7
6
Score Grade
Most important
5
Extremely important 4
Important
3
Relatively important 2
Fairly important
1
Somewhat important
Less important
Some influence
Small influence
No influence
The importance of the indices in the questionnaire is further divided into 10 grades, from 1 (no influence) to
10 (most important). See Table 2.4 for details of this classification. A total of 84 questionnaires were
collected for the 10 projects investigated.
The calculation formula for the importance of each index can be expressed as:
(2.46)
Relative weighting of the indices can be expressed as:
(2.47)
Where, μi1, μi2, μi3 and μi4 represent the weighting of educational background, age, working years and safety
education, respectively. Their values are determined as shown in Table 2.5; Ri is the specific score of indices
in a single questionnaire received. See Table 2.4 for the scoring standard; n is the number of questionnaires
collected; Si is the importance of each index in the same level index; Wi is the relative weight of each index in
the same level index. See Table 2.5 for the importance and relative weights of each index after processing.
Table 2.5 Weighting of individual differences.
μ1
μ2
μ3 (y) μ4 (h) Si
Below junior high school
20-29 1-2
Senior high school and secondary school 30-39 3-4
Junior college
Undergraduate and above
<10 0.90
10-20 0.95
40-49 5-10 21-30 0.98
50-60 >10 >30 1.00
2.7.5 Expert Scoring Results and Risk Evaluation Grades
The score value x(x ∈ [0,10]) of each evaluation index indicates the actual implementation status.
Questionnaires were distributed to managers, who were required to score all grades of the indices according
to the realities on site, in order to determine the score for each index.
A total of 120 questionnaires were collected from the 10 projects investigated. Respondents were asked to
evaluate the risk of formwork construction in the industry using a 10-point system. See Table 2.6 for the
averaged results of the experts’ scores. Uncertainties in the scoring of various evaluation indices of the
formwork support system are inevitable, since they reflect the inherent features of the framework support
system. When using random statistical methods, only the random uncertainty of the index score value can
be considered, not its own fuzzy uncertainty. Therefore, the score value of each index was fuzzed, as shown
in Figure 2.5. In Figure 2.5, the score was
. After introducing the λ horizontal truncation set, let
, and the fuzzy interval that can obtain the score value can be expressed as:
(2.48)
Table 2.6 Expert ratings.
Index
x
Index
Safety education
Personnel ability
Safety protection
Design scheme
8.50 Steel pipe quality
5.10 Fastener quality
7.55 Dynamic load
9.35 Stacking load
x
Index
x
6.65 Position selection 7.45
6.15 Proper torque
7.35
7.35 Aspect
8.35
7.00 Step distance
8.55
Inspection and acceptance 7.35 Support
7.20 Verticality
7.80
Dismantling time
8.55 Bottom horizontal tube 5.50 Connection mode 8.00
Component dismantling 8.55 Scissor support
6.65
Figure 2.5 Fuzzineation of score values.
Where, the value of c reflects the fuzzy boundary range of random variables, which is determined according
to project realities; in general,
. If the fuzzy interval of the score value does not belong to
[0, 10], then any interval lower than 0 is treated as 0, and any interval higher than 10 as 10.
According to the realities of construction risk evaluation for fastenertype steel pipe formwork, the
evaluation grade was divided into five grades. V={v1, v2, v3, v4, v5}={Grade 1, Grade 2, Grade 3, Grade 4,
Grade 5}={Least safe, unsafe, relatively safe, safe, and safest}, represented by a fuzzy function in Figure 2.6.
Figure 2.6 Membership function of the evaluation grade.
2.7.6 Evaluation of a Fastener-Type Steel Pipe Scaffold
Taking the actual data collected as an example, the fuzzy grey correlation method and the traditional
multilevel grey correlation analysis method were used, respectively, to describe the step-by-step data flow
process in the risk evaluation system for formwork construction, and to explain the calculation method of
multi-index multilevel grey correlation analysis. Superscript (1), (2) and (3) in the following table (Table 2.7
and Table 2.8) show the calculation results for the fuzzy grey correlation analysis method as represented by
plane distance, lattice distance, and traditional multilevel grey correlation analysis, respectively.
Table 2.7 Correlation matrix for each index in the D-layer; relative weight and correlation coefficient matrix
of fastener and pole index.
Fastener
installation
Position selection
W10
0.43 0.41, 0.49, 0.68, 1.00,
0.79
0.57 0.41, 0.49, 0.69, 1.00,
0.77
W11
0.34, 0.34, 0.45, 1.00,
0.48
0.34, 0.34, 0.46, 0.94,
0.47
0.40, 0.53, 0.78, 1.00,
0.62
0.41, 0.54, 0.80, 0.97,
0.61
Aspect
0.28 0.41, 0.48, 0.63, 0.94,
0.95
0.39, 0.39, 0.43, 0.78,
0.81
0.34, 0.43, 0.58, 0.90,
0.67
Step distance
0.29 0.40, 0.47, 0.62, 0.91,
0.98
0.21 0.43, 0.50, 0.68, 1.00,
0.86
0.39, 0.39, 0.42, 0.73,
0.90
0.39, 0.39, 0.48, 1.00,
0.63
0.33, 0.42, 0.56, 0.86,
0.69
0.36, 0.46, 0.65, 0.94,
0.60
0.22 0.42, 0.49, 0.66, 0.98,
0.89
0.39, 0.39, 0.46, 0.91,
0.69
0.35, 0.45, 0.62, 1.00,
0.62
Proper torque
Vertical rod
Verticality
Connection mode
Then the correlation matrix for the fastener installation indices D10
The correlation matrix for the vertical rod indices D11
Table 2.8 Relative weighting and correlation coefficient matrix of materials and erection indices.
Material
W7
Steel pipe quality
0.48 0.46, 0.57, 0.84, 1.00, 0.73 0.51, 0.51, 0.81, 1.00,
0.56
0.52 0.49, 0.61, 0.92, 0.91, 0.68 0.51, 0.51, 0.94, 0.82,
0.51
W9
0.36, 0.50, 0.83, 0.67,
0.44
0.38, 0.55, 1.00, 0.59,
0.40
0.44, 0.59, 0.90, 1.00,
0.63
0.48, 0.73, 1.00, 0.58,
0.41
0.43, 0.60, 1.00, 0.81,
0.52
Fastener installation
0.15 0.42, 0.50, 0.71, 1.00, 0.76 0.40, 0.40, 0.54, 1.00,
0.51
0.18 0.49, 0.64, 1.00, 0.77, 0.57 0.43, 0.43, 1.00, 0.56,
0.43
0.12 0.462 0.57, 0.84, 1.00,
0.51, 0.51, 0.81, 1.00,
0.73
0.56
0.27
Vertical rod
0.28
Fastener quality
Erection
Support
Bottom horizontal
tube
Scissor support
2) Correlation degree matrix for each index in level C
Then the correlation matrix for the material index C7
The correlation degree matrix for the erection index C9
The calculation for dismantling and load index is the same as above.
3) Calculation of correlation degree matrix for each index in Level B
The calculation method is the same as that for Level C and Level D, so the detailed calculation process can be
omitted. The result is shown in Table 2.9.
Correlation degree matrix for safety management index B1
Correlation degree matrix of field construction index B2
Table 2.9 Relative weight and correlation matrix of index.
Construction risk of formwork support W B(1) B(2) B(3)
Safety management
0.45
Onsite construction
0.55
4) Calculation of correlation degree matrix for each index in Level A
The correlation degree matrix for construction risk of a fastener-type steel pipe formwork can be expressed
as:
5) Determining the construction risk of fastener-type steel pipe formwork
The correlation degree is a numerical representation of the close relationship between sequences. When
studying the correlation between reference sequences and compared sequences of a system, special
attention should be paid to the order of the correlation degree between reference sequences and compared
sequences, that is, the ranking of the correlation degree, rather than the magnitude of the correlation degree
in numerical terms[2-21]. According to the principle of maximum correlation degree identification, the results
of the three methods are consistent, and the construction risk grade for fastener-type steel pipe formwork
investigated on site is Grade 4, which falls within the safety scope. The ranking for the correlation degree
obtained by the first two methods where fuzziness is considered is completely consistent, but slightly
different from that obtained by the traditional method of multilevel grey correlation analysis. In the former
two methods, where the fuzzification of standard index values and score values are considered, the ranking
of the correlation degree better complies with the objective requirements.
2.7.7 Discussion and Summary Analysis
In this paper, fuzzy numbers were introduced into grey relational multilevel and multi-index analysis theory,
and a grey comprehensive evaluation method based on fuzzy numbers was proposed. The distance of the
fuzzy numbers is represented by plane distance and grid distance. Regarding the standard index value and
the index score value as fuzzy numbers enables the complexity and inherent fuzzy uncertainty of objective
items to be considered more easily, thus making multilevel grey correlation analysis more scientific and
bringing it closer to reality. This method is applied to the construction risk evaluation of fastener-type steel
pipe formwork. The results show that the grey correlation ranking obtained via the improved grey
correlation analysis method (based on two fuzzy number distances) is completely consistent, but slightly
different from the traditional grey correlation analysis method. This method can facilitate the
comprehensive quantitative evaluation of fastener-type steel pipe formwork safety, avoid subjective
randomness during risk evaluation, and help managers understand and master site safety in a timely
manner, thus improving enterprises’ ability to carry out proper accident control.
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3
Reliability Analysis Method
The reliability of engineering structures can be characterized by reliability probability
or failure probability. Structural reliability can be measured by means of the theory of
reliability. Structural reliability is defined as the probability of fulfilling a certain preset
function within the prescribed time under the specified conditions. In contrast, if the
structure fails to fulfill a certain preset function, then the corresponding probability can
be called structural failure probability. Generally, the reliability and failure are two
events which are incompatible with each other. Therefore, structural reliability
probability and failure probability are complementary [3-1].
For the convenience of calculation and expression, structural failure probability is often
used to measure structural reliability during structural reliability analysis. The core of
structural reliability analysis lies in calculating structural failure probability according
to the statistical characteristics of random variables and the limit state equation of the
structure.
In structural reliability analysis, the working state of the structure is generally
described by a function. When there are n random variables
affecting structural reliability, the working state of the structure is expressed by
Equation (3.1).
(3.1)
The safety probability that the structure fulfills the present function under the specified
conditions is represented by Ps; at the same time, if the structure fails to fulfill the
preset function, then the corresponding probability is called structural failure
probability, represented by Pf. Structural reliability and failure are two events
incompatible with each other, and there is a complementary relationship between them,
i.e., Ps + Pf = 1.
Because structural failure is an event of small probability (Pf is usually less than 0.001),
structural failure probability is often used to measure structural reliability, for the
convenience of calculation and expression.
Let the joint probability density function corresponding to the basic random variable
in the structure be
. The structural function
is shown as Equation (3.1). According to the definition of structural reliability and the
basic principle of probability theory, structural failure probability can be expressed as
(3.2)
In practical calculations, when there are multiple basic random variables in the
function, the limit state function is nonlinear, and variables are not independent of one
another, making the above equation difficult to solve directly. Therefore, this direct
integration method is not usually adopted. Instead, a simple approximation method is
applied, and for all random variables, only their digital eigenvalues are considered, with
their statistical characteristics described through mean and variance. Therefore, the
reliability index β is introduced and calculated in order to compute the corresponding
failure probability.
First, suppose the function variable Z obeys a normal distribution; its mean is μz and its
mean-square error is σz. Therefore, its probability density function is as follows
(3.3)
The structural failure probability is represented by the dashed area in Figure 3.1. The
expression of the failure probability is as follows
(3.4)
Figure 3.1 Diagram of structure failure probability.
Notation β is introduced. Let
(3.5)
Equation (3.4) can be converted into
(3.6)
The relationship between β and reliability can be expressed by the following equation
(3.7)
The dimensionless coefficient β in the equation is the above-mentioned reliability index.
For structural reliability, it can be expressed as
(3.8)
(3.9)
where Z = G(X) is the limit state function, Z < 0 (failure), while Z > 0 (safety), and Z = 0 is
the critical state. Df represents the failure zone corresponding to Z < 0, and f(X)
represents the joint probability density function (PDF) of random variables. Φ(•) is the
cumulation probability function (CDF) of standard Gauss distribution.
The structural reliability theory should therefore be used to solve the following four
problems:
1. Various parameters involved in the design are processed into random variables
according to the random theory. Also, the corresponding probability distribution
(distribution type and statistical parameters) is determined. The random variables
are not statistically independent, it is necessary to determine their joint
distribution; these are problems of statistical analysis.
2. Functions composed of basic variables and definite quantities are determined
according to the design requirements in order to establish a relevant limit state
equation. That is, Z = G(X) where G( ) represents a function space composed of
multiple limit state functions, and g( ) means that it is composed of a single limit
state function; Z < 0 represents failure while Z ≥ 0 represents safety; this is a
problem of setting up a structural failure model.
3. Corresponding failure probability can be obtained when a multi-dimensional
integral problem is solved within Df; this is a problem of numerical analysis.
4. An allowable failure probability value will be set for different structures and their
corresponding limit states, which may involve economic and social benefits; this is
a comprehensive problem.
A probabilistic design method fully considering the above four problems is called level
IV, while a method considering the first three methods only is called level III. This is an
ideal aspect of structural design; after simplification in different ways, it can be called
level II, level I, etc. The current structural design code remains at level II.
3.1 First-Order Second-Moment Method
The first-order second-moment method is a method by which the linear term in Taylor’s
expansion and the first two moments of random variables (first moment μ and second
moment σ) are adopted for calculation. Common first-order second-moment methods
include the central point method [3-2] and the checking point method.
3.1.1 Central Point Method
For the limit state function of a structure
(1) When Z is a linear function
(3.10)
where ai(i = 0, 1, 2, ……, n) is a constant. So,
.
As n increases, the distribution of Z becomes asymptotically normal. Therefore
(3.11)
(2) When Z is a nonlinear function
Z is expanded into Taylor’s series at the central point, with the linear term taken.
(3.12)
(3.13)
(3.14)
(3.15)
(3) When Z is non-normally distributed
Let
transform X space into U space. According to the central limit
theorem, U is normal space and the above conclusion can still be drawn. There are the
following cases.
Case 1:
If Z = R − S, let the transformation
,
, so
. Then, the safety index is
(3.16)
Currently, the R, S plane represents the distance from the central point to Z′ = 0,
respectively. Apparently, there is an error between the distance from the central point to
the Z′ = 0 plane and the distance to Z′ = 0. This error increases with the increasing
nonlinearity.
Case 2:
If both R, S obey lognormal distribution, respectively, and the limit state function is
expressed as Z = ln R − ln S, then, Z obeys normal distribution, and its mean and
variance are
(3.17)
(3.18)
When both δR, δS are less than 0.3, respectively, or nearly equal,
(3.19)
(4) Evaluation
1. The calculation for the central point method is easy to perform because β has a
clear physical concept.
2. The distribution pattern of the variables is not considered, so “probability”, a
reasonable index, cannot be used to measure structural reliability; nonetheless, the
distribution pattern of the variables has an impact.
3. For the case of the limit state function with nonlinear problem, there is a large
error.
4. For different mathematical descriptions of the same problem, the results are not
the same, since linear results are achieved for nonlinear problems at the central
point.
Table 3.1 Relationship between reliability index and failure probability Pf.
β 1.0
1.5
2.0
2.5
3.0
3.5
4.0
5.0
Pf 1.587×10- 6.681×10- 2.275×10- 6.21×10- 1.35×10- 2.326×10- 3.167×10- 2.867×101
2
2
3
4
4
5
7
3.1.2 Checking Point Method
The linearization point of the checking point method is located at the failure boundary
to overcome the problems with the central point method. It is also the location of the
design checking point X* (where the distance from the origin of the coordinates to the
limit state surface is the shortest) corresponding to the maximum possible failure
probability of the structure. This kind of first-order second-moment method is known
as the checking point method [3-3] or improved first-order second-moment method, and
is the basis for structural reliability index calculations. The checking point method could
be used in X-space or U-space.
(1) X-space
A linearized limit state equation can be established by selecting the design checking
point
.
(3.20)
Because X* is situated on the failure boundary, there is
So
Because
The sensitivity coefficient,
, there are
, represents the relative
influence of the ith variable on the entire standard deviation. And
.
So
(3.21)
This can also be expressed as
It has
(3.22)
There is a total of n equations in the above equation, where
and β are unknown
numbers, totaling n + 1. This needs to be solved by an iterative method. The specific
steps are as follows:
1. Assume a β value
2. Set the initial value of the checking point, generally taken as
3. Calculate
4. Calculate αi
5. Obtain
from
6. Repeat 3) ∼ 5) so that
7. Substitute
8. If
into Z(X), and calculate
, and
, go to 9); otherwise, calculate the Δ-value
and estimate a new β value according to
repeat 3) ∼ 7) until
,
, and then
.
9. Calculate the failure probability according to Pf = ϕ(−β)
(2) U-space
Let
, so E(ui) = 0. We have the limit state function from Equation (3.20)
(3.23)
The proposition of βmin may be calculated, or it can be expressed in optimal form in U
space.
Condition
(3.24)
Where f(U) represents the probability density function of U.
Thus, Equation (3.24) can be expressed as
, so f(u) → max.
Therefore, we can construct a function, Z(u) = f(u) + λg(u), where λ is a Lagrange
coefficient
where
So
When g(u) = g(u0) + gu(u0)(u−u0), we have
So, the iteration for the design checking point is
(3.25)
where
represents direction cosines of design checking point u*. fu(u)
and gu(u) is the derivative of f(u) and g(u), respectively.
.
Because u* is a design checking point on Z=0, so
(3.26)
We have
(3.27)
(3.28)
The above solution process is:
1. Select u* randomly
2. Calculate α,
3. Calculate
4. Compare u(m+1) −u(m) < ε; if it is unsatisfactory, repeat 2) ∼ 3), otherwise
5.
, and X* = μX + αβσX
3.1.3 Evaluation
Comparison with the central point method, there are
1. The calculation process needs to be iterated. This is a complex undertaking, but the
checking point is located on the failure surface, making it good for structural
design.
2. The basic variables are only considered to obey normal distribution, not other
forms of distribution.
3. There remains an error for nonlinear problems.
4. The sensitivity coefficient α can express the importance of this variable.
3.2 Second-Order Second-Moment Method
3.2.1 Breitung Method
The Breitung method [3-4] [3-5] could treat with random variables by mapping
transformation and analyze the problem in standard space. Let Y be an independent
standard normal random variable, expand the function Z into Taylor’s series at the
checking point, and take the first and second terms to obtain:
(3.29)
(3.30)
Let the unit vector be
(3.31)
Because
, and
, Equation (3.30) can be
written as
(3.32)
So
(3.33)
αX is used to construct an orthogonal matrix H, HT · H = I, αX is a certain column of H, set
to n for orthogonal transformation of Y space to U space.
(3.34)
, αX = (0,
Equation (3.34) is substituted into (3.32). It is noted that
0, ⋯, 0, 1)T, so Equation (3.32) can be rewritten as
(3.35)
where (HTQH)n−1 is an n-1-order matrix generated from HTQH with the nth row and nth
column ruled out,
The joint probability density function of Y is
.
(3.36)
By substituting Equation (3.34) into (3.36), we get
(3.37)
Structural failure probability
(3.38)
For Equation (3.35), according to Equations (3.37) and (3.38), the failure probability of
the second-order second-moment method is
(3.39)
Let
, and expand InΦ(t – β) by Taylor’s series at t = 0 and take
the first term to obtain
(3.40)
The above equation can also be written as
(3.41)
By substituting Equation (3.41) into (3.39), we get
(3.42)
The integrand in Equation (3.42) is compared with the normal joint probability density
function with a mean value of 0 and a covariance matrix of
.
Equation (3.42) can be simplified as
(3.43)
If the reliability index of the first-order second-moment method is obtained, then the
failure probability of the second-order second-moment method can be obtained from
Equation (3.43).
(3.44)
Where κi is the eigenvalue of the real symmetric matrix
, and
approximately describes the principal curvature of the limit state surface in the ith
direction.
The following steps can be applied in the above derivation process:
1. Use the first-order second-moment method to calculate the reliability index.
2. Calculate the unit vector
in Equation (3.31)
3. Determine the orthogonal matrix H in Equation (3.34)
4. Calculate Q in Equation (3.33)
5. Calculate failure probability in Equation (3.42)
3.2.2 Laplace Asymptotic Method
The Laplace method is a type of second-order second-moment method, in which the
second-order partial derivative of nonlinear functions is adopted. When Y obeys
independent standard normal distribution, the structural failure probability is
(3.45)
When the Laplace asymptotic integral method is used to calculate the above multiple
integrals, the Laplace integral containing large parameters is adopted:
(3.46)
The properties of integral Equation (3.46) are fully determined by the properties in the
neighborhood of the integrand maximum. If functions h(x) and g(x) are twice
continuously differentiable, while p(x) is continuous and h(x) only reaches its maximum
at x*, a point on the boundary of the integral domain, Equation (3.46) can be
approximately expressed as
(3.47)
where
(3.48)
Matrix
is an adjoint matrix of matrix
(3.49)
A large number, λ(λ → +∞), is chosen for transformation
(3.50)
The Jacobi determinant of the transformation is det JYV = λn.
By substituting Equation (3.50) into Equation (3.47), we get
(3.51)
Equation (3.51) is also a Laplace integral shown in (3.46), and
,
.
If the function is twice differentiable, then the asymptotic integral value of Equation
(3.51) is
(3.52)
where
(3.53)
Matrix B1(v*) is an adjoint matrix of matrix C1(v*)
(3.54)
By substituting Equation (3.54) into (3.53), we get
(3.55)
where
(3.56)
is an adjoint matrix of
(3.57)
Considering that β is generally a relatively large positive value, φ(β) ≈ βΦ(−β), Equation
(3.54) can be written as
(3.58)
The following steps can be applied in the above derivation process:
1. Calculate β, y* and x*
2. Calculate
3. Calculate
in Equation (3.30)
in Equation (3.57)
4. Calculate
det C
5. Calculate J in Equation (3.56)
6. Calculate PfQ, Equation (3.58)
3.2.3 Maximum Entropy Method
In 1948, Shannon introduced the concept of thermodynamics into the information
theory. If a random variable has n possible results, the probability of each result is pi. To
measure the uncertainty of this event, the following function is introduced:
(3.59)
where c is a constant greater than 0, meaning H is also greater than 0. H is called the
Shannon entropy. A certain event has only one result, pi=1, and there is no uncertainty,
so H = 0.
If random events obey continuous distribution with the probability density function of
f(x), then the Shannon entropy is
(3.60)
The Shannon entropy is used to measure the uncertainty of an event before it occurs;
after the occurrence of the event, the Shannon entropy is used to measure the
information obtained from it. This is used to measure the uncertainty of an event or the
amount of information it contains.
Under given conditions, there exists a distribution among all possible probability
distributions that enables information entropy to reach a maximum. This is called
Jaynes maximum entropy principle. The information entropy is maximized under
known additional information constraints, and the obtained probability distribution is
minimally biased, thus finding a way to construct an “optimal” probability distribution.
We consider taking the first m–order origin moment of a random variable x as
constraints, i.e., the information entropy is maximized by meeting the conditions in the
equation below
(3.61)
Using the Lagrange multiplier method, as well as Equations (3.60) and (3.61), a
modified function is introduced
(3.62)
At the stable point there is
,
, i.e.,
. Let
and the maximum entropy probability density
function can be obtained as follows,
(3.63)
Equation (3.62) is equivalent to the central moment of a given x
(3.64)
Usually, we can get the first four-order central moments of X, as follows
(3.65)
where Csx is the skewness coefficient and Ckx is the kurtosis coefficient.
In Pearson’s system, it is considered that the probability density function fX(x) of a
random variable x is determined by the ordinary differential equation below:
(3.66)
By integrating Equation (3.66), we can obtain a family of curves.
Equation (3.66) is a general form of Pearson’s family of curves, in which the parameters
can be expressed by the first four-order central moment of X, as follows
(3.67)
The following recursive relation exists between all-order central moments of the family
of curves:
(3.68)
Each order may be totally different from one another in terms of μxi, so x is converted
into a standard random variable,
, to avoid interrupting the solving process
during calculation. There is a relationship between x and y in terms of their every-order
central moment
(3.69)
According to Equation (3.69) and (3.70), it is noted that μy = 0, σy = 1, vyi = μyi. The first
four moments of y are as follows
(3.70)
For the standard random variable y, and with determined Pearson system parameters,
Equation (3.67) can be written as
(3.71)
Equation (3.71) can be used to derive higher-order moments if x in Equation (3.71) is
replaced by y.
Let the structural function be Z = g(X), where the statistical parameters of Xi in
are
moments are
, while the first four-order central
.
By expanding Z into Taylor’s series at the checking point and taking the quadratic term,
we can get
(3.72)
According to Equation (3.72), the first four moments of ZQ can be calculated as follows:
(3.73)
(3.74a)
(3.74b)
(3.74c)
Z is standardized as
. The maximum entropy probability density function
of random variable Y meeting the constraint condition remains in the form of Equation
(3.63). By substituting Equations (3.73) and (3.74) into Equation (3.61), we get a
system of integral equations
(3.75)
The coefficient in f(y) can be calculated.
The structural failure probability is
(3.76)
The following calculation steps can be applied in the above derivation process:
1. Calculate μZ in Equation (3.73)
2. Calculate μZi in Equation (3.74)
3. Calculate CsZ and CkZ in Equation (3.65)
4. Calculate vyi in Equation (3.70)
5. Calculate higher-order vyi in Equation (3.70)
6. Solve ai in Equation (3.75)
7. Calculate Pf in Equation (3.76)
3.2.4 Optimal Quadratic Approximation Method
If the moments of two random variables correspond to each other, then they also have
the same probability distribution and eigenvalue. With the moments as constraints in a
given inner product space, the undetermined coefficients of probability density function
polynomials can be calculated, thus making it possible to determine the probability
distribution form of Z and calculate the structural failure probability.
Let the function f(x) be continuous in [a, b], and pi(x)(i = 0, 1, ⋯, m) be m+1 linearly
independent continuous function in [a, b]. Its linear combination
is utilized to approximate f(x) so that the, integral
where λi(i = 0, 1, ⋯, m) is the coefficient and ρ(x) is the weight function in [a, b]. This is
the least square approximation problem.
According to
, the essential condition for the extreme value of the multivariate
function, linear equations with determined coefficients can be obtained as follows
(3.77)
Or rewritten as
(3.78)
where the elements of matrix A and the components of vector b are as follows
(3.79)
(3.80)
Matrix A is an m+1-order nonsingular matrix, meaning Equation (3.78) has a unique
solution.
Let the probability density function of random variable x be f(x). If pi(x) = xi, ρ(x) = 1,
then the following can be derived from Equations (3.78), (3.79) and (3.80)
(3.81)
(3.82)
If the every-order origin moment of x is known, λi(i = 0, 1, ⋯, m) can be worked out by
solving Equation (3.77), with the polynomial of least square approximation, p(x),
obtained, leading to
(3.83)
(3.84)
The following steps can be applied in the above derivation process:
1. Calculate μZ in Equation (3.73)
2. Calculate μZi in Equation (3.74)
3. Calculate CsZ in Equation (3.65)
4. Calculate
in Equation (3.70)
5. Calculate higher-order vyi in Equation (3.70)
6. Calculate the matrix A and the vector b in Equation (3.78)
7. Calculate the coefficient Aij and bi in Equations (3.81) and (3.82)
8. Solve λ in Equation (3.83)
9. Calculate Pf in Equation (3.84)
3.3 Reliability Analysis of Random Variables
Disobeying Normal Distribution
The random variables in the reliability calculation of engineering structures do not
always obey a normal distribution, whereas the definition of reliability index is based on
the premise that random variables obey a normal distribution. Therefore, when the
random variables do not obey normal distribution, it is usually necessary to transform
these variables to solve the reliability index. The following methods are usually used to
convert random variables into normal random variables.
3.3.1 R-F Method
The R-F (Rackwitz & Fiessler) method [3-6] is designed to solve the problem of arbitrary
distribution of random variables. The following conditions are proposed in this method:
(1) At the design checking point, F(x*)=Fnormal(x*), and the probability distribution
functions before and after transformation are equal; (2) at the design checking point,
f(x*)=fnormal(x*), and the probability density functions before and after transformation
are equal; under this condition, β and Pf can be calculated by the checking point method
for normal variables.
So, from condition (1), the following can be obtained
From condition (2), the following can be obtained
So
(3.85)
The following steps can be applied in the above derivation process:
1. Given β
2. For all i values, select the initial value of the design checking point,
3. Calculate
4. Calculate
5. Calculate the sensitivity coefficient αi
6. Calculate
7. Repeat (3) ∼ (6) so that
8. Work out the value of β that meets the following condition:
9. Repeat (3) ∼ (8) so that
3.3.2 Rosenblatt Transformation
The Rosenblatt transformation [3-7] is designed to solve the problem of random
variables with arbitrary distribution and correlation. According to the principle of
conditional probability, the non-normal random variables are transformed into
independent standard normal random variables.
Conditions:
1. Dependent independent X
2. Arbitrary distribution om variables with distribution
3. Calculate β and Pf by means of the checking point method
According to condition (1), if ri is an independent standard normal variable, then R =
TX, where T refers to the Rosenblatt transformation
(3.86)
where
.
Conditional probability density function
where
Therefore, the following can be derived by the inverting method
(3.87)
The above inverting method usually needs to be applied in combination with the
numerical method, making it rather difficult to use; here,
, there
are i possible combination.
For example, n = 2
Apparently, this free combination leads to a difference in the method of solving for X.
But fortunately, the conditional probability density function or distribution function is
not always known in engineering practice. Often, some estimates or correlations can be
utilized. Particularly, when xi is independent,
is linear.
The Rosenblatt transformation can transform one type of variable distribution into
another type, so
(3.88)
In particular, when u is an independent standard normal variable, there is
(3.89)
Transformed by Invert into
(3.90)
In U-space,
(3.91)
where: Jij is a coefficient of Jacobian matrix, and the Inverse matrix is as follows:
(3.92)
So, for the linear limit state equations, the relationship between them is
(3.93)
After the above transformation, the checking point method can be used to calculate β
and Pf. This is usually an iterative solution process.
The following steps can be applied in the above derivation process:
1. Given the initial value of the checking point y*(0)
2. Calculate
3. Calculate the reliability index β
4. Calculate αYi
5. Calculate the new checking point y*(1)
6. Judge whether the error meets the requirements, and complete the calculation if
this is the case; otherwise, go to (2) to continue iteration.
3.3.3 P-H Method
The P-H (Paloheimo-Hannus) method [3-8] lies between the central point method and
the checking point method. Its basic starting point is as follows: (1) arbitrarily
distributed random variables; (2) the limit state function (LSF) of multiple random
variables.
For LSF is g(X), if there is
where
is the probability function for the random variable x1. So, the quantile
can be determined by Pf or 1 − Pf.
For
For
β is the safety index, which can be worked out according to LSF g(X) = 0.
Under multivariable conditions, it is considered that is generally not at the quantile
point. αi can be used as a weighting coefficient to adjust the influence of each variable,
so,
(3.94)
And
where βi can be obtained by
The following steps can be applied in the above derivation process:
1. Given β, calculate Pf = Φ(-β), 1 - Pf, and let x* = μx
2. Calculate
3. Calculate βi,
4. Calculate αi
5. Calculate
6. Repeat steps (3) ∼ (6) so that
7. Check
, adjust the value of β, and repeat step (3) for iterative
computation
For the modified P-H method [3-9], which is based on the P-H method,
the equivalent normal random variable xi, and
is replaced by
(3.95)
3.4 Responding Surface Method
In a complex structure, when the relationship between a function g(X) and a random
variable X cannot be expressed explicitly, an appropriate and explicit functional
expression can be used to approximately express g(X). That is, the smallest number of
deterministic finite numerical values are used to fit a response surface to replace the
unknown real limit state surface so as to calculate its reliability by any known method
(as shown in Figure 3.2). This is known as the responding surface method, proposed
and applied by Box and Wilson [3-10].
The responding surface method is a comprehensive statistical test technique, in which
an inference method is used to reconstruct the limit state equation near the checking
point. For the reconstruction of a structurally complex approximate function, this means
that a series of variable values are designed, with every group of variable values forming
a test point. The structure is then calculated point by point to obtain a series of
corresponding function values. These variable and function values can be used to
reconstruct a clearly expressed functional relationship to calculate the structural
reliability or failure probability [3-11][3-12].
Figure 3.2 Responding surface function.
For n random variables x1, x2, …, xn, a lot of research findings show that due
consideration needs to be given to any of the following: simplicity, flexibility, calculation
efficiency and accuracy. A quadratic polynomial exclusive of cross terms is usually
adopted as the analytical expression of a responding surface, as follows
(3.96)
where a, bi and ci are all undetermined coefficients, with a total number of 2n+1.
For each group of random design variables x1, x2, …, xn corresponds to a response
.
There are a total of 2n+1 undetermined coefficients used to determine a, bi, ci (i=1, 2,⋯,
n) on the right side of the equation. 2n+1 groups of experiments can be used to
determine 2n+1 groups of responses
. Then, the linear equations can be solved to
work out a, bi, ci (i=1, 2, ⋯, n). Thus, the limit state equation of the structure can be
determined.
For the responding surface method, the key is to fit the limit state function to the
structure. A clear functional relationship can be established by fitting according to
variable values and function values to calculate the structural reliability or failure
probability. The traditional responding surface method can be used to reconstruct the
approximate limit state to work out the checking point X* and reliability index β. The
steps are as follows
Given an initial value point
. Usually, the average point is
taken.
The function values are worked out by finite element simulation, as follows:
and
,
with 2n+1 point values obtained, where f is set to 3 during the first iteration process,
and then set to 1 for iterative computation.
1. 2n+1 point values are substituted into the equation to solve the 2n+1 equations,
obtaining 2n+1 undetermined coefficients a, bi, ci, thereby establishing a function
for quadratic polynomial approximation; according to this function, the JC method
can be used to solve the checking points X*(k) and reliability index β(k).
2. Judgment of convergence conditions
(3.97)
If the conditions fail to be met, a new initial value point needs to be selected using
the interpolation method
(3.98)
3.
is substituted into step (2) for the next iteration until the set convergence
accuracy is satisfied.
During the reconstruction of LSF by the responding surface method, a rough
approximate quadratic function is reconstructed based on the initial test results. Then,
under the condition that the convergence conditions are met, the function is expanded,
obtaining a new initial value point. According to the new test results, the reconstructed
function is constantly adjusted so that the initial value point gradually approaches the
checking point. The expression that meets the convergence conditions represents the
real surface behavior near the checking point. At present, there are more responding
surface reconstruction methods than polynomial methods. AI methods, such as the
neural network method and the SVM method, can also be used for responding surface
construction.
3.4.1 Response Surface Methodology for Least Squares Support
Vector Machines (LS-SVM)
1) Principle of function estimation
On the basis of a set of fixed training sample sets {(xj, yj}; j=1, 2, …, l}, xj∈Rn and yj∈R,
the samples from the original space Rn are mapped to the feature space Rnh using a
nonlinear mapping Ψ(·), Ψ(x) = {φ(x1), φ(x2), ⋯, φ(xl)}. Optimal decision functions y(x)
= wTφ(x) + b, w ∈ Rnh and b ∈ R are constructed in this high dimensional feature
space. Then, the principle of Structural Risk Minimization (SRM) is used to find the
weight vector w and deviation b, i.e., by minimizing the objective function
(3.99)
where, the error vectors ej ∈ R and γ are adjustable hyper-parameters.
Defining the Lagrange function to solve the optimization issue above
(3.100)
Where, the Lagrangian multiplier αj ∈ R; according to Karush-Kuhn-Tucker (KKT)
conditions:
(3.101)
The following equation can be established:
(3.102)
The symmetric functions[3-13] defining the Mercer conditions for kernel functions:
(3.103)
According to Equation (3.103), the optimization issue is transformed into linear
equations:
(3.104)
Where,
,
,
,
, Kji = K(xj, xi) and j, i = 1, 2, ⋯, l. Solving Equation (3.104)
with the least squares method, and obtaining α and b, enables the predicted output to
be obtained.
(3.105)
Different support vector machines can be constructed by using different kernel
functions. Common kernel forms include:
1. Linear kernel:
;
2. Polynomial kernel of order d:
3. Radial basis kernel:
;
;
4. Two-layer perceptron neural network kernel:
,
where σ, κ and θ are adjustable constants.
Compared with the standard SVM, the least squares support vector machine (LS-SVM)
replaces inequality constraints with equality constraints. With its fast solution speed,
this algorithm can be transformed for solving linear equations.
2) Response Surface Methodology Combined with LS-SVM
A response surface analysis method for structural reliability based on LS-SVM is hereby
proposed to solve the problem of weak function approximation of the response surface
method. As a coupled form of the LS-SVM, FEA and Monte-Carlo numerical simulation
methods, this method creates a new concept for structural reliability analysis. The LSSVM program and Monte-Carlo simulation are compiled using MATLAB; the response
(such as stress or displacement) of the structure is calculated using the finite element
method, with all three integrated into a structural reliability analysis system. See Figure
3.3 for the program block diagram.
First, the structure is analyzed to determine the main failure modes; then the input
vectors of the learning samples are obtained by an orthogonal experimental design
method based on the probability distribution parameters of basic random variables.
The structural response values of the input vectors of the learning samples are then
obtained using standard finite element programs, such as ANSYS, SAP, or similar, to
finally determine the learning samples. Next, the hyper-parameters and kernel
parameters are determined using the cross validation method, so that the estimated LSSVM value for the learning samples approaches the calculation results using the finite
element method. The test samples are generated by means of sampling; a check is then
required of whether the detection requirements of LS-SVM have been met or not. If not,
the values of the hyper-parameters and kernel function parameters need to be adjusted
until the requirements above are met, thus establishing a nonlinear mapping
relationship between structural action and response. Finally, the failure probability of
the structure is determined by the Monte-Carlo method and the LS-SVM nonlinear
estimation function.
Figure 3.3 Response surface method based on LS-SVM.
3) Generation and initialization of learning samples
To establish the nonlinear mapping relationship between structural action (input) and
response (output) using LS-SVM, the learning samples of the LS-SVM must be selected
or designed based on the specific purpose. If the complete combination method is used,
there will be an excessive number of training samples, which will in turn lead to
excessive workload. In this test, learning samples were selected using an orthogonal
experimental design. Each random variable in the design sample is evenly selected at n
levels within the range [mi − 3σi, mi + 3σi], with the value of n depending on the number
of random variables. To be more specific, mi and σi are the mean and standard
deviations of each random variable. For a normally distributed random variable, the
probability that its value deviates from the mean by more than 3 standard deviations is
not greater than 0.13%. According to this statistical theory, the above learning samples
are highly representative.
To eliminate the influence of various factors caused by differences in dimensions and
units, the input and output of the sample are normalized by the following equation:
(3.106)
Where, zi and yi are the variables before and after normalization, and zmin and zmax are
the minimum and maximum values of z.
3.4.2 Examples
1) Example 1: Linear limit state equation
The linear limit state function is
, where the basic variable xi
follows the rule of standard normal distribution. The exact solution of the reliability
index for the limit state function is β=3. When n=2, 5, 7 and 10, SVM with a linear kernel
function is used to reconstruct the response surface, with the reliability indexes
analyzed by the SVM response surface method and the improved SVM response surface
method converging to form accurate solutions. See Figure 3.4 to see how the number of
FEM calculations varies with the number of variables n.
According to Figure 3.4, as the number of variables n increases, the effect of reducing
the number of valid element calculations using the improved method becomes more
pronounced, indicating that the actual effect of the improved method is related to the
number of variables.
Figure 3.4 Number of FEM calculations.
2) Example 2: Nonlinear limit state equation
If we choose the following three nonlinear limit state equations and calculate the
corresponding reliability using different kernel functions. The results are shown in
Table 3.2.
Example 2-1:
, where x1 ∼ N(10, 22), x2 ∼ N(2.5, 0.3752).
Example 2-2: g(X) = 1 + x1x2 − x2, where x1 ∼ LN(2, 0.42) and x2 ∼ N(4, 0.82).
Example 2-3:
N(2.18, 0.06542) and x3 ∼ LN(32.8, 0.9842)
, where x1 ∼ N(0.6, 0.07862), x2 ∼
3) Example 3 [3-14] : Portal plane frame
In the portal plane frame shown in Figure 3.5, the elastic modulus of each unit is E =
2.0×106 kN/m2, and the relationship between inertial moment and cross-sectional area
is
. The random variables are the sectional areas A1 and A2 of the
elements and the external load P. See Table 3.3 for the random characteristics. If the
horizontal displacement u3 (unit: cm) of Node 3 is taken as the maximum deformation
of the structure to be controlled, calculate its failure probability.
Table 3.2 Comparison of results.
Examples Calculation FORM
Response surface method
Improved response
content
method
surface method
of
SVM
Improved Improved
original Quadratic ANN
ANN
SVM
equation polynomial
Example Reliability 2.330
2.331
2.350 2.331 2.330
2.333
2-1
index
Design
(11.186, (11.012,
(11.137, (11.016, (11.183, (10.940,
point
1.655)
1.647)
1.645) 1.647) 1.655)
1.643)
Iteration
5
4
5
6
6
(times)
Finite
29
20
26
11
11
element
calculation
(times)
Example Reliability 4.690
4.690
4.690 4.690 4.691
4.690
2-2
index
Example
2-3
Design
point
Iteration
(times)
Finite
element
calculation
(times)
Reliability
index
Design
point
Iteration
(times)
Finite
element
calculation
(times)
(0.797,
4.929)
(0.798,
4.949)
6
(0.794, (0.797, (0.800,
4.837) 4.931) 5.00)
5
4
6
(0.797,
4.926)
3
35
25
21
11
8
1.965
1.965
1.965
1.965
1.965
1.965
(0.46,
2.16,
33.42)
(0.46, 2.16, (0.46,
33.42)
2.16,
33.43)
3
5
(0.46,
2.16,
33.43)
3
(0.46,
2.16,
33.43)
6
(0.46,
2.15,
33.43)
3
23
22
13
10
35
Figure 3.5 Portal frame calculation diagram.
Table 3.3 Probabilistic characteristics of the random variables in Example 3.
Random variables Average mi Standard deviation σi Distribution type
ai
A1(m2)
0.36
0.036
Logarithmic normal 0.08333
A2(m2)
0.18
0.018
Logarithmic normal 0.16670
P(kN)
20
5
Extreme value type I —
The limit state equation can be expressed as
(3.107)
Where, [u]=1cm is the maximum allowable horizontal displacement, and the
relationship between u3 and each random variable cannot be clearly expressed.
Firstly, learning samples need to be designed, and random variables selected from 5
levels {mi −3σi, mi −1.5σi, mi, mi + 1.5σi, mi + 3σi} to create a 3-factor, 5-level orthogonal
experimental design. 25 groups of random variables are selected, which are also taken
as the input of the finite element program for calculating the horizontal displacements
of Node 3, so as to obtain a learning sample for the LS-SVM.
The 25 learning samples are normalized according to Equation (3.105), and then input
into the prediction model. LS-SVM is used for sample learning. In consideration of the
favorable statistical performance of the radial basis function [3-15], this is selected for LSSVM learning as
. Kernel parameters σ and hyper-
parameters γ exert a significant influence on the generalization performance of LS-SVM.
After considering the fast solution speed of LS-SVM, the cross-validation method is used
to select parameters γ and σ. The parameter set is determined for γ and σ, from which
parameters are selected for combination. The LS-SVM is trained to select the best
parameter combination of the model. Their generalization ability is at its best when
hyper-parameter γ = 4 × 108 and kernel parameter σ = 5. The estimated mean error
εRMS of the learning sample and the test sample is calculated as follows:
(3.108)
Where, the εRMS of the learning sample is 2.027×10-5 and the εRMS of the test sample is
3.9×10-3. See Table 3.4 for the comparison between the FEM calculation results of OFEM
and the estimated LS-SVM function values
for the learning samples and the 10
test samples, which were all sampled according to the probability distribution of each
random variable. To be specific, ε is the relative error. Due to space limitations, Table 3.4
only lists the training results of certain learning samples and test samples with
relatively large errors. As shown in Table 3.4, the estimated values of the LS-SVM
function of the learning samples and test samples are quite close to the results of the
FEM calculation, indicating that LS-SVM can establish a correct nonlinear mapping
relationship between the functions.
After learning and detecting, A1, A2 and P will generate random sampling points
according to their probability distributions and the Monte-Carlo principle. Here, N=
100,000 sampling points are taken, and A1, A2 and P form the input vectors of the LSSVM. By inputting these vectors into the learned estimation function relational
expression, 100,000 displacement values of Node 3 can be obtained. If this is
substituted into the function Z shown in Equation (3.107), then the number of samples
nf for Z < 0 can be obtained, and the failure probability is
. A failure probability Pf = 2.25×10−3,
obtained by the traditional response surface method is found in the literature [3-14],
while a failure probability Pf = 2.322×10−3 is obtained when using 2,000 iterations of
the importance sampling method.
Table 3.4 LS-SVM learning results.
Sample no.
Learning
Samples
Tested
Samples
FEM calculation
results (cm)
1 0.4297
2 0.1488
3 0.1791
4 0.4880
5 0.3371
6 0.3371
7 0.0753
8 0.2029
9 0.2980
10 0.54280
1 0.8050
2 0.5938
3 0.3694
4 0.3471
5 0.4784
LS-SVM estimated
value (cm)
0.42977
0.14878
0.17912
0.48795
0.33707
0.33707
0.075295
0.20291
0.29799
0.54278
0.80499
0.59382
0.36936
0.34714
0.47835
Relative error
(%)
0.0159
-0.0123
0.0112
-0.0096
-0.0095
-0.0095
-0.0072
0.0053
-0.0037
-0.0031
1.3336
0.9208
-0.7098
0.4757
-0.3223
4) Example 4 [3-14]: Plane frame structure
In the calculation diagram of a plane frame structure for a 3-span 12-storey building (as
shown in Figure 3.6), the elastic modulus of each unit is E = 2.0×107kN/m2, and the
relationship between the unit section inertial moment and the section area is
. See Table 3.5 for the sectional characteristics of each unit.
The random variables are the sectional area A1 of the unit and the external load P. The
statistical parameters are shown in Table 3.5.
Figure 3.6 Calculation diagram for Example 4.
Table 3.5 Probabilistic characteristics of random variables in Example 4.
Random variables Average mi Standard deviation σi Distribution type
ai
A1 (m2)
0.25
0.025
Logarithmic normal 0.08333
A2 (m2)
0.16
0.016
Logarithmic normal 0.08333
Random variables Average mi Standard deviation σi Distribution type
ai
A3 (m2)
0.36
0.036
Logarithmic normal 0.08333
A4 (m2)
0.2
0.02
Logarithmic normal 0.26670
A5 (m2)
0.15
0.015
Logarithmic normal 0.20000
P (kN)
30.0
7.5
Extreme value type I —
Table 3.6 Effect of sample numbers on calculated results.
l
γ
25 7×104
σ εRMS
Pf
Learning samples Test samples
9 0.051
0.3305
7.259×10-2
49 1.5×105 11 0.0594
0.2437
7.680×10-2
Assuming normal conditions and a maximum allowable deformation [u]=9.6cm, the
following limit state equation can be established according to the code:
(3.109)
No explicit relationship is established between uA and the random variables.
To study the influence of the number of learning samples l on the calculation results for
failure probability, 5 and 7 levels of each random variable were taken, i.e. {mi − 3σi, mi −
1.5σi, mi, mi + 1.5σi, mi + 3σi} and {mi-3σi, mi-2σi, mi-σi, mi, mi + σi, mi + 2σi, mi + 3σi}; an
orthogonal experimental design with 6 factors and 5 levels, and one with 6 factors and 7
levels were carried out. 25 and 49 learning samples were selected, respectively, and the
failure probability of the structure was calculated using the response surface method
based on LS-SVM. See Table 3.6 for the results. To be specific, Pf was obtained following
N=100 000 iterations, and the average error εRMS was calculated using Equation
(3.108).
As learnt from Table 3.6, a correct mapping relationship is established by LS-SVM
between 6 random variables and the structural response uA. The response surface
method based on LS-SVM is insensitive to changes in sample number. It can thus be seen
that this method has strong learning ability for small samples and can greatly reduce
the workload of FEA. The failure probability Pf = 7.309×10−2 obtained by the traditional
response surface method is used in Literature [3-14], while the failure probability
obtained after 2,000 iterations is simulated using the importance sampling method.
References
[3-1] Zhao, G.-F., Jin, W.-L., Gong, J.-X., Structural Reliability Theory[M], China Building
Industry Press, Beijing, 2000, (in Chinese).
[3-2] Yao, J.-T., Reliability Assessment of existing structures based on uncertainty
reasoning[M], Science Press, Beijing, 2011, (in Chinese).
[3-3] Hasofer, A.M. and Lind, N.C., Exact and invariant second-moment code format [J]. J.
Eng. Mech. Div., 100, 1, 111–121, 1974.
[3-4] Breitung, K., Asymptotic approximations for multinormal integrals pl. J. Eng. Mech.,
110, 3, 357–366, 1984.
[3-5] Breitung, K., Asymptotic Approximations for Probability Integrals[M], SpringerVerlag, Berlin, 1994.
[3-6] Rackwitz, R. and Fiessler, B., Structural reliability under combined random load
sequences [J]. Comput. Struct., 9, 489–494, 1978.
[3-7] Rosenblatt, M., Remarks on a multivariate transformation [J]. Ann. Math. Stat., 23,
470–472, 1952.
[3-8] Paloheimo, E. and Hannus, M., Structural design based on weighted fractiles [J]. J.
Struct. Div. ASCE, 100, ST7, 1367–1378, 1974.
[3-9] Zhao, G.-F. and Wang, H.-D., A practical analysis method for structure reliability in
generalized random space[J]. China Civ. Eng. J., 29, 4, 47–51, 1996, (in Chinese).
[3-10] Box, G.E.P. and Wilson, K.G., On the experimental attainment of optimum
conditions[J]. J. R. Stat. Soc., 13, 1–45, 1951.
[3-11] Jin, W.-L. and Yuan, X.-X., LS-SVM-based responding surface method for Structural
Reliability[J]. J. Zhejiang Univ. Eng. Ed., 41, 1, 44–47, 2007, (in Chinese).
[3-12] Jin, W.-L., Tang, C.-X., Chen, J., SVM-based responding surface method for
Structural Reliability[J]. Chin. J. Comput. Mech., 24, 6, 713–718, 2007, (in Chinese).
[3-13] Vapnik, V.N., The nature of statistical learning [M], Spring, Berlin, 1995.
[3-14] Guofan, Z., Reliability theory and its applications for engineering structures[M],
Dalian University of Technology Press, Dalian, 1996.
[3-15] Chappelle, O., Vapnki, V.N., Bousquet, O., et al., Choosing multiple parameters for
support machines [J]. Mach. Learn., 46, 131–159, 2002.
4
Numerical Simulation for Reliability
The numerical simulation of reliability is an important method for reliability
calculation. The Monte-Carlo simulation, also known as random sampling,
probability simulation or statistical test, is a method by which stochastic
simulation is performed to study objective phenomena [4-1]. Based on the
statistical sampling theory, this method is a numerical calculation used to study
random variables by means of a computer. Because it is based on probability
and mathematical statistical theory, some physicists named it after Monte-Carlo,
a city famous for gambling located on the borders of France and Italy, to
highlight its randomness.
The basic idea of the Monte-Carlo method is that if the probability distribution
of state variables is known, then according to the limit state equation of
structure,
, a group of random numbers that obeys
the probability distribution of state variables, x1, x2, ⋯, xn, is generated by the
Monte-Carlo method and substituted into the state function
to work out a random number of the state function.
Then, N random numbers of the state function are generated in the same way. If
M of the N random numbers are less than or equal to 1, or less than or equal to
zero, then when N is large enough, and according to the law of large numbers,
the frequency is approximate to the probability. This means that failure
probability can be worked out as follows:
(4.1)
If necessary, the mean μg and standard deviation σg can be calculated according
to the N known g(X) values to obtain the reliability index β.
For the Monte-Carlo method, a group of random samples is generated based on
the probability distribution of basic variables and substituted into the limit state
function (LSF) to confirm or rule out structural failure. Finally, the structural
failure probability is calculated. This method has the following characteristics:
1. The proficiency of simulation has nothing to do with the dimensions of the
basic random variables;
2. The complexity of the LSF has nothing to do with the simulation process;
3. It can solve problems directly, without the need to linearize the LSF and
perform “equivalent normalization” of random variables;
4. The error of numerical simulation can be easily determined, making it easy
to determine the amount and accuracy of the simulation;
5. Considering failure probability, the frequency and amount of the simulation
can be very large.
The first four characteristics are not possessed by other reliability calculation
methods, while the fifth characteristic becomes less and less prominent as
computer technology has continued to improve. Therefore, simulation methods
have achieved rapid development in recent years.
However, when using the Monte-Carlo method in practice, we must consider:
1. Developing a systematic method that supports numerical “sampling” of
variable X;
2. Selecting an appropriate, economical and reliable simulation technique or
“sampling strategy”;
3. Considering the effect of the simulation technique on the calculation
complexity of the LSF and the basic variables;
4. Determining the “sampling” size for a given simulation technique to achieve
an acceptable effect;
5. Allowing the total or partial correlation between basic variables.
4.1 Monte-Carlo Method
The failure probability of engineering structures can be expressed as
(4.2)
Whose structural reliability index is
(4.3)
where X = {x1, x2, ⋯, xn}T is a vector with n-dimensional random variables; f(X) =
f (x1, x2, ⋯, xn) is the joint probability density function of basic random variable
X, and when X represents a group of random variables independent of one
another,
; G(X) represents a set of LSFs. When
G(X)<0, it means that structural failure has occurred; when this is not the case,
the structure is safe; Df is a failure zone corresponding to G(X); Φ(·) is a
cumulative probability function that obeys standard normal distribution.
Thus, Equation (4.2), expressed by means of the Monte-Carlo method, can be
written as
(4.4)
where N represents the total amount of sampling simulation; when
, or otherwise,
; the mark “^”
represents the sampling value. So, the sampling variance in Equation (4.4) can
be expressed as
(4.5)
When a 95% confidence interval is selected to limit the sampling error for the
Monte-Carlo method, we obtain
(4.6)
Or, expressed as relative error ε:
(4.7)
Considering that
approximated to:
is usually a small quantity, the above equation can be
(4.8)
Given ε=0.2, the number of samples taken must satisfy
(4.9)
This means that the number N of samples taken is inversely proportional to
when
is a small quantity, i.e.,
;
, Pf cannot be estimated reliably
enough unless N=105; however, the failure probability of engineering structures
is usually low, indicating that N must be large enough to make a correct
estimation. Obviously, it is very difficult to use the Monte-Carlo method for the
reliability analysis of engineering structures in this way. Only by using the
variance reduction technique to reduce N can we use the Monte-Carlo method
for reliability analysis.
4.1.1 Generation of Random Numbers
There are several methods for random number generation, as follows:
(1) Linear multiplicative congruence method
① Multiplicative congruence method
(4.10)
where a is the multiplier, m is the modulus, and (mod m) means taking
remainders after division by m. That is,
, where m is the
remainder. ri is a random number uniformly distributed in [0,1].
② Mixed congruence method
(4.11)
where C is the coefficient of correlation between increments xi and xi+1, and its
upper bound is
.
When c = 0,
, and its upper bound is a minimum.
(2) Generalized congruence method
(4.12)
is a deterministic function, xi is an integer in
where
, and ri is a uniform random number in
If
method.
.
, we call it the quadratic congruence
If
, we call it the additive
congruence method.
(3) Random number sequences
① Multiple sequences of random numbers that do not overlap with one another
Different initial values are given so that the sequences do not overlap with one
another.
② Dual random number sequence
For two sequences
,
,
,
they must satisfy
(4.13)
③ Reverse order random sequence
For two sequences
,
they must
satisfy
(4.14)
where c represents the cycle length of the random number sequence.
4.1.2 Test of Random Number Sequences
When a random sequence is generated by one method, the randomness of this
sequence cannot be ensured, making it necessary to check whether it is
significantly different from the real uniform random numbers in [0, 1] in terms
of properties. If the difference is significant, the samples obtained from the
random variables based on the random numbers generated by this random
number generator cannot reflect the properties of the random variables,
thereby making it impossible to achieve reliable results for random simulation.
If a pseudo-random number sequence that has been generated passes a certain
randomness test, it could only be said that it is not contradictory to the
properties or law of random numbers. We cannot reject it, but it cannot be said
that they possess the properties and law of random numbers. Therefore, for the
testing of the pseudo-random number sequence, the more tests it passes, the
more reliable the random number sequence is. The following are methods for
random number testing:
1. Parameter test: Used to test the significance of difference between the
observed value and the theoretical value of its distribution parameters.
2. Homogeneity test: Also known as frequency test, used to test whether the
difference between the empirical frequency and theoretical frequency of
the pseudo-random number is significant.
3. Independence test: Used to check whether the independence and statistical
correlation of pseudo-random numbers generated are abnormal.
4. Test by combination rule: According to the generated sequence of random
numbers and a certain combination rule, this test is conducted to check
whether there is a significant difference between the observed value and
the theoretical value of the combination.
5. Discontinuity test: Used to test whether each random number appears
continuously (such as continuous rise or continuous decline).
4.1.3 Generation of Non-Uniform Random Numbers
(1) Inverse transformation method
For the continuous random number x of the distribution function F(x), if there is
a uniform random number r in (0, 1), there is also an inverse transform for the
generation of x
(4.15)
That is, F(x) is a strictly monotonic increasing function, which has an inverse
function, so
(4.16)
where for the random numbers in
.
(2) Acceptance-rejection sampling method
Let the PDF f(x) of random numbers be bounded, with a value range of
,
so
(4.17)
Steps:
① Generate random number x in
② Generate random number y in
③ When
, x is accepted as the required random number
④ Otherwise, repeat (1) ∼ (3)
Conditions:
① It is good if the area under the f(x) curve occupies a larger proportion of
the rectangular area.
② Applicable when it is hard to find the inverse function, and only when
qualified sampling is used.
4.2 Variance Reduction Techniques
The Monte-Carlo method is extremely useful. However, if the structural failure
probability of an actual engineering structure is less than 10-3, the Monte-Carlo
method requires a very large amount of simulation, which occupies a lot of
calculation time. That is the main problem with this method in terms of
structural reliability analysis. Therefore, during the practical application of the
Monte-Carlo method, it is often necessary to use sampling techniques to reduce
the variance [4-2] and thus reduce the frequency of simulation.
The variance reduction technique is a very important method. It can be divided
into dual sampling, conditional expectation sampling, importance sampling,
stratified sampling, edge number control and correlated sampling [4-2].
4.2.1 Dual Sampling Technique
If U represents a set of samples uniformly distributed in [0, 1], and the
corresponding basic random variable is X(U), where X obeys the distribution of
the probability density function (PDF) f(x1, x2, … xn), there exist I-U and X(I-U) as
well, which are negatively correlated with U and X(U). Therefore, the simulation
estimation of Equation (4.4) is
(4.18)
Apparently, Equation (4.18) is the unbiased estimation of Pf, and the variance of
simulation estimation can be expressed as
(4.19)
where
is negatively correlated with
,
. Therefore, the variance of the simulation
estimation is always smaller than the sampling variance obtained by the MonteCarlo method. It should be noted that the dual sampling technique does not
change the original process of sampling simulation estimation, but merely
makes use of the negative correlation among subsamples to reduce the number
N of sampling simulations. Therefore, the dual sampling technique can be
combined with other variance reduction techniques to further improve the
efficiency of the sampling simulation.
4.2.2 Conditional Expectation Sampling Technique
If there is a conditional expectation E(Pf | xi), then as long as a basic random
variable xi. exists, and it is also a random variable, then its sampling simulation
estimation can be expressed as follows:
(4.20)
Correspondingly, the variance of the simulation estimation is
in
is a variable expected to be estimated. So
(4.21)
where Y = {xl, x2, …, xi-1, xi+l, …, xn}T; so
(4.22)
Therefore, not only does the conditional expectation sampling technique reduce
the variance of the sampling simulation, but it is also very beneficial to the
calculation of truncated distribution probability in Equation (4.20).
4.2.3 Importance Sampling Technique
If there exists a sampling density function h(X), and the following relation is
satisfied
(4.23)
then Equation (4.2) can be written in the form of importance sampling
(4.24)
Then the unbiased estimation of Equation (4.24) is
(4.25)
where
represents a sample vector taken from the sampling density function
h(X); the variance of its sampling simulation is
(4.26)
When the sampling density function is
(4.27)
the simulated variance of Equation (4.26) reaches a minimum. It should be said
that Equation (4.27) just provides a way to select h(X), but h(X) is very difficult
to select because this depends on the distribution form of the random variables,
the LSF and the sampling simulation accuracy [4-3][4-4]. However, Equation (4.26)
has both upper and lower bounds, which can be obtained by means of the
Cauchy-Schwary inequality, i.e.,
(4.28)
Equation (4.28) can also be expressed as the ratio of the PDF f(X) of random
variables to the sampling probability density function h(X), i.e.,
(4.29)
This expression provides the boundaries of the sampling function h(X).
Obviously, Equation (4.27) is the upper boundary of the sampling function h(X),
requiring that h(X) should satisfy the
ratio in the failure zone Df.
Moreover, all samples should fall within Df (see Equation (4.23)). At the same
time, FU [4-5] has proved the upper boundary of Equation (4.29), as follows
(4.30)
where X⋆ is the maximum likelihood point on the LSF. Considering that the ratio
of Equation (4.26) is always greater than or equal to , and that f(X) can always
find such a point in the gradient direction of X⋆ to meet the conditions of
Equation (4.27), this point is selected as the subdomain center of the failure
zone so that at a given confidence level, the sampling mean obtained from this
subdomain meets the conditions of Equation (4.27). This observation is very
useful for the construction of sampling function h(X), implying that the sampling
center may be in the gradient direction of X⋆ and near X⋆ in the failure zone.
The above observation has been partially reflected in the existing importance
sampling methods [4-6][4-7][4-8][4-9][4-10][4-11]. However, there is still the tricky
problem of how to effectively determine the importance sampling density
function (type and parameters).
4.2.4 Stratified Sampling Method
The concept of stratified sampling [4-12][4-13] is analogous to that of importance
sampling because both of them require the samples which contribute a lot to Pf.
However, stratified sampling does not change the original density function, but
simply divides the sampling interval into further subintervals, keeping different
numbers of sampling points in the subintervals so that more samples should be
taken from those subintervals which make greater contributions.
The domain of integration G(X)<0 is divided into M mutually disjointed
subintervals Lj, and from each subinterval, Nj uniform random number vectors r
uniformly distributed within this subinterval are taken. Here, Nj represents not
only the number of uniform random number vectors generated in the j-th subinterval, but also the frequency of the uniform random number vectors falling
into this subinterval from [0, 1]. In this way, the simulation results for stratified
sampling can be written as:
(4.31)
The simulated variance is:
(4.32)
where
; correspondingly, the
estimated value of simulated variance is:
(4.33)
In the simulated variance Equation (4.33), it can be proved that if
(4.33)
and
(4.34)
then
(4.35)
Therefore, in order to reduce the simulated variance, the number of samples
taken from each subinterval should be directly proportional to the product of
the standard deviation and of this subinterval and its volume.
4.2.5 Control Variates Method
Suppose Equation (4.2) can be divided into two parts [4-2]:
(4.36)
where
Suppose
has an analytical solution. Therefore, only
simulation method, so
can be solved by the
(4.37)
The corresponding simulated variance is:
(4.38)
The estimation formula for simulated variance is:
(4.39)
As can be seen from Equation (4.38), if y(X) is very close to f(X), i.e.,
, then
. So, y(X) is called the control
variate of f(X).
The significance of the control variates method is that if part of the known
analytical solution is contained in the problem to be solved, then the variance of
sampling simulation can be significantly reduced by removing this part and
calculating the difference using the simulation method.
4.2.6 Correlated Sampling Method
Usually, one of the main purposes of simulation is to ascertain the effect of slight
changes that take place in the system. Thus, it is necessary to carry out
simulation repeatedly. If two tests are independent of each other, then the
variance of the difference between the simulation results is the sum of the
variances of each simulation; if the same random number is used in the two
tests, the test results are highly correlated with each other, thus making it
possible to reduce the test variance of the difference between the test results [42]
.
Consider the following two integrals:
(4.40)
(4.41)
Their difference is
and
.
are estimated by means of the expectation estimation method. So
(4.42)
where X1 and X2 are random number vectors with the density function of f1(X)
and f2(X), respectively. They are generated by the same set of uniform random
number vectors in [0, 1], so
(4.43)
and
are positively correlated with each other. So
(4.44)
Therefore
(4.45)
To facilitate the calculation of the estimated value of test variance, the same
number of tests, N, is taken for both tests. So
(4.46)
then
(4.47)
4.3 Composite Important Sampling Method
4.3.1 Basic Method
For the LSF of a structure
(4.48)
there exists
(4.49)
so
(4.50)
or
(4.51)
where
is the probability value of the random variable xk at
correspondingly, the PDF in Equations (4.52) and (4.53) is
;
(4.52)
or
(4.53)
If there exists an importance sampling density function h(X), as follows
(4.54)
Then the failure probability [4-4] in Equations (4.2) and (4.24) can be expressed
as
(4.55)
or
(4.56)
Equations (4.55) and (4.56) are the expression (4.52) of conditional expectation
sampling. According to the results of Equation (4.53), Equations (4.55) and
(4.56) also have good sampling efficiency.
If a normal distribution probability density function is chosen as the sampling
density function of xi(i≠ k) in Equation (4.54), then Equations (4.55) and (4.56)
will have higher sampling efficiency; during the selection of parameters for the
sampling distribution probability density function
, the sampling
function h(X) is required to satisfy at least the first and second moments of the
random variable in the failure zone, i.e.,
(4.57)
(4.58)
or
(4.59)
(4.60)
For the random variable xi in X ∈ Df, this can be selected according to Equation
(4.49) or by the Taylor expansion of Z = G(X); here, xi is allowed to take its value
near the boundary domain of Df, and the accumulated deviation will be
eliminated in Equation (4.49); therefore, for the selection of xk, it is necessary to
consider the univariate form in LSF, as well as the variable with high
discreteness in probability distribution.
After Equations (4.59) and (4.60) are determined, the importance sampling
density function becomes
. Therefore, Equation (4.55)
can be rewritten as
(4.61)
where
represents the sample
generated by the importance sampling density function; φ[·] is a standard
normally distributed probability density function.
4.3.2 Composite Important Sampling
For the extremal distribution function,
can be obtained by the
deterministic method. For other types of PDF,
can be obtained by
sampling. An effective importance sampling method will be given here.
If the importance sampling density function is set to a truncated distribution
function [4-2], as follows
(4.62)
where d represents the truncated value, as shown in Equation (4.49); σ1 is a
parameter that varies with the probability distribution of the basic variable x.
When the basic variable obeys the normal distribution N(μ, σ2), σ1 can be
expressed as
(4.63)
Figure 4.1 shows the form of the truncated distribution probability density
function; Table 4.1 shows the change of σ1 with σ when G(X)=3.0 - x; as can be
seen, the results obtained in Equation (4.63) are obviously better than other
results; when the basic variable obeys lognormal distribution, it can be
transformed into normal distribution first, before the value of σ1 can be worked
out using Equation (4.63). It can then be inverted to lognormal distribution.
Therefore, the sampling simulation of
and
can be expressed as
(4.64)
where is a sub-sample generated by hT(x); M represents the number of
samples. Table 4.2 shows a comparison of the results achieved by different
sampling simulation methods. ISM is the important sampling method, whose
sampling function obeys normal distribution; its mean is the “design checking
point” and the mean-square error of sampling is set to unit value. IFM is an
iteratively faster Monte-Carlo method; IISM is an improved numerical
simulation method; AIISM is the IISM with dual sampling technique. As can be
seen, IISM reflects the simulated sampling results of Equation (4.64).
Figure 4.1 Probability density function with truncated distribution.
Table 4.1 Simulation results of σ1 versus σ when G(X)=3.0 - x.
N
σ = 0.5
σ = 0.1
σ1 = 0.5 σ1 = 1.0 σ1 = 2.0 Eq.
(4.64)
(×10(×10(×1010
10
10
(×10)
)
)
10
)
σ1 =
σ1 =
σ1 =
Eq.
(4.64)
0.5
1.0
2.0
-3
(×10- (×10- (×10- (×10 )
3
3
3
)
)
)
9.577
1.504
1.316
0.997
1.316
7.940
18.480 9.992
50
100
10.480 10.127 8.481
10.420 9.281
7.327
9.832
9.826
1.187
1.236
1.329
1.326
1.544
1.339
1.329
1.326
200
9.630
9.880
1.326
1.343
1.394
1.343
9.692
9.297
N
σ = 0.5
σ = 0.1
σ1 = 0.5 σ1 = 1.0 σ1 = 2.0 Eq.
(4.64)
(×10(×10(×1010
10
10
(×10)
)
)
10
)
σ1 =
1.0
(×103
)
1.343
1.348
σ1 =
2.0
(×103
)
1.319
1.334
Eq.
(4.64)
(×10-3)
500
1000
9.834
9.755
9.898
9.588
10.080 9.870
9.430
9.862
σ1 =
0.5
(×103
)
1.234
1.370
2000
9.902
9.878
10.130 9.866
1.337
1.349
1.340
1.349
5000
9.796
9.751
10.110 9.860
1.340
1.350
1.384
1.350
10000
9.822
9.691
9.994
9.864
1.378
1.350
1.352
1.350
1000000 9.864
9.849
9.901
9.866
1.344
1.350
1.350
1.350
Note: 1. LSF G(X)= 3.0 - x, x ∼ N (0, o2), Pf = 9.866×1G-10
2. When σ = 0.5, σ1| Equation (4.64) = 0.25, exact value of Pf = 9.866×10-10;
When σ = 1.0, σ1| Equation (4.64) = 1.00, exact value of Pf = 1.350×10-10.
1.343
1.348
Table 4.2 Results of different sampling simulation methods.
N
ISM
IFM
IISM
AIISM
Pf(×10- Var
Pf(×10- Var
Pf(×10- Var
Pf(×10- Var
10
)
1.161
)
0.790
)
1.316
)
1.364
50
1.239
1.020
(-7)
1.233
2.005
(-8)
1.329
2.040
(-10)
1.353
2.793
(-11)
100
1.339
7.007
(-8)
1.234
9.233
(-9)
1.326
1.173
(-10)
1.351
2.166
(-11)
200
1.464
3.449
(-8)
1.476
2.989
(-9)
1.343
5.757
(-11)
1.352
1.052
(-11)
50
1.421
1.368
(-8)
1.386
6.252
(-9)
1.343
2.230
(-11)
1.351
4.234
(-12)
1000
1.333
1.375
1.039
(-11)
5.488
(-12)
1.350
1.323
2.168
(-9)
9.838
(-10)
1.348
2000
6.311
(-9)
3.030
(-9)
2.125
(-12)
1.025
(-12)
5000
1.368
1.235
(-9)
1.336
2.952
(-10)
1.350
2.164
(-12)
1.350
3.895
(-13)
10000
1.360
6.119
(-10)
1.342
1.332
(-10)
1.350
1.087
(-12)
1.350
1.888
(-13)
1000000 1.350
6.169
(-12)
1.346
2.218
(-12)
1.350
1.070
(-14)
1.350
1.915
(-15)
3
2.892
(-7)
3
1.363
2.520
(-8)
3
1.349
1.906
(-9)
3
1.350
Note: (1) The number in brackets after the variance represents the exponent of 10;
2) LSF
when n=1.
Thus, the sampling simulation of Equation (4.61) can be expressed as
1.048
(-11)
(4.65)
To sum up, in order to reduce the number of sampling simulations in the MonteCarlo method and improve sampling efficiency, conditional expectation
sampling can be combined with importance sampling to perform a composite
sampling simulation to ensure the effectiveness of sampling every time. This
also avoids the complexity caused by the use of an optimization method to
determine sampling parameters. As a new way, sampling parameters can be
determined by the moment method; the effectiveness of sampling simulation
can be improved by adopting a normal distribution function for importance
sampling and a truncated distribution function for conditional expectation
variables; various examples confirm the effectiveness and wide applicability of
this improved simulation method.
4.3.3 Calculation Steps
1. Select the basic variable xk of conditional expectation from the LSF
expression of the structure;
2. Determine parameters
for the importance sampling density
function according to Equations (4.57) and (4.58);
3. Generate simulation sample
from
, and calculate the truncated
value di through Equation (4.49);
4. Determine sampling parameter σ1 by Equation (4.63) according to the
probability distribution of di and xk;
5. Generate simulation sample
;
from Equation (4.62), and calculate
6. Repeat step (5) M times, and calculate
7. Calculate
and
using Equation (4.64);
;
8. Repeating steps (3) to (7) N times;
9. Calculate the failure probability of the structure based on Equation (4.65).
4.4 Importance Sampling Method in V Space
4.4.1 V Space
If X space is an original basic random variable space, U space, a standard
normally distributed variable space, can be obtained by means of the Rosenblatt
transformation [4-14][4-15] x=Txu (u). In U space, the joint PDF f(u) of random
variables is monotonous, and u⋆ is the maximum likelihood point on the failure
surface, i.e., the minimum distance from the origin of coordinates, which can be
obtained by FORM/SORM. This also means that the curvature at u⋆ is always
greater than or equal to −1/β [4-16]. P is the reliability index, and β=(u⋆T u⋆)1/2.
To make full use of the quadratic effect on the failure surface, the failure function
is expanded by Taylor series at u⋆, with the quadratic term taken. So
(4.66)
where Gu(u⋆) and Guu (u⋆) are first and second derivatives of G(u) at u⋆. Equation
(4.66) is orthogonally transformed, meaning
(4.67)
so that zn is parallel to the direction cosines of -Gu(u⋆)/|Gu(u⋆)| at u⋆. H can be
obtained by the standard Gram-Schmidt Equation [4-17]. Thus, Equations (4.66)
can be expressed as
(4.68)
where
,
. The principal
curvature of the failure function at u⋆ can then be taken into consideration.
According to the characteristic equation, we get
(4.69)
κ is the eigenvector of matrix , i.e., the diagonal matrix of principal curvature.
is the orthogonal eigenvector matrix of , while is a (n-1)×(n-1)-order
matrix obtained from the matrix A with the nth row and nth column deleted;
therefore, the new coordinate system for these variables corresponds to the
main curvature coordinates. So
(4.70)
This can also be expressed as the relationship with U space, as follows
(4.71)
where Tu, x = P · H is the orthogonal transformation matrix for U space and V
space. Therefore, V space is only the linear transformation of U space, and is also
a standard normally distributed variable space [4-15].
By substituting Equation (4.71) into Equation (4.66), we can establish a failure
function for V space:
(4.72)
where ann is an element on the nth row and nth column of the matrix A, and An is
a vector on the nth row or nth column of the matrix A that does not contain ann. It
should be said that Equation (4.72) is a general expression of the failure
function with a quadratic effect in V space. For simplification, an approximate
parabolic function is usually adopted to replace Equation (4.72). This type of
approximate parabolic function is effective in describing the quadratic effect of
the surface at u⋆, as follows
(4.73)
or
(4.74)
Figure 4.2 Approximate parabolic surface of V space.
Figure 4.2 shows the approximation of the parabolic surface in V space at u⋆.
4.4.2 Importance Sampling Area
The importance sampling zone refers to the fact that the samples in this zone
have an important contribution to the estimation of Pf, and it is composed of the
sampling center and a region with a certain confidence level. In V space, such an
importance sampling zone can be represented as an ellipse, as shown in Figure
4.3. According to the symmetry of the approximate parabolic surface and the
observation of the importance sampling method, the sampling center is located
on the vn axis and within the failure region Df, while the semi-axis length of the
ellipse is related to the variance of sampling variables and changes within the
geometric properties of the failure surface.
Figure 4.3 Important sampling area of V space.
(1) Sampling center
As shown in Figure 4.3, the position of the sampling ellipse center changes with
the principal curvature κi, while the sampling center is located on the vn axis. Its
distance from the maximum likelihood point v⋆ on the failure surface is kibi (i.e.,
in Figure 4.3), where bi represents the semi-axis length of the sampling
ellipse on the vn axis, and parameter ki should satisfy the following conditions:
① When κi= 0, ki = k0; this means that, when the failure surface is a plane,
the sampling center is near v⋆;
② When κi= -1/β, ki = 0; this shows that, when the failure surface is a
sphere in V space or a circle with the radius of β in the vi - vn plane, the
sampling center is located on v⋆;
③ When κi= +∞, ki = 1. This is a corner case, i.e., the failure surface is a very
convex surface, and the distance from the sampling center to the maximum
likelihood point v⋆ is equal to the semi-axis length of the sampling ellipse.
Therefore, the parameter ki can be constructed into the following function:
(4.75)
where k0 is related to the ratio δ0 of the effective sample region Aeff to the whole
sample region Awhole when κi=0. This can be obtained by the following equation:
(4.76)
Table 4.3 Relationship between area ratio of sampling ellipse δ0 and k0.
δ0 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 0.99 1.00
k0 0.000 0.079 0.158 0.238 0.320 0.404 0.492 0.585 0.687 0.805 0.934 1.000
Table 4.3 shows the numerical values of k0 corresponding to different δ0 values.
(2) Axial length of sampling ellipse
As can be seen from Figure 4.3, the semi-axis length of the sampling ellipse on
the vn axis can be expressed as
(4.77)
where Δβ is a parameter related to the simulation accuracy. Reference [4.20]
provides a suitable range of Δβ, i.e., 0.7∼ 1.4. The length of the other half axis of
the sampling ellipse can be obtained depending on the geometric properties of
the failure surface:
① For the convex surface, i.e., when –β-1 ≤ κi≤ 0, this is
(4.78a)
② For the flat surface, i.e., when 0 ≤ κi ≤ κ1,cr, this is
(4.78b)
③ For the concave surface, i.e., when κi,cr < κi, this is
(4.78c)
where κi,cr can be obtained by equating Equation (4.78b) and (4.78c).
Figure 4.4 shows the relationship between the principal curvature κ of the
failure surface and the elliptic parameters a, b and k.
Figure 4.4 Relationship between principal curvature k and sampling elliptic
parameters a, b and k (β=3, Δβ=1.0, δ0=0.8).
4.4.3 Importance Sampling Function
Choosing an appropriate sampling function is one of the main parts of
importance sampling. Because V space is standard normally distributed variable
space, an n-dimensional normal distribution probability density function
can be selected as the importance sampling function h(v). The
mean of the sampling function is
(4.79)
The standard variance of the sampling function involves the confidence level of
the samples generated in the importance sampling zone. Let α be the confidence
coefficient, and the double-axis length of the sampling ellipse be a confidence
interval. The standard variance of sampling will be
(4.80)
where αS = 1/Φ-1 [(1+α)/2], and Φ-1[·] is the reversal of the standard normal
probability function. Figure 4.5 shows the effect of different confidence
coefficients α on the simulation results. As can be seen, an appropriate α value
ranges from 0.90 to 0.999, but it should be noted that α also depends on the
geometric characteristics of the failure surface.
Figure 4.5 Influence of different confidence a on simulation results.
Therefore, the importance sampling expression of V space can be expressed as
(4.81)
where
is the sample vector generated by the sampling function h(v).
4.4.4 Simulation Procedure
We can now make a comprehensive comment on the above process of
constructing V space by the importance sampling method. As can be seen, the
construction of h(v) is related not only to the geometric parameters (maximum
likelihood point, gradient and curvature) of the failure surface, but also to
parameters of the sampling function (effective sampling zone ratio δ0, effective
sampling zone Δβ, confidence coefficient α). The former directly affects the
accuracy and efficiency of sampling, but it can be expressed by V-space
sampling; the latter plays an important role in the process of constructing h(v),
but it seems to be less sensitive to the simulation results given suitable
parameters. When parameters are set to δ0 = 0.8, Δβ = 1.0 and α = 0.95 in the
following examples, ideal V-space sampling results can be achieved under all
failure function conditions.
Based on the contents outlined in the previous sections, the simulation process
of V-space importance sampling (ISM-V) can be described as follows:
1. Use FORM/SORM to determine the maximum likelihood point u⋆ on the
failure surface in U space;
2. Calculate the principal curvature κ at u⋆ (Equation (4.69) and the
transformation matrix for V space and U space (Equation (4.71);
3. Determine the center and axial length of the sampling ellipse in the vi ∼ vn
plane (Equations (4.75), (4.77) and (4.78);
4. Calculate the mean and standard variance of the importance sampling
function h (v) (Equations (4.79) and (4.80);
5. Generate a set of samples
6. Transform
into
and
from h (v) (Equation (4.81);
by Tuv and Rosenblatt transformation;
7. Check whether the sample is within the failure zone Df, and work out the
ratio;
8. Repeat step (5)∼(7) N times;
9. Calculate the failure probability
using Equation (4.81).
4.4.5 Evaluation
A random variable space (U space) can be transformed into another random
variable space (V space) through a linear orthogonal transformation matrix so
as to reflect the quadratic effect of the failure surface at the maximum likelihood
point more accurately. Thus, an importance sampling method (ISM) is
established for V space. This type of ISM fully considers the geometric
properties of the failure surface (e.g., maximum likelihood point, gradient,
curvature, etc.), thus ensuring the validity of samples and improving calculation
efficiency. With a wide variety of applications, it is not only used for the convex
failure surface, but also for flat or concave failure surfaces; moreover, with
increased simulation times, the simulation results show very stable
convergence, close to the exact value.
4.5 SVM Importance Sampling Method
For the establishment of a structural LSF by regression SVM, a regression SVMbased ISM can be constructed for structural reliability based on the ISM [4-18][4-
19]
. By the SVM responding surface equation, a regression SVM-based ISM is
developed for the calculation of failure probability:
(4.82)
where N represents the number of sampling points; represents support
vectors; l represents the number of support vectors; N represents the sampling
frequency; f(X) is the joint density function of variables; h(X) is the importance
sampling density function; Xi is represents the samples generated by the
importance sampling function. K() is the Kelvin function.
The calculation steps for regression SVM-based ISM are as follows:
1. Given an initial expansion center point
with
average point taken normally;
2. Calculate the function at the current expansion center point and each
expansion point by FEM: g(x1, x2,…, xn) and g(x1, x2, …, xi ± fσi, …, xn), where
the f value generally ranges from 1 to 3;
3. Combine the above 2n+1 sampling points and the k checking points
obtained during calculation into 2n+k+1 training sample sets, and
normalize the input/output data;
4. Solve the quadratic programming problem with respect to the normalized
2n+k+1 sampling point data to work out the value of α⋆, α and b, thus
establishing an SVM responding surface equation;
5. Use the Rackwitz-Fiessler method to calculate design point and reliability
index;
6. Judge whether the variation amplitude |β(k)-β(k-1)| of the reliability index
and the value g(X⋆) of the checking point function meet the given
requirements for accuracy. If the conditions are met, we will obtain a
reconstructed LSF and proceed to the next step. If the conditions fail to be
met, we will return to step 3 and restart the iterative computation.
7. Take the design point obtained by the above iterative computation as the
sampling center, and let the sampling variance be a certain multiple of the
original variable variance to generate a set of samples with capacity N from
the importance sampling function h(X);
8. According to the reconstructed structural LSF, judge whether the sampling
point is located in the failure zone. If it is in the failure zone, work out the
value of
;
9. Calculate the failure probability using Equation (4.82).
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5
Reliability of Structural Systems
An actual engineering structure is composed of many components, and the failure of a single
structural component does not necessarily cause the failure of the whole structure. Considering
that the failure of a structure is marked by overall failure, the probability of overall failure is an
issue related to the system reliability of the structure. It is generally believed that the key to
solving the problem of system reliability lies in identifying the main failure mode of the
structural system and calculating the system reliability.
Usually, to identify the main failure mode of a structural system, it is necessary to repeatedly
search the failure section, failure path and failure mode of the structure to generate the trunk
and branches of the structural system failure tree. Therefore, in the early days, system reliability
researchers focused on how to identify the main failure mode of a structural system quickly and
accurately. Cornell[5-1] proposed a wide bound formula for the interval estimation method
applied to system reliability calculation. Ditlevsen[5-2] proposed a narrow bound formula. Ang et
al. [5-3] applied the idea of fault tree analysis to the reliability analysis of structural systems, and
proposed a point estimation method, i.e., the probabilistic network evaluation technique
(PNET) for system reliability analysis. With the establishment of the above theory, an available
solution can be developed for system reliability analysis under the condition that the main
failure modes are known. Dong [5-4] and Li [5-5] reviewed and summarized the research findings
of many academic researchers on system reliability, putting forward a new method for the
overall reliability analysis of engineering structures. They then explored how to use this method
in engineering practice. In references [5-6] [5-7], the system reliability of offshore structures is
analyzed based on the principle of most unfavorable load combinations from the perspective of
engineering applications. In practice, because there are an infinite number of structural system
failure modes, system reliability is difficult to calculate.
5.1 Failure Mode of Structural System
5.1.1 Structural System Model
The analysis of a structural system can be simplified by: (1) load modeling: the magnitude and
order of applied loads; (2) system modeling: the relationship between the structural system and
its components, as well as the relationship between components; and (3) material modeling:
material response and strength characteristics. In addition, it is necessary to define criteria for
limit state design.
(1) Load simulation
If the load mode of the entire structure is taken into consideration, a part of the structure will
enter a (local) limit state before the whole structure reaches its limit. Therefore, the structural
failure mode is likely to depend on an exact loading process. This problem is defined as a “loadpath relationship”, i.e., the structural failure probability is estimated using the path generated
according to the (random) vector of the loading process [5-8],[5-9]. As can be seen from Figure
5.1(a), there are two loading paths for the cylindrical specimen: the horizontal path and the
vertical path. In other words, there are two loading sequences. For the structure to enter the
limit state at point A, a vertical load must be applied first and then a horizontal load; if a
horizontal load is applied first, the structure will fail to enter the limit state at point A.
However, for actual structures, the load-path relationship is not so critical [5-10], because (1) the
combined effect of internal forces at key locations causes the internal force to propagate along a
path similar to OBC in (b), where OB represents the dead weight and a permanent live load
path, while BC represents a limit load path; and (2) a ductile failure mode can be designed for
many actual structural systems in order to avoid brittle failure, so that these structural systems
are insensitive to the external loadpath. Therefore, most structures are characterized by
plasticity. For this reason, the rigid plastic theory provides an approximate and accurate force
analysis of structural systems [5-11]. Furthermore, the bearing capacity of a simple, ideal rigidplastic system is not directly related to the loading path, and the failure deformation of this
system is only under the control of the “orthogonal flow rule”. In the reliability analysis of
structural systems, loading path-dependency is not taken into serious consideration, whereas in
many practical cases, the load is idealized as a time-independent random variable.
Figure 5.1 Load-path relationship.
(2) Material simulation
In contrast to the complexity of actual structures, material characteristics are often idealized in
structural engineering practice. When structural section characteristics are considered, the
relationship between components can be assumed as shown in Figure 5.2. The elastic
characteristics (Figure 5.2(a)) conform to the concept of maximum allowable stress. Based on
this idealized treatment, it can be considered that the failure of any part or component of a
structure is consistent with the failure of the structural system. For most structures, such
idealized treatment is impractical, but it is very useful. Because of structural redundancy, the
brittle failure of components does not mean structural failure. Therefore, the loading
characteristics of actual components can be better considered as being “elastic-brittle”. This
shows that components can still deform when the bearing capacity is zero, even after reaching
their final bearing capacity (Figure 5.2(b)). Elastic-plastic components (Figure 5.2(c)) allow a
single structural component or specific part to continue to deform while bearing maximum
stress. The structure is rigid-plastic when the stiffness Ki of elastic components is infinitely
great. The elastic-brittle and elastic-plastic characteristics can be summarized as the
characteristics of elastic-residual strength (Figure 5.2(d)), and further as elastic-hardening (or
softening) properties (Figure 5.2(e)). The latter can be considered approximate equivalence to
the overall performance, including post-buckling behavior. Even without considering the
concept of reliability, it remains difficult to consider the loading characteristics of the latter in
structural analysis, not to mention the analysis of the overall nonlinear (i.e., curvilinear)
strength-deformation relations (Figure 5.2(f)).
Figure 5.2 Different strength-deformation (R-A) relations.
(3) System simulation
Usually, the actual structural system needs to be simplified before it can be analyzed. For
example, the centroid of the components of a frame structure needs to be idealized, the joints
need to be regarded as dots, and only a few determined dots should be taken to check the
strength or stress of the key sections. Similarly, the load is applied by means of nodal load or
sustained load in a limited form. If the load is not a nodal load, the key points for structural
safety checking may change with differing load combinations and load strength.
When a structural system fails, this needs to be defined from the following different angles,
since it is different from the failure of a single component or material:
Maximum allowable stress occurs at any location;
A (plastic) failure mechanism is formed (e.g., structural stiffness is zero: |K|=0);
Structural stiffness reaches a limit (|K|=Klimit);
Deflection reaches a maximum (A—Alimit);
Cumulative failure reaches a limit (such as fatigue).
Structural failure modes include the combined failure effect of two or more components (or
sections), such as statically indeterminate structures. The structural failure model has attracted
special attention for determining the reliability of structural systems.
After all failure modes of a structural system are defined, the concept of a “fault tree” can be
used to systematically enumerate the failure modes (failure of components or sections) that
cause failure. An example of the “fault tree” is shown in Figure 5.3 (b), corresponding to the
structural component in Figure 5.3 (a).
The specific process involves decomposing a structural failure mode into sub-events and then
further decomposing these sub-events. The lowest sub-events on the fault tree correspond to
the components or sections of the structure that have failed. A local limit stress equation can be
obtained in this layer. The fault tree method is mainly used for overall reliability analysis, but it
is also applicable to structural reliability analysis [5-12],[5-13],[5-14]. These methods illustrate the
feasibility of simplifying a structural system, such as limiting the number of potential failure
modes, i.e., the number of limit states in the structural system.
For rigid-plastic structures in special cases, the traditional method for identifying failure modes
is to consider the combination of the mechanical properties of the mechanism. In the reliability
analysis of structural systems, different methods of system simulation can be introduced into a
single element. By introducing a modeling error, we can use an idealized model (e.g., rigidplastic stress) to simulate the actual system [5-15].
Figure 5.3 Fault tree.
5.1.2 Solution
For the reliability analysis of a multi-component structure, there are at least two methods that
complement each other, i.e., “failure” mode method and “valid” mode method [5-16].
(1) Failure mode method
The failure mode method is based on identifying all potential failure modes of a structure. Each
structural failure mode generally starts from the “failure” of one component after another (e.g.,
the components enter their corresponding limit state) until there are enough components that
have failed to cause the whole structure to enter its limit state. The possible ways in which
structural failure is caused can be represented by the “event tree” in Figure 5.4 or the “failure
graph” in Figure 5.5. Each “branch” in the failure graph represents the failure of a structural
component. Any complete path starts from the joints of the “complete structure”, involving all
“failure” points and indicating the possible failure sequence of components. Such information
can also be obtained from the event tree.
Because the failure in any failure path may cause structural failure, “structural failure” FS is the
union of all potential failure event modes:
(5.1)
where Fi represents the ith failure mode. For each failure mode, there must be enough failing
components (or failure joints of the structure). So
(5.2)
where Fij represents the jth failing component in the ith failure mode; n represents the number of
components causing the ith failure mode.
Figure 5.4 Event tree of structure.
Figure 5.5 Failure diagram of structure.
(2) Valid mode method
The valid mode method is based on identifying all states (or modes) in which the structure is
valid. For the structure in Figure 5.3 (a), each joint A, B, C, D, E, F or G (except H) in the failure
graph of Figure 5.5 represents a structurally valid mode (as can also be seen from Figure 5.4).
For each valid mode, the structure may suffer local failure, but still retains the ability to bear
loads (it is a statically indeterminate structure).
For the structure to be effective, at least one valid mode is required, or
(5.3)
where SS represents “structurally valid”; Si represents the ith structurally valid mode, i = 1…k is
not equal to the final number of joints.
From Equation (5.3), we get:
(5.4)
where
represents structural failure in the ith valid mode. Obviously, in order that the
structure is valid in any valid mode, all the components that contribute to the valid mode must
be valid. The failure in the structurally valid mode is equivalent to the failure of enough valid
components, or:
(5.5)
where Fij represents the jth failing component in the ith valid mode; lij represents the number of
components causing the ith valid mode.
(3) Upper limit and lower limit
According to Equations (5.1) and (5.5), if all failure modes have been analyzed, pf will be
underestimated when the failure mode method (Equation (5.1)) is used to estimate the failure
probability of a structural system; conversely, if all the valid modes have been analyzed, pf will
be overestimated when the valid mode method (Equation (5.4)) is used. For a rigid-plastic
structure, a similar conclusion can be drawn according to the bound theorem of ideal plastic
materials (limit analysis). Therefore, if the intensity of all loads acting on the structure is related
only to one parameter, the probability that the structure suffers plastic failure {Ek} in the kth
failure mode can be expressed as
. Obviously, if there are n failure
modes for the structure,
represents the failure
probability of the whole structural system (relative Equation (5.8)). If y is used to represent a
subset of all plastic failure sets n, then this set of failure modes can be used to represent system
failure. However, the occurrence probability of this set is lower than that of the union, as follows
(5.6)
This represents the upper bound theorem (maneuvering conditions).
In classical plastic limit analysis, the “statically determinate” method or balancing method
suggests that if there is at least one statically determinate allowable stress field (i.e., the stress
field is in a state of equilibrium under load, and the material does not suffer local yielding), then
the structure will not fail.
Now, suppose there is at least one statically determinate allowable stress field in the structure.
If D means that there is no allowable stress field, then according to the theory of static
determinacy, it represents a system failure. The probability of this event occurring is Prob{D} =
Pϕ(w). So
(5.7)
This represents the lower bound theorem (static conditions) in probability limit analysis.
5.1.3 Idealization of Structural System Failure
A structural system or its auxiliary system can be idealized into two kinds of systems: series
systems and parallel systems. Some structural systems can be composed of these two systems,
though sometimes they may be more complex.
(1) Series system
The series system is represented by a chain, and is known as a “weakly connected” system. The
limit state of any structural component can cause structural failure (Figure 5.6). In this ideal
model, the exact material properties of structural units or components do not play a critical role
anymore. If a structural component is brittle, then the fracture of this component will cause
structural failure; if the components have elastic deformability, then the failure may be
determined by excessive yield. Thus it can be seen that a statically determinate structure
comprises a train of series systems, since the failure of any subunit may lead to the failure of the
whole structure.
Figure 5.6 Series system.
Therefore, any unit is a possible failure mode. The failure probability of a weakly connected
structure composed of three units is:
(5.8)
Compared with Equation (5.1), the series system in Equation (5.8) is a form of failure mode.
If the failure mode F(i = 1, m) is represented by the limit state equation Gi(X) = 0 in the basic
variable space, the basic reliability problem can be directly extended to:
(5.9)
where Ek represents the vector form of all basic random variables (load, unit strength, unit
attribute, size, etc.) and D belongs to the domain X, used to define system failure. This can be
defined by several failure modes, e.g., Gi(X) ≤ 0. In two-dimensional x space, Equation (5.9) can
be defined by the dashed area composed of D and Gi(X) ≤ 0 in Equation (5.7).
Figure 5.7 Two-dimensional failure region for reliability problem of structural system.
The safe zone is defined as . As can be seen in Figure 5.7, it is a complementary zone to the
failure zone, marked as D and D1, and can be expressed by the following equation:
(5.10)
where
is defined as the “validity of the ith mode” or Gi(X) ≥ 0. Thus, the validity probability is:
(5.11)
It can thus be seen that this equation is equivalent to the “valid mode” presented in the section
of 5.1.2.
For the chain shown in Figure 5.6, the load effect S is consistent with the load Q at every step. If
FRi(r) is the cumulative distribution function (CDF) of intensity at the ith step, the CDF of the
entire chain, then FRi() can be expressed as:
(5.12)
where the strength of a single material can be converted to:
(5.13)
This equation is the basis for the resistance probability distribution of brittle materials. When
each Ri obeys normal distribution and m approaches infinity, the minimum value of R obeys
Extreme value type III distribution.
For the series system model, the failure of one component causes redistribution of the internal
force and leads to the failure of another component, and so on. Therefore, the failure probability
of the whole structural system can be approximated to the failure probability of the first set of
(brittle) components [5-17]. However, when there are many redundant brittle components, this
approximation method is not applicable, because the residual strength becomes critical.
(2) Parallel system
When the elements of a structural system (or subsystem) are interconnected in such a way that
one or more elements can enter a limit state without there being an important effect exerted on
the failure of the whole system, then such a system is called a parallel system, or a redundant
system. Figure 5.8 shows two simple parallel systems. There are two forms of system
redundancy. When the redundancy unit plays a role as soon as the structure is subjected to a
small load, we call this “active redundancy”. If the redundancy unit does not start functioning
until a certain number of structural units fail or degenerate, we call this “negative redundancy”.
This shows that the redundancy unit increases system reliability.
Whether active redundancy is helpful or not depends on the characteristics and failure
definition of components or units. For an ideal plastic system, the “static theory” ensures that
active redundancy does not reduce the reliability of the structural system [5-17].
Under the action of active redundancy, the failure probability of a parallel (sub-) system
composed of three units can be expressed as:
(5.14)
where Fi represents the failure probability of the ith structural component. Thus, it can be
directly determined that Equation (5.14) is equivalent to Equation (5.4) and can be expressed in
X space as:
(5.15)
Figure 5.8 Two simple parallel systems.
In contrast to the series system, the parallel system will not fail unless all the active components
enter their respective limit states. This means that the performance of system components is
very important to the definition of system failure.
(3) Parallel system
For ideal plastic materials in the perfect plasticity, there are totally different. Taking a steel
frame structure as an example (Figure 5.8(b)), the failure of each component can be expressed
by the following equation:
(5.16)
where Qi(i = 1, …) represents external load; Δi represents the deflection relative to Qi (as a
function of θj and size); Mj represents the plastic resisting moment at j = 1, …; θj represents the
plastic rotation angle at j = 1, …. Equation (5.16) clearly reflects this parallel system, because all
Mj can be superimposed together to resist external load Qi.
A series system comprises a set of failure modes consisting of a single failure equation like
Equation (5.16). This is because structural failure is bound to occur when any failure (or
collapse) mode occurs. Thus, there may be more than one failure mode of plastic bending
moment in the system, indicating that there may be no difference in the structural bearing
capacity obtained from different failure modes.
(4) Conditional combination system
A complete system is usually composed of several series and parallel subsystems so that it can
realize its function. The connection between components and the failure mode of subsystems
have an important impact on the limit state of the whole structure. For a complex structure, this
is not a simple exercise. Not only will the structural internal force be redistributed after the
failure of components, but loads will also change with structural response over time (structural
deflection).
5.1.4 Practical Analysis of Structural System Failure
An actual structural system model may also require the use of a conditional system or its
subsystem. If the failure of an individual unit or a group of units affects the failure of other units
or groups of units, then the latter will suffer associated failure. For example, if the top beam
collapses in Figure 5.9, this may affect the performance and reliability of the bottom beam
(because the bottom beam may be destroyed and come under additional load). In this case, the
possibility of structural unit failure depends on the performance of the structure in extreme
conditions. If the sequence of events can be enumerated, then the structure in a conditional
event can apparently be reduced to a group of units or subsystems that contain both “series”
and “parallel” components.
Therefore, the complex system can be decomposed into several subsystems or subunits to
constitute the evaluation process of system reliability. Jin [5-6][5-7] proposed a simplified model
considering pipe-pile-soil interaction, and used it to evaluate the reliability of the jacket
platform system. In this case, the jacket platform system was divided into three subunits: jacket
structure, pile and soil. The failure probability of each subunit in the given load mode was
calculated, and the reliability of the whole structure system calculated according to the failure
mode of the series system. During the evaluation of system reliability, the nonlinear push-over
structural analysis method and importance sampling method were used to calculate the
reliability of the jacket platform system.
Figure 5.9 Condition system.
5.2 Calculation Methods for System Reliability
5.2.1 System Reliability Boundary
Similar to component reliability, structural system reliability can also be calculated using the
immediate integration method. However, there is an alternative method, by which upper and
lower bounds are established for failure probability of the structural system for analysis. It is
assumed that a certain structural system, undergoing a series of loads, may fail in any possible
failure mode under the action of any load. So, the total probability of structural failure can be
expressed by the failure probability in this mode:
(5.17)
where Fi represents “structural failure in the ith mode under the action of all loads”; Si
represents “structural survival in the ith mode under the action of all loads (i.e., survival rate of
structure). These are complementary events. So
(5.18)
It can be written in the following form:
(5.19)
where
represents structural failure occurring under the joint action of Mode 1 and
Mode 2.
(1) Linear series system
The probability of structural failure can be expressed as P(F) = 1 − P(S), where P(S) represents
the probability of structural survival. For mutually independent failure modes, P(S) can be
expressed by the probability of survival in each mode, or directly by
:
(5.20)
where
, as before, represents the probability of failure in that mode. The above equation
can be expanded, achieving results consistent with that of Equation (5.18). In addition, as can be
seen from Equation (5.18), if
, then
can be ignored, and Equation (5.20) can
be approximated to Equation (5.21):
(5.21)
When all failure modes are completely correlated with one another, the weakest failure mode is
the most likely to directly cause structural failure when the contingency of material strength is
ignored. So
(5.22)
Equations (5.19) or (5.20) and (5.21) can be used to define a relatively rough boundary for the
failure probability of a structural system when any failure mode is completely independent from
or completely correlated with the rest:
(5.23)
However, for most engineering structure systems, the boundary set Equation (5.23) is too large
to be practical.
(2) Quadratic series boundary
The quadratic series boundary can be obtained by maintaining the form of
, etc., in
Equation (5.18). For the sake of description, it can be expressed as follows:
(5.24)
Because the number of the terms that are alternatively positive and negative increases with the
number of terms in the equation, the upper bound of P(F) can be obtained by considering the
linear conditions only (e.g., P(F1)). The lower bound can be obtained by considering the linear
and quadratic failure conditions only. The upper bound can be obtained by considering the
linear, quadratic and cubic failure conditions, and so on.
It should be noted that the probability of structural failure cannot be reduced if an additional
failure mode is considered. Therefore, every complete row in Equation (5.24) has a nonnegative effect on
. When
is not considered, if
is retained, then a lower boundary can be obtained from
Equation (5.24) to ensure that each part exerts a non-negative effect [5-18] :
(5.25)
and
can be alternated to combine all the terms containing k in Equation
(5.24) to work out the maximum value of the lower boundary [5-19]:
(5.26)
In these two forms, the results depend on the sequence of the marked failure modes. Operation
rules [5-20] are established for an optimal sequence of events in order to obtain an optimal
boundary. By means of an effective rule we can sort modes from high to low in the order of their
importance. For a given sequence, a more rational boundary can be obtained by using Equation
(5.25) than by Equation (5.26); if all possible sequences are considered, then the boundaries
obtained by these two equations will be consistent with each other [5-21].
An upper boundary can be obtained by simplifying each line in Equation (5.24). As pointed out
above, a standard line, such as the fifth line, has a non-negative effect on
, and Pijk can be
used to express
, as follows:
(5.27)
With P5 removed, the other lines can be written as:
(5.28)
where Eij represents event ij. For any events, A and B, if
,
then they will meet the equation below:
(5.29)
Since V5 has a non-negative effect on U5, boundary Equation (5.29) will increase the right
boundary value of Equation (5.28). So
(5.30)
However, because Pj5 = P(Fj ∩ F5), and since the fifth line is a typical line, it satisfies:
(5.31)
This result may also depend on the sequence of failure events.
The calculated results of Equations (5.29) and (5.31) with respect to rigid and rigid-plastic
frames have been compared with the results of Monte-Carlo simulation [5-22]. In a certain
distribution form and within a certain variation range, the boundary obtained by calculation is
generally quite close to the simulation results. However, they are not always close to each other.
(3) Quadratic series boundary under sequential loads
Load sequence and correlation determine the failure probability boundary of systems.
According to Equation (5.17), under load vector Q1, Q2, Q3…, the probability of structural failure
can be expressed as:
(5.32)
where Fi represents “structural failure that occurs under the action of the ith load”; S means that
“the structure is valid under the action of the ith load. They are complementary events,
.
In general, failure under the action of the ith load is correlated with survival before the
application of the ith load. The equation below can express this “transfer” of probability:
(5.33)
Moreover, because
(5.34)
Similarly, for other parts consistent with Equation (5.17), by using Equation (5.32), P(F) can be
expressed as:
Or, for the application of n sequential loads, we obtain:
(5.35)
The above equation is consistent with Equation (5.24), where
. Therefore, considering this change,
boundary Equations (5.24) and (5.30) can be regarded as the boundaries of structural failure
probability
under sequential loads.
(4) Series boundaries and sequential loads in multiple modes
The boundary Equations (5.25) and (5.31) of failure modes can be summarized as failures in
multiple modes within the range of load sequence. Equations (5.25) and (5.31) can be
interpreted as the probability
of structural failure caused by the kth load under
sequential loads. Equations (5.25) and (5.31) can also be used to work out the total failure
probability under a complete sequence of loads, but at present, they are only interpreted as
what exists under sequential loads.
In practice, Fi, Fj are usually related to each other, making it difficult to work out
by
estimation. One approach is to obtain a simplified solution by setting estimation undertow limit
conditions: Fi, Fj are completely independent of each other or completely related to each other. If
events Fi, Fj are completely independent of each other, Equation (5.32) can be simplified to
Equation (5.20).
Similarly, if events Fi, Fj are completely related to each other, and the critical state is known, then
, simplified as
. This means that Equation
(5.18) can be simplified to Equation (5.22). With both load sequence and multiple failure modes
considered, the linear series boundary Equation (5.23) can be expanded, so
(5.36)
where
represents the failure probability under the action of the jth load in the ith mode.
For the terms on the right side, let
:
(5.37)
When it is known that load sequence and multiple failure modes are independent of each other,
then the maximum value of the terms on the left side can be replaced by the summation
formula; similarly, if the load sequence is known to be completely correlated with multiple
failure modes, then the summation formula for the terms on the right side can be replaced by
the maximum value.
If the composition N of load sequence is known to be continuous, and there is a shared
probability density function (PDF) for the alternating independent loads, then the right
boundary can be replaced by the following formula:
(5.38)
(5) Optimization of series boundaries and parallel system failure boundaries If the results of
quadratic series boundaries gradually deteriorate when (linear) LSF dependency increases, the
problem can be transformed into a low-dependency LSF for boundary optimization.
Alternatively, if higher-order parts can be preserved, the boundary of the series system can be
optimized. For example, if
, which is obtained from
is known, then
. Thus, the
cubic failure boundary can be transformed into the following formula, where Fi(i = 1, …, m)
represents m possible failure probabilities [5-23]:
(5.39)
in where is
,
,
where
means that this part is not included unless it is positive.
In addition, the sequence of some failure events is of great significance for obtaining an optimal
boundary. However, it is most difficult to estimate Pijk, since it is jointly affected by all three
parts. When the event Fi is expressed as a linear function, the nonlinear lower limit can be
obtained from the following equation [5-24]:
(5.40)
It is known that ρkj > ρjk ρij > 0, where ρij is the correlation coefficient of the linear failure
function for events i, j. Feng [5-25] has proposed a good method for solving Pij and Pijk in linear
LSF.
For the failure probability of parallel systems, it can be calculated by Equation (5.2) or Equation
(5.14). For this kind of boundary, it can be created by applying series boundaries to the terms on
the right side of the identical equation:
(5.41)
The boundary value obtained by the above equation is low, since for a highly reliable system, the
second term on the right side of the equation is close to 1.0.
Dependency can also enhance load dependence, in particular the dependence between the load
and a single structural component [5-26] [5-27]. The correlation between basic variables in
second-order reliability analysis is determined by transforming the correlation set into a noncorrelation set. Two static loads are applied simultaneously to the correlation effect of rigidplastic portal frames, and the set of its typical results is shown in Figure 5.10 [5-28]. However,
there is a lack of actual experimental data on the dependence between intensity (and load). In
general, a conservative assumption needs to be put forward.
Figure 5.10 The impact of correlation on system security indications.
5.2.2 Implicit Limit State—Response Surface
In practical applications, it may not be possible to express the limit state surface explicitly by
one or more implicit limit equations, while it is often implicitly understood in the process of
finite element analysis. G(X) is used to represent the structural response, and G(X) is the implicit
function of random variable X. When calculating G(X), let represent a series of points in X
space, and then use the “responding surface method” to look for
, an equation which is
perfectly in line with the discrete value in
. Normally, let
be an n-order polynomial.
The undetermined coefficients in this polynomial can be determined by minimizing the
approximation error, especially in the zone around the design point.
The discrete points can be fitted to determine the order of n in the polynomial, and the results
will affect the number of evaluations and the number of estimated derivatives. Also, for a wellconditioned system of equations,
must be of equal order or lower in order than G(X).
Higher-order
will generate an ill-conditioned system of equations containing uncertain
coefficients. It is also a non-steady-state equation.
If an actual limit state equation is established by the results of discretization only, it will be
impossible to obtain its form and order, let alone estimate its design point. This means that
there will be no guidance on the selection of the estimation function
. However, quadratic
polynomials are often used to estimate the response surface [5-29] [5-30] [5-31] [5-32] ;
(5.42)
where
,
are uncertain (regression)
coefficients.
The (regression) coefficients can be obtained by a series of simulation “experiments”, i.e., a
series of structural analyses can be conducted by inputting selection variables depending on the
“experimental design”. For approximate experimental design, it is necessary to consider the
reasonable and accurate estimation of failure possibility. This shows that the focus needs to be
on the maximum probability area, including the failure zone, i.e., the design point. However, as
mentioned above, this is unknown at first. Therefore, a value near the average variable can be
selected as an input variable for experimental design. A simple experimental design is shown in
Figure 5.11. Rajashekhar and Ellingwood [5-33] have discussed more complex problems.
Once the experimental design is selected and the equation G(X) is calculated, it can be seen that
there is an error between the actual value of
, X = x, and the value estimated by Equation
(5.42). This error is caused by inherent randomness and the “test for lack of fit” brought about
using a simpler Equation, (5.42), to represent the actual limit state surface. It is impossible to
separate them without performing an accurate calculation. However, an important step is to try
to select regression coefficients A, B, C to minimize the total error.
Figure 5.11 Simple experimental design of two variables.
Let vector D represent the regression coefficients A, B, C. In this case, Equation (5.42) can be
expressed as G(D, X). Similarly, for each point
in the experimental design, let Ei represent the
value of the actual limit state equation, meaning the error represented by approximate surface
Equation (5.42) will be
. This can be done for each point in the experimental design.
Now, the total error can be minimized. A very simple way is to use the least square fitting
method, and use D to minimize the quadratic sum of errors
(5.43)
(1) Simplification of large-scale system
For some practical problems, it is impractical to apply Equation (5.42) because there are too
many variables. For example, this will happen when finite element analysis (FEA) is adopted for
structural response. Therefore, the number of random variables involved can be reduced in
many ways.
The first way is to replace free variables with formulas for determining uncertainty.
The second way is to reduce the set of random variables, X, to the minimum set, XA, while
describing their mean spatial value. For example, we can use the same (e.g., spatial averaging)
yield strength at each point in the stress plane, rather than exactly representing different yield
strengths from point to point (or from finite element to finite element).
The third way is to simplify the error effect of random variables (or spatial averages), i.e., they
may contain only a single additional effect to response, rather than a more complex relationship.
In this way, we can establish an Equation (5.44) for the additional error effect:
(5.44)
where ej, ejk, … represents the error term caused by spatial averaging (it is assumed that
spatially averaged random variables are independent here), and ε represents the retained error
(e.g., those caused by the degree of freedom). If ej, ejk, … can be obtained separately, this form is
useful, e.g., by analyzing the change of the spatial averaging effect. The error can be fitted by
G(X), and ε can be restricted.
(2) Iterative solution method
The optimum fitting point between the approximate responding surface and the actual (nonexplicit) LSF is uncertain in advance. It has been suggested that these points can be determined
by iterative search. Let Equation (5.42) contain sufficient undetermined coefficients to
accurately estimate
i.e., an experimental design based on complete saturation, so that the
surface should accurately coincide with the evaluation point £, especially with the mean point
Xm, and point Xi = Xmi ± hiσi, where hi is any coefficient and σi is the standard deviation of Xi.
According to these points Xm, an approximate surface
can be accurately obtained at the
postulated mean point. If the approximate surface falls in an optimal position, then the mean
point Xm will coincide with the maximum likelihood point (design point), and the distance from
the origin to these points will be the shortest in standard normal space.
If Xm is a design point, other points can be found on the approximate surface
and named
XD. These points are closer to the origin, so they are the best estimation points for the design
point. When XD is known, the new mean point
can be obtained by interpolation based on
the linear relation between Xm and XD, as follows:
(5.45)
The speed of this calculation method, which also boasts high fitting accuracy, is related to the
selection of the design point and the shape of the actual (non-explicit) LSF in the search area.
Obviously, when the design point representing resistance is on the left truncated section of this
random variable, while the design point representing load is on the right, the calculation results
will be improved [5-34]. Moreover, in general we cannot ensure that all possibly related design
points converge. It is possible to modify the search algorithm so that the search deviates from
the design points included.
(3) Responding surface and finite element analysis
The responding surface is not a method explicitly required for reliability analysis. When FORM
is used, we suggest taking the responding surface as a tool to apply FEA to structural reliability
analysis.
It should be noted now that in standard normal space, the quadratic responding surface is
essentially similar to the second-order second-moment theory. This is because a quadratic
surface can be used to fit either a known nonlinear limit state surface or a known discrete point
on the responding surface. The result is that these two methods feature extremely similar
discussions. When the responding surface method is used for FEA, the repeated results contain
a series of selected points determined by the finite element program, such as those [5-35] that
apply to the responding surface. If FORM is used, the fitted surface is linear, and only a small
number of points are required to determine the responding surface. This is because, generally
speaking, the fitting of the responding surface has nothing to do with the reliability calculation.
One requirement is that the gradient of the design point should be determined or estimated.
Some FEA programs can be modified to meet this requirement. Alternatively, an approximate
finite difference method can be adopted, but this will come with a higher calculation cost.
However, the responding surface method has been combined with FORM in some commercial
finite-element codes [5-36].
Importantly, the finite element model represents a random field in a reliability environment,
e.g., statistical variables that may be required to represent performance, such as cross-plate
Young’s modulus. This is quite a specialized topic, and many methods have been proposed [5-37]
[5-38][5-39][5-40]
.
5.2.3 Complex Structural System
(1) Truncation enumeration method
For a complex structural system, the best way to determine all structural failure modes is to
enumerate all combinations of joint failure. This is a huge amount of work, as all the sequential
probability of joint failure needs to be considered. All joints are analyzed in turn and then added
to the sequence of joints that failed previously. Every new sequence can be checked to find out
whether a feasible system failure mode has been obtained. If not, more joints will be added to
the sequence. If structural failure is identified as a collapse, then the failure mode must be
kinematically feasible. When a failure mode is identified, we can enumerate the next failure
mode by tracing back until a new joint combination appears, as shown in Figure 5.12.
Figure 5.12 Systematic enumeration process.
If those failure modes that have a small probability effect on the system can be identified, then
the calculation will be greatly simplified. If the error is insignificant, it can be eliminated from
subsequent calculations [5-41]. Obviously, it will be advantageous if this can be done at the early
stage of enumeration, preferably before the mode failure probability is calculated. Many
techniques have been proposed[5-42] [5-43] [5-44] [5-45] [5-46] [5-47], with some more intuitive than
others. One rational technique known as “truncated enumeration” will be described below.
Because all failure modes have been determined, it is not important to reveal their order. So, for
the convenience of enumeration, structural elastic stress analysis should be performed. Of
course, the specific order of failure modes obtained depends on the enumeration method used.
After specific modes have been eliminated, this order will have an impact on the calculation of
failure probability. For elastic and rigid-plastic joints, structural behavior is generally quite
similar. Moreover, for rigid-plastic behavior, the failure mode and corresponding stress field are
irrelevant to the elastic properties of any such structure.
In an order containing n joints, the probability of the kth joint can be calculated as follows:
(5.46)
where Ei represents the “resistance breakthrough of the ith joint” in the event. As shown by 1, 2
and 3 in Figure 5.12, there are many such modes in existence. The number n of joints required
for structural failure depends on the criteria for structural failure judgment. For structural
failure, n must be large enough to find a feasible failure mode.
If pjk ≤ δPf for all modes, then the number of failure modes to be considered will be reduced.
Here, δ is an appropriate “truncation criterion”, and Pf represents the estimated failure
probability of the nominal structural system. Apparently, if δ = 0, it will be exhaustive.
Unless objective estimation has been carried out before analysis, it may be possible for Pf to be
obtained from the currently known maximum Pf value. For example, in the previous estimation,
the mode failure probability is the current value, defined as :
(5.47)
Apparently,
, so the use of Equation (5.47) is conservative, because fewer failure modes
will be discarded. The mode failure probability calculated by Equation (5.46) may be bounded.
For q ≤ n, the inequalities below are adopted:
(5.48)
For any diversity of Equation (5.48), q ≤ n, the truncation condition
will be
conservatively evaluated, and any mode that satisfies it will be ignored immediately.
Theoretically, a joint sequence can be selected in any order. However, for an efficient algorithm,
the joint with the greatest contribution must be selected as soon as possible. As a result, the
estimated value of Pf in Equation (5.47) will be optimized, and therefore, trivial modes will be
removed as early as possible. A reasonable way is to select the next joint, so that the probability
of occurrence of some joint sequences, including selected joints, can be maximized. From the
first joint to the qth joint, this means that joint selection is:
The first joint:
(5.49a)
The second joint:
(5.49b)
Or in general, for the qth joint:
(5.49c)
Maximization is applicable for all “eligible joints” at the level of selection (see Figure 5.12).
Therefore, at the qth layer, events E1 ∼ Eq−1 are fixed, and a decision can be made to select event
Eq according to the remaining joints.
The simplification of Equation (5.49) helps to save time. For Equation (5.49c), approximation is
suggested to cover the upper limits:
(5.50a)
(5.50b)
These are called “one-dimensional” and “two-dimensional” branch conditions, respectively. For
example, Equation (5.50a) can be interpreted physically as follows: It is practical to select the
joint with the maximum failure probability rather than determine the maximum probability of
occurrence by searching the entire sequence. It does not need to be the same as Equation (5.49),
for the reason, when there is no difference in stress distribution at each joint, there is
correlation among events E1, E2, …, Eq.
For any level q, the mode failure probability conservatively estimated (overestimated) in
Equation (5.48) may be conservatively estimated (overestimated) again in Equation (5.50b).
Since Equation (5.50b) is now already overestimated, Equation (5.47) may also be
overestimated. Therefore, some modes may be unreasonably excluded from the truncation
criteria. However, these modes don’t have to be important unless they have a failure probability
like
. The most important mode (the one that contributes the most to the estimation of
system failure probability) is unlikely to be affected.
For the truncated enumeration method, it is necessary to establish the sequence of joint failure
events and calculate their probability. This requires the structural stress to be analyzed in steps.
Once various system failure modes have been determined, a limit state equation can be
established for structural failure. Two analysis processes are proposed in order to establish the
limit state equation. It is assumed in both cases that the failure sequence of joints is known, and
that the failure mode is the kth mode.
(1) Joint substitution (“simulated load”) method
In this method, the failure joint is replaced by its resistance after failure. For all subsequent
external load increments, this resistance can be considered as a locally applied simulated load,
whose random characteristics are the same as those of the resistance after failure [5-47].
Therefore, if the joint is fully plastic, the plastic resistance can be applied as a simulated “load”;
if it is elastic-brittle and there is no brittle strength, no simulated “load” will be applied.
Apparently, with the increase of external load, it is necessary to seriously check the stress state
of all joints. Although this method looks outstanding, there is still a lack of proper judgment
criteria to make it work.
(2) Incremental loading method
If the components show elastic-plasticity, and there is a known failure model k, then the failure
probability can be obtained by comparing the system resistance RS with the single-parameter
loading system Q1:
(5.51)
In case of structural failure, the system resistance can always be expressed as the maximum of
all accumulated load increments:
(5.52)
where rj represents the jth load increment related to the mode failure event Ej. Therefore, a
structure in which three components must fail before structural failure, r1 represents the load
when the first component has failed (starting from 0), r1 + r2 represents the total load when the
second component has failed, r1 + r2 + r3 represents the total load when all three components
have failed. Considering that the load may disappear after a component fails, the structural
resistance RS is expressed by the maximum load, i.e., Equation (5.52).
The corresponding probability of failure (kth) is:
(5.53a)
It can also be expressed as:
(5.53b)
The system resistance R is controlled by the nodal force, which can be related to the load
increment rj through a matrix equation:
(5.54)
where
is the vector of nodal force,
can be called the “utilization matrix”, where Aij represents
the number i of joints generated by pressure, because Q1 = 1 is the jth increment, such as the
corresponding load increment rj. For a given failure model, given the joint sequence, Aij can be
determined by traditional structural analysis. The inversion of Equation (5.53) generates an
incremental effect rj, so that Equation (5.54) can be evaluated:
(5.55)
Equation (5.55) can be extended to be suitable for more generalized joint stress behavior, such
as elastic unloading strain hardening, or for more than one load system (each model serves as a
random variable and each one is applied only once)
(5.56)
where B is an “unloading matrix” related to the unloading part corresponding to deformation;
CQ represents the random action of components under (no increment), and Ar is defined by
Equation (5.55). It should be obvious that any load system can be chosen to determine the load
increment system for load increment r. The incremental process has no other purpose, because
the order of joint failure is known, causing it to be canceled out.
Unloading and strain hardening (or post-buckling) behavior bring about various limitations and
problems in the deterministic analysis of (incremental) structural reliability. Therefore, it is
necessary to consider the range of the hardened zone, the possible reversal stress and
concomitant change of local rigidity in incremental analysis.
(3) Possibility estimation of system
To calculate the (nominal) failure probability of the structural system, we must also calculate
the failure probability of each dominant failure mode by Equation (5.57), as follows:
(5.57)
as well as the series combination of pfk. These calculations are usually different from those for
branch selection and truncation unless the probabilities have been precisely determined (in this
case, it is necessary to consider the possibility of correlation effects).
If the probability of the dominant mode is known, then the nominal failure probability of the
structural system can be calculated theoretically by means of Equation (5.1)
(5.58)
where Fi = (i = 1, …, m) represents failure in the ith mode. If only the dominant mode is
considered, a small probability will be underestimated (see Section 5.2.2).
There is an upper limit for the error introduced into the estimation of probability through
truncated (i.e., abandoned) failure modes. It can be obtained by estimating the probability
related to the unbranched joints at the first layer [5-48]:
(5.59)
Although the truncated enumeration method is commonly applicable to single load parameters
or single loads (i.e., time-independent cases), and is useful for determining failure modes [5-49],
in general, it (and other similar methods) requires a lot of calculation. Fortunately, for common
ideal rigid-plastic structures, there are alternative methods for determining failure modes.
Sigurdsson [5-50] proposed to use joints in place of (artificial loads) (but not by truncation) for
system reliability analysis to calculate rigid-plastic structures. For rigid-plastic joints only,
Murotsu et al. [5-51] and Thoft-Christensen and Sorensen [5-50] have proposed applicable
calculation steps similar to the truncated enumeration method.
5.2.4 Physically-Based Synthesis Method
The research on the overall reliability of traditional structures remains limited to the scope of
framed structures, i.e., it is still under the limits of ideal elastic-plasticity. For complex structures
composed of plates, walls, blocks, etc., as well as for other common nonlinear physical problems,
the above methods are basically unusable. For traditional analysis of overall structural
reliability, problems are studied from a phenomenological perspective according to the
consequences of failure. Therefore, it is essentially unable to reflect the real physical
development process of the structural state from linear to nonlinear.
Although nonlinear structural analysis is performed during the above-mentioned process for
the branch-and-bound method, the position and failure path of plastic hinges cannot be
determined yet due to randomness. Thus, structural failure and failure path must be physical
processes “modified by probability” rather than real physical processes. This error in analytical
methodology is the fundamental cause for the secular stagnation of traditional research on
overall reliability. Correct research should be based on an analysis of the structural stress
process and an investigation on the law of propagation of randomness within the physical
system.
For the comprehensive physical method, probability dissipation conditions need to be
established for generalized probability density evolution equations in accordance with physical
criteria. That is, for a given failure limit,
, when it satisfies the Z function: Zext > [Z], the
probability density of the representative point up is as follows
(5.60)
The representative point up is a point at which the overall state of the structure is observed. It
can be flexibly selected according to the object of specific analysis. For example, the fixed-point
displacement of a structure can be used for a high-rise building.
The reliability analysis of a structural system is the same as that of its components, but it is
necessary to consider multiple failure paths related to the structural system and their
corresponding limit states. For other methods, it is beneficial to connect systems in series or in
parallel, or to classify more complex subsystems, e.g., the system reliability boundary theory can
be used in combination with FORM.
Structural failure modes are not always known. To some extent, they can be tested by MonteCarlo simulation, by the method of exhaustion or by other simplified methods. Among these, the
failure mode method, which contributes the most to failure probability, is of special concern.
As mentioned above, all the methods discussed in this chapter have limitations on structural
loads. Proportioned (i.e., single load parameter) loading conditions are equivalent to the
problem of reliability arising under a single ultimate load (i.e., the maximum load that can be
borne by the system during its service life). For a complex structure, the critical limit state may
change with a sequence of multiple loads. In this case, the system itself depends on the loading
path. It should be noted that, although there have already been many studies carried out on this
problem, there is as yet no widely accepted method. A practical way is to define the combination
of loads in reliability analysis as being like the one used for traditional structural design, i.e.,
only a limited number of load combinations are considered [5-15]. Although it is not a
fundamental approach, this method can at least be used to analyze conditional probability
under clearly defined conditions.
The comprehensive physical method can better solve the problem of reliability.
5.3 Example: Reliability of Offshore Fixed Platforms
5.3.1 Overview
The offshore jacket platform is a pile platform supported by piles driven into the seabed. It can
be divided into group pile type, pile foundation type and leg pile type, and is composed of a
jacket, a pile, a jacket cap and a deck [5-52]. Cazzulo [5-53] studied the reliability of pile foundation
structures for offshore jacket platforms, substituting the pile foundation analysis results as
input conditions into the reliability analysis of the structural system. Amdahl [5-54] studied the
final bearing capacity of jack-up offshore platforms with a foundation effect. Hamse [5-55] used
the stiffness matrix for pile heads to describe the influence of pile-soil interaction on offshore
jacket platforms. However, these methods do not consider the pile foundation and
superstructure, but adopt a simplified analysis method to deal with it, which involves a certain
amount of approximation. Therefore, if the pile-foundation offshore platform system can be
taken, while considering both the pile-soil interaction and the influence on the offshore
platform structure, the final bearing capacity of the entire system can be studied. Moreover,
considering the discreteness of soil parameters, this research will become very meaningful for
studying the range of influence and mode of action of soil parameters on the bearing capacity
and reliability of an offshore platform system.
5.3.2 Calculation Model and Single Pile Bearing Capacity
(1) Pile-soil calculation model
The final bearing capacity of a single pile under vertical load is composed of the friction force Qf
of the pile body, and the supporting force Qp of the pile end, where:
(5.61)
Where: As and Aq are the surface area of the pile body and the gross area of the pile end,
respectively; f is the surface friction force per unit area. For cohesive soil, f can be expressed by
Olson’s calculation [5-56]:
(5.62)
(5.63)
For sandy soil, there will be
(5.64)
(5.65)
Where: Su is to describe the shear strength of cohesive soil, the effective covering pressure of
soil at the calculation point; K is the lateral earth pressure coefficient, the internal friction angle
of sandy soil, and Q is the bearing capacity of unit pile end; Nc and Nq are dimensionless bearing
capacity coefficients for cohesive soil and sandy soil, respectively.
Considering the transfer of a pile under an axial load and the displacement of a pile, Kraft [5-56]
put forward the axial load transfer curve of a pile based on practical experience and a full-scale
pile loading test. As shown in Figure 5.13 and Figure 5.14, t is the bond between dynamic pile
and soil, tmax = f, Q is the bearing capacity of the dynamic pile end, D is the outer diameter of the
pile, and z is the local displacement of the pile.
The final bearing capacity of soil to pile under a lateral load can be determined based on the
resistance of soft cohesive soil, hard cohesive soil and sandy soil. The expression for resistance
of soft cohesive soil under a shortterm static load is as follows:
(5.66)
Figure 5.13 t ∼ z curve.
Figure 5.14 Q ∼ Z curve.
Where J is a dimensionless empirical coefficient between 0.25 and 0.50, which can be
determined by field testing, and x is the depth of soil layer at the calculation point; for hard
cohesive soil, this becomes:
(5.67)
The load-displacement (p-y) curves for the two cohesive soils are shown in Figure 5.15a and
Figure 5.15b, where y50 is the displacement value corresponding to the strain value, that is, y50
= 2.5ε50 D, ε50 it is the strain that occurs at 50% of maximum stress in an undrained test for
undisturbed soil.
Figure 5.15 P-y curve of soil.
The resistance of sandy soil under a short-term load can be expressed by the following formula
[5-57]
(5.68a)
And
(5.68b)
The Equation (5.68a) is suitable for shallow soil, while the Equation (5.68b) is suitable for deep
soil;
is the empirical adjustment coefficient. K0 = 0.4 is the static earth
pressure coefficient, Ka = tan2(45° − φ/2) is the Rankine active earth pressure coefficient,
is the internal friction angle of sandy soil, β = 45° + φ/2, α = φ/2. The relationship
between lateral load and deformation (p-y) of sandy soil can be calculated based on the
following expression [5-56]:
(5.69)
where K is the initial modulus of the soil. The calculation model for pilesoil interaction can
therefore be determined using the foundation reaction coefficient method. Figure 5.17 shows
the model for calculating the reaction coefficient at each point on the pile, which uses the τ ∼ z
curve of Figure 5.13, the Q ∼ z curve of Figure 5.14, the p-y curve of Figure 5.15, and Figure 5.16
the expression described in Equation (5.68). In the model shown in Figure 5.17, the distortion
of the pile and the deformation of the rotation angle are ignored, which is acceptable in most
practical application.
Figure 5.16 p-y curve of sandy soil.
(2) Uncertainty of soil
The strength and stiffness characteristics for each layer of soil samples used here are shown in
Table 5.1. Due to the discreteness of the soil samples, the layer properties listed in Table 5.1
reflect the average value of the measured data, but in practice they show great variability. The
reasons for this uncertainty of soil properties include soil classification, soil sampling, soil
testing methods and calculation models, which are manifested as statistical uncertainty, testing
uncertainty and knowledge uncertainty. The former two are mainly used for soil parameters,
while the latter comes from the calculation model for soil, which includes the formulas for
calculating axial tension (compression) pile, lateral load pile and pile tip bearing capacity.
Referring to AP2A-LRFD [5-56] and the Beihai site data [5-57], Table 5.2 gives the deviation,
coefficient of variation and probability distribution characteristics of various soil parameters,
while Table 5.3 shows the uncertainty of the soil calculation model, which will be used as the
basis for probability analysis of pile bearing capacity.
Figure 5.17 Calculation model of pile.
5.3.3 Probability Analysis for the Bearing Capacity of a Single Pile
The single pile selected here has a diameter of φ2438 and a pile length of 60m, and is buried in
the calculated soil as shown in Table 5.2. The pile is divided into an average of 40 finite
elements, with each joint comprising two horizontal nonlinear spring supports and one vertical
nonlinear spring support (see Figure 5.17). The stiffness of the nonlinear spring can be
determined using the pile-soil interaction model described in the previous section, while the
bearing capacity of a single pile can be calculated using the nonlinear progressive failure
analysis method. As a comparison, the mean value of each variable is selected for calculation
during the deterministic bearing capacity analysis of a single pile, while 500 random samples
are generated for each variable during the probabilistic bearing capacity analysis. Every
nonlinear progressive failure analysis includes three load conditions, namely axial tension load,
axial compression load and lateral load.
Table 5.1 Soil parameters.
Depth Soil
r
Limit side
ϕ(o) Su
(m)
property
(kPa)
friction flim
(kP)
(kPa)
End supporting
force qlim (kPa)
0.0 ∼
2.0
Sand
33
109.0 15
1
5.5
2.0 ∼
6.6
Sand
35
10.0
100
10
34.6
6.6 ∼
7.7
Sand
35
9.0
40
1
5.5
7.7 ∼
9.2
Clay
30
7.0
1.0
9.2 ∼
30.4
Clay
75 ∼
115
9.0
0.5
30.4 ∼
48.5
Sand
48.5 ∼
50.0
Clay
50.0 ∼
69.3
Sand
69.3 ∼
72.8
Clay
34
10.0
120
12
150
34
34.6
0.7
10.0
200
ε50(%) k
(MN/m3)
9.5
120
15
34.6
0.7
Table 5.2 Uncertainty of soil parameters.
Soil parameter Deviation Cov
Distribution type
φ
1.00
0.15
Normal
Su
1.00
0.20
Lognormal
α
1.00
0.10
Lognormal
Nc
1.00
0.10
Normal
r
1.00
0.10
Normal
K
1.00
0.10
Lognormal
flim
1.00
0.15 ∼ 0.20 Normal
qlim
1.20
0.20
Normal
Nq
1.20
0.20
Normal
k
1.00
0.40
Lognormal
ε50
1.00
0.40
Lognormal
Table 5.3 Understanding the soil calculation model.
Computation
model
Axial tension Axial
compression
Lateral
displacement
Soil property
Deviation Cov Deviation
Cov
Deviation
Clay
1.00
0.15 1.00
Sand
1.00
0.15 1.10
End bearing
Cov
Deviation Cov
0.15 1.00
0.30
−
−
0.15 1.00
0.30
1.20
0.15
Figure 5.18 Load-bearing capacity under axial compression.
Table 5.4 shows the deterministic analysis results and the statistical results of the probability
analysis on the bearing capacity of a single pile under axial and lateral loads, where Δ
represents the bearing capacity under lateral displacement of the pile top.
Figure 5.19 shows the load-bearing capacity under axial compression and tension, respectively,
as well as the probability distribution simulation for load-carrying capacity. The lateral bearing
capacity of a pile top under fixed end and hinge constraints is shown in Figure 5.20.
Table 5.4 Bearing capacity of a single pile.
Load case
Deterministic
analysis
Probability analysis
QD (MN)
Mean
value
(MN)
Coefficient Distortion Steepness
of variation
39.6275
39.2930
0.1965
0.5355
3.0203
0.9916
Compress 130.3967
127.3500
0.1728
0.1558
3.1079
0.9766
Lateral Δ = 0.5 m 54.1108
52.3160
0.0707
0.2234
2.8828
0.9668
Δ = 1.0 m 64.4857
Δ = 1.5 m 68.0915
62.3260
65.559
0.0642
0.0683
0.3569
0.3345
3.097
2.9919
0.9665
0.9628
Δ = 2.0 m 69.5433
66.7710
0.0699
0.2964
3.1358
0.9601
Axial
Tension
QD/Qmax
Figure 5.19 Load-bearing capacity under axial tension.
From the above analysis of the single pile bearing capacity, it can be seen that: (1) for large
diameter piles, the effect of pile tip bearing capacity is significant. However, when the ratio of
pile length to pile diameter is greater than 20, the effect of length can be ignored. (2) The ratio
of axial tensile bearing capacity to compressive bearing capacity of a pile is 0.30, which means
that when the upper vertical load of the structure is small, especially for a tripod jacket,
structural volume damage may result from the axial tensile bearing capacity of the pile. (3) The
uncertainty of axial bearing capacity distribution of a single pile due to probability analysis
mainly comes from the uncertainty of the pile-soil calculation model; the coefficient of variation
of the axial tension pile is 0.20, which is greater than the 0.17 of the axial compression pile. (4)
The axial bearing capacity of a single pile obtained by deterministic analysis is very close to the
mean value obtained by probability analysis; the probability distribution of the axial tensile
bearing capacity obeys lognormal distribution (see Figure 5.19), while the axial compressive
bearing capacity obeys normal distribution (see Figure 5.18). (5) The determination of lateral
bearing capacity of a single pile is related to the constraint conditions of the pile head. The
bearing capacity of articulated constraint of the pile head is only about 50% of that of the fixed
end constraint, but the actual constraint of the pile head lies between the fixed end and the
articulated end. (6) The average value of lateral bearing capacity of a single pile obtained by
probability analysis is about 0.96 of that obtained by deterministic analysis, but its coefficient of
variation is small and obeys lognormal distribution (see Figure 5.20).
Figure 5.20 Lateral bearing capacity with different pile top constraints.
5.3.4 Bearing Capacity and Reliability of Offshore Platform Structural
Systems
(1) Structural model
A tripod jacket platform is selected as the type of structure, as shown in Figure 5.21. The
platform is in an ocean area with a water depth of 70m, and supported by three piles with a pile
length of 60m; as a result, the main form of load it needs to bear is wave loads. Depending on
the direction of wave incidence and the possibility of combination with other loads, there are a
total of 8 calculation conditions for the structural system [5-57]. The entire platform structure
can be discretized by the finite element method, for which the jacket structure has 29 nodes and
78 beam elements, and the pile foundation has 93 pile nodes, 90 beam elements and 96 spatial
nonlinear spring elements. The pile-soil interaction is simulated by means of a nonlinear spring
element and calculated using NLSPRINT software. The load-bearing capacity of the whole
offshore platform system is calculated by nonlinear progressive failure analysis. This method
enables the calculation of deformation and mechanical properties of the offshore platform in
various states under given load conditions, reflecting the non-linear material and geometric
characteristics of the structure and the final bearing capacity.
Figure 5.21 Deterministic analysis of computational structure model.
To analyze the influence of the structure-pile-soil interaction on an offshore platform structure,
three kinds of supporting boundary conditions are considered during bearing capacity analysis.
These are the fixed end support at the mud surface line, the linear spring at the mud surface line
(to consider the effect of soil action), and the pile support (to consider the nonlinear effect of
the pile-soil). The stiffness of the linear spring at the mud surface line is determined by the
displacement caused by a unit force acting on the pile top.
Figure 5.22 Shear and bending bearing capacity and structural placement diagram.
(2) Bearing capacity of pile foundation
The final bearing capacity (shear bearing capacity) of an offshore platform structure under an
environmental load is shown in Figure 5.20, with the shear and flexural bearing capacity and
structural load-displacement curve under operating condition 1 shown in Figure 5.22. It can be
seen that: (1) For a jacket platform structure, the influence of the supporting boundary
conditions on the bearing capacity of the structural system is significant. The bearing capacity
of an offshore platform structure with fixed end support is slightly greater than that of a linear
support on a mud surface, but significantly greater than that of a nonlinear spring support. For
both linear and nonlinear spring supports, the displacement value corresponding to the
maximum bearing capacity will be greater than that of the fixed end support. (2) In terms of the
failure state of the structure, the failure of the structure with a nonlinear spring support is
mainly composed of the bearing capacity of the pile foundation, that is, the axial tensile failure
of the pile can also show changes in stress and strain in the process of reaching the final bearing
capacity of the structure. In contrast, the consolidated structure on the mud surface or linear
spring support is composed of the compressive buckling or joint failure of the members of the
structure itself. This shows that the analysis method for bearing capacity of an offshore platform
structure with linear spring support is able to consider the influence of the structure-pile-soil
interaction. (3) Because of the effect of soil, the bearing capacity of the structure supported by a
fixed end is approximately equal to that of the structure supported by a linear spring, which is
only manifested in the difference of structural displacement. For the structure supported by a
nonlinear spring, the stiffness of the spring changes with every load, which absorbs the energy
generated by an external load and leads to an increase in structural displacement and a
decrease in bearing capacity. This shows that, if the bearing capacity of the whole offshore
platform structure can be improved, it will be necessary to coordinate the bearing capacity of
the pile foundation and structure.
(3) Probability analysis
According to the deterministic analysis results of the bearing capacity of the offshore platform
structure system, and to better study the influence of structure-pile-soil interaction on offshore
platform structures, a nonlinear spring support is used for the probabilistic analysis of bearing
capacity. The random variables used in probability analysis include soil parameters and
structural parameters. The uncertainty of the soil parameters can be determined using Table 5.2
and Table 5.3, while the uncertainty of structural parameters primarily considers the yield
stress of structural materials, with an average value of 325.0 MPa and a coefficient of variation
of 0.03, which obeys a normal distribution. The analysis procedure uses sampling simulation for
statistical distribution 500 calculation samples are included for each calculation condition, and
the results of each step-by-step failure analysis, such as the maximum bearing capacity and the
corresponding displacement value, are then extracted and statistically analyzed. In this way, the
probability analysis of the bearing capacity of the offshore platform structure can be obtained.
Tables 5.5 and 5.6 shows the statistical results for bearing capacity of the structural system,
while Figure 5.23 shows the fitting and comparison of the deterministic results with the
statistical results of the probability analysis for calculating the probability distribution of the
bearing capacity in operating condition 1.
Table 5.5 Bearing capacity under different supporting boundary conditions.
Load
case
Fixed end at mud
surface (MN)
Linear spring support
(MN) on mud surface
Nonlinear spring support
along pile length (MN)
1
30.3290
30.2160
18.8050
2
29.4800
28.4510
20.7260
3
30.4120
30.3600
18.3790
4
5
25.3900
26.0330
24.6530
24.9970
21.3690
21.7120
6
24.2760
23.7760
16.2270
7
27.4280
26.7620
17.8520
8
26.8507
26.5166
24.3330
Figure 5.23 Statistical results and probability analysis of bearing capacity.
Table 5.6 Statistical results for shear capacity and simulation of the structure.
Load
case
Base shear QDeterm/MN Average/MN Simulation results
Qmean/QDeterm
forceQ0/MN
Cov
Skewness Steepness
1
2
4.7212
4.8076
18.8050
20.7260
18.3810
20.2020
0.1468 0.4547
0.1458 0.3471
2.9568
2.6038
0.9775
0.9747
3
4.3052
18.3790
17.9678
0.1469 0.4540
2.9526
0.9776
4
3.2770
21.3690
20.5430
0.1237 -0.1722
2.0835
0.9613
5
3.3761
21.7120
20.8200
0.1229 -0.1896
2.1231
0.9589
6
7
2.3665
2.9338
16.2270
17.8520
15.8360
17.4000
0.1490 0.4471
0.1484 0.4428
2.9055
2.8899
0.9759
0.9747
8
4.7057
24.3330
22.7930
0.0997 -0.5977
2.5059
0.9367
Average
0.1354
0.9672
From the results of the above formula, we can see that: (1) The average values taken from the
probability analysis are very close to those calculated using the deterministic analysis, with a
ratio of about 0.9672, while the average value of the coefficient of variation is 0.1354. The
simulation results to a confidence (μ-1.96 σ) are given in Figure 5.23, where it can be seen that
the discreteness of the calculation results is relatively large. (2) The variability in bearing
capacity of the offshore platform system primarily depends on the uncertainty of the soil
calculation model, while the variability in operating condition 8 is caused by the crushing failure
of the brace; nonetheless, the probability of this operating condition is relatively small
compared with other operating conditions. (3) The probability distribution of the bearing
capacity of the system can be fitted to the lognormal distribution, which has a relatively
consistent coincidence.
(4) Reliability analysis
In order to study the influence of structure-pile-soil interaction on the reliability of an offshore
platform structure, and to remain consistent with the extreme state conditions of the structural
design code, only the safety and reliability of the system in its extreme state is considered here.
In this case, the reliability calculation model of an offshore platform structure system under
extreme load can be expressed as [5-58]
(5.70)
where SC is the resistance (final bearing capacity) of the offshore platform, Q is the external
action, and BSC and BQ are the deviation coefficients of SC and Q, respectively. According to the
analysis results of reference, BSC obeys a lognormal distribution lnN (1.034, 0.086), while BQ
also obeys a lognormal distribution with lnN (1.060, 0.265). Structural resistance SC can refer to
the above statistical results, while external action Q is a function of wave height, displaying a
Weibull distribution[5-7]. Therefore, the reliability of the offshore structure under various
operating conditions can be calculated by Equation (5.70). Table 5.7 uses three different
methods for reliability calculation: the primary reliability analysis method (FORM), the
secondary reliability analysis method (SORM) and the importance sampling method (ISM-V) [559]
.
Table 5.7 Failure probability obtained by different reliability calculation methods.
Case FORM
Pf
β
SORM
Pf
β
ISM-V
Pf
β
Cov
1
1.2219 (−5) 4.2199 1.1693 (−5) 4.2298 1.1619 (−5) 4.2313 0.0483
2
3
6.3284 (−6) 4.3660 6.0412 (−6) 4.3761 6.0098 (−6) 4.3772 0.0492
6.8924 (−6) 4.3473 6.5818 (−6) 4.3574 6.5466 (−6) 4.3585 0.0491
4
1.3328 (−7) 5.1457 1.2562 (−7) 5.1568 1.2553 (−7) 5.1569 0.0545
5
1.5367 (−7) 5.1189 1.4490 (−7) 5.1300 1.4478 (−7) 5.1301 0.0543
6
7
7.8598 (−8) 5.2439 7.3963 (−8) 5.2551 7.3957 (−8) 5.2551 0.0551
2.6681 (−7) 5.1038 2.5200 (−7) 5.0248 2.5163 (−7) 5.0251 0.0536
8
1.2663 (−6) 5.0138 2.5200 (−7) 5.0248 2.5163 (−6) 4.7168 0.0516
• Note: Exponents of 10 in brackets
5.4 Analysis on the Reliability of a Semi-Submersible
Platform System
5.4.1 Overview
It is of great importance to evaluate the reliability of the overall structure of a deep-water semisubmersible platform, which is exposed to a complex marine environment for long periods of
time. Finite element method (FEM) greatly facilitates reliability analysis and is constantly
developing. In particular, FEM has been used by many researchers in the reliability evaluation of
marine structures [5-60][5-61][5-62]. In this chapter, the reliability of a typical deep-water semisubmersible platform structure in one of China’s ocean areas will be evaluated. See Figure 5.24
for a 3D FEM model of the platform [5-63].
Wave parameters were first calculated based on statistical environmental parameters of the
South China Sea and the random method specified in the ABS code. Then the typical wave load
parameters corresponding to Group 6 were also calculated, as listed in Table 5.8. Working
Condition 1 refers to the horizontal force under the effect of a horizontal wave, Working
Condition 2 refers to the horizontal torque, Working Condition 3 refers to the vertical shear,
Working Condition 4 refers to vertical bending moment, Working Condition 5 refers to vertical
acceleration, Working Condition 6 refers to horizontal acceleration, while Working 7 refers to
general conditions. The amplitude is the maximum amplitude value from Working Conditions
1∼6.
Figure 5.24 3D FEM model of a semi-submersible platform.
Table 5.8 Wave parameters for a 100-year-return period.
Working Conditions
Case Load parameters 1
2
3
4
5
6
7
Cycle (s)
Amplitude (m)
9
7
8
9.6 7
6
9
8.89 6.33 8.15 8.21 7.32 7.01 8.89
Incident angle (°)
90
Phase angle (°)
-33.6 160.2 108 133 40.1 -108 50
120
135 180 180 90
180
Structural FEM calculation was carried out for the target semi-submersible platform using the
wave calculation parameters given in Table 5.8. The results were analyzed using ANSYS postprocessing tools. The main sectional force and bending moment of various working conditions
were then calculated (see Table 5.9 for the statistics of calculation results). The sectional forces
were calculated using ANSYS and results for extreme bearing capacity were also summarized
(see Table 5.10 for details).
Table 5.9 Data on sectional force and bending moment of each working condition.
Working
Condition
Sectional force
Fs (×107
N)
Mt (×108 N.
m)
Fs (×107
N)
Mb (×108 N. Fs (×107
m)
N)
FTa (×107
N)
1
2
10.2
2.66
6.28
24.3
1.21
1.38
2.23
3.76
0.0025
0.24
2.92
0.0056
3
1.48
2.81
2.56
2.62
0.00075
0.0035
4
1.41
8.22
1.91
9.07
0.35
0.000031
5
1.57
5.79
0.7
2.87
6.43
0.060
6
7
4.06
5.40
4.06
13.144
0.014
0.040
4.13
1.80
0.00036
0.0030
11.05
0.0014
Table 5.10 Data for limit state parameters in each working condition.
Parameters
Working Case
Still water 1
Wave amplitude/m
—
Amplitude extremum (×10 ) 3.42
8
2
3
4
5
6
25 24 20 19 26 19
2.84 75.4 0.74 16.4 2.30 3.00
5.4.2 Uncertainty Analysis
The ocean contains a great deal of uncertainties. In terms of structure, these so-called
uncertainties can be mainly reflected in: (1) uncertain ocean environment factors, such as wind,
waves and currents; (2) uncertain structural resistance; and (3) uncertain calculation models.
The uncertainties for final strength and load effect are discussed in this chapter. As a deepwater semi-submersible platform is exposed to the ocean environment for a long period of time,
wave load is the primary environmental load factor in the platform structure.
The wave load of small-scale components can be calculated using the Morison formula. In
contrast, no widely available analytic formula exists for the calculation of large-scale
components. The wave load of a deep-water floating platform was calculated using the
hydrodynamics software AQWA. The probability parameters for wave loads of large-scale
components were also calculated using the annual extreme wave height of a particular ocean
area in the South China Sea over a 50-year-return period. In addition, the actual probability
distribution model for extreme annual wind speed and extreme annual wave height at 12 sites
in the South China Sea were also verified.
The probability parameters of various sea areas were obtained by means of statistical analysis,
which was carried out against the wind load of a deep-water floating platform and the wave
load of small-scale components using data for perennial annual extreme wind speed and annual
extreme wave height in the South China Sea. The wave load on large-scale components was also
calculated using many-body dynamics software, based on which, statistical analysis was then
carried out to obtain the probability characteristic wave load parameters for large-scale
components [5-64].
After evaluating the reliability of a ship structure, and by considering the influence of the
interaction of static water and waves, Mansour and Hovem [5-65] concluded that the influence of
water and waves can generally be ignored. Table 5.9–5.11 shows the data for variable
distribution type in the limit state equation, and a method for selecting the variable coefficient.
According to the results obtained by such researchers as Teixeira[5-66] and Faulkner[5-67],
Scotland’s Glasgow University and Strathclyde University [5-68][5-69] recommended a random
variable for a floating production system (FPSO) and a similar random model was adopted.
Furthermore, the value of both the average and variable coefficient was adjusted to a certain
extent. See Table 5.11 for the statistical distribution and values of the calculation variables.
Table 5.11 Calculated variable distribution types.
Variable Distribution type
Variable coefficient Average
Ru
Lognormal distribution 0.15
Calculation
Sw
Type 1 extremum
0.10
Calculation
Ssw
Normal distribution
0.02
Calculation
γu
Normal distribution
0.1
1
γw
Normal distribution
0.1
1
γm
Normal distribution
0.1
1
5.4.3 Evaluation of System Reliability
5.4.3.1 Analytical Process and Evaluation
A total of six typical main failure modes for platform structures were obtained after analyzing
the structural limit state of the semi-submersible platform [5-70]. According to analytical results,
failure was ascribed to different sectional forces, from which four types of limit state equations
were obtained. The six limit states are respectively: pulling pressure along horizontal stay bar,
overall horizontal twisting of the platform, shear generated by relative movement of two
floating boxes, vertical bending effect of floating box, and horizontal and vertical accelerations
of the overall structure. In summary, all of the failure modes above are related to one specific
component. As shown in Figure 5.25, Mode 1 is the limit state of horizontal force, which is
caused by cross brace yield; the limit state of horizontal torque in Mode 2 is reflected in the
form of torque yield failure of the deck. Mode 3 shows the failure of overall column cutting
under the effect of vertical and horizontal accelerations. Mode 4 is the yield failure after the
structure of the floating box under the effect of a bending moment. Modes 5 and 6 show
horizontal and vertical accelerations, respectively. The failure of each part of the platform will
lead to the failure of the whole structure. Therefore, each different kind of force exerts an
influence on platform reliability; if the platform is affected by one load working condition,
special attention should also be paid to the mutual influence between different failure modes.
Figure 5.25 Analysis of structural reliability of a semi-submersible platform.
The platform will be either used or left idle during its service life. Once left idle, the platform
will be exposed to rainstorms, making the structure less safe. Therefore, system reliability is
calculated supposing the platform is in an idle state.
The complexity and variability of the ocean environment make the load standard for calculating
the platform’s reliability a key issue. In other words, reliability index may differ along with wave
load during different periods of reoccurrence. The wave load during a 100-year-return period,
which is commonly used in design, was used as the load model for reliability evaluation. Overall
reliability was analyzed by taking the semi-submersible platform as an example. The overall
strength reliability of the semi-submersible platform was evaluated using a Level 1 reliability
calculation model and by considering the structural calculations for a platform under the effect
of typical wave loads. See Figure 5.26 for the reliability evaluation process.
Figure 5.26 Reliability evaluation procedure for a semi-submersible platform.
5.4.3.2 Reliability Calculation of Main Components
The reliability level of various failure modes was calculated based on the limit state equation
form of a semi-submersible platform structure and the probability distribution form of the
variables in Table 5.12. The reliability index of the platform’s main structure and corresponding
failure probability can be calculated using the procedure for system reliability and self-prepared
reliability calculations given in Figure 5.15. See Table 5.13 below for the calculated results.
The overall reliability index and overall failure probability of the seven kinds of working
conditions under different failure modes were calculated. Based on the analysis results, the
reliability associated with different failure modes was basically the same as the calculation
results for the failure mode corresponding to main structure response. Generally speaking, the
overall structural reliability under all working conditions, except for the working condition
featuring a vertical bending moment, were fairly reasonable across a wave load for a 100-yearreturn period. The minimum value of structural reliability obtained was 2.38, which
corresponds to the working condition of a vertical bending moment. The structural reliability
index corresponding to Working Condition 7 is the highest, with a reliability index of 6.18. The
calculated results show that the main structure of the platform, i.e., the anti-bending capacity of
the floating box, is relatively low.
Table 5.12 Stochastic models of calculated variables.
Variable Distribution
Average Standard deviation Variance
Fu
Lognormal distribution 2.84×108 4.26×107
0.15
Fw
Type I extremum
1.02×108 1.02×107
0.1
γu
Normal distribution
1
0.1
0.1
γw
Normal distribution
1
0.1
0.1
γm
Normal distribution
1
0.1
0.1
Table 5.13 Reliability index and failure probability of a semi-submersible platform.
Working conditions Reliability index of various failure modes Main structure (β & Pf)
1
a
b
c
4.00 8.77 6.66
d
e
7.28 12.91
f
7.88
4.00
3.16×10-5
2
8.44 4.38 6.23
5.56 110
9.89
4.38
5.93×10-6
3
9.39 9.59 4.11
6.73 12.96
10.00
4.11
2.01×10-5
4
9.43 7.98 5.13
2.38 11.43
9.85
2.38
8.70×10-3
5
9.35 8.98 8.38
6.46 4.87
9.98
4.87
5.58×10-7
6
9.24 9.95 10.00
5.24 12.87
3.23
3.23
6.18×10-4
7
6.18 6.47 20.32
7.87 12.89
10.00
6.16
3.6×10-10
5.4.3.3 Reliability Calculation for Local Nodes
There are a total of two key node layout models for the deep water semi-submersible platform
[5-30]
: connection of lower column and buoy, and connection of upper column and deck. In the
application to the overall platform, there are a total of four failure modes: shear destruction of
lower buoy, bending destruction at the end of the cross brace, bending destruction of lower
column end, and shearing destruction of upper column end. To calculate the reliability of
various failure modes, the internal force of the sectional area corresponding to various working
conditions should be calculated first. See Table 5.14 for the statistical results calculated using a
similar sectional force calculation method, and Table 5.15 for the ultimate bearing capacity of
various sections.
The limit state equation of the main structure of the semi-submersible platform can be referred
to. However, in the course of reliability calculation, the failure equation of local noes can be
centrally expressed as:
(5.71)
Table 5.14 Calculated values for sectional force of semi-submersible platform node.
Sectional force
(×106N)
Working conditions
1
2
3
4
5
Lower buoy shear
(×106N)
Bending of cross brace
end(×105N. m)
10.4 12.9 5.90 1.17 5.23 6.25 1.12 Type I
extremum
0.10
39.2 181 4.10 40.1 32.1 26.4 12.2 Type I
extremum
0.10
Bending of lower
column end(×107N)
37.5 6.20 3.44 54.8 35.9 34.3 4.83 Type I
extremum
0.10
Shear of upper column
end(×106N. m)
9.54 2.57 12.8 6.81 6.31 8.56
0.10
6
7
Distribution
type
Type I
extremum
Variable
coefficient
Table 5.15 Resistance parameters for semi-submersible structural joints.
Extreme value of sectional resistance Average Variable coefficient Distribution type
Lower buoy shear (kN)
Lognormal
2.45×107 0.15
Lognormal
Bending of lower column end (kN*m)
3.35×107 0.15
1.12×109 0.15
Shear of upper column end (kN)
2.27×107 0.15
Lognormal
Bending of cross brace end (kN*m)
Lognormal
Where, R is the ultimate bearing capacity of the section, which can be tension and pressure or
bending moment. S is the internal sectional force generated by the load. γu is the uncertainty
parameter for ultimate strength calculation. γw is the uncertainty parameter for load
calculation. γm is the uncertainty parameter of the model. γu, γw and γm follow a normal
distribution, and their value and variable coefficients are 1.0 and 0.1 respectively.
The ultimate bearing capacity of the node in Table 5.15 was calculated based on the internal
sectional force when the overall statistical structure has failed. Strictly speaking, the node does
not fail totally when the overall structure has failed. In other words, if the sectional force
reaches the statistical value specified in the table, the corresponding control section is not
considered a total failure, since other parts of the structure may have failed and the node cannot
be loaded continuously. Therefore, the value of the ultimate bearing capacity of the node may be
relatively small, which may also lead to a small calculated value of reliability. To get more
accurate calculations for reliability, the precise value of the ultimate bearing capacity of the
failure node used for control should be calculated further. See Table 5.16 for the calculated
results of local reliability for various nodes of the semi-submersible platform as calculated
based on the reliability in Formula (5.3) and Table 5.16. The lower buoy shear is numbered as 1,
bending of cross brace end as 2, bending of lower column end as 3 and shear of upper column
end as 4.
Table 5.16 Reliability data for local nodes.
No. Cases Working Condition
1
2
3
4
5
6
7
1
2
3
3.39 2.57 5.39 9.45 5.80 5.19 9.48
7.76 2.47 9.87 7.68 8.37 8.92 9.60
4.24 9.35 9.67 2.85 4.40 4.56 9.51
4
3.42 7.86 2.30 4.63 4.90 3.82 7.12
5.4.3.4 Calculation of Overall Platform Reliability
After analyzing the component reliability of the target semi-submersible platform and the
reliability of key connection nodes, the simplified system reliability calculation method 5 used
in this paper is used to calculate the system reliability of a target platform under different
working conditions. See Table 5.17 for the results.
According to Table 5.17, the calculated results for the target platform’s reliability is relatively
low under the effect of Working Conditions 2, 3 and 4. To be specific, the control factors of
Working Conditions 2 and 3 are node reliability, while the platform’s main structure is the main
control factor for the reliability index of Working Condition 4. The platform’s reliability under
all three working conditions is lower than 3. Generally speaking, the system reliability index of
the target semi-submersible platform is relatively low.
Table 5.17 Overall reliability of target platform.
Parts
Main body
Node
Overall
situation
β&
Pf
β
Cases
1
4
Pf
3.16×10- 5.93×10- 2.01×10- 8.70×10- 5.58×10- 6.18×10- 3.6×10-10
β
3.21
2
4.38
5
3
4.11
6
5
2.26
-
4
2.38
3
2.30
7
2.85
4
4.381
3.81
3.20
2.26
-
3
2.30
6
2.29
2
-
5
4.36
2
-
-
13
3.20
2
-
6.16
6.87×10 1.19×10 1.07×10 1.10×10 6.50×10 6.87×10 3.6×10-10
4
-
-
7.12
β
Pf
2
-
7
6.16
6.64×10 1.19×10 1.07×10 2.19×10 5.91×10 6.95×10 5.40×102
-
6
3.23
Pf
4
-
5
4.87
6
-
4
References
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6
Time-Dependent Structural Reliability
The external load borne by an engineering structure would change constantly,
while structural resistance shows a time-dependent change trend, too. All this
may cause a change in the reliability of the engineering structure. Usually, a basic
random variable cluster may be defined as a function of time. As can be seen,
whether the applied load changes with time (even for a quasi-static load, such as
most floor loads) or the material strength changes with time (caused by load
effect or corrosion), the structural response is bound also to change constantly
with time. Fatigue and corrosion are two typical examples of structural strength
degradation.
The problem [6-1] of time-dependent structural reliability at a certain moment
can therefore be expressed as Equation (6.1):
(6.1)
where R(t) and S(t) represent time-dependent structural resistance and timedependent load effect, respectively. If the transient probability density functions
(PDF) fR(t) and fS(t) of R(t) and S(t) are both known, then the transient failure
probability pf(t) can be worked out by convolution integral computation.
Strictly speaking, Equation (6.1) makes sense only when the load effect increases
suddenly at t or when the random load effect is reapplied at this moment.
Otherwise, the structure will fail at an earlier time. The structure will not fail at a
certain moment before t, i.e., it is assumed that the structure is always safe
before t. Therefore, the load or load effect must change, and one of the following
two situations needs to occur:
1. The value of variable load is discrete;
2. If the time-dependent load effect is continuous, t can be replaced with an
arbitrarily small time increment δt.
So, Equation (6.1) can be expressed as
(6.2)
For any given time t, the failure probability pf(t) can be expressed by the
calculation method described in Chapter 3; X(t) in the equation is a vector
composed of a set of random variables. Theoretically, the time interval [0, t] can
be integrated by the transient failure probability expressed by Equations (6.2)
and (6.1) to work out the failure probability of this interval. But in fact, the
transient value of pf(t) is often related to the value of pf(t + δt), (δt → 0), since
the random process X(t) is auto-correlative in terms of time. Figure 6.1 shows a
typical sample function for the random process of load effect.
The traditional processing method is to consider integrating the sample function
presented in Figure 6.1, thus transforming it into a load or a load effect process,
and expressing it in the form of extreme value distribution throughout the entire
time interval. However, for this processing method, it is effectively considered
that the structural resistance does not change with time. The improved method
is to consider solving problems using the extreme value theory within a
relatively short time interval (e.g., the duration of a windstorm or one year). This
is like using the concept of a return period to calculate the failure probability of a
structure in its service life. This method is very effective in calculating the failure
probability of important structures such as offshore platforms and towers under
discrete loads.
Figure 6.1 Sample function of random process of load effect.
Another way to calculate the failure probability of structures is to establish a
limit state function (LSF) based on Equation (6.1):
(6.3)
Then, we can calculate the probability that Z(t) will be less than or equal to zero
within the service life of the structure. This is the so-called “crossing problem”.
The moment at which Z(t) becomes lower than zero is known as the “moment of
failure” in Figure 6.2, which is a random variable. The probability that Z(t) will
become lower than zero within the service life of the structure is known as the
“probability of first transcendence”.
Figure 6.3 shows the situation in 2-D space. The “probability of first
transcendence” refers to the probability that the random process vector X(t) will
leave the “safe zone” G(X) > 0 within the service life of the structure. Because
structural failure is defined as G(X) > 0, the “probability of first transcendence” is
equal to the probability of structural failure.
The concept of first transcendence is more universally applicable than the
traditional calculation methods, and there is no formal restriction on the form of
G(X). However, the knowledge related to random processes is required for
calculating the first transcendence and accurately understanding its concept [6-2]
[6-3]
.
Figure 6.2 Sample function and failure time of safety limit state process Z(t).
Figure 6.3 Transcendence of random process vector X(t).
6.1 Time Integral Method
6.1.1 Basic Concept
In the time integral method, the entire range [0, tL] of the service life of the
structure is a unit, and the statistical characteristics of all random variables must
be related to it. Therefore, the probability distribution of a load is the
distribution of the maximum load within the service life of the structure.
Similarly, the probability distribution of structural resistance is more concerned
with the minimum value. But in general, the maximum load effect rarely occurs
at the same time as the minimum resistance. Figure 6.4 shows the typical sample
functions of R(t) and S(t).
Figure 6.4 Sample function of nonstationary load effect and resistance.
For most cases, structural resistance is time-independent, and in such
circumstances, R(t) is a straight horizontal line, as shown in Figure 6.5. If R is a
random variable, its actual value will be determined by PDF fR( ). In this case,
Equation (6.1) may become
(6.4)
where Sm(tL) refers to the maximum value of load effect within the service life of
the structure, [0, tL]. The PDF of Sm( ) can be obtained directly by fitting
previously observed extremum data with a proper PDF. The effectiveness of this
method assumes that the statistical characteristics of load effect will not change
in the future. Under normal circumstances, however, data will not always be
valid for a long time, making it necessary to comprehensively analyze the
extreme value distribution using updated data.
Using the time integral method, we can apply a single-parameter load system Q
to the structure at every other fixed time interval. In this case, the failure
probability of the structure can be obtained by counting the number N of
structural failures caused by the application of every independent load to the
structure:
(6.5)
Figure 6.5 Sample function of load effect and resistance (when resistance is
constant).
where n represents the number of times a certain load is applied; FN(n) is the
cumulative distribution function (CDF) of N; LN(n) is defined as a “reliability
function”. Because N is discrete, PDF fN(n) can be established as:
(6.6)
Similarly, a “risk function” can be defined as follows:
(6.7)
This function represents the probability that the structure will not fail under the
first n − 1 loads, but will fail under the nth load. It can usually be expressed as a
function of time.
Suppose load application is time-independent, i.e., the ith time of application and
the jth time of application are independent of each other. For the sake of
convenience, we also suppose the PDF and CDF of resistance R and load effect S
are known, and that both resistance R and load effect S (load system Q) are
stationary random processes, i.e., PDF fR( ) and fS( ) are time-independent.
Therefore, for the mutually independent loads applied to the structure, the
probability that the maximum load effect will be less than a certain value can be
expressed as:
(6.8)
If n is very large, Equation (6.8) will gradually approach an extreme value
distribution. If the failure probability of the structure can be expressed as
(6.9)
after integration by parts, the following equation can be obtained
(6.10)
Equation (6.10) expresses the extreme value distribution of the maximum load
effect. Obviously, Equation (6.10) does not contain the number of times loads are
applied, but is reflected in the distribution of S*.
6.1.2 Time-Dependent Reliability Transformation Method
In engineering practice, many types of loads are often applied to the structure
together. It is usually assumed that loads are independent of each another (e.g.,
live load and wind load), but in many cases this is not actually so (e.g., wave load
and wind load). In addition, the peaks of different loads do not appear at the
same time. Therefore, it appears very conservative to take the extreme value of
each load. Usually, a loading process is selected to represent an equivalent load
effect combination. The problem of load combination will be further discussed in
Chapter 7.
Essentially speaking, a problem of time-dependent reliability can be converted
into a problem of time-independent reliability and then solved. First, it is
necessary to investigate the maximum combination of various loads within the
service life of the structure, and then calculate the probability that the limit state
equation of the structure can no longer be satisfied under these loads. Equation
(6.4) can be summarized as follows:
(6.11)
where S( ) represents the load effect generated in the load process Q(t); R is a
structural reactance vector defined in the PDF of structural reactance, fR( ).
Because Equation (6.11) is difficult to solve directly, Wen and Chen[6-4] proposed
using r in place of resistance R in Equation (6.11); the probability thus becomes
a conditional failure probability pf(tL|r) and the failure probability can be
expressed as
(6.12)
where conditional failure probability pf(tL|r) is a function of a function of the
random process of vector load Q(t). Importantly, it lists conditional failure
probability under a given structural resistance.
Now we can consider the problem to be the crossing problem mentioned above.
Given the boundary of structural resistance R = r, we are concerned about the
crossing probability of random load process Q(t). Suppose pf(tL|r) is related to
the crossing rate. The previous crossing theory can now be directly used to
calculate the failure probability.
Because R and Q(t) are independent events, Equation (6.12) can be expressed as
(6.13)
where
life.
is the CDF of maximum load combination Smax during the service
Another similar way is to convert the problem of time-dependent reliability into
a problem of time-independent reliability by means of an auxiliary random
variable. It is first necessary to establish a limit state equation for timeindependent reliability, and then to perform analog computation using FORM or
the Monte-Carlo method. See Chapters 3 and 4.
These methods have a shortcoming, however: conditional failure probability
pf(tL|r) must be effective for all R and r, i.e., the replacement of the actual loading
process with equivalent load effect is only effective within the linear elastic
range (based on the principle of superposition) or ideal plastic range of the
structure.
Although the time integral method seems very simple to use, under the
combined action of multiple loads, the random process theory is actually a better
method for solving practical problems.
6.2 Discrete Method
When the discrete method is used to deal with the time-dependent reliability of
a structure, the design reference period of the structure will be divided into
several basic time intervals. These time intervals can be one year, or the duration
of a special event, such as a windstorm. Thus, the problem of reliability is
converted into a calculation of the failure probability of nL events that occur
within a given time interval or within the service life of the structure.
Once the service life of the structure, tL, is determined, the number of basic time
intervals, nL, can also be determined. For a windstorm, although the frequency of
its occurrence may be known, it is still impossible to accurately know its
duration in advance. The discrete basic time interval could be one day, one
month or one year, and it is most common to set it to one year.
Thus, the calculation of failure probability discussed in the previous chapters is
transformed into the problem of calculating the annual probability of failure. The
resistance and load effect, variables people are concerned about, are
transformed into the extreme value of the appropriate PDF for each year. The
distribution of resistance and load effect needs to be determined according to
observed data (e.g., the data on wind load and wave load), and the specific type
of distribution depends on the length of the basic interval selected before.
Therefore, the PDF of maximum annual wind force is different from that of
maximum daily wind force, and these distributions can only be related to one
another through some specific assumptions.
The proof of the discrete method is analogous to that of the time integral method
presented in the previous section, and can be divided into two cases:
1. nL is a known variable;
2. nL is a random variable.
6.2.1 Known Number of Discrete Events
The structure may be subjected to various random loads at the same time. Given
a time interval (e.g., one year), the value of nL can be very large, and the loads to
be independent of one another. So, the extreme value distribution of annual
load effect S can be obtained by Equation (6.8). Then, the annual failure
probability
can be obtained by Equation (6.10). It is assumed that the failure
events in each year are independent of one another. So, S in Equation (6.10) just
needs to be replaced with to calculate the failure probability in nL:
(6.14)
By Taylor series expansion
(6.15)
where
, with the second-order term ignored. It can get:
(6.16)
or
(6.17)
The ignored second-order term in Equation (6.15) does not need to be included
in the calculation unless the value of y makes
. Obviously, Equations
(6.16) and (6.17) are more accurate when σS >> σR. In the process of
approximate calculation, it is necessary to ensure that
and pf(nL) have a
minimum. Equation (6.17) shows that, in the approximate calculation of the
failure probability pf(nL) of the structure within its service life, it is only
necessary to multiply the annual failure probability
by nL within the service
life of the structure, tL.
Compared with the service life of the structure determined according to the
number of loads, it is obviously more reasonable to take time as a parameter. The
failure probability of the structure in time interval [0, t] can be expressed as
(6.18)
where FT( ) is the CDF of time T, and LT( ) is the time-based expression of the
reliability function.
Let pi be the failure probability of the structure in the ith time interval, it can
obtain
(6.19)
Suppose pi is independent from time interval to time interval, pi = p and pt is
sufficiently small. The failure probability can thus be approximated to
(6.20)
It is similar to the results of Equation (6.17). Obviously, time can be determined
arbitrarily. In most cases, the time interval is set to one year. The failure
probability of the structure within the service life [0, tL] can be written as
(6.21)
6.2.2 Unknown Number of Discrete Events
The failure probability pi of the structure in the ith time interval depends on the
number of loads applied during this time interval. If a basic time interval is
regarded as an event, such as a windstorm, then the actual number of loads in
this event can be ignored, and an approximate extreme value probability
distribution is used to describe the maximum load effect for this event.
In order to obtain the failure probability pf(tL) of the structure within its service
life, the frequency of event occurrence needs to be given in advance. Suppose the
probability that k events will occur in the time interval [0, t] is pk(t), then the
failure probability of the structure in the time interval [0, t], like Equation (6.14),
can be expressed in time a:
(6.22)
where all possible frequencies of the occurrence of the event are considered.
Suppose pk(t) obeys the Poisson distribution:
(6.23)
where ν represents the average rate at which events occur. According to the
characteristics of the Poisson distribution, this means that the e specimens are
independent of one another and do not overlap. This assumption will be quite
reasonable when ν is small enough. By substituting Equation (6.23) into
Equation (6.22), we get:
(6.24)
where
represents the failure probability of a given event, and can be
obtained by replacing S* in Equation (6.10) with . If it is considered that
is the average failure probability during the basic time interval, Equations (6.25)
and (6.21) are consistent with each other.
It must be noted that some external information needs to be taken into
consideration for the calculation of the failure probability
of a given event [65]
. For an event that an offshore structure is affected by a rainstorm, the
maximum load effect will depend on the characteristic wave height Hk of the
rainstorm.
is used to represent the maximum load effect at the given
characteristic wave height Hk.
represents the probability of the
characteristic wave height between h and h + δh(δh → 0). Conditional failure
probability
can be calculated by Equation (6.12):
(6.25)
(6.26)
Suppose Hk > 0 in the equation, then the PDF
can be obtained directly by
field observation. However, for the value of
, it is necessary not only to
give load data under Hk, but also to analyze the load effect
load
.
under the given
6.2.3 Return Period
The return period refers to an average time interval within which an event that
exceeds a certain level or load recurs within a certain statistical period.
According to this concept, the return period can be regarded as the reciprocal of
failure probability in the limit state. So, in a broad sense, the calculation of the
return period can be defined by the following formula:
(6.27)
where
represents the failure probability in the limit state in the basic time
interval (usually set to one year) calculated by Equation (6.24), and
is also
defined in the same unit interval. If there are nL basic time intervals within the
designed service life tL of the structure, and it is assumed that the events in each
basic time interval are independent of one another, then the failure probability of
the structure within its service life can be obtained by Equation (6.28):
(6.28)
Taking this idea further, if there are m mutually independent random loads
affecting the failure probability, then:
(6.29)
or
(6.30)
This equation shows that the reciprocals of the return period can be added
together, provided that the events in each basic time interval are independent of
each another.
6.2.4 Risk Function
Another f analysis index frequently used in classical reliability theory is the “risk
function”. This can be expressed either by the number of loads applied or by
time. As can be seen from Equation (6.18), the structural lifetime expressed with
time as a parameter is P(T < t) = FT(t), and the PDF of the designed service life is:
(6.31)
This is also called “unconditional failure rate”, because it reflects the failure
probability between time interval t and t + dt when dt → 0. The risk function
(also known as the “year-specific failure rate” or “conditional failure rate”)
expresses the failure probability between time interval t and t + dt when dt → 0.
Furthermore, it is assumed that there is no failure occurring before t. So
(6.32)
or
(6.33)
This equation shows that, when FT(t) is very small, hT(t) is very close to fT(t).
Figure 6.6 shows some typical risk functions. By substituting Equations (6.18)
and (6.20), we can transform Equation (6.33) into:
(6.34)
and
(6.35)
where the integral term in the brackets represents total failure probability.
Therefore, if any of fT(t), FT(t) and hT(t) is known, the remaining two can be
obtained by calculation. For a typical structure, the risk function is usually a
“bath-tub” curve (see Figure 6.7). The figure shows that the risk is highest at the
initial stage, and reduces rapidly with the gradual application of load. This stage
is mainly a period during which a test load is applied. As time goes by, the
structure gradually deteriorates, while risk rate increases.
Figure 6.6 Typical risk function.
Figure 6.7 Variation trend of risk function in different structural stage.
6.3 Calculation of Time-Dependent Reliability
6.3.1 Introduction
In the time-dependent reliability method, the failure probability of the structure
can be directly estimated by calculating the probability of first transcendence.
For a highly reliable system, this is a feasible approach. If the random process
S(t) has a safe zone DS within the service life of the structure, [0, tL], then the
structural reliability can be expressed as:
(6.36)
where [0, tL] represents the service life of the structure or other time intervals
concerned; R(t) represents the structural resistance at t; S(t) is also the random
process of load effect with time as the independent variable. For the sake of
operation, it can be expressed by the formula below:
Usually, there is no simple corresponding relationship. At present, for the most
commonly used method, based on the random process theory, a upper bound
relation between the failure probability and crossing rate is put forward as
follows:
(6.37)
where pf (0) represents the failure probability of the structure at t = 0 or at the
exact moment when load is applied. The results are valid for the random process
of a stationary vector load. If the random process of the vector load is smooth,
the term
will be replaced by the term
.
Obviously, there is an error in the calculation formula because the reference
source is not found! Three problems need to be paid attention to. The first
involves calculating the term pf (0), which is a non-time-dependent probability
and can be calculated directly by any of the methods presented in Chapter 3. The
second problem involves calculating the crossing rate. The method discussed in
Section 6.5 can be used. After these conditional terms are obtained, the integral
term in Equation (6.12) is needed to calculate the unconditional failure
probability. The third problem concerns the degree of closeness between the
upper boundary in Equation (6.37) and the real results. For the random process
of a narrow-band load, crossing phenomena are rather concentrated, and in this
case, the upper boundary expression will be overestimated; conversely, if the
random load vector is not a narrow-band random process, then the upper
boundary estimation will be quite accurate.
Another way to evaluate the time-dependent reliability is to bypass the
calculation of the crossing rate and evaluate the probability of the random mvector process in the given failure domain by simulating the random process of
every continuous vector [6-6]. In particular, the random process of every standard
multivariate Gaussian can be represented by the trigonometric series of random
coefficients [6-7]. Then, by using directional simulation, the probability for the
random process to cross the security domain is calculated by maximizing the
service life of each directional sample. Although this method is straightforward,
it still has a disadvantage, i.e., a heavy calculation burden, because it needs at
least s(t +1) simulation runs; s represents the number of components of the
random vector process and t represents the number of parameters used to
express the random process. An optimal value for t has not yet been determined,
but it may be quite important. This method for evaluating the time-dependent
failure probability is often used to check the results obtained by other methods
[6-8]
.
6.3.2 Sampling Methods for Unconditional Failure Probability
The integral of Equation (6.12) may hardly obtain an exact analytical solution.
The same problem exists in Equation (6.37). The trick is to use numerical
simulation, FORM, etc., to solve these.
(1) Importance sampling and conditional expectation sampling
Importance sampling is adopted for Equation (6.40) while conditional
expectation sampling is adopted for Equation (6.39) [6-9]. Importance sampling is
expressed as follows:
(6.38)
The failure domain D is first integrated. The limit state equation is G(x) = 0. m
independent limit state equations constitute
probability, the multiple integral can be written as:
. Using conditional
(6.39)
(6.40)
The term { } in Equation (6.39) is replaced with
; X1 and X2 in Equation
(6.39) are subsets of random vector X;
is the conditional PDF of X2 given
X1; D1 and D2 are sample spaces of X1 and X2, respectively. D1 and D2 must be
mutually exclusive and belong to D.
As can be seen, Equation (6.40) is of the same form as the time-dependent
reliability function Equation (6.12). The m-integral on D1 can be obtained by
Monte-Carlo simulation based on importance sampling PDF hv( ).
If the term { } in Equation (6.39) can be calculated numerically, then every
sample in the importance sampling method can be calculated as well. In this way,
the conditional expectation gap can be narrowed, thereby reducing the number
of Monte-Carlo samples required.
When the directional simulation in the load random process space is closely
related to obstacle crossing, as shown in Figure 6.8, then the random process
space will be the load random process Q(t). In load random process space, the
security domain SD can be interpreted as structural resistance, denoted as R = r.
The traditional limit state equation Gi(q, x) = 0, i = 1, …, k can be interpreted as
the boundary.
Let the joint PDF of resistance R be fR( ), and be the function of other random
variables, R= R(X). The components of X can be structural components or the
components of resistance, material properties, etc. (X contain the uncertainties
arising from load random processes. Q(t) can contain stochastic processes other
than load random processes. Its PDF must reflect the independence of load
components.)
Figure 6.8 Sample functions of vector stochastic processes.
Structural failure probability can be provided in Equation (6.12). Given the
structural resistance R = r and the conditional failure probability pf(tL | r) of the
highly reliable structural system in the time interval tL, a calculation can be made
by means of Equation (6.24) or (6.38) according to the crossing rate
and
initial failure probability pf(0):
(6.41)
where pf(0,s | a) is the failure probability at t = 0, and
represents the crossing
rate at which vector random processes Q(t) leave the security domain D.
Equation (6.42) can be adopted for directional simulation in vector random
processes:
(6.42)
where A is the directional cosine vector, fA( ) is the PDF, S is the radial distance
representing structural resistance, and fS|A(s | a) is the conditional PDF.
6.3.3 First-Order Second-Moment Method
For a given limit state equation, Gi[X(t)]≥0,i = 1,...,n, the security domain DS can
be obtained. Moreover, pf(0) and
can be obtained by the first-order secondmoment method. For every independent limit state equation, the conditional
failure probability can be calculated using the upper boundary formula for
crossing rate presented in Equation (6.41). If the load random process is a
completely normal vector random process, the upper boundary of the first
transcendence probability can be calculated by the first-order second-moment
method with respect to Equation (6.38).
The main problem that remains is how to calculate the integral of the resistance
R, which is a random variable. An important way presented in Section 6.2 is to
introduce an auxiliary random variable to transform the problem of timedependent reliability into one of time-independent reliability. However,
experience shows that, if the limit state equation is time-dependent, or if the
random process is non-stationary when an auxiliary random variable is adopted
for calculation, then the calculation will take much more time [6-10]. In addition,
the Laplace integral method can be used for calculation [6-11].
The traditional first-order second-moment method is highly effective in
analyzing problems of time-independent reliability, but ineffective in calculating
the crossing rate with respect to the problems of time-dependent reliability. The
main reason for this is that there must be structural resistance R, which must be
a time-independent random variable during the process of integration. For this
reason, the importance sampling method and Monte-Carlo simulation have
become the most practical methods.
6.4 Structural Dynamic Analysis
6.4.1 Randomness of Structural Dynamics
It is very important to perform a dynamic structural analysis when the structure
bears a time-dependent load which affects the structure’s response to the load
(e.g., deformation and stress). In traditional structural dynamic analysis methods
based on time-domain analysis, the structural dynamic equations are integrated
over time. The load changes over time, and structural response is also a function
of time. This whole calculation process can be accurately applied to a very
complex structure. Material and structural properties can also be nonlinear, but
iteration is required during calculation and is highly time-consuming.
If the load is stochastic, then the time-domain-based calculation results will have
limitations since the load can no longer be described based on time. A common
practice is to analyze structural response by a sample function of load, and then
to observe the statistical characteristics of the response by constant repetition. If
there is a maximum allowable stress and deformation set under the design
standards, then the limit state equation G(x) = 0 can be established. Therefore,
theoretically at least, the structural reliability can also be calculated by the
Monte-Carlo method. Unfortunately, this method is not so practical in practice,
because it requires a lot of time-domain-based analyses, and each analysis is
highly time-consuming. Therefore, it is quite difficult to combine traditional
dynamic analysis with structural reliability problems unless the random process
theory can be further simplified. For this reason, the time domain method
remains poorly applicable to the theory and application of structural reliability.
When the structure is linear, i.e., when it is an elastic structure based on the
assumption of infinitesimal deformation, or when the input-output (e.g., loadstress) transformation equation is linear, a calculation method based on the
frequency domain can be used. This method is widely used to solve random
vibration problems [6-12].
6.4.2 Some Problems Involving Stationary Random Processes
A stationary random process X(t) can be decomposed into a finite number of
sine and cosine components, each of which has its stochastic amplitude and
natural frequency. This is the famous Fourier integral transform. The coefficients
of cosine terms are determined by the integral of the production of the random
process X(t) and cosωt. The same is true for the coefficients of sine terms. Every
term is a Fourier transform of X(t). Because the randomness of X(t) is
determined by the autocorrelation function RXX(τ), the Fourier transform of X(t)
can also be expressed as a function of RXX(τ). The coefficients of the cosine terms
can be expressed as a continuous function of ω:
(6.43)
Because RXX(τ) is a symmetric function, the coefficients of the sine terms
corresponding to Equation (6.43) are zero; of course, SX(ω) is also a symmetric
function (see Figure 6.9 (a)). SX(ω) is what we usually call the (mean squared)
spectral density. It is not hard to see that, if X(t) is completely disordered, RXX(τ)
obtained from Equation (6.43) will be zero (except where τ = 0), and SX(ω) is a
constant for all ω. This is what is known as white noise. Due to the lack of
prominent frequency components, its power is the same across all frequency
bands (see Figure 6.9 (b)). Similarly, if there is a dominant frequency near ω0 for
the random process, then the form of the spectral density is shown in Figure 6.9
(c)). This is the narrow-band process, which is of great importance for structural
dynamic analysis because most structures have a major vibration mode or
response frequency. This mode usually corresponds to the (minimum) natural
frequency of structures.
Let τ in Equation (6.43) be zero, and integrate both sides in (−∞, ∞), so
. Let uX = 0, and substitute it into Equation (6.44) and
Equation (6.45), obtaining Equation (6.46):
(6.44)
(6.45)
(6.46)
The area enclosed by the curve SX(ω) is the mean square value of the stationary
random process X(t), equivalent to its variance (see Figure 6.9 (a)). Let uX = 0,
since randomness is the main concern here. The analytical results for uX ≠ 0 can
be directly superimposed on the stochastic results at ux = 0.
In addition to the variability of random processes, the probability distribution of
peaks is also a key concern. It is denoted as Fp(a), i.e., the probability that the
peak of random process X(t) lies below the horizontal line X(t) = a, when uX = 0.
The results achieved in the previous section can be used directly here. For a
narrow-band random process that is smooth enough, it will have a maximum at
X = 0 only. The ratio of X > a is
.
is the probability of transcendence
offered in Equation (6.47);
is the probability of transcending X = 0; when 0 ≤
a ≤ ∞, Equation (6.48) can be obtained:
(6.47)
(6.48)
Figure 6.9 Sample function and spectral density of random process.
Figure 6.10 Probability density function of Rayleigh distribution.
For the horizontal line x = a, the following is obtained after differentiation:
(6.49)
Equation (6.49) represents a Rayleigh distribution (Figure 6.10). Obviously, fp(a)
reaches its maximum at a = σX. There is no peak at a = 0. If the random process is
not completely smooth, e.g., if there are multiple extreme values every time it
transcends zero, or if the random process X(t) does not obey normal
distribution, it is more appropriate to describe the probability distribution of the
peaks of the random process by the more generalized extreme value distribution
(e.g., EV-III).
6.4.3 Random Response Spectrum
It should be noted that the spectral density of structural deflection or stress at a
certain point can be obtained by considering the excitation-response
relationship of the linear structure in the frequency domain, and the
corresponding relationship will be as follows:
(6.50)
Figure 6.11 Analysis on the relationship between input and output spectral
density function of offshore platform structure.
For a single input source or a load random process X(t), a single output Y(t) can
be generated. H(ω) is a frequency-response function. Equation (6.50) can be
generalized as a linear superposition of multiple independent inputs:
(6.51)
When there is a correlation between inputs, there will be a more complex
relationship between input and response. Figure 6.11 shows the main principle
by which the frequency-response function is used to obtain the output spectral
density from input spectral information. The input spectral information can be
inferred by observation and physical process analysis. H(ω) can be obtained
based on the system analyzed or by integrating the structural response in a given
excitation mode
6.5 Fatigue Analysis
6.5.1 General Formulas
A very important problem related to time-dependent reliability is fatigue. This is
also a typical limit state, i.e., it is controlled by the number and intensity of
imposed loads rather than by a single load. The limit state equation of fatigue
can be described as:
(6.52)
where Xa represents the actual structural strength or performance. Xr represents
the structural performance that must be satisfied within the designed service
life. Therefore, if the fatigue life is expressed by the number of stress cycles, Xa
becomes the number of stress cycles needed to cause structural failure and Xr
becomes the number of stress cycles that can still meet the design requirements
within the given service life. Correspondingly, there are some cases in which the
fatigue life is described by means of crack size, damage index, etc. In general,
both Xa and Xr are uncertain, and their accuracy depends on the fatigue model
adopted.
6.5.2 S-N Model
The equation below provides a traditional model that can be used to describe the
fatigue life Ni of a component or structure under constant stress:
(6.53)
where K and m are usually constants, but here, they are random variables; Ni
represents the number of stress cycles at constant stress amplitude Si. The value
of K and m can be evaluated by experimentation. In general, a conservative value
is adopted in Equation (6.53) to ensure that the fatigue life Ni can be assessed
safely. For structural reliability analysis, Equation (6.53) needs to be closer to
reality rather than being a conservative prediction. Therefore, the value of K and
m presented in the reference literature may not be applicable in practice. In
addition to the applicable expected value, it is also necessary to consider the
impact of the uncertainty of these parameters. As for the analysis method, please
refer to traditional structural reliability analysis.
For the safe domain, Equation (6.52) can be rewritten as:
(6.54)
where N0 represents the number of cycles that the structure must be able to bear
in order to meet the design requirements; N0 also contains uncertainties.
In engineering practice, the stress amplitude is not a constant, but a random
variable. If the amplitude in each cycle can be recorded, then the linear damage
accumulation rule can be applied:
(6.55)
where ni represents the actual number of cycles under the action of stress
amplitude Si; Ni represents the fatigue life under the action of stress amplitude Si.
If stress amplitude varies, Equation (6.55) is turned into:
(6.56)
where N represents the total number of amplitude cycles. Traditionally, the
damage parameter Δ is a uniform value, but in fact, it ranges from 0.9 to 1.5.
Therefore, Δ reflects the fact that great uncertainty is caused in practical
application due to the empirical problem of Equation (6.56). It is roughly
reasonable to adopt a lognormal distribution with a mean of 1.0 and a coefficient
of variation of 0.4∼0.7 [6-12][6-13].
When equation (6.56) is used, it is necessary to determine the N value and the
probability distribution of Si. There are many different methods for extracting
the N value. The safety domain can be redefined with damage parameters:
(6.57)
where X0 is used to solve the uncertainty problem caused by failure to accurately
determine stress amplitude Si during the application of Equation (6.53) in the
model.
The main difficulty encountered in applying Equation (6.57) is that the N value is
uncertain. One way is to divide the stress amplitude into l groups (where l is a
given number), and the number of cycles in each group, Ni, is uncertain, so
(6.58)
6.5.3 Fracture Mechanics Model
Another way of using the fatigue model is to consider crack propagation under
the action of cyclic or random load, but this method is not always effective [6-14]
[6-15]
. The experimental results show that the crack propagation rate da / dN
depends on the change amplitude Δk of the stress intensity factor:
(6.59)
where a represents the current crack length; N represents the number of stress
cycles; C and m are testing constants, usually dependent on the frequency of load
application, the average load and the testing environment. During reliability
analysis, C and m must be processed as uncertainties. Generally speaking, the
stress intensity factor does not change significantly with stress, so the change
amplitude of the stress intensity factor can be calculated by:
(6.60)
where ΔS represents the applied stress amplitude; K(a) is a function of the
current crack length a; ΔKth is the threshold of ΔK, and ΔK = 0 at ΔKth.
After N stress cycles, the variability of the current crack length a can be obtained
by integrating Equations (6.59) and (6.60):
(6.61)
where a0 represents the initial crack length. After obtaining the statistical
characteristics of the parameters in the equation, we can obtain the mean and
variance of a(N). The variable-amplitude load ΔS depends on the load sequence,
and is a random variable. This problem can be solved by the incremental method
according to Equation (6.60) or by using an effective ΔK [6-14]. For either method,
the limit state Equation (6.52) can be rewritten as:
(6.62)
where aa represents the limitation of the crack length under certain functional
conditions after the structure has borne N load cycles within the designed
service life tL. Another method for the limit state equation is to express it by
means of the crack tip displacement, which represents the material stiffness [614]
.
6.5.4 Example: Fatigue Reliability of an Offshore Jacket
Platform
(1) Probability model of fatigue load
Under the long-term action of random loads such as wind, waves and currents,
marine structures will produce alternating stress, micro-cracks and continue to
expand, resulting in fatigue failure of these structures. Existing engineering
experience shows that once the fatigue problem of an offshore structure occurs,
it will lead to serious consequences. Before carrying out fatigue life analysis of a
platform structure, it is therefore necessary to understand the fatigue load of the
structure first. The mooring force and motion parameters of a BZ28-1 SPM
(Single Point Mooring) system are derived from the model test conducted on the
Dutch ship model test pool in 1986. By analyzing the test data [6-16], the
nonlinear relationship between mooring force and wave environment
parameters can be obtained, which is convenient for the fatigue life analysis of a
jacket structure.
① Statistical characteristics of environmental loads
Because the symmetry of the platform structure and the statistical
characteristics of the ocean state in each direction are quite similar, 11 ocean
states in the three wave directions of 0°, 45° and 90° (corresponding to the east,
northeast and north, see Figure 6.12) are considered for fatigue calculation and
analysis. Each direction has the same probability of occurrence P, and it is
assumed that the probability of occurrence of waves and current in the same
direction and wave and currents perpendicular to each other in these three wave
directions account for half of each other, as is shown in Table 6.1 and Table 6.2.
Figure 6.12 Coordinate system of a single point mooring offshore jacket
platform.
Table 6.1 Statistical standard deviation of high frequency mooring force range.
HS/m p
TZ/S Waves and current flow
in the same direction
FUS/kN
FZS/kN
Waves and current are
perpendicular to each
other
FUS/kN
FZS/kN
0.25 0.15000 4.0
0.4
0.1
0.6
0.1
0.75 0.17500 4.0
1.25 0.11,
4.2
000
1.75 0.03750 4.7
5.5
17.5
1.3
5.2
7.4
23.6
1.6
6.5
37.4
13.1
50.6
16.3
2.25 0.01700 5.2
2.75 0.00600 5.6
66.1
104.0
26.2
45.5
89.3
146.0
32.6
56.6
HS/m p
TZ/S Waves and current flow
in the same direction
Waves and current are
perpendicular to each
other
FUS/kN
FZS/kN
FUS/kN
FZS/kN
3.25 0.00300 6.1
3.75 0.00100 6.6
4.25 0.00030 7.0
151.7
209.7
278.3
72.1
106.8
150.8
205.2
283.5
376.3
89.7
133.0
187.6
4.75 0.00015 7.4
5.25 0.00005 7.8
357.8
448.7
208.0
269.6
483.9
606.8
254.8
335.5
Table 6.2 Statistical standard deviation of low frequency mooring force range.
HS/m p
TZ/S Waves and current flow
in the same direction
FZS/kN
Waves and current are
perpendicular to each
other
FUS/kN
FZS/kN
0.25 0.15000 90.0 3.9
0.75 0.17500 90.0 43.1
2.25
15.4
2.0
26.5
1.4
9.0
1.25 0.11
90.0 131.3
000
1.75 0.03750 90.0 272.4
37.4
80.4
22.0
67.2
167.6
39.4
2.25 0.01700 90.0 470.4
2.75 0.00600 90.0 727.2
3.25 0.00300 90.0 1044.7
104.2
147.8
197.7
289.1
446.9
642.9
61.0
86.5
115.8
3.75 0.00100 90.0 1426.9
4.25 0.00030 90.0 1872.8
253.6
315.4
878.1
1152.5
148.6
184.6
4.75 0.00015 90.0 2384.3
5.25 0.00005 90.0 2964.5
382.8
455.7
1467.1
1823.8
224.2
267.0
FUS/kN
② Distribution of environmental load
The distribution of load during the whole lifetime of a structure is usually
referred to as the long-term distribution of fatigue load, but the fatigue life of a
structure is normally unknown when fatigue life analysis is carried out.
Therefore, the representative distribution of fatigue load over a certain
appropriate time length is usually regarded as the long-term distribution, which
is known as the recovery period of a load spectrum. The long-term distribution
of the load spectrum is composed of a number of short-term ocean conditions,
while the distribution of wave height and mooring force of each short-term
condition is assumed to be a Rayleigh distribution, which can be expressed as:
(6.63)
Where P (H) is the probability density function of the corresponding wave
height. This means that the number of cycles between the wave height H and H
+ΔH is
(6.64)
where: n is the number of cycles of ocean conditions during the recovery period.
According to the Rayleigh distribution, every effective wave height Hs can be
discretized into a series of regular waves with wave height Hd. The discrete wave
train contains the maximum wave height that may occur within 100a. After
discretization, the corresponding number of cycles of wave height in a 1a
recovery period is shown in Table 6.3.
Following the Rayleigh distribution and if the mooring force and wave height
have the same probability density function, the discrete mooring force sequence
will be as shown in Table 6.3. In this way, the high- and low-frequency mooring
forces corresponding to each discrete wave train can be obtained, where Fu and
Fz are the horizontal and vertical discrete mooring force sequences, respectively.
The number of cycles of the discretized high-frequency mooring force in 1a
recovery periods is the same as that of the waves, as shown in Table 6.4. The
number of cycles of low-frequency discrete mooring force sequences in 1a
recovery periods is shown in Table 6.5.
(2) Fatigue life assessment
① Numerical simulation of a platform structure
The substructure method is used to analyze the pile-soil interaction of an
offshore jacket platform, which involves the total structure being subdivided into
upper jacket structure, pile, and foundation soil subsystems at the pile head.
After this, the single reaction is combined to satisfy the interaction conditions,
after which the reaction of the whole structure can be obtained.
Table 6.3 Mooring force discovery series under different effective wave heights.
Hd/m Waves and current flow
in the same direction
FUS/kN
Waves and current
perpendicular to each
other
FZS/kN
FUS/kN
FZS/kN
6.29
14.16
7.83
1.00 52.35
1.75 115.17
2.50 177.98
31.45
69.20
106.94
70.79
155.75
240.70
39.14
86.11
133.08
3.25 240.80
4.00 303.62
144.69
182.43
325.65
410.60
180.05
227.02
4.75 366.44
5.50 429.26
220.17
257.92
495.55
580.51
273.99
320.96
6.25 492.07
7.00 554.89
295.66
333.41
665.46
750.41
367.93
414.90
7.75 617.71
8.50 680.53
371.15
408.89
835.36
920.31
461.90
508.84
9.25 743.35
10.00 806.16
446.64
484.38
1005.27
1090.22
555.81
602.78
10.75 868.98
522.13
1175.17
649.75
High frequency
mooring force 0.25 10.74
Hd/m Waves and current flow
in the same direction
Waves and current
perpendicular to each
other
FUS/kN
FZS/kN
FUS/kN
FZS/kN
High frequency
mooring force 0.25 70.58
10.85
43.42
6.36
1.00 352.92
1.75 776.42
54.25
119.35
217.12
477.66
31.78
69.72
2.50 1199.92
3.25 1 623.42
184.45
249.55
738.20
998.74
108.05
146.19
4.00 2 046. 92
4.75 2 470.42
314.65
379.75
1 259.28
1 519.28
184.32
222.46
5.50 2 893.92
6.25 3 317.42
7.00 3 740.92
444.85
509.95
575.05
1 780.36
2 040. 90
2 301.44
260.60
298.73
336.87
7.75 4164. 42
8.50 4587. 92
640.15
705.25
2 561.98
2822. 52
375.0
413. I4
9.25 5011. 42
10.00 5434. 92
770.35
835.45
3083. 06
3343. 60
451.28
489.41
10.75 5 858. 42
900.55
3604. 14
527.55
The Winkler foundation beam model is used to simulate the dynamic interaction
between pile and soil. The soil is treated as a Winkler foundation, and the pile
consists of a long beam buried in the soil. When analyzing the bearing capacity of
a single pile, the finite element model is used to carry out the calculation. The
construction can be seen in Chapter 5.
The deterministic analysis method (also known as the discrete wave method) is
used to analyze the effects of wave and mooring loads on the platform jacket
structure. Generally speaking, this includes the calculation and analysis
described hereafter. The quasi-static analysis of the structure can be carried out
under different ocean conditions. Depending on the fatigue load condition
determined at the front, linear wave theory can be applied to calculate the wave
force acting on the conduit frame according to each wave height and period for
the wave condition. For the mooring force, the discrete horizontal and vertical
mooring force components are directly applied to the top of the column, after
which the internal forces and stresses on the members can be obtained by finite
element analysis of the platform jacket structure. Considering the stress
concentration of the corresponding welded tubular joints in the structure, the
hot spot stress range of the tubular joints can be obtained by multiplying the
calculated nominal stress by the corresponding stress concentration factor,
which can then be used for fatigue life analysis.
Table 6.4 Wave ocean state distribution.
Hd/m HS/m
0.25
0.25
0.75
1.25
1.75 2.25
340851 91641 21167 3354 838
2.75 3.25 3.75 4.25 4.75 5.25
185 61 14 3
1
0
1.00
1.75
2.50
53349 355122 177599 36867 10378 2433 831 197 44
0
13129 71085 32300 12901 3635 1383 351 82
0
0
5370 9935 7340 2854 1312 376 95
17
33
40
4
9
1
3.25
4.00
0
0
0
0
92
0
1331 2380 1467 883 300 86
0
468 526 450 191 63
39
32
12
10
4.75
5.50
0
0
0
0
0
0
0
0
57
4
135 178 99
25 55 43
40
1
23
14
8
6
6.25
7.00
0
0
0
0
0
0
0
0
0
0
3
0
4
3
5
5
10
4
8
4
4
2
7.75
8.50
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
0
2
1
1
1
9.25 0
10.00 0
10.75 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
② Cumulative fatigue damage and life assessment method
The long-term distribution of stress range is composed of a series of shortterm
Rayleigh ocean conditions. The fatigue loads of the single-point mooring jacket
platform structure include wave forces, high-frequency mooring forces and lowfrequency mooring forces. The fatigue life of a BZ28-1 SPM jacket structure has
been analyzed by the China Classification Society [6-17], where the fatigue
damage caused by these three forces is calculated separately. After this, the total
damage can be obtained by adding them together, and the fatigue life of the
structure can be evaluated. It is, however, unreasonable to calculate it in this way,
since it will lead to an overestimation of the fatigue life of the structure. Fatigue
analysis of the bow structure of a floating production and storage ship was
carried out by Shanghai Jiao Tong University under the combination of wave
frequency and low-frequency mooring forces [6-18]. The Norwegian Classification
Society Offshore Standard manual [6-19] introduces the fatigue damage
calculation method for the combined stress of wave frequency and lowfrequency mooring forces:
(6.65)
(6.66)
Table 6.5 Low-frequency moving force cycles.
Hd/m HS/m
0.25
0.25 0.75
15 149 4073
1.25 1.75 2.25 2.75 3.25 3.75 4.25 4.75 5.25
988 175 48 12 4
1
0
0
0
1.00
1.75
2371
0
15 783 8288 1925 600 151 56
584
3317 1687 745 226 94
14
26
3
6
1
3
0
1
2.50
3.25
0
0
0
0
251 519 424 178 89
4
70 138 91 60
28
22
7
7
3
3
1
1
4.00
4.75
5.50
0
0
0
0
0
0
0
0
0
4
0
0
27
3
0
33
8
2
31
12
4
14
7
3
5
3
2
3
2
1
1
1
1
6.25
7.00
0
0
0
0
0
0
0
0
0
0
0
0
1
3
1
0
1
0
1
0
0
0
7.75
8.50
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
9.25 0
10.00 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
10.75 0
0
0
0
0
0
0
0
0
0
0
Where the fatigue damage dij calculated by wave, low frequency and high
frequency combines to a stress range Si for the first subocean state the i-th sea
case and in the j-th wave direction; ni is the number of cycles of combined stress
Si; Nj is the fatigue life under the given combined stress range Si in the j-th wave
‘direction based on the X curve recommended by the API specifications; ND is the
number of wave directions, with a value set to 3; Ns is the number of subocean
states in each wave direction, set to a value of 90.
(6.67)
where SWi, SLi and SHi are the hot spot stress amplitudes of wave fwi, low
frequency fLi and high frequency fHi tube joints under i working conditions
respectively. wiiLii
(6.68)
Where
the average wave zero-crossing rate,
low-frequency
mooring force zero-crossing rate,
high-frequency mooring force zerocrossing rate.
Thus, the cycle number of combined stresses is
(6.69)
Where nwi, nLi and nHi are the cycle times of wave, low-frequency and highfrequency mooring forces, respectively.
The total cumulative fatigue damage consists of 3 wave directions, with each
direction being the sum of fatigue damage under 180 suboperating conditions:
(6.70)
In this formula, P represents the same direction of wave and current, V
represents waves and current running perpendicular to each other, and the
fatigue damage D of tubular joints is calculated with a recovery period of 1.0a,
while the required years when D=1. 0 are calculated as the fatigue life of the
tubular joints:
(6.71)
Table 6.6 False damage and false life of the main pipe joints.
Node Bar Type of
structural
components
FSC
Axial In-plane
bending
moment
In-plane
bending
moment
36
36- CHD
32
4.524 1. 833
3.261
0.0207 48.3
36- BRC
35
4.555 2.011
2.895
0.0219 45.6
3640
3639
3640
3637
3632
3633
CHD
6.104 2.027
4.297
0.0159 62.7
BRC
6.420 2.087
4.145
0.0181 55.4
CHD
6.104 2.027
4.297
0.0159 62.9
BRC
6.420 2.087
4. 145
0.0180 55.5
CHD
4.524 1. 833
3.261
0.0207 48.4
BRC
4.
555
2.011
2.895
0.0219 45.7
37- CHD
41
6.104 2.027
4.297
0.0163 61.2
37- BRC
38
6.
420
2.087
4.145
0.0186 53.9
37- CHD
41
6.104 2.027
4.297
0.0163 61.2
37- BRC
36
6.420 2.087
4.145
0.0185 54.0
37
D
Tf/a
Node Bar Type of
structural
components
FSC
Axial In-plane
bending
moment
In-plane
bending
moment
38
CHD
4.524 1.833
3.261
0.0206 48.6
BRC
4.555 2.011
2.895
0.0218 45.9
CHD
6.104 2.027
4.297
0.0155 64.4
BRC
6.420 2.087
4.145
0.0176 56.8
CHD
6.104 2.027
4.297
0.0156 64.2
BRC
6.420 2.087
4.145
0.0176 56.7
38- CHD
34
4.
524
1.833
3.261
0.0206 48.5
38- BRC
33
4.
555
2.011
2.895
0.0219 45.7
39- CHD
43
6.104 2.027
4.297
0.0165 60.4
39- BRC
36
6.420 2.087
4.145
0.0188 53.2
39- CHD
43
39- BRC
38
6.104 2.027
4.297
0.0163 61.3
6.420 2.087
4.145
0.0185 54.0
39
3834
3835
3842
3839
3842
3837
D
Tf/a
Therefore, in order to prolong the service life of the platform structure and
ensure it can be used for 20 years, the total service life of the structure can be
extended to 25 years. Considering the strict requirements of the API (American
Petroleum Institute) code on the fatigue design life of welded pipe joints on the
offshore jacket platform structure, the design life of a single-point mooring
jacket platform needs to be increased to 50 years. Table 6.6 lists the cumulative
fatigue damage and fatigue life analysis results of the main components of the
platform jacket structure caused by wave force, as well as high-frequency and
low-frequency mooring forces, where FSC is the stress concentration factor, with
the corresponding pipe joint numbers shown in Figure 6.13 [6-20].
Figure 6.13 Structural model of a BZ28-1 SPM platform.
6.5.5 Example: Fatigue Reliability of a Submarine Pipeline and
Analysis of its Parameters
6.5.5.1 Introduction
Fatigue of submarine pipelines has long been the research focus of academics
from all over the world. For example, Xu [6-21] evaluated the vortex-induced
vibration response and fatigue life of linear submarine pipelines. Nguyen and
Kocabiyik [6-22] used a numerical simulation method to simulate vortex-induced
vibrations. Cui Weicheng et al. [6-23][6-24] studied vortex-induced vibrations and
fatigue damage of flexible risers under the influence of ocean currents. Shu
Hengmu [6-25] carried out random vibration analysis of a 3D submarine
suspended pipeline. Yu Jianxing [6-26] studied the fatigue reliability of vortexinduced vibrations of a pipeline in a suspended state on the basis of considering
the second-order square damping term of a wave load, and thus established a
nonlinear vortex-induced vibration equation for pipeline span. Li Xin [6-27]
studied factors including span, proximity from the seabed, support conditions at
pipeline ends, and the influence of fluids in pipelines on the dynamic response of
suspended pipelines using dynamic model experiments.
The random vibration response and fatigue reliability analysis methods for
suspended span pipelines can be further divided into time-domain simulation
and frequency-domain analysis, of which the former cannot be practically
applied to actual engineering situations due to its complicated calculation
process and low calculation efficiency. The latter cannot accurately consider the
influence of geometric nonlinear factors in spite of its simple calculation process,
leading to certain errors in practice. In particular, when the pipeline has a large
span, this analysis method will lead to significant errors. In consideration of the
above, it is of great necessity to improve the random frequency domain vibration
response method for suspended pipelines, and to take into account the influence
of geometric nonlinear factors on the random vibration response [6-28][6-29].
6.5.5.2 Analytical Process
A structural model was established by referring to the design parameters of a
submarine pipeline in the Bohai Sea. A random motion response analysis of
pipelines with different span lengths was carried out using a random vibration
analysis method where nonlinear factors were taken into consideration. Finally,
the fatigue damage caused by transverse vortex-induced vibrations of a pipeline
was calculated based on the Palmgren-Miner linear cumulative damage criterion
[6-30][6-31]
and via the S-N curve method. The fatigue life and fatigue reliability of
the structure were also predicted, and the influence of span length, wave height,
water depth, outer diameters of pipelines and initial stress on fatigue life and
fatigue reliability of the structure were discussed.
6.5.5.3 Finite Element Model
A structural finite element model was established based on the design
parameters of a submarine pipeline in the Bohai Sea. See Figure 6.14 and Table
6.7 for the structural parameters.
6.5.5.4 Random Lift Model
As pointed out in the literature [6-32], the longitudinal resonance of the pipe span
has a small amplitude, with the vibration stress range generally lying within the
fatigue limit of the pipeline. Therefore, fatigue failure does not need to be
considered. In contrast, the transverse resonance of the pipe span has a large
amplitude, with a vibration stress range far greater than the fatigue limit of the
pipeline. Therefore, the transverse vibration fatigue failure of the span serves as
the main mode of vortex-induced vibration failure of a submarine pipeline.
Finally, only the lifting force of the lateral wave force caused by the wave force
load ascribed to the lateral vibration is considered, and can be expressed as:
(6.72)
Figure 6.14 Prototype cross section of an oil pipeline.
Table 6.7 Design parameters of a submarine pipeline.
Materials
Size
Remarks
Pipeline
Steel (API 5L X65 Diameter 168.3mm wall E=2.07×105MPa
SML)
11.0mm thick
Anti-corrosive
coating
Epoxy powder
0.4mm thick
Density
940kg/m3
Insulation layer Polyurethane
foam
40mm thick
Density 60kg/m3
Protective layer Polyethylene
Weight coating Concrete
8mm thick
40mm thick
Waterproof
Density
2950kg/m3
Where,
; ρ is the density of seawater; D is the outer diameter of
the pipeline; u(t) is the horizontal velocity of the wave water quality point at the
depth of the pipeline axis; CL is the lift coefficient.
It can be seen from the equation above, and from linear wave theory, that the
lateral wave force is proportional to u(t) | u(t)|. By linearizing Morison’s
equation, the lateral wave force can be expressed as:
(6.73)
Where, σu is the mean square deviation of the random process of wave velocity
u(t).
The horizontal velocity u(t) of a water point is a random process, as is the
transverse wave force fL(t). According to linear wave theory, the maximum
horizontal velocity of a single cosine component wave water point can be
expressed as
(6.74)
Where, k is the wave number, H is the wave height, T is the wave period, d is the
water depth, and z is the height of wave water particles relative to the wave
surface. The equation can be expressed as:
(6.75)
Where,
is the circular frequency of the waves; η(t) is the wave height
function.
Based on the random process theory, the horizontal velocity spectral density of
the wave water point at depth z can be expressed as:
(6.76)
Where,
is the transfer function for the
horizontal velocity of the wave water point; Sη(ω) is the spectral wave density
function.
Therefore, according to the spectral analysis method, the mean square deviation
u(t) of the random wave velocity process σu can be expressed as
(6.77)
The relationship between the autocorrelation function of the transverse wave
force
and the autocorrelation function of water particle velocity Ru(τ)
can be expressed as:
(6.78)
Therefore, the transverse spectral wave force density can be expressed as:
(6.79)
6.5.5.5 Structural Modal Analysis
According to statistical data for submarine pipeline overhang length in Chengdao
Crude Oil taken from the literature [6-2], three different lengths were selected for
calculation on the basis of considering the influence of different overhang
lengths on the calculated results. See Table 6.8 for the specific case.
The finite element pipeline model was established based on the pipeline-related
parameters in Table 6.7 and considering different overhanging span lengths.
Modal analysis was also carried out for the model. Table 6.9 shows the first 10
modal frequencies of lateral vibration in various cases. Using the calculation
method for the modal contribution coefficient, the modal contribution ratio of
each order in Working Condition 1 was calculated after entering random wave
force values. Because span length only changes in other cases, the contribution
rate of each mode will not change greatly, so calculation was ignored and the 1st,
2nd and 3rd order modes were used.
Table 6.8 Calculated case.
Case
1
2
3
Span Length 50.00 40.00 30.00
Table 6.9 Structural modal analysis results.
Mode Case 1
Frequency
(Hz)
1
0.400
Modal contribution
rate
0.8379
Case 2
Frequency
(Hz)
0.625
Case 3
Frequency
(Hz)
1.111
2
3
1.102
2.160
0.1107
0.0289
1.722
3.373
3.059
5.989
4
5
3.568
5.327
0.0106
0.0048
5.571
8.314
9.886
14.742
6
7
7.435
9.890
0.0025
0.0014
11.598
15.421
20.548
27.294
8
9
12.691
15.836
0.0009
0.0006
19.799
24.666
34.967
43.555
10
19.323
0.0004
30.078
53.041
6.5.5.6 Random Vibration Response of Suspended Pipelines
(1) Displacement response spectrum
The lift coefficient is 1.0, which was determined based on the lift coefficient
value method described in the literature [6-33][6-34], and by virtue of the model
and environmental parameters given in this chapter. Figure 6.15 shows the nodal
force spectra for various operating conditions on the basis of considering the
motion response of a pipeline under the action of 3m effective wave height and
20m water depth. The whole pipeline is divided into 100 equal parts, with the
random lifting force of each node simplified to be completely correlated. It can
be seen from the figure that the peak frequency of the random lifting force is 1.1
rad/s.
The calculated results for Working Condition 1 in the frequency range 0∼50rad/s
are shown in Figure 6.16a, from which it can be seen that the vibration of the
suspended pipeline is mainly concentrated in two spectral bands. To be specific,
the first spectral band is the frequency peak of the random lifting force
spectrum, while the second is the first-order frequency band for pipeline
vibration. Figure 6.16b shows the displacement spectra for the 0 - 8rad/s range
both with and without non-linearity.
(2) Stress response spectrum
The stress power spectrum can be obtained using the finite element method
based on the known displacement response spectrum. In light of pipeline
characteristics, beam elements can be used to obtain the stress values of pipes.
The displacement at any point within the pipe element, assuming local
coordinates, can be expressed as:
(6.80)
Figure 6.15 Foree spectrum of pipeline nodes.
Figure 6.16 Power spectrum of pipeline midspan displacement response.
Where, u is the displacement vector of any point in the element, de is the
displacement vector of all nodes of the element,
, and N is
the displacement interpolation function matrix, [N1,N2].
(6.81)
Where, L is the length of the cell and x is the coordinates of the calculation point.
The strain at any point in the element can be obtained by geometric equations,
which can be expressed as
(6.82)
Where, H is the linear operator of the coordinates.
For beam elements, the strain consists of axial and bending components
following axial and bending deformation of the structure.
(6.83)
Where, B is the strain matrix, which can be expressed as
(6.84)
Where, γ is the distance from the neutral axis to the stress calculation point.
After obtaining the strain matrix, the stress at any point in the element can be
calculated by physical equations, which can be expressed as
(6.85)
Where, D is an elastic matrix. de is the displacement vector in unit local
coordinates, meaning a transformation between local and global coordinate
systems is required. If the coordinate transformation matrix is expressed as T,
then the global coordinate system and the local coordinate system obey the
following transformation relationship, which can be expressed as
(6.86)
In Equation (6.14), the correlation coefficient of the stationary random process
stress spectrum at any point in the element can be expressed as
(6.87)
Figure 6.17 Linear and nonlinear calculation of maximum stress spectrum of
midspan section of suspended pipeline for various cases.
The random stress spectrum can be expressed as:
(6.88)
The stress spectrum at the critical midspan section of a suspended pipeline can
be calculated by the method above. To calculate the spectrum, the midspan
element was used to obtain the displacement response spectrum of the two
nodes of the element, which was then substituted into Equation (6.22). The
stress spectrum at the corresponding position was also calculated using the x
coordinates. Figure 6.17 shows the stress spectrum of the midspan critical
section for various cases.
6.5.5.7 Random Fatigue Life and Fatigue Reliability Analysis of a
Suspended Pipeline
The following parameters are required to calculate structural fatigue damage in
the frequency domain:
1. m-th moment of stress spectrum:
(6.89)
Where, ω is the frequency and S(ω) is the stress power spectrum. Therefore,
the 0th, 2nd, and 4th moments of the stress spectrum can be expressed as:
(6.90)
2. Characteristic parameters of stress spectrum
(6.91)
3. Spectral width parameter of stress spectrum
(6.92)
4. Max. frequency
(6.93)
5. Positive crossing zero frequency of stress process
(6.94)
For a narrow-band Gaussian random process, the cumulative fatigue damage
over time T can be expressed as [6-35][6-36]:
(6.95)
Where, vp is the maximum frequency of the stress process, while K, m is the
fatigue parameter of the material, satisfying the relationship of Equation (6.25).
Γ is the gamma function.
(6.96)
The wideband random Gaussian process can be approximated by multiplying the
cumulative fatigue damage of a narrowband process with the same σ by an
equivalent coefficient λ, i.e.
(6.97)
Wirsching [6-37] proposed an empirical formula for calculating λ through a large
number of simulations against various stress spectra:
(6.98)
Where, both a and b are the functions of the fatigue parameter m, and can be
calculated empirically by:
(6.99)
According to the Palmgren-Miner hypothesis, the probability of lateral pipeline
vibration fatigue failure [6-38] can be expressed as:
(6.100)
Where, N(s) is the material fatigue parameter relation and Pp (s) is the
probability distribution density function of random stress.
Suppose the displacement response can satisfy a Gaussian normal distribution,
then the random stress process at the critical cross section under lateral
vibration also satisfies Gaussian normal distribution. Pp (s) can be expressed as:
(6.101)
Table 6.10 Fatigue life and failure probability of a suspended pipeline in
different cases.
Cases Span
length
(m)
Annual
cumulative
damage
Fatigue
life (y)
Reliability
index
Failure
probability
1
50.00
0.239
4.18
2.87
0.207×10-2
2
40.00
0.200×10-1
0.50×102
3.58
0.173×10-3
3
30.00
0.843×10-3
0.692
0.12×104
1.45
4.34
0.730×10-5
2.51
0.599×10-2
240.00
Linear
0.342×10-1
0.29×102
3.43
0.296×10-3
330.00
Linear
0.942×10-3
0.11×104
4.20
0.815×10-5
150.00
Linear
The cumulative damage and failure probability of the structure can be obtained
by Equations (6.95) and (6.100).
Based on the S-N curve in the API [6-39] specification, the fatigue parameter m =
4.38, K = 1.15×1015 was selected. See the results in Table 6.10, from which the
following can be seen: The reliability index of pipeline fatigue life and fatigue
decreases rapidly with increased suspension span length; in Case 3, the linear
and nonlinear results differ only slightly, but in Case 2, they are significantly
different, while in Case 1, this difference becomes very significant indeed. To
conclude, if geometric nonlinearity is not considered, the evaluation of pipeline
fatigue performance will lead to large errors.
6.5.5.8 Sensitivity Analysis of Random Vibration Influencing Factors of a
Suspended Pipeline
The random vibration of suspended pipelines are mainly influenced by such
factors as span length, wave height, water depth, outer pipeline diameter and
residual stress.
(1) Span length
Table 6.11 shows the fatigue life and fatigue reliability index of different pipe
lengths. On the basis of considering the influence of different span lengths (30,
35, 40, 45 and 50m) with an outer diameter of 0.345m, the span length can be
found to exert a very significant influence on fatigue life and fatigue failure
probability.
Table 6.11 Fatigue life and failure probability of a pipeline under different span
lengths.
Span
Annual cumulative Fatigue life Reliability
length (m) damage
(y)
index
Failure
probability
30.00
0.843×10-3
0.12×104
4.34
0.730×10-5
35.00
0.427×10-2
0.23×102
3.96
0.370×10-4
40.00
0.200×10-1
0.50×102
3.58
0.173×10-3
45.00
0.567×10-1
0.239
0.18×102
4.18
3.30
0.491×10-3
2.87
0.207×10-2
50.00
(2) Wave height
In light of the big difference on marine environments in different ocean areas in
China, special attention must be paid to the influence of different effective wave
heights. A pipeline with a span length of 40m and an outer diameter of 0.345m
was used to analyze the motion response, fatigue life and fatigue reliability at
different effective wave heights of 2m, 3m, 4m and 5m. Table 6.12 shows the
fatigue life and fatigue failure probability of pipes with different wave heights. As
indicated by the analysis, the relationship between wave height and reliability
index, as well as that between wave height and stress spectrum peak, is close to
linear. This conclusion was made by considering wave lift alone. The influence of
effective wave height on a suspended pipeline will be more significant than the
result analyzed in this section, if further adverse effects are considered, such as
sea bottom scour and increased span length due to the increase in wave height
and current velocity.
Table 6.12 Fatigue life and failure probability of a pipeline at different wave
heights.
Wave
Annual cumulative Fatigue life Reliability
height (m) damage
(y)
index
Failure
probability
2
0.122×10-1
0.82×102
3.71
0.105×10-3
3
0.200×10-1
0.50×102
3.58
0.173×10-3
4
0.430×10-1
0.23×102
3.37
0.372×10-3
5
0.823×10-1
0.12×102
3.19
0.712×10-3
(3) Water depth
The motion response and fatigue reliability of a pipeline with a 40m span and an
outer diameter of 0.345m at 3m effective wave height, and at different water
depths of 15m, 20m, 30m and 40m were analyzed. See Figures 6.18-6.20 for the
results, which indicate that the response spectra of displacement and stress at
15m water depth are much higher than those at the other three water depths,
while the response spectra at 20m, 30m and 40m water depths are similar to
each other. Table 6.13 shows the fatigue life and failure probability of pipelines
at different water depths.
Figure 6.18 Vibration displacement response spectrum of pipeline at different
water depths.
Figure 6.19 Vibration stress response spectrum of pipeline at different water
depths.
Figure 6.20 Pipeline reliability index and peak stress spectrum at different
water depths.
Table 6.13 Fatigue life and failure probability of pipeline at different water
depths.
Pipeline
depth (m)
Annual
cumulative
damage
Fatigue life Reliability
(y)
index
Failure
probability
15
0.200×101
0.50
2.11
0.174×10-1
20
0.200×10-1
0.50×102
3.58
0.173×10-3
30
0.252×10-5
0.40×106
5.48
0.218×10-7
40
0.251×10-9
0.40×1010
6.93
0.217×10-11
(4) Outer pipeline diameter
Figure 6.21 shows the motion displacement response spectrum of different pipe
diameters, while Figure 6.22 shows the corresponding stress spectrum and
Figure 6.23 shows the corresponding relationship curve between outer pipe
diameter, reliability index and stress spectrum peak value. Table 6.14 lists the
fatigue life and fatigue failure probability of different pipe diameters, in order to
study the influence of different outer diameters on the motion response and
fatigue analysis of suspended pipelines.
Figure 6.21 Vibration displacement response spectra of pipelines with different
diameters.
Figure 6.22 Vibration stress response spectrum of pipelines with different
diameters.
Figure 6.23 Reliability index and peak stress spectrum of pipelines with
different outer diameters.
Table 6.14 Fatigue life and failure probability of pipelines with different
diameters.
Pipeline
Annual
diameter (m) cumulative
damage
Fatigue life Reliability
(y)
index
Failure
probability
0.25
0.241×10-1
0.41×102
3.53
0.209×10-3
0.30
0.168×10-1
0.60×102
3.62
0.145×10-3
0.345
0.200×10-1
0.50×102
3.58
0.173×10-3
0.40
0.125×10-3
0.80×104
4.74
0.108×10-5
0.45
0.112×10-3
0.89×104
4.76
0.971×10-6
0.55
0.142×10-4
0.70×105
5.16
0.123×10-6
(4) Residual stress
Axial residual stress remains on the pipeline due to the influence of load and
temperature, and as a ersult of the pipelaying ship method being employed.
Figure 6.24 shows the vibration displacement response spectra of pipes at
different residual stresses, while Figure 6.25 shows the corresponding stress
response spectra. Figure 6.26 shows the corresponding curves between different
initial stresses and reliability indexes as well as stress spectrum peaks. Table
6.15 lists the fatigue life and fatigue failure probability of pipes at different initial
stresses. It can be seen from the above that the fatigue life and fatigue failure
probability also change rapidly.
Figure 6.24 Vibration displacement response spectrum of pipeline at different
residual stresses.
Figure 6.25 Vibration stress response spectrum of pipeline at different residual
stresses.
Figure 6.26 Pipeline reliability index and peak stress spectrum at different
residual stresses.
Table 6.15 Fatigue life and failure probability of pipeline at different residual
stresses.
Axial force Annual cumulative Fatigue life Reliability
(kN)
damage
(y)
index
Failure
probability
-100
0.219
0.46×101
2.90
0.189×10-2
0
0.200×10-1
0.50×102
3.58
0.173×10-3
100
0.315×10-2
0.32×103
4.04
0.273×10-4
200
0.812×10-3
0.12×104
4.34
0.703×10-5
300
0.247×10-3
0.40×104
4.44
0.443×10-5
6.5.6 Example: Fatigue Reliability of Deep-Water SemiSubmersible Platform Structures
Based on the analysis on the fatigue life of deep-water semi-submersible
platform structures, the fatigue reliability of the connection between column and
transverse brace of deep-water semi-submersible platform structures was
studied using the S-N curve method and the fracture mechanics method (P-M).
The fatigue reliability of each key node of a semi-submersible platform structure
in different sea areas was analyzed in this section by means of the S-N curve and
fracture mechanics method respectively, using data on sea conditions for 12
areas in the South China Sea. The sensitivity of fatigue parameters for the two
methods was analyzed as well.
6.5.6.1 Analytical Process for Fatigue Reliability
The fatigue reliability analysis of deep-water semi-submersible platforms was
carried out using the S-N curve and fracture mechanics methods. The analytical
process is similar to that of the fatigue life analysis for a deepwater semisubmersible platform structure, as shown in Figure 6.27.
6.5.6.2 Fatigue Reliability Analysis of Key Platform Joints
(1) Structural finite element model
From the structural strength and fatigue analysis results of the deepwater semisubmersible platform, it can be seen that the fatigue life of the connection
between the platform column and the cross brace is the smallest, as shown in
Figure 6.28. Therefore, this study is mainly focused on the fatigue reliability of
key nodes at 8 connection positions between platform column and cross brace,
as shown in Figure 6.29. The calculated results for hot point stress was obtained
by establishing a local submodel of the connection between the column and the
cross brace of the platform, selecting the wave dispersion diagram of a certain
ocean area in the South China Sea (see Table 6.16) and analyzing the stress
response of the submodel under different cases. The stress parameters of the
key fatigue nodes were also calculated.
Figure 6.27 Fatigue reliability analysis process for deep-water semisubmersible platform structure.
Figure 6.28 Fatigue reliability analysis for a deep-water semi-submersible
platform.
Figure 6.29 Schematic diagram of the connection between the platform column
and the transverse brace.
(2) Calculation of stress parameters
According to the hotspot stress extrapolation method recommended in ABS [640]
, the hotspot stress amplitude at the weld toe was obtained using Lagrange
interpolation and linear interpolation.
It can be learnt based on the S-N curve method and Miner’s rule that stress
parameters are related to probability distribution and frequency of stress range.
The stress amplitude transfer function for different periods and different wave
directions, as well as the first principal stress amplitude transfer function of each
hotspot can be obtained based on the hotspot stress amplitude of each node for
different cases. The fatigue stress amplitude transfer function was also employed
to determine the fatigue stress energy spectrum, and calculate the spectral
moment. On the basis of considering the rain flow correction, fatigue stress
parameters of the platform structure can be expressed as
(6.72)
Table 6.16 Wave dispersion map of the South China Sea (ΣP=100).
HS (m) TZ (s)
<=3 3∼4 4∼5 5∼6 6∼7 7∼8 8∼9 9∼10 >10
0∼0.5
0.29 1.13 5.35 8.20 1.47 0.12 0.01 0.00 0.00
0.5∼1.0 0.05 0.67 7.05 6.61 4.62 1.64 0.30 0.03 0.00
1.0∼1.5 0.00 0.02 1.46 8.34 5.05 2.06 0.97 0.37 0.03
1.5∼2.0 0.00 0.00 0.00 2.24 7.91 2.21 1.09 0.69 0.16
2.0∼2.5 0.00 0.00 0.00 0.01 3.72 4.68 0.86 0.56 0.33
2.5∼3.0 0.00 0.00 0.00 0.00 0.13 4.29 1.67 0.34 0.34
3.0∼3.5 0.00 0.00 0.00 0.00 0.00 0.91 3.17 0.38 0.20
3.5∼4.0 0.00 0.00 0.00 0.00 0.00 0.05 1.81 1.00 0.23
4.0∼4.5 0.00 0.00 0.00 0.00 0.00 0.00 0.33 1.38 0.30
4.5∼5.0 0.00 0.00 0.00 0.00 0.00 0.00 0.03 0.81 0.46
5.0∼6.0 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.17 0.84
>6.0
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.02 0.86
Where, pi is the probability of occurrence of the combination of significant wave
height HS and mean zero-crossing period TZ; M is the total number of ocean
states in the wave dispersion diagram; f0i is the stress response zero-crossing
frequency for each short-term ocean state; it is expressed by the formula
(6.103).
(6.73)
Where, m0 and m2 are zero-order and second-order spectral moments; λ (m, εi)
is the rain flow correction factor, which can be expressed by Equation (6.74).
(6.74)
(6.75)
Where, a(m) = 0.926-0.033m; b(m) = 1.587m-2.323.
Fatigue stress parameters in fracture mechanics should be calculated based on
such factors as the total days (calculated by 300 days) when the platform
operates at sea every year and the specific marine environment, the type of the
structure and its dynamic performance. Suppose the longterm distribution of the
stress range during the service life of the platform obeys the Weibull
distribution, then the probability density function can be expressed as:
(6.76)
Where, ξ is the shape parameter; SL is the maximum stress range for each sea
state; NL is the total number of stress cycles.
The fatigue stress parameters of the platform can be expressed as
(6.77)
Where, Γ is the gamma function; m is the exponent.
(3) Calculation and selection of random variables
On the basis of the studies conducted by Torng [6-41], Kung [6-42], Jiao [6-43], Moan
[6-44]
, Siddiqui [6-45,6-46], etc., and by considering relevant data for platform
structure material characteristics and S-N curves, m, A and its variation
coefficient in the E curve were selected according to the recommended S-N curve
of ABS [6-47]. The median value and variation coefficient of Δ were also
determined based on studies conducted by Wirsching et al. [6-48,6-49]. The median
value and variation coefficient of parameter B were also selected. Table 6.17
shows the parameter values for fatigue reliability analysis using the S-N curve
method. Table 6.18 shows the parameters for fracture mechanics in fatigue
reliability analysis.
Table 6.17 Fatigue reliability analysis parameters in the S-N curve method.
Random variables
Values
Fatigue strength coefficient, A
4.16×1011 0.4
1.0
0.25
1.0
0.25
Structural fatigue damage degree, Δ
Uncertainty random variable in stress
calculation, B
Fatigue index, m
3.0
Coefficient of
variation
-
Table 6.18 Fatigue reliability analysis parameters for fracture mechanics.
Random variables
Values Coefficient of
variation
Initial crack size, a0 (mm)
0.5
1.00
Critical crack size, ac (mm)
25
-
Paris coefficient, C
2.3×10- 0.25
12
Random uncertainty variable in stress calculation, 1.0
B
Random uncertainty variables in the calculation of 1.0
the geometric correction coefficient, BY
0.25
Paris coefficient, m
0.09
-
3.0
Threshold of fatigue crack growth, ΔKth (MPa·m1/2) 2
0.15
The analysis on fatigue reliability of the platform structure using the fracture
mechanics method has to be carried out by considering the uncertainties in the
calculation of stress intensity factor range, crack propagation behavior of
materials, initial crack state, and structural response of the finite element
calculation model. The initial crack depth a0 was determined according to DnV
[6-50]
and ABS [6-41] specifications, while the critical crack size ac was calculated
by subtracting a0 from the plate thickness. The crack propagation parameters m
and C were determined by referring to ABS. The ξ and NL for fatigue reliability
analysis of a semi-submersible platform were determined according to the
method of determining the shape parameter ξ and the total number of stress
cycles NL proposed by Hu Yuren [6-51] et al. The geometric correction coefficient
Y(a) and the random uncertainty variable BY in this calculation were calculated
by referring to the Newman-Raju [6-52] formula and BS7910 [6-53].
(4) Fatigue reliability calculation
Fatigue reliability analysis was carried out on key fatigue parts of the deep-water
semi-submersible platform structure via the cumulative damage-based fatigue
reliability analysis method and through computer calculation. The reliability
index of key fatigue nodes was calculated using Equation (6.78) [6-54].
The reliability index of the key nodes was calculated by iteration using the firstorder second-moment method of the crack propagation–based fatigue reliability
analysis method. To facilitate calculation, new variables were defined according
to Equation (6.79). Let
; R2 = t1−m/2, then
(6.78)
(6.79)
(6.80)
Where, R1, R2 and R3 are lognormally distributed random variables, all of which
are independent from each other. Their median and variable coefficient can be
expressed as:
(6.81)
(6.82)
(6.83)
(6.84)
(6.85)
(6.86)
Let Ri=lnXi (i=1,2,3), then X1, X2 and X3 are normally distributed random
variables. The mean and standard deviation can be expressed as
(6.87)
(6.88)
Then the safety margin is:
(6.89)
Normal transformation was carried out based on iterative steps. Let
, then the limit state function can be transformed into
(6.90)
The partial derivatives of the limit state functions are as follows:
(6.91)
(6.92)
(6.93)
The initial value is the iterative calculated result of
. The
calculation above was carried out via computer programming in order to
calculate the fatigue reliability index of the key fatigue node.
(5) Calculated results and analysis
The fatigue reliability index and failure probability of key nodes at the
connection between column and cross brace of a deep-water semi-submersible
platform were calculated by using the S-N curve method and the fracture
mechanics method. See Table 6.19 and Table 6.20 for the results, which indicate
that the maximum values of fatigue reliability index are 7.354 and 5.553, while
the minimum values are 3.018 and 2.977, respectively. Since the connection
between the platform column and the cross brace is structurally complex, and
since they are mostly connected through welding, the stress response of each
part of the structure does vary under different conditions, which leads to
different reliability indexes for the key nodes of each connection. The reliability
index of Node B at Joint #5 is the smallest, which indicates that the failure
probability is the largest; this is consistent with the results of the fatigue life
analysis. Therefore, when the platform is in service in the above-mentioned
ocean areas, special attention should be paid to the fatigue changes of these
joints, while regular inspection of welds at the corresponding joints should be
strengthened to ensure the safe and reliable operation of the platform structure.
Figure 6.30 shows a comparison of the calculated results. Since the cumulative
damage to the structure before crack initiation is not considered in the fracture
mechanics method and the critical crack size ac was calculated as the plate
thickness minus a0, the fatigue reliability index calculated in this way is lower
than that by the S-N curve method. The maximum and minimum values for
fatigue reliability index calculated by the two methods appear at the same
position, which suggests reasonable results were obtained by analysis. The two
methods can therefore be used to analyze the fatigue reliability of deep water
semi-submersible platform structures.
Table 6.19 Fatigue reliability index and failure probability of key nodes based on
the S-N curve method.
Connection
part no.
Reliability index
and failure
probability
S-N curve method
A
B
C
D
1
β
6.840
5.997
5.736
4.825
Pf
3.960×10- 1.005×10- 4.847×10- 7.000×1012
9
9
7
β
7.354
6.565
5.625
4.689
Pf
9.618×10- 2.602×10- 9.275×10- 1.373×1014
11
9
6
β
6.423
3.976
5.318
5.219
Pf
6.681×10- 3.504×10- 5.246×10- 8.995×1011
5
8
8
β
5.453
3.496
5.075
5.812
Pf
2.476×10- 2.361×10- 1.937×10- 3.087×108
4
7
9
β
5.644
3.018
5.244
5.485
Pf
8.307×10- 1.272×10- 7.857×10- 2.067×109
3
8
8
β
6.158
4.139
5.196
5.421
Pf
3.683×10- 1.744×10- 1.018×10- 2.963×1010
5
7
8
β
5.549
5.415
4.891
3.924
Pf
1.437×10- 3.064×10- 5.016×10- 4.355×108
8
7
5
β
5.830
5.096
5.631
5.358
Pf
2.771×10- 1.735×10- 8.958×10- 4.207×10-
2
3
4
5
6
7
8
9
7
9
8
6.5.6.3 Sensitivity Analysis of Fatigue Parameters
For the fatigue reliability analysis of a deep-water semi-submersible platform,
many different parameters are employed and the rationality and importance of
selected parameters need to be analyzed. Different parameters influence fatigue
reliability in different ways, so they must be calculated and determined
according to specifications and related research. This section mainly discusses
the influence of parameters using the S-N curve method and the fracture
mechanics method on the fatigue reliability of a platform structure through
calculation [6-55].
Table 6.20 Fatigue reliability index and failure probability of key nodes based on
fracture mechanics.
Connection
part no.
1
2
3
4
5
6
7
8
Reliability index
and failure
probability
β
Fracture mechanics method
Pf
3.304×10- 1.874×10- 8.351×10- 1.332×107
6
6
3
β
Pf
5.553
4.962
4.376
3.396
8
7
6
4
β
Pf
4.666
3.372
4.313
3.849
6
4
6
5
β
Pf
4.581
3.389
3.728
3.985
6
4
5
5
β
4.402
2.977
4.517
4.003
Pf
5.351×10- 1.457×10- 3.130×10- 3.123×106
3
6
5
β
5.510
3.439
4.263
4.320
Pf
1.793×10- 2.915×10- 1.010×10- 7.786×108
4
5
6
β
4.506
4.357
3.919
3.482
Pf
3.302×10- 6.588×10- 4.437×10- 2.492×106
6
5
4
β
4.661
3.782
4.882
3.991
Pf
1.571×10- 7.786×10- 5.261×10- 3.284×10-
A
B
C
D
4.973
4.625
4.305
3.004
1.401×10- 3.482×10- 6.056×10- 3.415×10-
1.539×10- 3.728×10- 8.038×10- 5.928×10-
2.312×10- 3.503×10- 9.633×10- 3.380×10-
6
5
7
5
Based on the evaluation principles described above, the fatigue sensitivity
analysis was performed by parameter sensitivity analysis in the S-N curve
method.
(1) Sensitivity analysis of parameters in the S-N curve method
1. Influence of different S-N curves on fatigue reliability
In consideration of different S-N curves taken from the classification societies of
various countries, three groups of S-N curves most widely used in shipping and
ocean engineering were used in this section to analyze the influence of different
S-N curves on fatigue reliability. Table 6.21 shows the parameters for B, C, D, E, F,
G and W curves in S-N curves given by ABS under different environments.
Figure 6.30 Comparison of calculated results for fatigue reliability index.
The fatigue reliability index for key nodes on the platform’s #1 connected part in
air, cathodic protection in seawater, and free corrosion in seawater, were
calculated respectively using the fatigue reliability analysis method based on the
S-N curve, as well as on curves B, C, D, E, F, G and W in the S-N curves. See Table
6.22 for the calculated results.
Table 6.21 S-N curves in different environments.
Curve
type
In air
A
Curve
type
Cathodic
protection in
seawater
m
Curve
type
A
m
Free corrosion
in seawater
A
m
B
1.01×1015 4
B
4.04×1014
4
B
3.37×1014
4
C
1.69×1013
3.5 C
1.41×1013
3.5
D
4.23×1013 3.5 C
1.52×1012 3 D
6.08×1011
3
D
5.07×1011
3
E
1.04×1012 3
E
4.16×1011
3
E
3.47×1011
3
F
F
2.52×1011
3
F
2.10×1011
3
G
6.30×1011 3
2.50×1011 3
G
1.00×1011
3
G
8.33×1010
3
W
1.60×1011 3
W
6.40×1010
3
W
5.33×1010
3
Table 6.22 Fatigue reliability index of key nodes at the No. 1 connection for
different S-N curves.
Curve
type
In air
1
2
3
4
Cathodic protection in
seawater
Free corrosion in
seawater
1
1
2
3
4
2
3
4
B
8.87 7.94 7.53 6.34 8.87 7.94 7.53 6.34 8.71 7.77 7.36 6.17
C
D
8.29 7.40 7.06 5.98 8.29 7.40 7.06 5.98 8.10 7.21 6.87 5.80
7.28 6.43 6.17 5.26 7.28 6.43 6.17 5.26 7.07 6.23 5.96 5.05
E
6.84 6.00 5.74 4.83 6.84 6.00 5.74 4.83 6.63 5.79 5.53 4.62
F
6.26 5.42 5.16 4.25 6.26 5.42 5.16 4.25 6.05 5.21 4.95 4.04
G
W
5.20 4.36 4.10 3.19 5.20 4.36 4.10 3.19 4.99 4.15 3.89 2.97
5.74 4.90 4.64 3.73 4.69 3.84 3.58 2.67 4.48 3.63 3.37 2.46
In the three environments of cathodic protection in air, cathodic protection in
seawater, and free corrosion in seawater, the fatigue reliability index calculated
using the B curve is the highest, while that calculated using the W curve is the
lowest. The results for other curves lie between the two values above, and the
calculated values of each node follow the same trend. It can be seen from Tables
6.10-6.22 that the fatigue reliability index calculated by B, C, D, E, F, G and W
curves in air is higher than that calculated by cathodic protection in seawater
and free corrosion in seawater; the fatigue reliability index calculated using the
B, C, D, E, F, G and W curves with cathodic protection in seawater is higher than
that calculated for free corrosion in seawater. By comparing the calculated
results of the E curve in different environments, the fatigue reliability index
calculated by the E curve in air is 13.31% higher than that by the E curve with
cathodic protection in seawater, and 16% higher than that by the E curve with
free corrosion in seawater. The fatigue reliability index calculated by the E curve
with cathodic protection is 3.10% higher than that by the E curve with free
corrosion in seawater. Therefore, in terms of fatigue analysis, suitable curves
should be selected based on the environment of the platform structure nodes.
2. Influence of parameters in the S-N curve method on fatigue reliability The
application of the S-N curve method to fatigue reliability analysis of platform
structures needs to consider the uncertainty of statistical data for wave
dispersion diagrams in the relevant sea areas, as well as the uncertainty of wave
environment simulated by P-M spectrum, the uncertainty of structural response
of the finite element calculation model, and the uncertainty of wave load
calculations. Therefore, it is of great importance to determine the random
variables, since these are related to the accuracy and rationality of the analysis
results. In consideration of the influence of material performance, temperature,
environment, stress concentration and load sequence, the parameters given by
different codes and related studies vary as well. In order to study the influence of
each parameter on the fatigue reliability of the platform structure, the fatigue
reliability index of the key node A at the #1 connection part in the S-N curve
method was calculated for different values of each parameter, and based on the
codes and related studies of various countries.
The variable coefficient CA of A increases from 0.30 to 0.60, while that of β
decreases by 12.95%; fatigue index m increases from 2.5 to 5.0, while β
decreases by 41.61%. In terms of stress calculation, the variable random
uncertainty median B increases from 0.8 to 1.2, while β decreases by 18.40%.
The variable coefficient CB of B increases from 0.15 to 0.35, while that of β
decreases by 42.76%. The results show that CA, m, B and CB exert a significant
influence on the fatigue reliability of the structure, while the fatigue reliability
index β decreases rapidly with the increase in CA, m, B and CB. The analysis
shows that the parameters Δ and CΔ exert little influence on the fatigue reliability
of the structure. The fatigue reliability index β increases slightly as Δ increases,
but decreases slightly as CΔ increases. Therefore, the sensitivity parameters CA,
m, B and CB should be carefully selected when the S-N curve method is applied to
fatigue reliability analysis of platform structures.
(2) Parameter sensitivity analysis using the fracture mechanics method Many
uncertainties in calculation parameters need to be considered when analyzing
the fatigue reliability of platform structures using the fracture mechanics
method. This includes uncertainty in the calculation of the stress intensity factor
range, uncertainty of material crack propagation behavior, uncertainty of initial
crack state, and uncertainty of the structural response of the finite element
calculation model. Many methods for calculating and evaluating parameters are
provided in various specifications and related literature, but none of these have
yet been unified. To study the influence of parameters in fracture mechanics
method on the fatigue reliability of a platform structure, fatigue reliability
indexes of the #1 Joint A are calculated for different parameter values based on
national codes and related research.
Conclusions following analysis: Initial crack size a0, the crack propagation
parameters of the materials, the variable uncertainty random medians in
geometric correction coefficient calculation, and in the stress calculation, the
variable uncertainty random variable median B and geometric correction
coefficient Y(a), all have a significant influence on the fatigue reliability of the
structure. The fatigue reliability index β decreases rapidly along with the
increase in a0, C, m, BY and B and Y(a).
The analysis shows [6-56] that the parameters ac, ξ and NL exert an insignificant
influence on the fatigue reliability of the structure, and that the fatigue reliability
index β increases slightly with increased ac, and ξ, and decrease slightly with
increased NL. Therefore, the sensitivity parameters a0, C, m, BY and B and Y(a)
should be calculated and determined carefully when the fracture mechanics
method is applied to fatigue reliability analysis of platform structures.
(3) Analysis of the influence of design life on fatigue reliability
The design life of a deep-water semi-submersible platform exerts a direct
influence on the fatigue reliability of its structure. By fitting the curve between
fatigue reliability index β and the platform’s design life TD calculated against
joint A of the #1 connection joint of a column and cross brace using the S-N
curve method and fracture mechanics method, respectively, it was learnt that β
decreases gradually as TD increases. In other words, the longer the design
service life of the platform, the smaller the fatigue reliability index. The
conclusion above better complies with reality.
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7
Load Combination on Reliability Theory
Random load combination plays an important role in engineering structural
design theory, and especially in reliability theory. Loads are combined to
find an equivalent load to replace two or more random processes acting on
the structure. The load combination rule provided in the code should be
simple enough to facilitate engineering application. The combination rule is
mainly based on the time integral method. The rule of simplification will be
discussed in the next chapter.
If X1(t) and X2(t) represent stationary, mutually independent and
continuous load random processes, the probability distribution of the linear
combination X = X1 + X2 in the time interval [0, tL] can be calculated by
considering the transcending rate of X(t) with respect to the horizontal line
x = a in Equation (7.1). The major problem involves calculating the
transcending rate of horizontal a, X, or the crossing rate at which (X1, X2)
leaves the area enclosed by the plane x1 + x2 = a.
(7.1)
where v is average cross rate of event occurrence.
If both X1(t) and X2(t) are normal random processes, while X = X1 + X2 is
also normal and stationary, then the mean and variance can be calculated.
For the crossing rate v of stationary normal random processes, these can be
directly obtained from the results of a single random process in Equation
(7.2).
(7.2)
Unfortunately, not all load random processes can be properly described. For
example, sufficient accuracy cannot be ensured for transient normal
random processes or the application of Equation (7.3) to non-normal
process calculation.
(7.3)
7.1 Load Combination
7.1.1 General Form
Generally, the crossing rate of random processes can be calculated by
Equation (7.4).
(7.4)
The joint probability density function (PDF)
must be obtained
before calculation. This can be made by the convolution of
and
.
(7.5)
where
, x1 = x – x2. If the integral order is changed, the triple
integral form of the crossing rate is turned into:
(7.6)
It is usually difficult to obtain an analytical solution. Once the integral
domain of and is changed, their boundaries will change, too. If the
integral domain of the components of is expanded to
,
, while the integral domain of the components of is
expanded to
,
, then the upper limit of the
integral is:
(7.7)
where vi(u) represents the crossing rate of random process Xi(t) at the
transcending rate of horizontal u.
is the PDF of Xi(t) at any time point
t, also known as the PDF at any time point. Equation (7.7) is sometimes
called the point crossing formula. The lower limit can also be calculated.
The results of calculation can be extended to a nonlinear combination of
multiple non-stationary load random processes [7-1].
Equation (7.7) offers a solution to the combination of random processes,
especially when the random processes X1(t) and X2(t) obey a discrete
distribution. Equation (7.7) also may apply to the combinations of random
processes that satisfy:
(7.8)
Figure 7.1 shows typical random processes that are consistent with this
situation.
If both X1(t) and X2(t) are stationary normal random processes, with a mean
of
and
, and a standard deviation of
and
, respectively,
then:
The crossing rate of each independent random process can be obtained
from:
Figure 7.1 The combination of random process.
After the above results are substituted into Equation (7.7), the upper limit
of crossing rate of X = X1 + X2 is turned into:
where
is
,
. When
,
. The error
, the maximum value is
load combinations, the error does not reach
[7-2]
. For most
.
7.1.2 Discrete Random Process
Let’s consider a combination of two non-negative rectangular update
processes. Figure 7.2 shows a typical trajectory (or implementation) chart.
The crossing rate of each random process can be calculated:
(7.9)
(7.10)
where the mean arrival rate of the random pulses is viqi = vmi, the
distribution at any time point is
, and δ is
the Dirac function (see Figure 7.2), qi = 1 − pi. Let Gi( ) = 1 − Fi( ); substitute
this into Equation (7.7), and let i = 1,2, so
(7.11)
Figure 7.2 Typical sample function of mixed rectangular update stochastic
process with given probability density function.
Fortunately, this result is very simple to explain, and can be solved using the
most basic theory.
For a random process, if the X(t) value is non-negative whenever it is
updated, then the probability value is q. Similarly, when the X(t) value is 0,
the probability value is p, regardless of the X(t) value within the current
time increment. Therefore, the random process has no aftereffect. Table 7.1
shows various possible states of transcending level a after two random
processes are combined.
The effect of each state change (e.g., from X1 + X2 < a to X1 + X2 > a) on the
crossing rate
is as follows:
(7.12)
Table 7.1 Different state combinations that cause crossing.
State change Random process 1 Random process 2 Crossing cause
(a)
Inactive
Active or inactive
Random process 2
(b)
Active
Inactive
Random process 2
(c)
(d)
Active
Active or inactive
Active
Inactive
Random process 2
Random process 1
(e)
Inactive
Active
Random process 1
(f)
Active
Active
Random process 1
Let Ω = •State Change Δt•. So, the term (a) in Table 7.1 becomes:
or
(7.13)
The above formula is consistent with the second term in Equation (7.11)
because v2q2 = vm2.
The term (b) in Table 7.1 becomes:
or
(7.14)
The above formula is consistent with the fourth term in Equation (7.11)
(v2q2 = vm2). The term (c) in Table 7.1 is turned into:
or
(7.15)
The above formula is equal to the sixth term in Equation (7.11). Other terms
in Equation (7.11) change with the change of states (d)∼(f) in Table 7.1. If
each pulse returns to zero before the start of the next pulse, then the term
p2 + q2F2(a) in Equation (7.13) is equal to 1 by definition, and the same is
true for the term p2 in Equation (7.14).
If the crossing rate is obtained by calculation, then the cumulative
distribution function (CDF) FX( ) of load X = X1 + X2 can be obtained by
calculation using Equation (7.1). Some academic researchers have studied
the calculation error between Equation (7.1) and Equation (7.2). By
comparing with certain known exact solutions [7-3][7-4][7-5][7-6], we find that
the calculation error for a high-level obstacle a is 20%, while the calculation
error for a low-level obstacle is 60%.
7.1.3 Simplified Method
(1) Directional simulation in load effect space
The load in pulse mode can be simplified by Equation (7.11). For example,
let
where G12( ) = 1 − F12( ), F12( ) is the CDF between two pulses. So, Equation
(7.11) can be simplified to:
(7.16)
where vmiμi is replaced with qi. vmi represents the average pulse arrival rate
of random process Xi(t). μi represents the duration of pulse Xi(t). If the
duration is short enough, i.e., μi → 0, and the relative probability of its
occurrence is very low, then pi → 1. The crossing rate of combined random
processes can be approximated to:
(7.17)
This equation was first proposed by Wen [7-7]. The first and second terms in
Equation (7.17) represent the crossing rate when each random process acts
alone, while the third term represents the crossing rate when two random
processes act simultaneously (i.e., pulse overlapping). The third term can be
ignored if the pulse duration is very short.
The existing research shows that, compared with the simulation results,
Equation (7.17) can be used to accurately estimate the crossing rate
,
even if the height of each random process is as high as 0.2, or in the case of a
very high obstacle level a [7-6]. The formula also considers other pulse
shapes than rectangular ones, as well as the correlation between pulses [7-8]
[7-9][7-10]
.
(2) Borges processes
The combination of Borges processes is of particular significance for code
validation, because it helps to accurately estimate the maximum
combination effect of load probability distribution [7-11].
When every random process Xi(t) in the random process combination X = X1
+ X2 + X3 +⋯ is a Borges process, then the pulse duration is τ1 < τ2 < τ3 < …,
and τi / τi−1 is an integer. See Figure 7.3. Therefore, the occurrence rate is
equal to the number ni of pulses in the random process Xi during the time
interval [0, tL] divided by the duration, i.e., vi = ni / tL. According to the above
crossing rate formula, the CDF of the maximum value of X can be obtained
by Equation (7.1). However, another method will also be briefly introduced
below.
The CDF Fmax X( ) of maximum X can be calculated by convolution
computation in the equation below [7-12]:
Figure 7.3 Borges process combination.
(7.18)
The above formula is a bivariate random process, τ2 = τ1 / m, where m is an
integer, and there are n pulses about τ1 within [0, tL]. For a combination of
two or more loads, this integral term is more complex, but a calculation can
be performed using the first-order second-moment method.
For a linear combination of three loads, the maximum value within [0, tL]
can be written as:
(7.19)
where Z2(t) is expressed as:
(7.20)
where
represents the maximum of X3 in time interval τ2.
The CDF
of Z2 can be obtained by the convolution integral of
Equation (7.18). The maximum value can be expressed as:
(7.21)
where
(7.22)
where
Similarly,
represents the maximum value of Z2 in time interval τ1.
can be obtained from Equation (7.18).
If the supplementary terms to all the random processes in
and
can be obtained, then the above combination method can be extended to a
combination of any number of loads. The calculation steps are summarized
below. First, the CDF
of each process can be approximated by a
normal distribution based on a series of checking points. The
transformation starts with X2, with the same checking point x*,
; m = τ2/τ3 represents the pulse count in pulse X2
with respect to X3.
The mean
and variance
of the equivalent normal distribution X2
after transformation can be obtained. Similarly, the mean
of
can be obtained. Therefore, the mean
and variance
and variance
of
Z2(t) can be calculated by means of Equation (7.22), where
,
. Thus, we can calculate
,
obtain the normal distribution equivalent to X1(t) and
, and the
normal distribution equivalent to Z1(t), as shown in Equation (7.20). This is
the structure that we are seeking, but it will depend on the selection of the
initial value of the checking point x*. A complete distribution function Fmax X
( ) can be obtained by taking different checking points x = x* and then
repeating the above steps. According to what is presented in Chapter 4, i.e.,
the joint PDF
can reach its maximum if the
independent checking point x* is selected for a random variable Xi(t), it can
then use the equivalent normal PDF
.
(3) Deterministic load combination—Turkstra’s combination rule
Many simplified methods are used for load combination calculations, but
they are still very complicated for conventional structural design and code
formulation. The most basic load combination method is to add up the loads
directly without considering the uncertainty of these loads. Then, a series of
coefficients are proposed to amplify or reduce the loads to an appropriate
extent. The key problem is how to ensure the rationality of these
coefficients.
Load combination rules can be extracted from a combination of Borges
processes [7-13] or from Equation (7.7). If the approximation
in Equation (7.1) is substituted into
Equation (7.7), it can be obtained:
(7.23)
where the maximum value is taken over the entire time interval concerned,
such as the service life [0, tL] of the structure. For combination Z = W + V,
where W and V are independent of each other, the CDF GZ( ) can be obtained
by means of the convolution integral:
(7.24)
Therefore, the two integrals in Equation (7.23) represent max
, where
represents the Xi value at any time point.
Similarly, if Z = max(W, V), then:
and
(7.25)
For a high failure level a, the last item can be ignored, so
(7.26)
This is the famous Turkstra combination rule. It can be expressed as: (1)
The sum of the maximum value of Load 1 within the service life and the
value of Load 2 taken when Load 1 reaches its maximum value; (2) The sum
of the maximum value of Load 2 within the service life and the value of Load
1 taken when Load 2 reaches its maximum value. This combination rule can
be extended to a combination of multiple loads or multiple load effects. For
a combination of n loads:
(7.27)
This combination is analogous to the load combination rules specified in the
current codes, and obviously, is also related to the probability theory
requirements in load combination. However, the load level is not specified
in Turkstra’s combination rule, but needs to be selected one by one. Chapter
8 will describe that if the random process Xi is stationary, then the value of
[max Xi] is usually equal to 95% of the load, and
is the mean value at any
time point.
Although Turkstra’s combination rule is very simple to apply to
standardized engineering applications, it is not very suitable for probability
calculations that have high requirements for accuracy.
7.2 Load Combination Factor
Load or load effect is a decisive factor in structural design, and its load
combination is one of the core issues, running right the way through
reliability design standards. For a structure that bears n loads, if the timedependent change of structural resistance is ignored, then its limit state
equation can be expressed as
(7.28)
where R represents structural resistance, Si(t) represents the random
process of the ith load effect, and S(t) represents the random process of the
combined effect of n loads.
Obviously, the key problem in structural reliability analysis is load
combination. From a mathematical perspective, the problem of load
combination involves the superposition of multiple random processes. In
engineering practice, the random processes are generally replaced with the
maximum load in the reference period (a variable random), for reliability
analysis. Therefore, the key to load combination is to determine the
maximum value SM of load combination, as follows
(7.29)
The theory and rules of random load combination include [7-14] (i) peak
superposition; (ii) crossing analysis; (iii) combinatorial theory Poisson
process as a simplified model; (iv) maximum value formed by the local
extreme values of certain time intervals from the perspective of engineering
practice; and (v) square root of the sum of the squares (SRSS).
7.2.1 Peak Superposition Method
This method assumes that all load processes reach a maximum
simultaneously, with these maximum values added up to obtain the
designed value. This is the most conservative combination model. Actually,
it should be considered as a deterministic analysis method. Currently, this
method is often used for offshore structure design.
7.2.2 Crossing Analysis Method
The up-crossing theory has been used by many academic researchers to
study nonlinear or dynamic load combination problems. Let X(t) be a
stationary random process, implemented as x(t) in t1,t2. Given a constant
threshold value X(t) = r, as shown in the figure, the up-crossing level X(t) is
called positive crossing, which is marked in the figure.
The derivative process
of the stationary random process X(t) can be
defined. If X(t) meets the following condition: the autocorrelation function
RXX(t) has a continuous second derivative
, and the above-defined
derivative process is also a stationary random process. Because X(t) is a
stationary random process, for τ = t1 – t2, there is
The step function H(x) needs to be used to derive the up-crossing rate
theory, and it is defined by Heaviside, i.e.,
(7.30)
(7.31)
For random process X(t) and fixed threshold x(t) = r, we define
(7.32)
so
(7.33)
then
, and the realization of the derivative process
depends on many unit pulses. The positive unit pulses cross the
threshold in a positive manner while the negative unit pulses cross the
threshold in a negative manner.
For the number of such unit pulses in the time interval [t1,t2], the total
number of times the threshold level x(t) = r is crossed can be obtained as
follows
(7.34)
The average number of all pulses in time interval [t1,t2] is as follows:
(7.35)
Consider the positive crossing rate per unit time, and let N(r, t). Then
(7.36)
(7.37)
so
(7.38)
The average crossing rate (also called up-crossing rate) per unit time is as
follows:
(7.39)
Taking two random processes as an example, let X1(t) and X2(t) be
independent continuous processes, then the probability distribution of X =
X1 + X2 in [0, t] can be obtained according to the assumption that its upcrossing obeys the Poisson distribution
(7.40)
where υ(x, t) represents the up-crossing rate corresponding to the
threshold at t. If X1,X2 are stationary random processes, then υ(x, t) = υ(x),
and υ(x) can be expressed by an integral, i.e.,
(7.41)
7.2.3 Combination Theory with Poisson Process as a
Simplified Model
This method is based on a combined model of Poisson processes, first
proposed by Hasofer [7-15], who assumed that there should be a linear
relationship between load and load effect, as follows:
(7.42)
(1) Hasofer combination method.
In this method, loads are divided into persistent variable loads (Class-a) and
temporary variable loads (Class-b). If the distribution style and probability
density of these two types of loads, as well as the average occurrence rate of
load effects, are known, then the maximum distribution of Class-b load
effects in the duration of Class-a loads can be obtained by probability
theory. Then, the extreme value distribution of combined effects during the
design reference period can be obtained. This method, which involves the
computation of multiple continuous integrals, is not a mature approach in
practice.
(2) Wen combination method [7-16][7-17] [7-18] [7-19].
It is basically the same idea as the Hasofer combination method. There is a
great difference between Class-a and -b effects in terms of frequency and
duration. If the average occurrence rate of the ith effect is υi, and the average
duration is
, then the size of the product
can actually represent the
probability of its occurrence within a certain time interval. If the product
, this means that the load will not appear for most of the time,
making it unnecessary to consider the possibility that this particular load
will occur at the same time as other loads. When
is large, it becomes
necessary to consider the possibility that they will meet. By taking the
composite load formed between them as a new process independent of the
original loads, and if it is still a Poisson process, we can derive the maximum
value distribution after load combination, as follows:
(7.43)
where Gi(X) = 1 − Fi(X), Gij(X) = 1 − Fij(X), Gijk(X) = 1 − Fijk(X), where Fi(X) is
the distribution function of load I; Fij(X) and Fijk(X) are, respectively, the
probability distribution functions of loads I and j and the combination of I, j
and k. υij and υijk represent the average meeting rate of I and j, as well as I, j
and k, respectively.
After further simplification, the maximum post-combination distribution
can be obtained by a mathematical method. Although such methods feature
a reasonable combination type, they require complete observed and
statistical data, such as the average occurrence rate of loads. These tend to
be difficult to obtain in practice. Moreover, the calculation process is
complex, making it difficult to apply them to ocean engineering designs.
7.2.4 Square Root of the Sum of the Squares (SRSS)
In a linear system, the maximum/RMS value is approximately identical at
time T for all responses, while the period and frequency are independent of
each other, so
(7.44)
where peak factor
;
It is simple to adopt this combination, i.e.,
, where
S1, S2 and are load effects involved in the combination.
7.2.5 Use of a Combination of Local Extrema to Form a
Maximum Value
This kind of method is not designed to calculate the probability distribution
of the maximum value of the combined effect from the perspective of
mathematical logic. Rather, it is used to work out the actual maximum value
of the combined effect generated on the envelope structure based different
combinations of the local extreme values of a single effect taken in a specific
time intervals. Common combination rules are as follows.
(1) Turkstra’s rule for random load combination [7-20] [7-21]
This is the combination model recommended in American National
Standard A58. At present, this method has been adopted in the Unified
Standard for Reliability Design of Hydraulic Engineering Structures [7-22]
(GB50199-94) and the Unified Standard of Reliability of Structure Design for
Harbor Engineering [7-23] (GB50158-92). According to the first load
combination rule to be proposed by Turkstra, the extreme value of a load
effect in [0, T] can be combined with the instantaneous value of the
remaining load effects in turn, after which the combination with the largest
load effect is selected as a control form. The combined maximum is then as
follows:
(7.45)
where t0 represents the moment at which Si(t) reaches its maximum, as
shown in Figure 7.4. Figure 7.4 shows three different loading processes.
According to Turkstra’s principle, the combined maximum to be evaluated is
determined by the most unfavorable of the three combinations. The
distribution function of the combined maximum is as follows:
(7.46)
where
represents the number of replications of variable loads
within design reference period T. The results achieved by Turkstra’s rule
are not too conservative or safe, while more unfavorable combinations may
theoretically exist.
Figure 7.4 TR combination diagram of three load combinations.
(2) Ferry Borges-Castanheta rule for random load combinations
The combination rule is an iso-temporal load combination model, proposed
in 1972 by Ferry Borges and Castanheta[7-24], with reference to Turkstra’s
rule. According to this combination rule, for every variable load xi, it can be
divided into ri equal sections τi, which are then used as basis intervals
throughout the service life of the structure, T. In each time interval, xi(t)
does not change with time, xi(t) is statistically independent in sequential
time intervals, and the probability of occurrence of xi(t) in each time
interval is Pi. ri represents the number of repetitions of loads during the
service life. For selection of τi, the maximum reached in sequential time
intervals should be used to build an independent hypothesis.
As can be seen from the above, the distribution of the maximum can be
described using a stationary binomial process. So
(7.47)
When considering a combination of several variable loads (random
processes), we can assume that the random processes are independent of
one another, and that the number of time intervals, ri, divided by time T, is
an integer. Also, let
be a positive whole number, and arrange it in
increasing order of ri. So
When n=3, the Figure 7.5 shows three rectangular oscillo-grams simplified
as equal time intervals.
Figure 7.5 Process combination of three rectangular wave.
Starting from the load xn that is repeated for the largest number of times,
we work out the maximum xn(t, τn−1) of xn(t) in time interval τn−1, and
obtain
(7.48)
According to Equation (7.48), max(x1 + x2 + ⋯ + xn) can be approximated as
max(x1 + x2 + ⋯ + xn−2 + Zn−1) so as to reduce the problem from a
combination of n loads to a combination of (n-1) loads, where the
distribution of Zn−1 is the convolution of the distribution function
of xn(t, τn−1) and the density function fn−1(x) of xn−1(t).
Therefore, the maximum value distribution can be determined by n-1
sequential convolution operations, as follows:
(7.49)
Equation (7.49) is the superposition process of this rule. Because Zi =
xi+1(τi) + xi obtained by superposition each time is a constant maximum in
this time interval, the results obtained are obviously more conservative,
especially when n is larger. Besides, the sequential convolution operations
in the previous formula are also very tedious. Figure 7.5 shows the
superposition process of the F-B rule.
There are three load effects, x1(t),x2(t),x3(t), where r1 < r2 < r3, and the
combined maximum is shown in Equation (7.50):
(7.50)
(3) JCSS combination rule for random load combination (pyramid
method)
This load effect combination model is recommended in Volume I [7-25][7-27]
of the International System of Unified Standard for Structures issued by the
International Joint Committee on Structural Safety (JCSS), composed of six
international organizations. It is also a combined probability model of load
effects adopted in China’s Unified Standard for Building Structures [7-26] [728]
. This model can be used to combine load effects in accordance with
engineering experience. This is compatible with the use of first-order
second-moment method considering the probability distribution types of
basic variables to analyze structural reliability. The main contents are:
i. Load Q(t) is an isochronous stationary random process.
ii. There is a linear relationship between load Q(t) and load effect S(t), as
follows:
Where CQ is a coefficient of Q(t).
iii. Mutually exclusive random loads cannot be combined. Only variable
loads that may meet in [0,T] are combined. Also, the types of loads
concerned must be determined based on experience.
iv. When a load reaches a maximum in the design reference period or
when a certain time interval is applied, other loads involved in the
combination only reach a maximum during the duration of this
maximum load; otherwise the load is taken at any time point. That is,
let n variable loads be involved in the combination, and let the total
number of time intervals with respect to various modeled loads Qi(t) in
the design reference period T be ri to arrange the loads from small to
large, i.e., r1 ≤ r2 ≤ r3 ≤ ⋯. Any kind of load effect can be combined to
obtain the maximum load effect (combined load effect) of n
combinations,
, i.e.,
(7.51a)
(7.51b)
where Si(t0) is a random variable at any time point of the ith load effect Si(t).
represents the maximum value distribution of the i − 1th load
effect in duration τi−1 of the ith load effect. The number of changes of Si in
time interval τi−1 is
is
. If pi = 1, then the distribution function of
. The reliability index of structural components
can be calculated by the PDF FMi(x) of the maximum load effect of various
combinations according to the limit state concerned, and then taken as a
minimum load effect combination, which is an optimal load combination for
control structure design.
(4) Comparison of common random load combination rules
Three examples of live load combinations: persistent live load Li, temporary
live load Lr and wind load W, are taken to compare the above random load
combination rules. The specific data is shown in Table 7.2.
The load meeting method, F-B combination, TR combination and JCSS
combination of these three random loads at different load-effect ratios are
calculated according to the above-mentioned random load combination
theory. For the convenience of comparison, the combination results are
achieved by the dimensionless ratio
represents
the mean of the maximum distribution of the combined effect; μsi
represents the mean of the truncated distribution of the effect of a single
load. μsM varies with combination rules, while
is only related to the
inherent law of the loads involved in the combination, and is not affected by
the combination method. The specific calculation results are as follows:
As can be seen from Figure 7.6, the calculation results of various
combination rules vary greatly with the load-effect ratio SL/SW; when the
load-effect ratio is 1, the minimum value is reached. The F-B combination
rule boasts the largest calculation results, indicating that this combination
method is more conservative. As can also be seen, the JCSS combination rule
and TR combination rule are less conservative when the load-effect ratio is
low, but become increasingly conservative as the load-effect ratio rises.
Therefore, various combination rules differ from one another in terms of
risk at different load-effect ratios.
Table 7.2 Parameters of various random loads.
Load type
Mean
Coefficient Mean
Mean
Duration of
2 of variation occurrence duration time
(N/m )
rate
interval
(Year)
Persistent
live load
128.7
0.46
0.1
10 year
Temporary 48.8
live load
0.69
1.0
0.01year 1
Wind load
0.45
1.0
10
minutes
66.67
10
1
Figure 7.6 Comparison of several combination rules.
7.3 Calculation of Partial Coefficient of
Structural Design
In engineering structure design, partial coefficients are adopted in the
expression of limit state design in order that the designed structure has the
specified reliability. These coefficients are called partial coefficients of
engineering structure design.
According to the probability limit state design theory, there are load partial
coefficients (including dead load and variable load) and resistance partial
coefficients in the expression of engineering limit state design. The material
properties, coefficient of variation of geometric parameters and uncertainty
of resistance calculation model in the design expression are all included in
resistance partial coefficients. The uncertainty of load calculation models is
also included in load partial coefficients.
The purpose of determining design partial coefficients is to implement the
target reliability index of the engineering structure to each partial
coefficient in the limit state design expression, so that complex numerical
operations of probability limit states can be converted into a simple
algebraic operation.
7.3.1 Expression of Design Partial Coefficient
When determining the partial coefficients in the limit state design
expression, we must consider the type of load combination, the type of
components, extreme working conditions, marine environmental loads, and
the relevant statistical characteristics of structural resistance.
Now, let’s show how partial coefficients are determined for the common
load combination form, i.e., (dead load + live load). According to the limit
state expression:
At checking point X⋆, the limit state equation is written as:
(7.52)
where , , are design checking point coordinates for dead load, live
load and resistance of structural components, respectively.
According to the analysis of the structural reliability index, the
determination of design partial coefficients is the inverse operation of the
structural reliability index calculation. Therefore, we must use the relevant
standard value and partial coefficients to replace the basic variables in the
probability limit state equation at design checking point X⋆, as follows
(7.53)
where SGK, SQK and RK are the standard value of dead load, the standard
value of live load and the standard value of the resistance of structural
components, respectively; γG, γQ and γR are partial coefficients of dead load,
live load and resistance of structural components, respectively. Therefore,
the following conditions must be met:
(7.54)
(7.55)
From the foregoing explanation, the partial coefficients γG, γQ and γR depend
on the design checking point coordinates of load and resistance. According
to the analysis of the structural reliability index, the checking coordinates
are related to the reliability index, indicating that design partial coefficients
are also related to the reliability index.
In addition, the checking point coordinates and corresponding reliability
index are both related to the ratio of the standard value of live load to that
of dead load (ρ), suggesting that design partial coefficients are also related
to ρ. However, in engineering practice, the ρ value is constantly changing. In
other words, both design partial coefficients and the corresponding
reliability index change with the ρ value in practical engineering. Obviously,
this does not meet practical requirements. The optimum partial coefficient
means that there is a minimum difference between the reliability index of
the structure designed with this coefficient at different load-effect ratios ρ
and the target reliability index. In other words, the value of the partial
coefficient should be determined according to the principle that the
difference between the standard value of structural resistance worked out
by the expression of partial coefficients and the standard value of structural
resistance are directly worked out by the target reliability index.
7.3.2 Determination of Partial Coefficient in Structural
Design
To determine the partial coefficients of engineering structure design, we
primarily consider two cases, i.e., the simple condition and the additional
combination (a combination of three loads), and follow the principles
below:
1. The partial coefficient of the dead load is taken for simple combination
and additional combination;
2. Different resistance partial coefficients are taken for the same type of
components, but the same resistance partial coefficient is taken for the
same component in the same combination under different working
conditions;
3. The value of a set of optimum load partial coefficients and resistance
partial coefficients should be taken based on this set of partial
coefficients;
4. The reliability index β of the designed components at different loadeffect ratios ρ is the closest to the target reliability index βT.
7.3.3 Determination of Load/Resistance Partial Coefficient
Load partial coefficients γG and γQ are determined by considering a simple
combination. Dead load and live load combinations are thoroughly
calculated and analyzed in order to determine γG and γQ. In order that γG
and γQ should be universally applicable to structural components made of
various materials, representative structural components and load-effect
ratios are selected as the basis for calculation and analysis. For example, the
target reliability index is 2.6∼2.9, and the following four different methods
are used to calculate γG and γQ.
(1) Least squares method for standard value of resistance
According to the given target reliability index, mean value and standard
value of random loads, and the coefficient of variation of structural
resistance, the mean μR of structural resistance can be worked out by the
checking point method based on the limit state equation. Then, the standard
value RK * of resistance can be worked out according to the statistical
parameters.
For a certain component, if the standard value RK* of resistance obtained by
target reliability is equal to the standard value of resistance obtained by the
limit state equation, i.e., RK* = RK, then the reliability index of the component
designed by this limit state equation must be equal to the target reliability
index. Therefore, the above principles for determining partial coefficients
can be converted into conditions for minimizing Hi below:
(7.56)
where
represents the standard value of resistance obtained by the
probability method according to the target reliability index given the jth
load-effect ratio of the ith structural component;
represents the
standard value of resistance obtained by the limit state equation based on
the selected partial coefficient in the same situation, i.e.,
.
The possible value ranges of partial coefficients for dead loads are γG =1.1,
1.2, 1.3, 1.4 and 1.5; the possible value ranges of partial coefficients for
variable loads are γQ =1.1, 1.2, 1.3, 1.4, 1.5, 1.6. Therefore, there are 30 value
ranges for γG and γQ. Given each value range of γG and γQ, for every
structural component, the least squares method can be used to calculate the
optimum resistance partial coefficient
, thus enabling the error Hi to be
minimized. The overall error I in the components and tubular joints of all
common engineering structures is:
(7.57)
The size of the I value reflects the difference between the target reliability
index and the implicit reliability index of the structural component
designed according to the design expression containing this set of γG and γQ,
and various optimized
values. Dimensionless relative error is used to
achieve a balance in the ratio of the error of each component to the overall
error.
This method can be used to determine partial coefficients according to the
principle of least squares. Essentially, this method is to fit the standard
value
of structural component resistance, which changes with the
load-effect ratio ρ, using the standard value
of the structural
component resistance determined by the design expression, given the
target reliability index and dead load effect or live load effect. A linear
function is used to fit it. If the error between the mean reliability index of
components obtained by reverse calculation and the target reliability index
is adopted for measurement, the effect is not so good when there are only a
few optimization variables, but calculation is simple.
(2) Least squares method for reliability index
By this method, partial coefficients are determined according to the
principle that the error between the structural reliability index in the limit
state design expression and the target reliability index is minimized.
(7.58)
where
is the target reliability index of the ith type of component; βij is
the reliability index of the ith type of component in the limit state design
expression at the jth ρ load effect ratio.
(3) Specification method 1
All the load partial coefficients obtained by Equations (7.57) and (7.58) are
results of optimization within a certain range. The load partial coefficients
obtained by calculation may be different from the partial coefficients
commonly used by designers. Thus, inconvenience is caused for both
engineering design and engineering practice. To ensure the continuity
between new and old codes, a new method called “Specification Method 1”
has been proposed in Figure 7.7: Under extreme conditions, the dead load
coefficient is kept unchanged when the dead load and live load are
combined, i.e., γG = 1.3. The resistance partial coefficients of all structural
components are calculated by the principle of least squares. Under the
condition that load coefficient is known, the live load coefficient can be
changed to calculate the partial coefficients of resistance. This is consistent
with both engineering practice and designers’ design habits.
(4) Specification method 2
Specification Method 2 is basically the same as Specification Method 1. To
ensure the continuity between the new and old codes, four common
resistance partial coefficients are known, i.e., axial compression is 0.85,
axial tension is 0.95, bending is 0.95, and shearing is 0.95, to calculate load
partial coefficients. Then, other resistance partial coefficients are calculated
under the condition that load partial coefficients are known in Figure 7.8.
Four common resistance partial coefficients are known in this method to
calculate the partial coefficients of load and resistance. This is consistent
with accepted engineering practice and design norms.
Figure 7.7 Flow chart of specification method 1.
Figure 7.8 Flow chart of specification method 2.
(5) Method for determining resistance partial coefficients
As mentioned earlier, during the process of selecting an optimal partial
coefficient of load, given each set of γG and γQ, for every component i, under
the condition that Hi is minimized, the least squares method can be used to
determine an optimal partial coefficient resistance,
, under the
combination of dead load and live load. The calculation is as follows:
(7.59)
where Sj = γG(SG)j +γQ(SQ)j, and the condition that ∂Hi/∂γi = 0 must be met.
So
(7.60)
Sj and
can be calculated for each load combination, and each loadeffect ratio ρ, and then substituted into the above equation to determine the
resistance partial coefficients in the design expression under the condition
that there is a minimum error between the designed reliability index of this
structural component and the target reliability index.
7.4 Determination of Load Combination
Coefficient and Design Expression
In addition to dead load, structural components only bear a variable load,
which is a simple combination. The maximum distribution
of this
variable load in the reference period is taken for probability limit state
analysis. Given multiple variable load combinations, if the maximum
distribution of each variable load in the reference period is still considered,
this will become too conservative and unrealistic. The above-mentioned
maximum distribution SM of load effect combination should be introduced
into the limit state equation for probability analysis. This contains the
maximum distribution of each variable load in a specific time interval.
Theoretically, if the statistical parameters of these basic variables are
known, then the resistance can be designed directly by the checking point
method according to the target reliability index; or if the resistance is
known, the reliability of the component can be checked when it is being
acted upon by multiple variable loads. But at present, it is impractical to
adopt probability design directly, while a partial coefficient design
expression still needs to be presented in the codes for design value control
[7-27][7-29]
. Domestic and overseas design expressions considering load
combination effect can be divided into the following two categories.
7.4.1 Design Expression Using Combined Value Coefficients
Given multiple variable loads, the standard values and partial coefficients of
each variable load in the design expression are equal to what they would be
when there is only one variable load. A coefficient less than 1 can be
adopted and multiplied by the load effect term. This is the combined value
coefficient. According to the multiplication, the combined value coefficient
can be divided into the following three types.
(1) All load terms (including dead load) are multiplied by the
combination coefficient.
For example, in the U.S. Building Code Requirements for Structural Concrete
and Commentary (ACI318-83)[7-28][7-30], while considering the concurrent
occurrence of multiple variable loads, the load effect terms with load
coefficients, including dead load, are all reduced by 0.75. They are then
compared with dead load plus a variable load term, and the most
unfavorable load condition and load effect term selected:
where D, L, W and E are the effects caused by dead load, live load, wind load
and earthquake, respectively. T is the effect caused by differential
settlement, creep, shrinkage or temperature variation.
(2) Reduction of all variable load terms
For example, the National Building Code of Canada [7-29] (2010) adopts the
following formula for basic load effect term selection:
(7.61)
where υ represents the effect of the various loads in the brackets; D, L, W
and T represent the standard value of dead load, live load, wind and
temperature, respectively; γD, γL, γW and γT are the corresponding partial
coefficients determined under the action of dead load and variable load
alone; ψ is the load combination value coefficient, which is set to 1.0, 0.7
and 0.6 when there are one, two or three loads in the internal brackets. In
China’s load code (TJ9-74) [7-30][7-32], the following formula is taken as the
design expression for load combinations. When wind load is combined with
other live loads, the standard value of wind load (W) and all other live loads
(Qi) apart from dead load (G) is multiplied by the combination coefficient ψ.
ψ = 0.9 for common buildings.
(7.62)
The greater of the above two equations is adopted for design control. K is a
single safety coefficient.
(3) Only a few variable load terms are multiplied by the combination
coefficient
According to the Unified Rules for Various Structures and Materials[7-31],
Volume I of the International System of Unified Standard for Structures
formulated by the International Joint Committee on Structural Safety (JCSS),
and Code for Design of Concrete Structures[7-32], Volume II of the
International System of Unified Standard for Structures formulated by the
Fédération Internationale de la Précontrainte (CEB-FIP), the major variable
loads involved in combination are not multiplied by the combination
coefficient, while only subordinate variable loads are reduced. For example,
in Code for Design of Concrete Structures, the following formula is adopted
for basic load combination.
(7.63)
where the partial coefficient of dead load γG is generally set to 1.35, or to 1.0
when G is favorable; the partial coefficient of prestress γP is set to 0.9 when
P is favorable; the partial coefficient of live load is generally set to 1.5; the
load combination value coefficient ψ0i can be set to 0.3 for residential
buildings, 0.6 for office buildings and parking garages, and 0.5 for wind and
snow; S represents the effect of each load.
For the calculation of such design expressions, it is generally necessary to
take every variable load involved in the combination as the main load (Q1)
one by one in turn, with the rest as subordinate load Qi, to calculate the
combined value and select the most unfavorable combination. According to
the Soviet Union’s load code, when more than two loads (long-term or
short-term live load) are applied simultaneously, the total internal force X
(bending moment, torque, longitudinal or transverse force) is determined
according to the following formula:
(7.64)
where
represents the internal force caused by the standard value of
live load (i.e., the standard value of load effect). When there are two or more
short-term live loads applied, the
of short-term live load should be
multiplied by a combination coefficient of 0.9; m represents the number of
live loads (including long-term and short-term live loads) acting at the same
time; ni represents the coefficient of various loads, set to 1.2 for live floor
loads and crane loads, 1.4∼1.6 for snow load, and 1.2∼1.3 for wind load; XG
represents the internal force generated by the standard value of dead load,
and the coefficient of dead load nG is set to 1.1, or to 0.9 when G is favorable.
7.4.2 No Reduction Factor in the Design Expression
In this kind of design expression, different load partial coefficients or load
eigenvalues are specified for variable loads according to their degree of
participation in the combination. For example, U.S. national standard
Minimum Design Load for Buildings and Other Structures (ANSI A58.1-1982)
[7-33]
stipulates that, when strength design is adopted for structure and
component foundations, the load effect term is the most unfavorable effect
to be selected in Equation (7.65).
(7.65)
where D, L, Lr, S, R, W and E represent the standard value of dead load, live
load, roof live load, snow load, rain load, wind load and earthquake,
respectively. The basic concept of load combination is as follows: except for
the dead load that is in permanent action, one of the variable loads is
“maximized in the service life”, while the other variable loads are assumed
to be at a “value at any time point”. Because the standard load value exceeds
the value at any time point, some load coefficients in the above equations
are less than 1.
American experts Galambos and Ravindra [7-34] suggested taking different
standard values of load partial coefficients for variable loads according to
different combinations, as follows:
(7.66)
where represents the effect caused by the mean value of dead load; LT,
, represent the effects caused by the mean value of the maximum
distribution of live load, wind load and snow load throughout the service
life;
represents the effect caused by the mean value of the distribution
of live load at any time point.
As can be seen from the above, there is a difference from country to country
in considering the load combination design expression. The first kind of
method, based on the combination coefficient, is simplest to use, with all
loads (including dead load) reduced by 0.75. This amounts to increasing the
bearing capacity by 1/3 after considering load combination. However,
because the dead load is theoretically a random variable that does not
change with time, combinational reduction of the dead load in this formula
is inconsistent with the statistical characteristics of load combination,
making it unsafe to make calculations in some cases. In Equation (7.66,
7.69), all variable loads involved in the combination are multiplied by the
combination coefficient in no order, and the method is easy to operate.
However, when the effect of one variable load is more significant than that
of the others, this load is still reduced, and obviously the value will be on the
low side. In Equation (7.67), it is more rational to take all the values of the
main loads and reduce subordinate loads only. Besides, there is no
combination coefficient in Equation (7.66), and Equation (7.67) intuitively
reflects the load combination mode. The mean value and corresponding
partial coefficients of variable loads that may meet one another are
introduced into this equation, with everything easy to understand, but this
method requires that the codes provide several design standard values for
each variable load. This is inconsistent with China’s design conventions.
According to the above analysis, as well as China’s design practice and the
principle that the standard value and partial coefficients of combined loads
must be equal to that of a simple combination, the first type of design
expression multiplied by the combination coefficient is adopted in China’s
Unified Standard [7-25]. In the process of compilation, regarding the design
expression for the two types of combination coefficients in the formula
below, the combination value coefficients ψ and ψc were calculated by
probability analysis. After analysis and comparison, the two formulas below
were finally adopted for the unified standard, thanks to their advantages of
providing a reasonable value and a simple design.
(7.67)
where γ is the load partial coefficient, which has the same value as that in a
simple combination; C is the effect coefficient of load conversion into effect;
GK, Qi, Qik represent the standard values of dead load, main variable loads
and subordinate variable loads, respectively; ψ and ψc represent the
combined value coefficients multiplied by all loads and by the subordinate
variable loads, respectively.
7.4.3 Method for Determining Load Combination Coefficient
in Ocean Engineering
At present, the probability limit state of offshore structures cannot yet be
designed directly based on the target reliability index. Instead, an
internationally accepted universal practical design expression, represented
by the standard value and partial coefficients of basic variables, is adopted
for probability limit state design. The standard value of various basic
variables is determined according to the high fractional value
corresponding to its maximum value distribution, while partial coefficients
are, as shown in the previous analysis, determined according to the target
reliability index and an optimized simple combination of random loads in
the marine environment. A change will take place in the probability
distribution of load combination when more environmental loads are
involved in the combination, and the probability is extremely low that
various load effects involved in the combination meet one another through
their standard value. The standard value of each load involved in the
combination must be adequately reduced in order that the structural
reliability index will not change after combination. This reduction factor is
also the load combination coefficient.
In the practical design expression for offshore engineering structures [7-35]
[7-36]
, the standard value of variable loads is reduced under the premise that
the partial coefficients are kept unchanged. The value of the combination
coefficient is determined after optimization according to the principle of β
equivalence, i.e., load partial coefficients are determined according to the
simple combination of environmental marine loads. When two or more
variable loads are combined, the standard value of these variable loads can
be reduced on the basis of the combination coefficient, so that the reliability
index of various components designed in accordance with the limit state
design expression remains consistent with that of the simple combination
as much as possible.
According to the above principle, when there are several variable loads
present, the ultimate state expression for structural bearing capacity can
take the following form:
(7.68)
where ψ is the combination coefficient multiplied by the sum of various
variable loads.
From the above formula, we obtain:
(7.69)
where γG, γQ, γR are all determined according to the simple combination of
environmental loads; RK* is the value of structural resistance at the checking
point. It can be determined by the aforesaid probability limit state design
method, in accordance with the reliability index of the structure designed
according to the limit state design expression under simple combination
and the load combination form that functions as a controller during the
design reference period.
Owing to the value ψ of the combination coefficient obtained by the above
formula, the reliability index β of the structure designed according to the
limit state design expression is the same as that in a simple combination.
However, the ψ value varies with the type of components, the type of loads
involved in the combination, the effect ratio ρ between variable loads and
the dead load, and the ratio ξ between variable loads. Although a reliability
index consistent with that in the simple combination can be obtained if the
ψ value is obtained under different conditions, it is extremely inconvenient
to use. For this reason, the ψ value of an optimal combination coefficient
applicable to various components and ρ values can be determined
according to the effect ratio ξ between different variable loads under a
certain load combination, so that the I value below is minimized:
(7.70)
where m represents m components and j represents j ρ values.
Take the reciprocal of the above formula, and let ∂I/∂ψ = 0. So
(7.71)
where C = γRm/R*K(mj), S’ = γGSGK,
But at this time, the ψ value still varies with the effect ratio ξ of variable
loads and cannot be a constant, making it inconvenient to use. For the
convenience of engineering application, the ultimate state design
expression for bearing capacity takes the following form:
(7.72)
where the practical load combination coefficient ψC can be a constant value,
which can be multiplied by all other variable loads except the dominant
variable load. Based on the combination coefficient ψ previously obtained
by optimizing various components and ρ values, the following relationship
exists between the practical combination coefficient ψC and the
combination coefficient ψ.
(7.73)
It is assumed that the partial coefficients of each variable load have the
same value in the practical expression, i.e., γQ1 = γQ2 = γQ3 = ⋯ = γQi. When
there are two types of loads involved in the combination, the above
equation can be simplified as:
(7.74)
where ξ = SQ2K/SQ1K represents the effect ratio between variable loads.
According to the above equation, ψC can be obtained if ψ is known.
7.5 Example: Path Probability Model for the
Durability of a Concrete Structure
7.5.1 Basic Concept
Reinforcement corrosion of concrete structures in a corrosive environment
has long been an issue of great concern for academics throughout the world
[7-37]
. Reinforcement corrosion is the main cause of deterioration in the
resistance and reliability of concrete structures. Throughout their service
life, concrete structures are affected by a variety of uncertain factors.
According to the damage mechanism of steel bar corrosion, the lifespan of
concrete structures can be divided into four stages: destruction of the steel
bar passivation film, corrosion and cracking of the protective concrete layer,
the width of concrete cracks reaching their upper limit, and the bearing
capacity of concrete components dropping below their lower limit.
According to existing research, the corrosion process of steel bars can be
divided into two stages [7-38][7-39], the initial rust stage and the corrosion
propagation stage, both of which are affected by many uncertainties. In the
initial corrosion stage, the diffusion coefficient, concentration on the surface
and critical concentration of chloride ion are all highly random, which in
turn leads to the high randomness of the initial corrosion time. During the
extension stage, the corrosion depth is also somewhat random due to the
random nature of the corrosion current density.
According to the basic concept of the Path Probability Model [7-38], the
corrosion process of steel bars can be divided into a series of paths, based
on the initial corrosion stage and the corrosion expansion stage, while
taking into consideration the influence of uncertain factors in both stages.
The unconditional probability distribution for steel bar corrosion under all
paths is obtained using Bayes’ probability summation formula, as shown in
Figure 7.9.
Figure 7.9 Corrosion path model.
In the Figure 7.9, TE is the given lifespan of the structure, tc is the time when
the steel bar initially corrodes, and the corrosion propagation time is TE − tc,
which defines a possible corrosion path in
. When TE
has been determined, the corrosion ratio of any corrosion path
is a function of time tc, and is recorded as
of the steel bar.
If
; η is the corrosion ratio
is further recorded as the probability density of the initial rusting
of the steel bar at time tc, then the probability of initial rusting of the steel
bar at time tc is
. According to Bayes’ probability formula:
(7.75)
Where,
is the conditional probability density function of the
corrosion ratio of the steel bar under the condition that the initial corrosion
occurs at time tc; a number of corrosion paths can be defined within
.
On the basis of considering n time points in
corrosion paths
paths in
. then there are n
; these can contain all corrosion
, as long as n is large enough. When initial corrosion occurs
at ti, then steel bar corrosion does not occur at any time before ti. Therefore,
these paths are mutually exclusive. According to the probability summation
formula, the unconditional probability density function
of a steel bar
corroding at time t can be expressed as:
(7.76)
Where,
is the probability density function of the steel bar
corroding under the condition that initial rusting occurs at time ti;
is the probability of initial rusting of the steel at time ti,
(7.77)
Where,
is the cumulative probability of initial rusting
occurring at time ti.
7.5.2 Multipath Probability Model
According to the structural damage mechanism caused by steel bar
corrosion, the lifespan of a concrete structure generally goes through four
stages: Initial rusting of steel bar, rust expansion and cracking of the
concrete protective layer, cracking limit being reached, and structural
bearing capacity dropping below its limit. Therefore, it is very important to
master the diffusion mechanism of chloride ions inside concrete, crack
development, and the degradation law of bonding properties between steel
bar and concrete, in order to evaluate durability and predict the residual
lifespan of structures or components.
Based on the path probability model mentioned above, the corrosion
process for a steel bar is divided into three stages in the literature [7-39], [740]
: Initial stage, rust expansion stage and late stage concrete cracking (from
cracking of the protective layer due to corrosion expansion, to the bearing
capacity of the structure or component dropping below its limit). The
Multipath Probability Model is thus established to predict corrosion ratio,
crack width and bearing capacity degradation of steel bars.
In the Figure 7.10, tc and tcr are initial rusting time and protective layer
cracking time, respectively. There are a total of three conditions for concrete
structures in service:
1. t0 < TE < tc, steel bar is not corroded
2. tc < TE < tcr, steel bar is corroded, but concrete protective layer has not
cracked due to corrosion expansion
3. tc < tcr < TE, cracking of concrete protective layer due to corrosion
expansion
In this paper, the model is established by the third case, with a corrosion
path
. It can be seen from the previous section that
can be defined as many paths, and that any n time points in
can be
considered. For any initial rusting time ti(i = 1,2 … n) and protective layer
cracking time
, there are
corrosion paths
in total.
Figure 7.10 Corrosion multi-path model.
It can be seen from Equation (7.76) that, as long as enough corrosion paths
are defined during the lifespan, and as long as tc and tcr of each path are
uniquely determined, these paths will constitute a group of mutually
exclusive events. According to Bayes’ probability summation formula,
unconditional probability density functions
, f(W) and f (L) for the
corrosion ratio of steel bars, crack width due to corrosion expansion, and
bearing capacity reduction coefficient in service time can all be calculated.
(7.78)
(7.79)
(7.80)
Where, F(ti) is the cumulative probability of initial rusting occurring at any
time ti;
and
are probability density functions of
the crack width and bearing capacity reduction coefficient under the
condition that ti and tk are determined, respectively;
is the
probability density function of protective layer cracking at time tk under the
condition of initial rust occurring at time ti.
With the increased service time of concrete structures, when trying to study
the time-varying characteristics of corrosion ratio or crack width, the
maximum service time
should be considered. The time domain is
divided into equal sections, with the time interval set as Δt;
corrosion paths can thus be obtained, whose initial corrosion time, time of
cracking due to corrosion expansion and late cracking time are all set as
(7.81)
Therefore, for any service time
altogether
, there are
corrosion paths, whose initial corrosion time, time
of cracking due to corrosion expansion and late cracking time are all set as
(7.82)
As j < m,
(7.83)
Therefore, the corrosion path for any lifespan
can be
derived from the path defined by the maximum service time, and only the
paths defined by
need to be simulated.
7.5.3 Probability Prediction Model Featuring Chloride
Erosion
When the concentration of chloride ion on the surface of a steel bar
gradually increases and exceeds critical concentration after a reinforced
concrete beam has been used in a chloride-containing environment for a
certain period, the passivation film on the surface of the steel bar will be
destroyed, and the steel bar will corrode. The transportation modes of
chloride ions in concrete include convection, electromigration, and diffusion
[7-41]
. In general, unsteady diffusion is considered the main transportation
mode, which also conforms to Fick’s second law. Then, the chloride ion
concentration at a depth of x from the concrete surface at time t can be
expressed as [7-42]:
(7.84)
Where,
is the chloride ion concentration (percentage of concrete
mass) at a distance x from the concrete surface kg / m3; C0 is the
concentration of initial chloride ion inside the concrete (usually 0); Cs is the
concentration of chloride ion on the concrete’s surface, Val [7-43] considers
that its distribution parameters are related to the environment, and it is
recommended to take values based on Table 7.3; Dcl is the chloride ion
diffusion coefficient mm / s. Stewart believes this is related to time,
temperature, humidity and stress level, and assuming a condition of no
measured data, it can approximately obey a normal logarithmic distribution
with a Mean Value of 2×10-6 and a coefficient of variation of 0.450 [7-44];
Table 7.3 Concentration of chloride ion on a concrete surface.
Environment types
Splash areas
Coefficient of
variation
7.35kg/m3 0.70
Distribution
types
Log-normal
distribution
Near-shore
environment
2.95
kg/m3
0.70
Log-normal
distribution
1km from coast
1.15
kg/m3
0.03
kg/m3
0.50
Log-normal
distribution
0.50
Log-normal
distribution
Normal atmospheric
environment
Mean
When the concentration of chloride ions on the steel bar surface reaches a
critical value, the steel bar will corrode. Then the cumulative probability
of initial rusting of the steel bar occurring at any time ti can be
expressed as:
(7.85)
Where, Ccr is the threshold value of concentration of surface chloride ion
that leads to the initial rusting of the steel bar, which is highly random. As
suggested by Stewart, normal distribution with mean 3.35 kg /m3 and
coefficient of variation of 0.375[7-45][7-46] should be obeyed.
7.5.4 Probability Prediction Model for Concrete Carbonation
Concrete carbonation generates CaCO3, reduces porosity, improves concrete
compactness and hinders CO2 diffusion. However, it also reduces Ca(OH2)
concentration and pH value, leading to destruction of the passivation film
on the steel surface and thus accelerating corrosion of the steel bar. The
variation of concrete carbonation depth is mainly determined by its own
variability and the environment[7-39]. The practical empirical carbonation
model considers various influencing factors, and is given in the literature [745] [7-47][7-48]
:
(7.86)
Where, k is the carbonation coefficient; Kmc is a calculation model
uncertainty variable; kj is the corner correction coefficient; kCO2 is the
coefficient influence CO2 concentration; kp is the correction coefficient of
the casting surface; kb is the coefficient influencing working stress; T is
ambient temperature in °C; RH is the relative ambient humidity %; fc is the
standard value MPa of concrete cube strength, which has a certain
randomness and obeys a normal distribution. See Table 7.4[7-47] for the
specific distribution.
1% phenolphthalein solution is generally used to measure the pH value of
concrete in engineering fields. The pH value from concrete surface to steel
bar tends to gradually increase. Based on the change in pH value, concrete
carbonation can be divided into three parts: completely carbonized part,
partially carbonized part, and uncarbonized part. The partial carbonation
area is supposed to change in a linear fashion [7-45], as shown in Figure 7.11:
Table 7.4 Distribution of the standard value of compressive strength of a
concrete cube.
Concrete C15
strength
grade
C20
C25
C30
C35
C40
C45
C50
C55
C60
Standard 22.86 28.33 33.75 39.29 21.54 50.00 55.83 60.91 67.27 72.00
value/MPa
Coefficient 0.21 0.18 0.16 0.14 0.13 0.12 0.12 0.11 0.11 0.10
of
variation
Figure 7.11 Carbonation diagram.
Where, The value of pH varies from 8.5 to 12.5. When the value reaches
11.5, the passivation film on the steel surface will be destroyed and the
initial rusting of the steel begins [7-39]. The influence of environmental
humidity, water to cement ratio, carbonation time, cement dosage and CO2
concentration on the length of the partial carbonation zone is described in
the literature [7-49], which also provides a calculation model for partial
carbonation zone length.
(7.87)
Where, RWC is the water-cement ratio, C is the cement dosage (kg / m3), RH
is the relative ambient humidity. When RH > 75%, the length of the partial
carbonation zone can be ignored.
According to the carbonation curve of the pH value in the partial
carbonation zone, the length from pH=11.5 to the front end of the complete
carbonation zone can be expressed as:
(7.88)
Therefore, from Equations (7.87) and (7.88), it can be seen that, under a
carbonizing environment, the cumulative probability
of initial
rusting occurring in a concrete steel bar at any time ti can be expressed as:
(7.89)
Where, Xc is the thickness of the concrete protective layer (mm);
is
the length of the complete carbonation zone at the point of initial rusting of
the steel bar (mm).
7.5.5 Probability Prediction Model under the Combined
Action of Carbonation and Chloride Ions
As shown in the literature, chloride ion diffusion in concrete is subject to
such factors as load, cracking, and carbonation. The effects of carbonation
on chloride ion diffusion behavior are summarized in this section, which
also presents a probabilistic prediction model for the joint action of
carbonation and chloride ions.
According to existing research on the interaction of chloride ions and
carbonation [7-45][7-49][7-50], it has been found that carbonation influences
chloride ions in two ways. To be specific, carbonation products reduce the
porosity of concrete, improve its compactness, and hinder the transmission
of chloride ions; the bound chloride ions released by carbonation lead to an
increase in free chloride ions at the front end of the carbonation, which in
turn accelerates the diffusion of the chloride ions within the concrete.
After studying free chloride ions in concrete, Glass [7-50] found that the
passivation film on the surface of steel bars grows weaker when the
alkalinity and pH value of the concrete are low, or when the concentration
of chloride ions is low. Jin Weiliang et al. [7-39][7-40][7-45][7-50] found the same
phenomenon after analyzing concrete engineering works in coastal area of
Ningbo City. The concentration of chloride ions on the surface of steel bars
in some concrete structures is relatively low, and the concrete carbonation
depth is very shallow, but the surface of the steel bar can be significantly
corroded. Therefore, it can be seen that the concentration ratio of chloride
ion to hydroxide ion serves as an important parameter affecting steel bar
corrosion. According to the pH value of the steel bar surface, the functional
relationship between pH value and critical chloride ion concentration can
be expressed as:
(7.90)
Figure 7.12 Curve of pH value and critical chloride concentration.
The change curve is shown in Figure 7.12.
It can be seen from the figure that a positive correlation does exist between
pH and critical chloride ion concentration; critical chloride ion
concentration increases with the increase in pH value. It is assumed in the
literature [7-39][7-40] that the diffusion behavior of chloride ion and carbon
dioxide in concrete do not affect each other. Therefore, in a concrete
carbonation environment featuring chloride ion corrosion, a steel bar is
considered corroded as long as there is a condition that meets the
requirements. That is, when the concentration of the chloride ion on the
surface of the steel bar reaches critical chloride ion concentration, or the pH
value of the steel surface reaches 11.5. Under the combined action of
carbonation and chloride ion, the cumulative probability of the initial
rusting of the concrete steel bar at any time
can be expressed as:
(7.91)
7.5.6 Corrosion Propagation in a Steel Bar
After the initial corrosion of the steel bar occurs, the corrosion propagation
time is TE − tc. Song Zhigang [7-38] et al. believe that the probability density
of the corrosion ratio in the corrosion propagation stage of a steel bar is
. The specific calculation process can be expressed as:
Assuming the steel bar is corroded uniformly, the calculation process for the
corrosion ratio can be expressed as:
(7.92)
Where, ΔAs is the area of the corroded steel bar (mm2), As is the initial area
of the steel bar (mm2), d is the diameter of the steel bar (mm), and Δd is the
corrosion depth of the steel bar (mm).
According to Equation (7.92), the relationship between corrosion ratio and
corrosion depth can be expressed as:
(7.93)
ΔD can be determined by the empirical formula (7.94).
(7.94)
Where, λ(t) is the corrosion ratio of the steel bar before cracking of the
protective layer (mm/a). According to the literature [7-44][7-51], the equation
for λ(t) can be expressed as:
(7.95)
Where, icorr is the corrosion current density of the steel bar μA/cm2, tc is the
corrosion time. After discovering the law that the corrosion current density
decreases with time, as demontrated by Liu et al., Stewart [7-51] et al.
established an empirical formula for corrosion current density:
(7.96)
Where, icorr(1) is the corrosion current density at the time of initial rusting
of the steel bar. In a typical environment, with a temperature of 20°C and a
relative ambient humidity of 75%, this is related to the water-cement ratio
of concrete and the thickness of the protective layer. The empirical formula
can be expressed as [7-51]:
(7.97)
The empirical formula for the water-cement ratio RWC is:
, Xc is the thickness of the concrete protective layer
(mm),
is the standard value of the compressive strength of the concrete
cylinder MPa [7-51], [7-52], and fc is the standard value of the compressive
strength of the concrete cube (MPa).
Thus, the empirical expression for the corrosion depth of the steel bar can
be expressed as:
(7.98)
7.5.7 Cracking of the Protective Layer and Determination of
Crack Width
The relationship between corrosion ratio and crack width is expressed by a
new model proposed by Vidal [7-53] et al., and used for predicting the loss of
steel sections. In this way, the crack width w after cracking due to corrosion
expansion can be expressed as:
(7.99)
(7.100)
(7.101)
(7.102)
Where, ΔAs is the area of corroded steel bar (mm2); ΔAcr is the critical
corrosion area of steel bar (mm2) when the protective layer cracks; K is the
correction coefficient, which is 0.0575; xc is the depth of the pit (mm); d is
the diameter of the steel bar (mm); ηcr is the critical corrosion ratio when
the protective layer cracks.
7.5.8 Bearing Capacity of Corroded Concrete Components
Based on the existing design calculation theory, the following calculation
models [7-54][7-55][7-56][7-57][7-58][7-59] can be used to analyze the bending and
shear capacity of corroded rectangular concrete section beams:
(7.103)
(7.104)
(7.105)
(7.106)
(7.107)
(7.108)
(7.109)
(7.110)
(7.111)
(7.112)
Where, ks is the collaborative work coefficient; α1 is the ratio of the stress
represented by the rectangular stress diagram of the concrete in the
compression zone to the design value of the compressive strength of the
concrete; fy and fc are the standard value MPa of yield strength of steel bar
and cubic compressive strength of the concrete, respectively; h0 and b are
the effective height and width of the section, respectively (mm); fyc is the
nominal yield strength of the corroded steel bar (MPa); Asc and Vvc are the
residual areas of the corroded longitudinal bars and stirrups, respectively
(mm2); β is the influence coefficient considering the reduction of concrete
shear strength caused by steel bar corrosion; M0 and V0 are the flexural and
shear capacities of the uncorroded reinforced concrete beams, respectively;
LM and Lv are the reduction coefficients of bending and the shear capacity of
beams, respectively.
Figure 7.13 Simulated flow diagram.
The Monte-Carlo numerical simulation flow chart based on the path
probability model is as follows in Figure 7.13:
7.5.9 Engineering Example
7.5.9.1 Corrosion of Steel Bars in a Chloride Environment
A certain bridge consists of a simply supported reinforced concrete
prestressed structure [7-56], as shown in Figure 7.14. The bridge has been
used for 16 years. Most piers are located in the water. The main steel bar
used in the pier sections is Φ22 while the stirrups are Φ10. The main steel
bar of the cap beam is Φ25 while the stirrups are Φ12. See Table 7.5 for
related values, such as the probability distribution form and distribution
parameters of the influencing parameters. The width of the rust expansion
crack of each concrete pier column in wave splash and tide range area is
within 0.2∼0.4mm. See Figure 7.15 for the statistical quantities.
Figure 7.14 Bridge structural status.
Table 7.5 Calculation parameters and distribution types.
Parameter Unit
type
Mean
Coefficient of Distribution Source
variation
type
Cs
kg/m3 7.35
0.7
Dcl
mm2·s- 2×10-6
0.45
Xc
mm
Ccr
kg/m3 3.35
0.375
fc
MPa
0.064
1
35.2/29.8 0.060/0.066
29.9
Logarithmic
normal
Stewart[7-
Logarithmic
normal
Stewart[7-
Logarithmic
normal
Measured
value
Logarithmic
normal
Stewart[7-
Logarithmic
normal
Measured
value
8]
8]
8]
Predict the probability distribution of corrosion degree and corrosion
expansion crack width of the pier and cap beam steel bar in different time
periods based on the data given in Table 7.5; estimate the percentage of
corrosion samples and cracked samples of steel bar components, as shown
in Figure 7.16-Figure 7.19.
Figure 7.15 Number of cracks in piers.
Figure 7.16 PDF of main rebars.
Figure 7.17 CPDF of main rebars.
It can be seen from Figure 7.16 that about 28% of the steel bar samples
have been corroded. According to the test data given in Figure 7.15, if the
number of rust expansion cracks on Pier 30 with large-area network cracks
(the number of cracks along the longitudinal steel bar is close to that of the
longitudinal steel bar), 35, is taken as the sampling number for each pier,
then there are 35×64=2,240 pier samples in total, with 367 cracked
samples, meaning about 16.4% (<28%) of the samples are cracked. After
considering the fact that the corrosion probability of steel bar is greater
than the probability of cracking due to corrosion expansion, the prediction
result is reasonable. It can also be seen from Figure 7.16 that the corrosion
ratio of steel bars in cap beams (5%<25%) is obviously lower than that of
piers in the wave splash area, which is also reasonable. The chloride ion
content of the bridge pier and superstructure samples shows that the
chloride ion content in the pier concrete is high, which will in turn induce
steel bar corrosion. In contrast, the chloride ion content in the
superstructure is low, which has an uncertain influence on steel bar
corrosion. According to the probability distribution Figure 7.17 of the
corrosion ratio of the corroded steel bar, the average corrosion ratio can be
up to about 5%. According to Figure 7.18, cracking samples account for
25% (<28%, reasonable), but the number of corrosion samples are similar
to the number of cracked samples, which also indicate that steel bar
corrosion has been taking place. The reasons why the crack rate of 16.4% is
lower than 25% can probably be ascribed to: 1) omissions in the test
sample data; 2) deviations of model parameters and error of the model
itself; and 3) other uncertainties.
Figure 7.18 PDF of corrosion-induced crack width.
Figure 7.19 CPDF of corrosion-induced crack width.
The sample distribution of the detected crack width should be classed as a
conditional probability problem. Statistical analysis of pier crack width
shows that the Mean Value is 0.34mm and the standard deviation is 0.16.
From the conditional probability density function for rust expansion crack
width shown in Figure 7.19, the mean and variance are 0.3, 642 and 0.1,
879 respectively, which are basically consistent with the measured values.
The probability distribution of the steel bar corrosion ratio and crack width
in wave splash zones of piers for every second year within a 10∼24 year
period were further analyzed. See Figure 7.20 and Figure 7.21 for the
results. It can be clearly seen that with an increase in service time, the
corrosion ratio, crack width and variability all increase too. If 20% of crack
samples reach 0.5mm, the durability limit state of a structure requiring
maintenance, then maintenance measures should be taken after 14 years of
service (i.e., 2 years before detection). Therefore, repair measures must be
taken immediately for all piers showing cracks, especially for any concrete
with seriously corroded steel bars, where crack width exceeds the limit, or
where the protective layer has been peeled off. High-performance
reinforced concrete should be used for the engineering repair design. For
concrete piers with no cracks, or featuring tiny cracks only, durability
protection, such as anti-corrosion coating, is recommended, if maintenance
funds can be ensured.
Figure 7.20 Time-dependent CPDF of main rebar.
Figure 7.21 Time-dependent CPDF of corrosion-induced crack width.
7.5.9.2 Corrosion of Steel Bar Under the Combined Action of
Carbonation and Chloride Corrosion
A certain bridge is a T-type simply supported structure, as shown in Figure
7.22. The bridge has been used for 36 years since its completion in 1968.
However, based on examination, the bridge suffers from serious durability
problems, including: 1) a large amount of spalling and exposed steel bars at
the bottom of the T-beam and the side concrete, with cracks found
everywhere; 2) some concrete sections were damaged at the T-beam resting
point at the top of the pier cap beam; and 3) the ends of adjacent beams and
slabs were squeezed and damaged due to no expansion joint being included
in the design. More than 70 cracks wider than 0.2mm and 28 locations with
spalling in the protective layer were found by means of site hole inspection.
T-beam cracks mainly consist of vertical cracks, most of which are long
cracks running through the whole web with widths up to 2mm. All the
observed beam bottoms featured problems such as horizontal cracks,
significant concrete spalling areas, exposed stirrups and main steel bars,
and serious corrosion.
Figure 7.22 Structural damage to the bridge.
Table 7.6 Calculation parameters and distribution types.
Parameter Unit
type
Mean
Coefficient Distribution Sources
of variation type
Cs
kg/m3 2.95
0.7
Logarithmic
normal
Stewart[7-8]
Dcl
mm2·s- 0.167×10- 0.45
1
6
Logarithmic
normal
Measured
value
Xc
mm
15
-
Constant
Measured
value
Ccr
kg/m3 3.35
0.375
Logarithmic
normal
Figure 7.15
ds
mm
2Φ20
-
Constant
Design value
dv
mm
Φ6@ 200 -
Constant
Design value
fy
MPa
365
0.05
Logarithmic
normal
Design value
fc
MPa
25
0.15
Logarithmic
normal
Measured
value
T
°C
20
-
Constant
Meteorological
survey
Parameter Unit
type
RH
%
%
Mean
75
0.03
Coefficient Distribution Sources
of variation type
Constant
Meteorological
survey
-
Constant
Meteorological
survey
Test results showed that the chloride ion content of the bridge is not high,
basically within 0.2%. The bridge is in a critical stage of induced steel bar
corrosion. The chloride ion diffusion coefficient of the concrete is less than
1.67×10-9 cm2/s, and the compactness of the concrete is good. According to
the carbonation depth test for the concrete core samples, some components
are already seriously carbonized. Therefore, the bridge is being attacked by
both carbonation and chloride ions. Further analysis showed that the joint
action of the concrete carbonation and chloride ion corrosion is the cause of
early depassivation and accelerated corrosion of the steel bars. See Table
7.6 for the probability distribution form and the distribution of influencing
parameters.
For the convenience of comparison, probability density functions of critical
chloride ion concentration, corrosion time of the steel bar, corrosion
expansion cracking time of the concrete, corrosion ratio of the steel bar, and
crack width at detection time are estimated under the conditions of single
chloride ion corrosion and chloride ion corrosion combined with concrete
carbonation.
According to Figure 7.23, under the condition of chloride ion attack alone,
the critical chloride ion concentration is a time-invariant random variable
with a mean of 3.5%. In contrast, under the combined effect of chloride ion
corrosion and concrete carbonation, the change of critical chloride ion
concentration is a random process as the carbonation progresses, while the
probability density function gradually approaches the vertical axis with
time. This shows that the critical chloride ion concentration decreases along
with the decrease in concrete pH value on the surface of the steel bar, which
conforms to the law of Figure 7.12. Because of the synchronous effect of
carbonation, the corrosion sample of the steel bar has reached more than
95% after the bridge has been used for 15 years (Figure 7.24). In addition,
the concrete protective layer is too thin, and the concrete cracks due to rust
expansion (Figure 7.25). However, for chloride ion alone, the steel bar
corrosion progresses more slowly, with extensive corrosion of the steel bar
specimens and the extensive cracking of the concrete only occurring after
the bridge has been used for 40 or 50 years. The corrosion ratio of the main
steel bar also shows the same characteristics, that is, the corrosion of the
steel bar is fully developed under the combined action, while the corrosion
ratio is generally very small under the action of chloride ion corrosion alone
(Figure 7.26). Even for corroded specimens, the corrosion of the steel bar
under the combined action is more complete, with 95% of corroded
specimens showing a corrosion ratio above 20% (Figure 7.27). The same
characteristic also applies to crack width (Figure 7.28). 95% of concrete
cracks are wider than 2mm (Figure 7.29), and the protective layer has been
peeled off. In fact, the actual project verified the investigation results that a
considerable area of peeled-off and exposed steel bars and cracks
commonly exist at the bottom of the T-beam and at the side protective layer
of the concrete.
Figure 7.23 PDF of chloride threshold value.
Figure 7.24 PDF of time to corrosion initiation of main rebars.
Figure 7.25 PDF of time to crack initiation of concrete.
Figure 7.26 PDF of corrosion ratio of main rebars.
Figure 7.27 CPDF of corrosion ratio of main rebars.
Figure 7.28 PDF of corrosion-induced crack width.
Figure 7.29 CPDF of corrosion-induced crack width.
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8
Application of Reliability Theory in
Specifications
Since the 1970s, the theory of structural reliability based on probability
theory and mathematical statistics has gradually ushered in a period of
practical application in the international civil engineering industry. In
1975 and 1979, Canada took the lead in issuing reliability-based design
codes for building and highway bridge structures [8-1][8-2]. In 1977, the
Federal Republic of Germany compiled the Basis for Confirming Building
Safety [8-3] as the foundation for the compilation of other codes. In
1980, the American National Standards Institute (ANSI) issued
Probability-based Load Criteria [8-4]. In 1982, the UK [8-5] added the
theory of structural reliability to the BS5400 Bridge Design Code. This
shows that the theory and method of structure design in civil
engineering have marked the dawn of a new era.
China did not start carrying out research on the theory and application
of structural reliability in the field of construction until the mid-70s. In
1984, the State Development Planning Commission approved the
implementation of the Unified Standard for the Design of Building
Structures (GBJ 68-84) [8-6]. This standard contains unified principles
for probabilistic limit state design based on reliability. Later,
referencing the International Standard of General Principles of
Structural Reliability (ISO2394)[8-7], based on China’s practical
engineering experience, and after soliciting opinions from the national
departments concerned, the country compiled a unified standard for
engineering structure reliability design [8-8][8-9][8-10][8-11][8-12]
pertaining to architecture, water conservancy, water transport,
highways and railway construction. These codes were all theoretically
based on structural reliability theory, while the probabilistic limit state
expressed by partial coefficients was adopted as the main criterion for
the revision of the structural design codes. Under the guidance of the
“Unified Standard”, structural design codes for architecture, water
conservancy, transportation and railway construction were revised or
compiled on a massive scale. This was known as the “transformation” of
codes in the engineering sector, i.e., the experience-based safety factor
method was transformed into a probability analysis-based limit state
design method. After a lot of hard work, a general national standard,
the Unified Standard for Reliability Design of Engineering Structures (GB
50153-92), was officially released in 1992 [8-13].
In the early days, the deterministic allowable stress analysis method
(WSD) was adopted for the design of engineering structures. As human
understanding of objective things deepened, and as reliability theory
continued to develop, it became necessary to consider the uncertainty
of things in design. Therefore, the resistance and partial coefficient
design method (LRFD), which also considered the uncertainty of
events, was thought of as a more reasonable design method and
therefore accepted by most designers. The core problem of LRFD is
how to reasonably determine partial coefficients for different kinds of
loads in combination. This method can be used to build a correlation
between the target reliability index and load partial coefficients so that
the target reliability can be reflected in the design process. It is
necessary to determine the reliability index of engineering structures,
and then to calculate the partial coefficients according to the
determined reliability index. As is explicitly stipulated in the new
“Unified Standard”, the basic function of load effect should be
considered in the process of bearing capacity limit state design, and the
combination of loads must be considered when necessary. Three types
of effect combinations can be selected in the process of normal
serviceability limit state design. These are standard combination,
frequency combination and quasi-permanent combination. Different
load combinations produce different partial coefficients for loads and
resistance. Therefore, load combination, which extends throughout the
entire “unified standard”, is both its foundation and core content.
After many years of theoretical research and engineering practice, the
unified standard for reliability design has developed to varying extents
with respect to professional fields. The corresponding code system has
also undergone changes along with the transition from understanding
structural reliability to applying structural reliability. During 2006 and
2014, the Unified Standard for Reliability Design of Engineering
Structures (GB 50153) [8-14][8-15] was officially published. In 2019, the
latest Unified Standard for Reliability Design of Building Engineering
Structures (GB50068-2018) [8-16] was implemented. The following
contents were modified in the new code. (1) The level of structural
safety was adjusted; the value of related partial coefficients was
improved, and for basic combinations of loads, that combination where
permanent load plays a controlling role is eliminated; (2) the
conditions for anti-quake design were added to the new code; for antiquake architectural structure design, the design idea that “the structure
is unbreakable in minor earthquakes, repairable after middle
earthquakes and indestructible in severe earthquakes” was introduced;
(3) the regulations on the reliability evaluation of existing structures
were improved; (4) relevant regulations on overall structural stability
design were added; and (5) relevant regulations on structural
durability limit state design were added. The most critical modification
in terms of structural design was as follows: the partial coefficient of
dead load was adjusted from 1.2 to 1.3, while the partial coefficient of
live load was adjusted from 1.4 to 1.5.
Structural design codes are formulated based on meeting structural
functional requirements. The safety, serviceability and durability
required by structural reliability are the basic functions that a structure
must possess. Structural reliability is a criterion for the establishment
of various design codes. At present, the design methods in international
structural probability are divided into three levels, depending on
accuracy: Level I, Level II and Level III.
(1) Level I—Semi-probability design method
Although the design method at this level considers probability in terms
of load and material strength, since it considers load and resistance
separately, and does not consider structural reliability from an overall
perspective of structural components, the core of structural reliability
—structural failure probability—remains untouched. Moreover, each
partial safety factor is primarily determined based on engineering
experience; hence it is known as the semi-probability design method.
(2) Level II—Approximate probability design method
This is a probability design method that has already been put into
practice all over the world. Based on probability theory and
mathematical statistics, it makes a relatively approximate estimation of
“reliability probability” for the design of engineering structures,
components or sections. Under China’s unified standard, the probability
theory-based first-order second-moment limit state design method is a
method at this level. Although it is already a probability method, the
time-dependent change of basic variables is ignored or simplified
during analysis; it features considerable approximation because it is
restricted by existing information when used to determine the
distribution of basic variables; moreover, some complex nonlinear limit
state equations are linearized in order to simplify the design
calculation, so it is still only an approximate probability method. But at
the present stage, it is indeed a reasonable and feasible method for
handling structural reliability.
(3) Level III—Total probability method
The total probability design method is an ideal method that is based on
probability theory. It not only describes various factors affecting
structural reliability by means of a random variable probability model,
but also considers time-varying characteristics and describes them via
a random process probability model. Moreover, based on accurate
probability analysis of the whole structural system, the structural
failure probability is taken as the direct measurement of structural
reliability. It is a real and complete probability method. At present, this
is just a field of research worth exploiting, it will take a long time to
realize any practical use. Of the last two levels, Level II approximates
Level III. From Level III, we can derive an optimal total probability
method based on the optimization theory.
At present, the probabilistic reliability design method is the one used in
most codes. Its basic design principle is as follows: within the specified
time (design reference period), the structure can satisfy specific
functions (durability, serviceability and safety) under various
environmental loads. The design reference period is a time parameter
used to determine variable loads and time-dependent material
properties. Design conditions are divided into three types: durable,
transient and accidental. Structural design should ensure that the
relevant functional requirements are met under these conditions.
8.1 Requirements of Structural Design Codes
8.1.1 Requirements of Structural Design
The general requirements [8-17][8-18][8-19] for structural design are as
follows: Structural resistance R should not be less than the
comprehensive structural load S. See below
(8.1)
Because the structural resistance and load effect are, in practice, both
random variables, the above equation cannot be strictly satisfied, and
can only be satisfied in a certain probabilistic sense, i.e.,
(8.2)
where PS represents structural probabilistic reliability. Therefore,
structural design helps to ensure that structural resistance is greater
than or equal to structural load under a certain reliability condition.
8.1.2 Classification of Actions
The most important function of an engineering structure is its ability to
withstand various environmental actions that may occur during its use.
Various forces directly acting on a structure caused by different
environmental factors are called loads. The loads acting on the
structure cause internal forces and structural deformation (called load
effect). The purpose of structural design is to ensure that the bearing
capacity of the structure is large enough to resist internal forces, so that
the deformation is controlled within the scope of normal structural use.
During structural design, it is necessary to consider not only the
various loads acting directly on the structure, but also indirect factors
causing internal forces and structural deformation. The factors that can
cause structural effects are called actions [8-20].
(1) According to time-dependent variability, the environmental
actions on the structure can be classified into:
① Permanent action: Its value does not change with time during
the reference period of structural design, or the change is
negligible compared with the mean value. Examples include
structural weight, soil pressure, water pressure, prestress,
foundation settlement, welding, etc.
② Variable action: Its value changes with time during the
reference period of structural design, and the change cannot be
ignored compared with the mean value. Examples include vehicle
gravity, people and equipment gravity, wind load, snow load,
temperature change, etc.
④ Accidental action: This does not necessarily appear in the
reference period of structural design, but once it appears, its
magnitude is large and its duration short. Examples include
earthquakes, explosions, etc.
Because the variability of a variable action is greater than that of a
permanent action, its relative value should be greater than that of a
permanent action, too. Owing to the small probability of accidental
action, the reliability of the structure resisting accidental action is
lower than its reliability against permanent and variable actions.
(2) According to the variability of spatial position, the
environmental actions on the structure can be classified as:
① Fixed action: This obeys a fixed distribution in the spatial
position of the structure. Examples include structural weight, fixed
equipment load, etc.
② Movable action: This can be distributed arbitrarily within a
certain range around the spatial position of the structure, such as
people in a house, furniture load, vehicle load on a bridge, etc.
Because movable actions can be distributed arbitrarily, it must be
ensured during structural design that their distribution does not
cause the most unfavorable effect.
(3) Classification by structural response:
① Static action: This does not cause acceleration in a structure or
components, or the acceleration is negligible, Examples include
structural weight, soil pressure, temperature change, etc.
② Dynamic action: This causes a non-negligible acceleration in a
structure or components. Examples include earthquakes, wind
load, impact load, explosions, etc.
For dynamic actions, structural dynamic effects must also be
considered. The structure has to be analyzed by a dynamic method, or if
the dynamic action is transformed into an equivalent static action, then
the structure can be analyzed by a static method.
8.1.3 Target Reliability
The reliability that must be ensured during structural design is known
as the target structural reliability [8-21]. This has a significant influence
on structural design. If the target reliability is slightly higher, then the
structure will be stronger, but the cost will increase; if the target
reliability is on the low side, the structure will be a bit weak, causing
insecurity. Thus, for the determination of target structural reliability, it
is necessary to fully consider the balance between structural reliability
and economic benefits. In general, the following factors need to be
considered: (i) public psychology; (ii) structural importance; (iii)
structural destructibility; (iv) economic bearing capacity of society.
For important structures, the target reliability should be a little higher;
for minor structures, the target reliability can be a bit lower. In many
countries, engineering structures are divided into three levels, in order
of importance: important structures, general structures and minor
structures. Normally, the target reliability of general structures is taken
as the basis for design, and for an important structure, its failure
probability is decreased by one order of magnitude, while for a minor
structure, its failure probability is increased by one order of magnitude.
For a brittle structure, there is almost no warning before it fails, and the
consequences of its failure are more serious than those of a ductile
structure. Therefore, it is generally requested in engineering practice
that the target reliability of brittle structures should be higher than that
of ductile structures.
The economic bearing capacity of society also has a certain influence on
the target reliability of engineering structures. The more developed the
social economy, the higher the public requirements for reliability of
engineering structures, and the higher the target reliability.
The major methods for determining target reliability are as follows:
accident analogy, economic optimization and empirical calibration.
1. Accident analogy is a method for determining appropriate target
reliability by comparing with various risks encountered in daily
life.
2. The idea of economic optimization is that for the determination of
target structural reliability, comprehensive measures should be
taken to minimize the consequences of structural failure and the
cost incurred by structural failure to cut down the total cost of the
structure during its service life.
3. The principle of the calibration method is that the reliability
calculated by the traditional design method is analyzed in
accordance with the theory of structural reliability, on the premise
that it is rational to stipulate requirements for structural safety
during traditional design. The reliability achieved by the
traditional structural design method is taken as the target
reliability for structural probabilistic reliability design. For
example, the target reliability currently adopted in China’s
deterministic probability design method is determined according
to the reliability level adopted for the semi-empirical and semideterministic method.
According to the structural safety class and failure type, the target
reliability index [β] for bearing capacity limit state design is specified in
the Unified Standard for Reliability Design of Building Structures [8-9]
shown in Table 8.1.
Failure structures and structural components are divided into two
types: ductile failure and brittle failure. Ductile failure be preceded by
an obvious warning sign, so remedial measures can be taken in time.
Therefore, the target reliability index can be set slightly lower. Brittle
failure is usually a sudden kind of failure, with no adequate warning. As
a result, the target reliability index should be set higher.
When the reliability index β is used for structural and reliability
checking [8-17][8-18], the objective variability of factors influencing
reliability can be taken into full consideration, enabling the structure to
meet its expected reliability requirements.
For empirical calibration, the reliability method is used to perform an
inverse analysis of the original structural design code to identify the
reliability level implied in the original structural design code. On this
basis, all factors are then taken into consideration to determine the
target reliability. The steps involved are as follows: (1) determine the
standard value of the resistance of components; (2) determine the
mean value, standard deviation and distribution type of various basic
variables; and (3) determine the reliability index by means of the JC
method.
In engineering practice, the load effect is often a combination of two or
more loads. Moreover, the load effect does not necessarily obey normal
or lognormal distribution, while the limit state equation of the
structure may also be nonlinear. At this point, the structural reliability
needs to be calculated by the checking point method according to the
iterative model. The solution steps are as follows: (1) determine the
limit state equation; (2) set the initial value of random variables; (3)
transform normal and non-normal random variables into standard
normal random variables; and (4) substitute the reliability index and
random variables into the equation, and perform an iterative
calculation using the checking point method until the results converge,
with all checking point coordinates obtained.
Table 8.1 Target reliability index of current building structures in
China.
Failure type
Class of structure
Important General Minor
Ductile Structure 3.7
3.2
2.7
Brittle Structure 4.2
3.7
3.2
8.1.4 Limit State of Structural Design
Structural functional requirements are divided into three aspects in the
Unified Standard of Reliability Design of Engineering Structures
(GB50153 [8-15]): safety, serviceability and durability.
Structural safety means that within its service life, a structure should be
able to bear various loads, imposed deformation (e.g., the differential
settlement of the bearings of statically indeterminate structures) and
constrained deformation (e.g., when temperature or shrinkage is
constrained) which may occur during normal construction and normal
use. When and after an accidental event (e.g., an earthquake or
explosion) occurs, the structure should be able to maintain its overall
stability, and shall not collapse or suffer continuous damage, resulting
in a great loss of life and/or property. As the most important quality
index of structural engineering, safety mainly depends on the level of
structural design and construction. It is also related to the correct use
(maintenance and inspection) of the structure. All these factors are
related to the reasonable formulation and correct use of civil
engineering laws and technical standards. For the design of an
engineering structure, structural safety is primarily reflected in the
security of the bearing capacity of structural components and the
overall firmness of the structure.
Structural serviceability means that a structure should maintain a good
working performance during normal use. For example, excessive
deformation (deflection, lateral displacement) or vibration (frequency,
amplitude) should not occur during normal use, or an excessively wide
crack that may make the user feel uneasy should not appear. The Code
for Design of Concrete Structures (GB50010-2010) requires that
serviceability should primarily be achieved by controlling deformation
and crack width. For the limit of deformation and crack width, the
structure must be subjectively acceptable to the user except where the
structural functional requirements need to be met to protect the
structural components and non-structural components against all
adverse actions.
Structural durability refers to the ability of a structure to permanently
resist performance degradation under proper maintenance conditions
within its pre-defined service life when it is used under the influence of
various actions that may cause a change in its performance (loads,
environment, internal factors of materials, etc.).
As can be seen from the concepts of structural safety, serviceability and
durability, they all have specific connotations. Structural safety is the
ability of a structure to resist various loads; structural serviceability
means a good and suitable working performance, with both primarily
representing structural functional issues. Structural durability refers to
the ability of a structure to resist performance deterioration caused by
long-term loads (environmental, cyclic loads, etc.). Durability problems
exist throughout the whole service life of a structure and have an
impact on structural safety and serviceability. They are the most
fundamental cause for the degradation of structural performance.
Figure 8.1 Limit state of structural design.
Therefore, the durability limit state has been newly added to the Unified
Standard for Reliability Design of Building Structures (GB50068) [8-16],
where it is listed as an essential requirement for structural design,
along with the bearing capacity limit state and the serviceability limit
state. Actually, the durability limit state (i.e., the condition limit state
described in ISO 2394, and the initiation limit state described in ISO
13823), serviceability limit state and bearing capacity limit state are
also equally specified in the international standards General Principles
of Structural Reliability [8-21] (ISO 2394-2015) and General Principles of
Structural Durability [8-22] (ISO 13823-2008). Obviously, the durability
limit state occurs before the serviceability limit state, and is one of the
control conditions for structural design. Figure 8.1 provides a
schematic diagram of the three limit states related to structural design
at different stages.
8.2 Expression of Structural Reliability in
Design Specifications
8.2.1 Design Expression of Partial Coefficients
According to the probabilistic limit state design theory, there exist
partial coefficients for loads (including dead load and variable load), as
well as resistance partial coefficients in the engineering limit state
design expression [8-17]. The material properties, coefficients of
variation of geometric parameters and the uncertainty of the resistance
calculation model in the design expression are all included in the
resistance partial coefficients. The uncertainty of the load calculation
model is also included within the load partial coefficients.
The purpose of determining design partial coefficients is to implement
the target reliability index of the engineering structure to each partial
coefficient in the limit state design expression, so that complex
numerical operations of probability limit states can be converted into a
simple algebraic operation.
From experience, for the same structural component, when the loadeffect ratio, i.e., the ratio between variable load effect and dead load
effect, changes, the reliability index changes drastically, indicating poor
consistency of the reliability. The reason is that the difference of
variable load is greater than that of the dead load. So, when variable
load occupies the dominant position, the reliability of structures
designed by the same design expression decreases. If a multi-coefficient
expression is adopted, the structural reliability will have good
consistency. Therefore, the partial coefficient design method is
proposed in order to overcome the shortcomings of the single
coefficient design method. By this method, we can decompose the
safety factor in the single coefficient design expression into load partial
coefficients and resistance partial coefficients, so that each load adopts
its own partial coefficient when the load effect is caused by multiple
loads [8-23].
The general form of the partial coefficient design expression is as
follows
(8.3)
where
γ0 —Factor for importance of structure;
γG —Partial coefficient for permanent load;
γQ1, γQi —Partial coefficient for variable load;
ψci —Combination value coefficient for variable load;
SGk —Standard value effect of permanent load;
SQ1k —Standard value effect of maximum variable load;
SQik —Standard value effect of other variable loads;
ψci —Combination value coefficient for the ith variable load;
R(·)—Resistance function for structural components;
γR —Resistance partial coefficient for structural components;
fk —Standard value of material properties;
ak —Standard value of geometric dimensions.
The design expression in the form of Equation (8.3) has excellent
applicability, and can be used to study various factors affecting
structural reliability separately. Different load partial coefficients can
be adopted according to the variability of the loads. However, different
numeric values can be assigned to resistance partial coefficients
according to the working performance of the structural materials.
Considering that different countries adopt different methods to
determine the standard value of load and resistance, there is also a
difference in the level of target reliability. Therefore, different values
are assigned to partial coefficients in the structural design expression
in different countries. For each country, the load partial coefficients and
resistance partial coefficients are used together with the standard value
of load and the standard value of resistance. As a whole, in the design
expression, they have precise probabilistic reliability significance. It is
not allowed to use a certain country’s design expression for structural
design with another country’s standard value of load or standard value
of resistance.
8.2.2 Design Expression of Ultimate Limit State
If any of the following events occurs to a structure or a structural
component, it shall be confirmed that the structure or structural
component has exceeded its bearing capacity limit state: (1) the
structural component or connection is damaged because it has
exceeded its material strength, or it cannot continue to bear loads due
to excessive deformation; (2) the whole structure or part of it is out of
balance as a rigid body; (3) the structure is transformed into a
maneuvering system; (4) the structure or structural component is
unstable; (5) the structure gradually collapses due to local damage; (6)
the bearing capacity of the foundation is weakened; and (7) the
structure or structural component suffers from fatigue failure.
For the bearing capacity limit state, structural design is carried out
according to the following design expression:
(8.4)
where γ0 is the factor for importance of the structure and is shown in
Table 8.2; S is the designed value of the load effect combination; R is the
designed value of the resistance of structural components and can be
determined according to the code for the design of the relevant
structure.
The general expression for the designed value S of load effect
combination is as follows
(8.5)
Table 8.2 Factor for importance of structure γ0.
Safety class
1
2
3
Designed Service Life 100 50 5
Factor for Importance 1.1 1.0 0.9
When the permanent load effect is bad for the structure, its partial
coefficient is set to 1.3; in general, the partial coefficient for variable
load is set to 1.5. when the permanent load effect is good for the
structure, its partial coefficient is set to 1.0, but to 0.9 for checking for
overturning, slippage or floatation of the structure; the partial
coefficient for variable load is generally set to 0.0. Load adjustment
coefficient of service life for structural design γL is shown in Table 8.3.
The combination value coefficient ψci should be determined for
variable loads according to the following principle: Given partial
coefficients γG and γQ for variable load and γR for resistance, then for a
combination of two or more variable loads, the determined
combination value coefficient should ensure optimal consistency
between the reliability index β of the structure or structural component
designed according to the partial coefficient expression and the target
reliability index.
The combination value coefficient of variable load can be determined
by the following steps:
1. On the basis of safety-class-II structures or structural components,
select the representative structures, structural components or
failure modes, a combination of a permanent load or two or more
variable loads, a ratio between the standard value effect of the
common dominant variable load and the standard value effect of
the permanent load, and a ratio between the standard value effect
of the) accompanying variable load and the standard value effect of
the dominant variable load;
2. Calculate the designed value of resistance given different
structures or structural components, different load combinations
and common load-effect ratios according to the determined partial
coefficients γG and γQ;
3. Calculate the standard value of different given structures or
structural components, different load combinations and common
load-effect ratios according to the determined resistance partial
coefficient γR;
4. Calculate the reliability index given different structures or
structural components, different load combinations and common
load-effect ratios;
5. For all selected representative structures or structural
components, action combinations and common load-effect ratios,
optimize and determine the combination value coefficient ψc so
that the reliability index β of the structures or structural
components designed according to the partial coefficient
expression has optimal consistency with the target reliability index
βt;
6. Judge the optimized combined value coefficient ψc in accordance
with past engineering experience, and adjust it as necessary.
Table 8.3 Load adjustment coefficient of service life for structural
design γL.
Designed service life (year) of the structure γL
5
0.9
50
1.0
100
1.1
Note: For a structural component with a designed service life of 25 years, γL should be set
according to the codes for the design of various structures.
8.2.3 Design Expression of Serviceability Limit State
If any of the following events occurs to a structure or structural
component, it shall be confirmed that the structure or structural
component has exceeded its serviceability limit state: (1) deformation
that affects serviceability or appearance; (2) local damage that affects
serviceability; (3) vibration that affects serviceability; and (4) other
specific states that affect serviceability.
For the serviceability limit state, it should be designed by the following
formula using the standard combination, frequency combination or
quasi-permanent combination of loads depending on different design
requirements:
(8.6)
where C is the specified serviceability limit for structures or structural
components, such as the limits of deformation, cracks, amplitude,
acceleration, stress, etc.
When there is a linear relationship between load and load effect, the
effect design value of the standard combination can be calculated as
follows:
(8.7)
The notation in Equation (8.1) is the same as the bearing capacity limit
state design expression. However, for the serviceability limit state [8-24],
the partial coefficient γM of material properties should be set to 1.0
unless otherwise specified in the design codes.
The reliability index of serviceability limit state design for structural
components should be set to 0∼1.5 according to the degree of
reversibility.
8.2.4 Design Expression of Durability Limit State
Environmental impact should be assessed during the design of building
structures. When the environment has a significant impact on
structural durability, the proper structural materials, design structures,
protective measures and construction quality requirements should be
adopted, depending on different environmental factors. Also, a regular
structural overhaul and maintenance system should be established so
that structural safety and serviceability is not affected by material
degradation within the designed service life. Environmental impact on
structural durability can be evaluated by engineering experience,
experimental investigation, calculation, inspection or comprehensive
analysis.
If any of the following events occurs to a structure or structural
component, it shall be confirmed that the structure or structural
component has exceeded its durability limit state: (1) the material
properties have deteriorated, affecting structural bearing capacity and
serviceability; (2) the material is cracked, deformed, chipped or
weakened, affecting structural durability; and (3) other specific states
occur, affecting structural durability.
In terms of the durability limit state, it should be designed by the
following equation depending on specific design requirements and
environmental conditions [8-25]
(8.8)
where C represents the limits of a structure or its components and their
connection, determined based on environmental erosion and other
material characteristics. This may include the corrosion of steel
structures, corrosion cracks on the surface of concrete components or
the decay of wooden structures caused by mold. The effect of durability
is different from that of the bearing capacity of components. This is a
combination of environmental impact strength over a certain time span
and the ability of components to resist environmental impact.
The signs or limits of the durability limit state of structural
components, as well as the damage mechanism, should be taken as the
basis for adopting various durability measures. The structural
durability limit state design should ensure that the signs or limits of the
durability of structural components are not less than the designed
service life. During the durability limit state design shown in Table 8.4,
the reliability index of structural components should be set to 1.0∼2.0
according to the degree of reversibility.
Table 8.4 Signs of durability limit state of various structures [8-16].
Structure type
Wooden Structures
Signs of durability limit state
1. Decay caused by mold;
2. Damage by worms;
3. Damage by termites;
4. Damp-proof course in
laminated structures fails or
glue fails;
5. Metal connectors on wood
structures are corroded;
6. Components suffer warpage,
deformation and shrinkage
cracking in the nodal region.
Steel tubes wrapped on steel
structures and concrete-filled
steel tube structures, and section
steel components in combined
steel structures
1. Components are corroded;
2. Anticorrosion coating fails;
3. Components suffer stress
corrosion cracking;
4. Special anticorrosion
protection measures fail.
Aluminum alloy, copper and
copper alloy components and
connections
1. The surface of components is
damaged;
2. Stress corrosion cracking
occurs;
3. Special protection measures
fail.
Structure type
Concrete reinforcement bars and
metal connectors
Signs of durability limit state
1. Prestressed reinforcement and
main reinforcement are
corroded;
2. Metal connectors are
corroded;
3. Corrosion cracks occur on the
surface of concrete
components;
4. Cathodic or anodic protection
measures fail.
Structural components of
inorganic nonmetal materials
such as masonry and concrete
1. Freezing-thawing damage
occurs on the surface of
components;
2. Medium erosion causes
damage to the surface of
components;
3. Sandy wind or human action
causes wear on the surface of
components;
4. There is cavitation erosion on
the surface caused by highspeed airflow;
5. Surface damage caused by
impact;
6. Biological damage.
Polymer materials and structural
components
1. Significant change in color,
cracking or obvious
Structure type
Signs of durability limit state
performance deterioration
caused by photoaging;
2. Significant change in color,
cracking or obvious
performance deterioration
caused by high temperature,
high humidity, etc.;
3. Significant change in color,
cracking or obvious
performance deterioration
caused by medium erosion.
Light-transparent glass
components
1. Cracks occur in structural
components;
2. Transparency is affected by
fretting corrosion;
3. Transparency is affected by
bird droppings.
8.3 Example: Target Reliability and
Calibration of Bridges
8.3.1 Basic Issues
There are several corresponding regulations in the unified reliability
standards and load codes for highway engineering, building structures
and port engineering. These involve value methods for various partial
coefficients and combined coefficients in structural design formulas.
However, it is not difficult to calculate these by comparing the various
regulations with each other:
1. The codes all stipulate that the frequency and quasi-permanent
values of variable loads are determined by the cross-threshold
rate, but different codes have selected different values for this
cross-threshold rate. In terms of the value itself, the building
structure code uses 0.1, while the port and highway codes use
0.05; for the determination of the quasi-permanent value of the
cross-threshold rate, neither code provides a clear standard, only
requiring that the cross-threshold rate should not exceed 0.5. As a
result, the values for variable load frequency coefficient and quasipermanent value coefficient using different code designs are quite
different. For example, for wind loads, the coefficients are taken as
0.4 and 0.0 in the building structure load code and as 0.75 in the
general code for highway bridge and culvert design. However, the
values are 0.8 and 0.6, respectively, in the load code for port
engineering. The difference between these values leads to
significant discrepancies in practice in terms of the cross-section
size or steel bar dosage of the same component, depending on the
design code that was used, even for the same load conditions.
2. The partial load coefficients and combined coefficients in the
design formula given in the code are obtained by means of
optimization and fitting. This is carried out by considering the
reliability calculation of the failure states, which includes tension,
compression, bending, shear and torsion, under the condition of a
common load-effect ratio. When full stress design is performed
using these coefficients, the difference between the nominal
reliability index and the design target reliability index within the
range of the common load-effect ratio and various failure modes is
generally minimal. However, for a specific component, it is
impossible to have multiple stress states (such as tension,
compression, bending, shear and torsion) at the same time, and
the load-effect ratio range will not necessarily be considered as
part of the code calibration. The applicability of existing codes to
special components and special structures is therefore in need of
further study.
3. For certain variable loads, all codes attach great importance to the
measured field data. If the measured field data are complete, all
codes suggest that variable load values should be determined
based on measured field data. For projects with limited observed
load field data, how related loads can be determined based on
statistical data also needs to be studied further.
8.3.2 Parameter Analysis
According to the Unified Standard for Reliability of Engineering
Structures, GB 50153-92 [8-26], the frequency value of the variable load
is taken as the load value of a 5% span-time rate, while the quasipermanent value of the variable load is taken as the load value for not
less than 50% span-time rate; in practice, this is often taken as the load
value of 50% span-time. The Feasibility Study for the Zhoushan
Continental Island Connection Project-Meteorological Observation and
Wind Parameter Research Report [8-27] gives the monthly extreme wind
speed data for Zhoushan, Shengsi and Beilun, and further provides
wind speed design parameters for Jintang Bridge. Based on this data,
this project provides an analysis of the wind speed cross-time rate
distribution curve combined with a Monte-Carlo simulation, as shown
in Figure 8.2.
In Figure 8.2, the monthly extreme wind speeds in Zhoushan, Shengsi
and Beilun are taken as the measured values. Based on the
Meteorological Observation and Wind Parameter Research Report, the
data for the Jintang Bridge is taken as the estimated value based on the
above measured values. It is suggested in the report that the measured
wind speed at Beilun should be used for estimation. In this study, the
monthly extreme statistical distribution is obtained based on the
estimated wind speed of the bridge position. On this basis, a MonteCarlo simulation for 100 years is carried out to obtain the monthly
extreme wind speed value of the bridge position for 100 years, after
which the cross-time rate distribution data can be calculated. The
statistical results are shown in Figure 8.2. In each of these cross-time
rate statistical distribution maps, the abscissa is the wind speed ratio,
that is, the ratio of the set wind speed to the maximum wind speed
across the total time, and the ordinate is the cross-time rate, that is, the
sum of the time exceeding the set wind speed and the total time. Based
on the results given in Figure 8.2, within 100 years, the wind speed of a
5% span time rate is 32.597 m/s, the wind speed of a 50% span time
rate is 26.223 m/s, and the design wind speed of a 100-year return
period recommended in the Meteorological Observation and Wind
Parameter Research Report is 40.16 m/s. This means that the ratio of
wind pressure value V0.05 of a 5% span time rate and V0.5 of a 50% span
time rate to the standard wind pressure value VK should be as follows:
(8.5)
(8.6)
Figure 8.2 Wind speed span-time rate distribution curve of a bridge.
8.3.3 Calibration Target Reliability
Target reliability refers to the reliability of a structure designed using
the design checking calculation formula. Since the structural reliability
standards of various countries are based on JCSS reliability model
specifications, the target reliability of this calibration is also
determined using these standards [8-28], as shown in Table 8.5 and
Table 8.6.
Because the sea-crossing bridge is an important project, the reliability
index of crack resistance for the bridge components can be selected
accordingly. In this calibration, three target reliability indexes are
selected as the calibration points: βT = 1.5, βT = 2.0 and βT = 2.3.
Table 8.5 Annual target reliability and failure probability of bearing
capacity limit state.
1
2
3
4
Relative
failure
loss
Moderate failure
with minor
consequences
Moderate failure Significant
consequences
consequences of
failure
high
β = 3.1, Pf ≈ 10−3
β = 3.3, Pf ≈
5×10−4
β = 3.7, Pf ≈ 10−4
medium
β = 3.7, Pf ≈ 10−4
β = 4.2, Pf ≈ 10−5
β = 4.4, Pf ≈
5×10−6
low
β = 4.2, Pf ≈ 10−5
β = 4.4, Pf ≈
5×10−6
β = 4.7, Pf ≈ 10−6
Table 8.6 Annual target reliability and failure probability of the
serviceability limit state.
Relative failure
loss
Target reliability index for unrecoverable limit
states
high
β = 1.3, Pf ≈ 10−1
medium
β = 1.5, Pf ≈ 5×10−2
low
β = 2.3, Pf ≈ 10−3
Table 8.7 Calibration operating condition.
Direction of Number
action
of load
case
Content of load case
Participation
Longitudinal 1
Permanent action +
maximum wind load
Dead load, wind
Horizontal *
2
Permanent action +
Dead load,
automobile + automobile temperature,
braking force +
vehicle
temperature force +
wind load on vehicle
3
Permanent action +
maximum wind load +
ultimate wave and
current force
Dead load, wind,
wave current
4
Permanent action +
maximum wind load +
maximum wave and
current force +
temperature force
Dead load, wind,
wave current,
temperature
5
Permanent action +
maximum wind load
Dead load, wind
8.3.4 Operating Conditions and Parameters
See Table 8.7 for the operating conditions to be calibrated, and Table
8.8 for the statistical characteristics of parameters under each
operating condition.
8.3.5 Load Effect Ratio
All load effects in this calibration are considered as tensile stress of
steel bars, and the four groups of load-effect ratios are determined as
follows:
In the formula, σGK is the standard value of steel bar stress caused by a
dead load, σWK is the standard value of steel bar stress caused by wind
(or wind + wave current), σWL is the long-term effect value of steel bar
stress caused by wind (or wind + wave current), σVGK is the standard
value of steel bar stress caused by vehicle, σVSK is the standard value of
steel bar stress caused by the automobile braking force and σTK is the
standard value of steel bar stress caused by temperature. Based on the
reinforcement stress calculations for pier and pile foundation from the
Zhoushan Island Connection Project, the calculation of various loadeffect ratios is shown in Table 8.9 below.
Table 8.8 Function distribution and parameters *.
Action
Coefficient of
variation*
Deviation
coefficient *
Dead load
0.05
0.924
Wind load
Annual extreme value 0.412
distribution: F (x) = exp
{− exp [−α (x − μ)]}
0.3266
Temperature
Annual extreme value 0.03
0.9614
distribution: F (x) = exp (maximum)
{−exp [−α (x − μ)]}
0.474 (lowest)
0.276 (tempera
ture difference)
Vehicle load
Distribution type
See specification GB/T502831999
Vehicle
Annual extreme value See specification GB/T50283braking force distribution: F (x) = exp 1999
{−exp [−α (x − μ)]}
Remarks: The form of distribution and the coefficient of variation for dead load, temperature,
vehicle and vehicle braking force are taken from the load specifications, while the coefficient of
variation for wind load is estimated based on the wind speed distribution characteristics given
in the Meteorological Observation and Wind Parameter Research Report. See the subsequent
explanation for the actual estimation process. The deviation coefficient is determined using the
ratio of the average action value for yearly distribution to the standard action value in the
design reference period.
Table 8.9 Load ratio.
Pier
Dead load G
number
Nz
Hx M y
Wind load W ρ1 = σWK / σGK
Hx
My
Compressive
stress ratio
Tensile
stress
ratio
Load (action) effect of pier shaft and pier bottom along bridge
direction
C24
20785 110 1379 306 1829
0.073
-8.22
×10-02
C25
52784 1443 12530 202 786
0.011
C26
54011 1463 14843 244 1130
0.015
-1.66
×10-02
-2.43
×10-02
C27
54911 1467 17801 303 1700
0.021
-3.81
×10-02
C28
55601 1458 21368 377 2593
0.031
C32
60589 1458 40258 786 10498 0.096
-6.20
×10-02
-3.67
×10-01
C33
62139 1467 45335 899 13489 0.116
-5.34
×10-01
C34
63277 1463 50044 1015 16913 0.139
C35
63882 1443 54112 1135 20772 0.164
-7.74
×10-01
-1.12
Load (action) effect of pier shaft and pier bottom in the transverse
bridge direction
C24
20785 110 1379 2186 30670 5.51 ×10-01
-5.80
×10-01
C25
52784 1443 12530 4973 67817 4.51 ×10-01
-5.42
×10-01
Pier
Dead load G
number
Nz
Hx M y
Wind load W ρ1 = σWK / σGK
Hx
My
Compressive
stress ratio
Tensile
stress
ratio
C26
54011 1463 14843 5000 75125 4.82 ×10-01
-5.96
×10-01
C27
54911 1467 17801 5036 85124 5.28 ×10-01
-6.78
×10-01
C28
55601 1458 21368 5083 97882 5.88 ×10-01
-7.91
×10-01
C29
54905 538 21449 5162 118148 7.17 ×10-01
-9.70
×10-01
C30
57259 185 1257 5232 135418 8.99 ×10-01
-9.14
×10-01
C31
57815 354 25314 5304 152948 8.68 ×10-01
-1.22
C32
60589 1458 40258 5358 165880 8.36 ×10-01
-1.41
C33
62139 1467 45335 5435 183901 8.86 ×10-01
-1.57
C34
63277 1463 50044 5510 202134 9.39 ×10-01
-1.76
C35
63882 1443 54112 5586 220579 9.99 ×10-01
-1.96
8.3.6 Reliability Calibration Process
Reliability calibration is carried out as shown in Figure 8.3.
The analysis and calculation for load-effect ratio shows that the loadeffect ratio features the following characteristics: (1) the load-effect
ratio of (wind or wind + wave current) and dead load varies
significantly (0.07 ∼ 30), and in a small range of load-effect ratio, the
steel bar is actually in a compressed state, making it unnecessary to
check its crack resistance; (2) dead load in steel bar produces tensile
stress in some cases and compressive stress in others, leading to
different symbols of dead load action in the limit state equation.
Therefore, the calibration method of hierarchy and situation is used,
which involves the following states:
1. In the initial instance, several component coefficients are set, with
the corresponding variation range of reliability index calculated
under different load-effect ratios. The range of load-effect ratio
with a reliability index greater than 5 is not regarded as part of the
key range of calibration (in these instances, the tensile stress of
steel bars is very small, or they tend to be in compression state).
2. After eliminating the non-critical range, the key range is analyzed
in detail. That is, several groups of different partial coefficients are
selected for numerical calculation of the reliability index of the
critical load-effect ratio range. These partial coefficients are
adjusted so that the calculated reliability index approaches the
target reliability index, thus determining the change range of the
partial coefficient.
3. Detailed optimization analysis is carried out within the variation
range of the partial coefficients, and finally, a group of partial
coefficients which deviate from the target reliability index can be
determined.
Figure 8.3 Calibration process.
8.3.7 Results of Reliability Calibration Calculation
For the combinations of “dead load + limit wind load” and “dead load +
limit wind load + limit wave current”, the reliability calculation can be
carried out using the following formula:
(8.7)
(8.8)
Among these σ WK is the standard value of wind load effect; this is the
standard value of wave and current load effect σ FK. Sum are the partial
coefficients which are initially determined and can be considered as the
approximate values of load frequency and quasi-permanent values r1,
r2, respectively.
Reliability calibration is performed by selecting a target reliability of
1.5, 2.0 and 2.3, respectively. For the combinations of “dead load +
maximum wind load” and “dead load + maximum wind load + ultimate
wave current”, the full stress expressions are still given based on
Equations (8.7) and (8.8) during optimization analysis. The main
calibration results provide the law of reliability index changing with
load-effect ratio and the corresponding checking points. It can be found
that: (1) when r1 = 0.54 ∼ 0.56 and r2 = 0.31 ∼ 0.4, the minimum
reliability index is around 1.5; (2) when r1 = 0.65 and r2 = 0.35 ∼ 0.4,
the minimum reliability index is near 2.0; (3) when r1 = 0.7∼0.72 and r2
= 0.38 ∼ 0.4, the minimum reliability index is around 2.3. Because of
these factors r1 and r2, the range of values is very small, so we can
basically think of them as values ψS and ψL.
On the basis of the optimization analysis described in Chapter 8, a
group of coefficients with the smallest deviation from the target
reliability and the smallest deviation from the partial coefficients
calculated according to the checking points are selected as the final
calibration results. For the two operating conditions of “dead load +
limit wind load” and “dead load + limit wind load + limit wave current”,
the recommended values for reliability calibration are shown in Table
8.10.
Short-term effect combination and long-term effect of internal force
combination are calculated according to the following formula:
(8.9)
(8.10)
Table 8.10 Recommended cost for reliability calibration.
βT P f
γ G ψS
ψL
1.5 7 × 10-2 1.0 0.55 0.35
2.0 3 × 10-2 1.0 0.65 0.35
2.3 1 × 10-3 1.0 0.70 0.40
Where GK is the standard value of the internal force (or steel bar stress)
under a dead load. This WK is the standard value of internal force (or
steel bar stress) under a wind load for the 100-year design reference
period. FK is the standard value of internal force (or reinforcement
stress) under wave and current load during the 100-year design
reference period.
8.4 Reliability Analysis of Human Influence
8.4.1 Parameters of Human Influence
Since the construction of buildings entails the interaction of humans
and machines, human factors should be considered an integral part of
the entire system. The working environment (size of structure, lighting,
ventilation, temperature, humidity and noise of working area, tools
provided, management level, etc.), human behavioral characteristics
(professional knowledge, working skills and experience, personal
conditions, working attitude and motivation, emotional changes, etc.)
and task complexity all exert a direct influence on human reliability. As
indicated by the investigation results into a number of different
construction accidents [8-29], these accidents are in fact mostly caused
by human error. Therefore, the reliability of reinforced concrete
structures taking human influence during construction into account
can be analyzed by establishing simulation models for the occurrence
of human error and its influence during the construction of concrete
structures and formwork support systems [8-30].
The influence of human error in the construction process refers to all
behaviors and results that are not in line with the requirements of
relevant codes, laws, and regulations. If there was no influence of
human error, the main reasons for deviation Ωa between actual size
and design requirements of structural components would lie in
measurement, machining error and formwork deformation after
concrete pouring. Component size discounting human obeys a normal
distribution, and its deviation falls within the range required by the
relevant construction codes. In this paper, statistical parameters from
relevant domestic surveys were taken as the distribution parameters of
component size without the influence of human error (see Table 8.11 [831]
, Table 8.12), while a truncation was made corresponding to
allowable deviation from relevant codes. See Figure 8.4 [8-32] for details
of the construction deviations from the stipulated parameters.
Table 8.11 Statistical data for geometric parameter uncertainty KA.
Structural type
component
Item
Reinforced concrete
components
Height and width of section
1.00 0.02
Effective height of section
1.00 0.03
Cross-sectional area of
longitudinal reinforcement
1.00 0.03
Concrete cover thickness
0.85 0.30
Average stirrup spacing
0.99 0.07
Anchorage length of longitudinal
reinforcement
1.02 0.09
Formwork support steel Outer diameter
pipe
Wall thickness
1.00 0.01
Formwork support
erection
Step distance
0.94 0.05
Spacing of vertical rod
1.05 0.21
0.94 0.09
Note: The statistical data for the KA of reinforced concrete components in this table were
sourced from the literature [8-31].
Figure 8.4 Human error event tree.
Table 8.12 Geometric size distribution of components without the
influence of human factors.
Item
Max. allowable deviation
Height and width of section 1.0 0.02 -5mm, +8mm
Concrete cover thickness
0.85 0.3 -5mm, +8mm
Table 8.13 Standard deviation of concrete strength.
Concrete strength grade C10∼C20 C25∼C40 C45∼60
σ (N/mm3)
4.0
5.0
6.0
The strength of concrete, when discounting any influence of human
error, obeys a normal distribution. By taking the construction
preparation strength of concrete as the average strength, while
neglecting human error, the concrete preparation strength fcu,0 can be
expressed as [8-33]
(8.11)
Where, fcu, k is the standard value of the designed concrete strength, and
σ is its standard deviation. The standard deviation shall be based on
recent data taken from the construction enterprise. See Table 8.13 [8-34]
for the standard deviation of an actual average level of Chinese
construction organizations as reflected by relevant statistical data.
For the erection of a formwork support system without the influence of
human error, cross bridging and sweeping rods shall be set according to
relevant building codes. The fastening torque of all parts shall be 40
N•m.
8.4.2 Influence of Human Error on Construction
Construction and operation complexity leads to a large number of
human errors during the construction process. The following common
human errors are discussed in this chapter:
1. Insufficient concrete strength;
2. Missed steel bars or fewer bars than required installed;
3. More steel bars than required installed;
4. Premature formwork removal;
5. Failure to set sweeping rod support;
6. Failure to set cross bridging support;
7. Insufficient fastening torque on horizontal and vertical rods at the
bottom of the formwork;
8. Insufficient fastening torque on vertical and transverse horizontal
rods and vertical support rods.
The errors above are caused by such factors as calculation errors by
technicians, issues of information exchange between technicians and
construction operators, and construction worker carelessness.
8.4.3 Human Error Rate, and Degree and Distribution of
Human Error Influence
(1) Human error rate
The likelihood of human error on a specific task can be expressed by
the human error rate (PE):
(8.12)
Where, NE is the number of operational errors in this task, while N is
the total number of times the task is performed. In consideration of
different error rates among different operators, even when performing
the same task, the error rate is not a constant even when the same
operator performs a specific task. Therefore, the human error rate
should be regarded as a random variable subject to a certain
probability distribution.
In light of the application of lognormal distributions to the description
of human error rate in human reliability analysis [8-35], a lognormal
distribution was also used in this paper to describe the distribution
model of human error. The distribution parameters of this error rate
reflect the randomness of an error occurring due to factors affecting
task completion, including different abilities, personalities, working
environments, and similar. The average human error rate
can be
obtained from survey data or the relevant literature, and this estimate
is also used as the median of the logarithmic human error rate
distribution model. Since it is difficult to calculate the parameter σ
(which represents the dispersion of human error rate distribution)
through investigation, the error coefficient EF is defined to determine
the parameter σ. EF can be expressed as:
(8.13)
Where, Pr(F90) and Pr(F10) correspond to distributions of 90% and
10% error rates respectively [8-36]. The standard deviation of the
logarithmic distribution of human error rate σ can be expressed as:
(8.14)
In the absence of other criteria, the error coefficient EF is estimated by
referring to the criteria for estimating the error coefficient of nuclear
power plant operation under normal conditions, as given by Stewart et
al. [8-37] (as shown in Table 8.14).
(2) Degree of influence of human error
Once an error occurs, the relative value of parameter deviation from the
design requirements caused by human error is expressed as the degree
of influence of human error mE. mE can be expressed as
(8.15)
Where, xE and xm are the parameter values neglecting and including
human error, respectively. Due to the complexity and uncertainty of the
human error mechanism, the degree of human error is also a random
variable subject to a certain probability distribution. No typical model
can be found in the descriptions of random variables describing human
error. In this chapter, a lognormal distribution was therefore chosen to
describe the distribution model of the degree of influence of human
error. Its mean estimate λBE and max. estimate λUB are the median of
the lognormal distribution and the distribution value corresponding to
90%, respectively [8-37]. Both λBE and λUB were obtained from survey
data. In particular, λBE is the statistical average of the average degree of
human error experienced by all respondents, while λUB is the statistical
average value of the maximum degree of human error experienced by
all respondents. The standard deviation of the lognormal distribution
can be expressed as:
(8.16)
Table 8.14 Estimation criterion for error coefficient EF.
Mean error rate estimate
Error coefficient
<0.001
10
0.001-0.01
3
>0.01
5
Table 8.15 Human error rate and distribution parameters for degree of
influence.
Error Error type
code
Average Error
error rate coefficient
EF
Average
error
degree
λBE (%)
Max.
error
degree
λUB (%)
E1(a) Reduction of
0.0077
reinforcement
area due to
incorrect
direction of steel
bar
3
—
—
E1(b) Reduction of
0.0154
reinforcement
area due to other
factors
5
-14.6
-52.9
E2
Increase of
reinforcement
area
0.012
5
11.3
39.8
E3
Reduction of
0.22
concrete strength
5
-12.3
-43.5
E4
Thickness error
of concrete slab
Normal distribution:
Average = standard,
coefficient of variation
=0.02
—
—
E5
Column size
error
Normal distribution:
Average = standard,
coefficient of variation
=0.02
—
—
E6
Thickness error Normal distribution:
—
in concrete cover Average =0.85 times
standard and coefficient
of variation =0.3
—
Error Error type
code
Average Error
error rate coefficient
EF
Average
error
degree
λBE (%)
Max.
error
degree
λUB (%)
E7(a) Premature
formwork
removal (5-day
construction
cycle)
0.0175
5
-20.0
-44.0
E7(b) Premature
formwork
removal (7-day
construction
cycle)
0.0374
5
-21.4
-40.0
E7(c) Premature
0.0396
formwork
removal (10-day
construction
cycle)
5
-22.0
-43.3
(3) Human error rate in the construction of buildings and the
distribution of the degree of influence of human error
Errors in the construction process of buildings include errors occurring
in the construction of concrete structural components and to the
erection of formwork support systems. Suppose the statistical
distributions of section height, thickness and effective height of
concrete components in the current concrete structure code include the
influence of human errors, then they can be applied to the analysis of
human reliability without truncation [8-38]. The rate of occurrence and
degree of influence of human error in steel bar placement, concrete
pouring strength and formwork removal are all taken from the
literature [8-33][8-37] [8-39] as shown in Table 8.15. Through field
investigation of human errors occurring during the erection of
formwork support systems, it was found that errors often include
failure to set cross bridging and sweeping rods, as well as insufficient
tightening torque on fasteners.
8.4.4 Simulation of Human Error in Construction
In light of the difficulties in obtaining data for human error on
construction sites, quantitative data for the occurrence of human error
and its influence on the main parameters of a structure can be obtained
by simulating human error in construction. This can provide a basis for
an analysis of structural reliability that considers human error during
the construction process. The Human Reliability Analysis (HRA)
method [8-35] was used to simulate human error and to establish
models for its occurrence and influence. The event tree method was
used within the HRA method to describe logic, while Monte-Carlo
numerical simulation technology was used to analyze the event tree.
This is a method that is commonly used for analyzing the influence of
human error on many complex systems, such as in the system
reliability analysis of nuclear power plants and electronic systems. It
has the advantage of being able to break down a complex system into a
series of simple tasks. Stewart [8-37] [8-38] [8-40] and Xu Maobo [8-37]
repeatedly applied the HRA method to analyze the effect of human
error on the reliability of concrete structural components.
An event tree was used to describe the whole construction process.
First, the construction process was broken down into several tasks
which occur in turn, after which each task and its results were
represented as nodes and branches of the event tree. There are two
possible outcomes for each task: Success (error) and failure (error
occurs). The task in Figure 8.4 consists of two operations, 1 and 2. In
this figure, Si(i=1,2) indicates the success of operation i, while Fi(i=1,2)
indicates the failure of operation i. A single error may trigger multiple
errors (i.e., the branch of the event tree). Therefore, the event tree has
multiple branches, which means that there will most likely be several
different paths and several different results. Finally, the event tree was
analyzed using Monte-Carlo numerical simulation technology. In other
words, if the test starts from the beginning and runs to the end of the
event tree, then we go through all possible paths on the event tree after
multiple tests to consider all combinations of human error, thus
ultimately calculating the structural reliability during construction.
Human errors in construction were simulated separately, and divided
into human errors in the construction of reinforced concrete structures,
and human errors in formwork support system construction.
(1) Simulation of human error in the construction of reinforced
concrete structures
Human error can be simulated based on the human error rate and the
distribution of its impact. See Figure 8.5 for the simulation of error
codes E3 and E7 in Table 8.5. The simulation process for error codes
E1(a), E2(b), and E2 is elaborated in the Literature [8-35] and shown in
Figure 8.6, where x2 is the reinforcement area in the other direction.
Therefore, Table 8.15 does not list the average error degree λBE of the
maximum error degree λUB for error code E1 (a).
xmin and xmax in Figure 8.5 and Figure 8.6 are the minimum and
maximum values of the parameters affected by human error. It can be
easily concluded that the minimum values of reinforcement area,
concrete strength and formwork removal time are 0 after the
occurrence of human error. Since the minimum spacing of the stressed
reinforcement slab is 70 mm and the maximum diameter of the
reinforcement used is 16 mm, then the maximum area of reinforcement
of the concrete slab affected by human error is the area corresponding
to a spacing of 70 mm and a diameter of 16 mm.
Figure 8.5 Block diagram of human error simulation program for E3
and E7.
Figure 8.6 Block diagram of human error simulation programs for
E1(a), E1(b) and E2.
(2) Simulation of human error in the construction of formwork support
systems
Table 8.16 Influence of different human errors on the buckling
strength of formwork support systems.
Design
Tightening torque on fastener bolts (N•m)
20
30
40
50
60
No E8 and E9 occur -0.111
-0.052
0
0.009
0.035
E8
-0.561
-0.488
-0.428
-0.418
-0.384
E9
-0.263
-0.242
-0.231
-0.227
-0.223
E8+E9
-0.693
-0.65
-0.619
-0.612
-0.597
Table 8.17 Occurrence of human error.
Item
Error rate
Setup of sweep bar
0.45
Setup of cross bridging
0.335
Verticality of vertical rod
0.22
Extension mode of vertical rod 0.2
Table 8.18 Distribution of tightening torque on bolts in different parts.
Statistical
parameters
Plate bottom fasteners Vertical cross
fasteners
Mean (N· m)
43.15
18.79
Standard
deviation (N·m)
15.84
16.65
Distribution
Normal distribution N
(43.15,15.842)
Exponential
distribution EXP
(18.79)
Common errors occurring in formwork support system during
investigation and construction include in Table 8.16:
E8: No cross bridging;
E9: No sweeping rod;
E10: The tightening torque on the fastening bolts of the horizontal
and vertical rods at the bottom of formwork does not conform to
the requirements of the code;
E11: The tightening torque on the fastening bolts of the vertical
and transverse horizontal rods and vertical rods of the support
does not conform to the requirements of the code.
See Table 8.17 for the human error rate for E8 and E9. The human error
coefficient is determined according to Table 8.14. See Table 8.18 for the
distribution of fastening bolt tightening torque corresponding to E10
and E11, assuming it obeys a logarithmic normal distribution. The field
measurement statistics show that the maximum value of the bolt
tightening torque is 90 N·m, while the minimum value is 0. Therefore,
the distribution in Table 8.18 has been truncated, and then human
error effect analysis was carried out against a calculated example.
For a formwork support with a step distance of 1.7m and a vertical rod
spacing of 0.75m, the influence of the fastening bolt tightening torque
on horizontal and vertical support rods on the buckling strength of the
formwork support system is calculated under four different conditions
respectively: (1) no cross bridging; (2) no sweeping rod; (3) cross
bridging and sweeping rod; and (4) no cross bridging or sweeping rod.
See Table 8.19 for the results, the values in which show the degree of
influence on buckling strength caused by different human errors.
Table 8.19 Average value of skid resistance for fasteners under
different bolt tightening torques.
Fastening tightening torque/N•m Average m/kN Ratio m/m(40)
20
7.94
0.56
30
11.33
0.80
40
14.13
1.00
50
15.29
1.08
60
16.77
1.19
Note: m(40) is the average value of anti-sliding bearing capacity of rightangle fasteners when
fastening bolt tightening torque is 40 N·m.
For fastening bolt tightening torques other than 20, 30, 40, 50 and 60
N•m, the cubic interpolation method was used, leading to extrapolation
and interpolation as reflected in the values shown in Table 8.19. See
Figure 8.7 for the simulation of the influence of different fastening bolt
tightening torques on buckling strength under the four kinds of
erection.
Figure 8.7 Human error simulation program block diagram for E8 and
E9.
In terms of the influence of human error E11 on the slide resistance
capacity of fasteners for a formwork support system, suppose the
human error E11 exerts an influence on the mean value of slide
resistance capacity for fasteners instead of on its variability. By means
of a fastener anti-sliding test, a probability model for fastener antisliding bearing capacity when fastening bolt tightening torque is
40N•m was studied, and the difference of fastener anti-sliding bearing
capacity at 20, 30, 50, 60N•m and 40N•m was analyzed, as shown in
Table 8.18. The third interpolation method was used to determine the
fastener anti-slide bearing capacity corresponding to other tightening
torques, with the Monte-Carlo numerical method used to simulate the
influence of different bolt tightening torques on the fastener anti-slide
bearing capacity.
8.4.5 Example: Support System for a Ten-Storey
Beamless Floor Structure
The Monte-Carlo method can combine the calculation of system
reliability with the simulation of human error occurrence and its effect
on structural parameters. This enables each calculation cycle to be
regarded as a numerical computer simulation experiment. The method
above was used to calculate structural system reliability under the
influence of human error during the construction period, as shown in
Figure 8.8.
A 10-storey beamless floor structure with a storey height of 3m,
column grid size 6,000mm×6,000mm, column size 550mm×550mm,
and slab thickness of 200 mm; concrete strength of the concrete slab
and column: C30; positive and negative moment reinforcement:
HPB235 steel and HRB335 steel. Area of reinforcement at support:
1,214 mm2/m; area of reinforcement at mid-span: 808 mm2/m.
Fastener-type steel pipe formwork is used for support. Steel pipe φ48
mm×3.5 mm; vertical rod spacing 750 mm; step distance 1,700 mm.
Section stiffness of the bracing system is 6.4×103 kN, determined after
considering the influence of wood keel on bracing stiffness and the
depreciation effect of steel uprights[8-41].
Both the construction period and concrete strength exert a significant
influence on the safety of the concrete slab during construction period,
but very little influence on formwork support system. EI-Shahhat et al.
[8-42]
and Eppaarachchi [8-43] studied their influence on the reliability of
concrete slabs during the construction period in detail. Therefore, the
Monte-Carlo numerical simulation method was used in this chapter to
calculate the reliability of a structural system during construction
under the following conditions: (1) three support models: 3-layer
formwork support (3S), 2-layer formwork support (2S) and 2-layer
formwork support + 1-layer secondary support (2S1R); (2) different
statistical parameters for the live construction load were used to study
the influence of the formwork erection model and construction load on
the reliability of the structural system during construction.
Figure 8.8 Flow chart for structural system reliability calculation in
construction period under the influence of human errors.
Since the research object is a structure with a storey height of only 3m,
human errors such as absence of cross bridging and sweeping rods and
incorrect fastening bolt tightening torque being used, as discussed in
the previous section, were also considered, but errors in extension and
verticality of vertical rods in the construction process of the formwork
support system were not. To study the influence of human error, the
system failure probability of the three support models (3-layer
template support 3S, 2-layer template support 2S, and 2-layer template
support 1-layer secondary support 2S 1R) were calculated, both for the
case where human error was taken into account and for the case where
it was neglected. See Table 8.20 for the results.
It can be seen from Table 8.20 that human errors during construction
exert a significant influence on the reliability of the formwork support
system and concrete structure. See Figure 8.9 for the overall structural
failure probabilities of the three support models with and without
human errors. Nonetheless, due to the inspection and acceptance link
in the actual construction process, the failure probability for the
structure under the influence of human error in Table 8.20 and Figure
8.9 is larger than that of the actual project. After inspection, the human
error deviating from the normal value will be detected and corrected,
and the failure probability of the structure will be greatly reduced.
Table 8.20 Comparison of failure probability.
Item
Pf × 10−2
2S
FE
E
2S1R
3S
FE
FE
E
E
Formwork support 0.66 5.23 0.66 5.16 0.87 5.80
Concrete slab
1.23 3.42 0.69 2.90 0.52 2.53
Overall structure
1.89 8.63 1.35 8.05 1.38 8.31
Note: FE means no human error while E means human error.
Figure 8.9 Influence of human error.
Under the influence of human errors, the formwork support system
rarely becomes unstable or destroyed. Most problems are caused by
anti-sliding failure of fasteners, while the destruction of concrete slabs
is mainly due to bending problems.
Because the bending failure of horizontal rods at the bottom of the
formwork is mainly affected by the distance between the vertical rods,
the human error in the construction process of the formwork support
system considered in this paper does not affect the bending failure
probability of horizontal rods at the bottom of the formwork. The
bending failure probability of horizontal rods can be effectively reduced
by shortening the distance between vertical rods.
Although human errors E8, E9 and E11 exert a significant influence on
the overall stability of the formwork support system, the formwork
support in this example was designed according to the requirements
for anti-sliding bearing capacity of fasteners, while the buckling
strength of the formwork support system is much higher than the
required value under the influence of human errors. Some instability
damage also occurs under the influence of human errors.
Human error E10 exerts a significant influence on the anti-sliding
capacity of right-angle fasteners. Compared with E10 without human
errors, the failure probability of the formwork support system
increases dramatically upon occurrence of human error. The
occurrence of human error in E10 must therefore be strictly controlled
in order to ensure the safety of the formwork support system and the
entire construction.
8.4.6 Discussion
1. Numerical simulation is a typical way of supplementing data when
there is insufficient information available on human error
investigation. Based on field investigations, testing and buckling
strength studies, the occurrence and influence of human error on
concrete structures and formwork support systems during
construction were simulated using the HRA method.
2. By considering the main failure probability of the concrete
structure itself and of the formwork support system, and by
comparing the reliability of the system during the construction
period with and without human error, the results show that human
error exerts a significant influence on the reliability of structures
during construction. Effective inspection measures must be
adopted to control and reduce the incidence of these human
errors.
3. The research on human error in the construction of formwork
support systems shows that, for multi-storey building structures, it
is of great importance to strictly control the tightening torque on
fasteners connecting horizontal and vertical rods at the bottom of
the formwork.
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Index
Accidental action, 357, 358
Applicability, 135, 364, 372
Artificial neural network, 45
Bayes, 37, 323–324, 326
Black system, 40
Borges process, 293, 295
BP neural network, 45, 46
Breitung method, 79
Brittle failure, 360
Calibration method, 359, 379
Central limit theorem, 72
Checking point, 3, 4, 6, 9, 53, 134, 294–295, 309–311, 315, 321, 361,
378, 380
Code calibration, 372
Conditional combination system, 159
Conditional expectation, 6, 7, 122–123, 131, 135, 227, 228
Conditional expectation sampling technique, 123
Confidence coefficient, 142–143
Constraints, 86, 91, 103
Convolution, 211, 286
Correlated sampling method, 128
Correlation effects, 179
Corrosion, 15, 211, 227, 228, 323–327, 329, 331–342, 344–347, 362,
369, 370
Covariance, 81
Crack growth, 272
Crossing analysis method, 298
Crossing rate, 218, 226, 229, 230, 246, 285–290, 292–293, 298–300
Direction cosines, 78, 136
Discretization, 169, 241
Ditlevsen, 7, 10, 147
Dual sampling technique, 122, 134
Durability, 1, 2, 16, 17, 41, 323, 325, 341–342, 355–356, 361, 362–363,
368–370
Durability limit state, 341, 355, 362, 363, 368–370
Dynamic action, 358
Dynamic analysis, 230, 231
Earthquake, 1, 13, 316, 318, 355, 357, 358, 361
Equivalent normal distribution, 295
Event Tree, 152, 382, 388
Failure consequences, 17, 180, 238, 359, 374
Failure diagram, 153
Failure probability, 3–10, 17, 19–20, 36, 41, 44, 47, 68–70, 73, 74, 76,
80–82, 90, 91, 99, 100, 104, 106, 109, 110, 112, 115–116, 118, 121,
130, 135, 143, 144, 145, 148, 154, 156–158, 160–162, 165–168, 174–
177, 179, 181, 196, 202, 203, 211–213, 215–227, 229–263, 265–267,
275–277, 355–356, 359, 374, 395–398
Fatigue life, 235, 236, 238, 240, 241, 244, 245, 246, 248, 249, 250, 257,
260, 261, 263, 265, 266, 267, 275
Fault Tree, 10, 147, 151
Ferry Borges-Castanheta rule, 303
Finite element analysis, 10, 16, 169, 171, 173
First-order second-moment method, 71, 74, 82, 229, 230, 273, 294
Fixed action, 358
Frequency domain, 231, 233, 249, 257
Fuzzy subset, 37, 38
Fuzzy uncertainty, 35, 59, 65
Hasofer combination method, 300–301
Human error, 381–389, 391–398
Human error impact, 389
Human error rate, 384–389, 392
Human factors, 381, 383
Human reliability analysis, 384, 387
Importance sampling method, 125, 132, 136, 138, 143–144, 160, 197,
228, 230
Improved numerical simulation method (IISM), 134
Incompleteness, 35, 36, 40, 41
Incremental loading method, 177
Integration, 3, 5, 126, 161, 216, 230
Inverse transform, 120
Jacket platform, 160, 181, 182, 191, 193, 238, 239, 241, 244, 248
Jacobian, 96
Jaynes maximum entropy principle, 86
JCSS combination rule, 12, 305, 307
Knowledge uncertainty, 35, 186
Lagrangian multiplier, 102
Laplace asymptotic method, 82
Laplace integral, 82, 83, 230
Limit state equation, 3–6, 9, 18, 47, 50, 74, 97, 99, 100, 105, 106, 108,
112, 115, 156, 169, 171, 200, 202, 204, 217, 218, 227–230, 234, 238,
268, 297, 309, 311, 315, 355, 360, 361, 378
Load adjustment coefficient, 366
Load combination method, 10, 13, 14, 295
Lower bound theorem (static condition), 155
Maximum entropy method, 85
Maximum likelihood point, 125, 136, 138, 139, 143, 144, 172
Modified P-H method, 98
Movable action, 358
Nominal failure probability, 179
Numerical simulation method, 5, 8, 103, 134, 249, 396
Objective uncertainty, 34, 35, 43, 44
Offshore platform, 1, 16, 182, 191–196, 213, 234
Parallel system, 155, 158, 159, 167, 168
Parameter uncertainty, 36, 43, 44, 382
Path probability model, 323, 325, 337
Permanent action, 318, 357, 375
P-H method, 97, 98
Pile-soil calculation model, 182, 190
Probability density function, 6, 18, 36, 68, 117, 122, 125, 131–133, 141,
167, 211, 233, 241, 271, 286, 289, 324, 326, 341, 344
Probability design method, 355–356, 360
Probability of failure, 162, 178, 219
Random field, 173
Random function, 18
Random process, 18, 212–214, 216–218, 225–234, 251, 256, 258, 285–
290, 292–294, 296–298, 300, 304, 306, 344, 356
Random uncertainty, 35, 59, 272
Random variable, 3, 4, 7, 8, 14, 18, 19, 33, 36, 42, 46, 60, 68, 70, 71, 79,
85–88, 90–94, 97–100, 103–106, 108–112, 115, 116–117, 120, 123–
125, 130–131, 136, 143, 149, 156, 169, 171, 172, 178, 194, 200, 295,
356, 361, 384, 387
Rayleigh, 233, 240, 241, 244
Rayleigh distribution, 233, 240, 241
Regression function, 49, 50
Regression support vector machine, 47, 48
Reinforced concrete, 1, 13, 327, 381, 382, 388
Reinforcement bars, 370
Reliability, 1–10, 12–22, 33–36, 40, 42, 44, 46–48, 50, 52, 53, 54, 56,
69–76, 115–119, 121, 123, 125, 127, 129, 131, 133, 135–137, 139, 141,
143–145, 147, 149, 151, 152, 153, 155–161, 165, 167, 168, 169, 171,
173, 175, 179–183, 185, 187, 189, 191, 193, 195, 197, 199, 211, 213,
217–221, 223, 225, 226, 228, 229, 230, 231, 234, 235, 237, 238, 247,
250, 285, 296, 297, 302, 308, 309, 307–315, 320, 321, 323, 353–361,
363–380
Reliability index, 4, 6, 9, 13, 19, 20, 53, 68, 70, 73, 74, 82, 92, 93, 97,
100, 105, 107, 115, 117, 136, 145, 201–204, 206, 260, 261, 264–266,
273, 275–281, 307–315, 320–321, 354, 360, 363, 366–368, 372, 374,
378–380
Response surface method, 101, 103–105, 107, 110, 112
Return period, 198, 199, 201, 203, 212, 373
R-F (Rackwitz & Fiessler) method, 93
Risk function, 216, 223–225
Rosenblatt transformation, 4, 94, 95, 136, 143
Safety, 1–4, 14–17, 44, 54, 55, 57–59, 63–65, 70, 150, 196, 213, 236,
274, 305, 316, 353–356, 359–368, 394, 395, 397
Safety index, 44, 79, 97
Safety margin, 274
Sample function of random process, 212
Second-order reliability method (SORM), 3
Semi-submersible platform, 197, 199, 201–206, 267–269, 272, 273,
275, 276, 281
Series system, 155–157, 159–161, 167
Shannon entropy, 43, 44
Single point, 238, 239
S-N model, 235
SRSS, 297, 302
Standard normal distribution, 82, 117
Static action, 358
Step function, 298
Stochastic processes, 228
Stratified sampling method, 126
Subjective uncertainty, 33, 35, 39, 40, 44
Support vector machine, 9, 47, 48, 101, 103
System reliability, 10, 15, 147, 158, 160, 161, 180, 181, 200, 201, 202,
206, 388, 394, 395
System uncertainty, 36, 43, 44
Target reliability, 13, 308, 310–315, 354, 358–360, 363, 364, 366, 367,
371, 372, 374, 379, 380
Time-dependent reliability, 15, 20, 217, 218, 225, 226, 228–230, 234
Time-independent reliability, 15, 20, 229, 230
Transfer function, 45, 46, 252, 268, 269
Truncated distribution function, 132, 135
Truncation enumeration method, 173
Turkstra combination rule, 296
Uncertainty, 2, 17, 18, 21, 33–37, 39–41, 43–45, 85, 86, 171, 186, 187–
188, 190, 194, 195, 199, 205, 235, 236, 272, 273, 280, 295, 308, 329
Upcrossing rate, 298, 300
Upper bound theorem, 154
V-space, 7, 143
Wen combination method, 301
X-Space, 74
Yield strength, 171, 337
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