applied
sciences
Article
Analytical and Numerical Reliability Analysis of Certain Pratt
Steel Truss
Marcin Kamiński *
and Rafał Błoński
Department of Structural Mechanics, Lodz University of Technology, 90-924 Lodz, Poland; 210495@edu.p.lodz.pl
* Correspondence: marcin.kaminski@p.lodz.pl
Citation: Kamiński, M.; Błoński, R.
Abstract: The main aim of this paper was to propose a new reliability index for steel structure
assessment and to check it using the example of a popular Pratt truss girder. Structural analysis was
completed in the finite element method system Autodesk ROBOT, and probabilistic analysis was
implemented in the computer algebra software MAPLE. The stochastic finite element method (SFEM)
was contrasted here with the Monte Carlo simulation and the girder span was selected as the input
structural uncertainty source. Both methods were based on the same structural polynomial response
functions determined for extreme deformation, for extreme stresses and also for the structural joint
exhibiting the largest effort. These polynomials were statistically optimized during the additional
least squares method experiments. The first four basic probabilistic characteristics of the structural
responses, the first-order reliability method (FORM) index, and as the new proposition for this
index were computed and presented. This new index formula follows the relative probabilistic
entropy model proposed by Bhattacharyya. The computer analysis results presented here show a
very strong coincidence of both probabilistic numerical techniques and confirms the applicability
of the new reliability index for the input coefficient of variation not larger than 0.15. These studies
should be continued for other engineering systems’ reliability and, particularly, for large-scale and
multiscale computer simulations. The results presented in this paper may serve in different applied
sciences, from biology through to econometrics, experimental physics and, of course, various branches
of engineering.
Analytical and Numerical Reliability
Analysis of Certain Pratt Steel Truss.
Appl. Sci. 2022, 12, 2901. https://
Keywords: reliability index; Monte Carlo simulation; stochastic perturbation method; stochastic
finite element method; relative entropy
doi.org/10.3390/app12062901
Academic Editors: Dario De
Domenico and Francesco Tornabene
Received: 27 January 2022
Accepted: 8 March 2022
Published: 11 March 2022
Publisher’s Note: MDPI stays neutral
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Copyright: © 2022 by the authors.
Licensee MDPI, Basel, Switzerland.
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distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
1. Introduction
Stochastic computational mechanics is still a very extensively explored research area
with many concurrent methods. However, many of them do not have widely accessible
computational implementations, especially in the context of the finite element method [1,2].
One could recall here not only the classical Monte Carlo simulation [1,3] or the stochastic perturbation technique [1,4–6] in its various order implementations. Moreover, still
very popular is the Bayesian approach [7], Karhunen–Loeve decomposition and polynomial chaos expansion [8], stochastic kriging models [9], different analytical and semianalytical techniques [1,10] and fuzzy stochastic analysis [11]. It should be mentioned
that various computational studies concerning numerical error for stochastic methods are
available [1,12]. Stochastic analysis has been implemented together with acceleration [13]
and reduction algorithms [14], and is recommended for the homogenization approach in
micromechanics [1]. The aforementioned numerical approaches may be used even for a
solution to the insufficient material data problem in solid mechanics [15]. Unfortunately,
practicing engineers must use some additional software or must create new small programs
to determine the reliability indices mandatory for modern engineering structures. A very
specific role is in the distribution and usage of the computer algebra systems such as
MAPLE, MATLAB or MATHEMATICA, where many stochastic methods can be and are
Appl. Sci. 2022, 12, 2901. https://doi.org/10.3390/app12062901
https://www.mdpi.com/journal/applsci
Appl. Sci. 2022, 12, 2901
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implemented. Let us note that the finite element method (FEM) modeling in the presence
of any uncertainty still remains a separate part of the engineer’s work. It is well known
that a huge portion of mathematical effort in the development of analytical formulas for
structural engineering analyses has been lost in the last few years thanks to a common and
fast methodology of “computing everything”. However, many relatively simple structures
may be quite efficiently designed using classical formulas, extended relatively easily with
mathematical software towards reliability assessment.
The guidelines for structural safety are based upon the very simplified reliability
index formulated by Cornell many years ago. It belongs to the first-order reliability
method, so that even if the engineers follow Eurocode 0 statements [16], their analyses
have rather limited importance. This index is based upon the assumption that structural
resistance R and the effect of external loadings E on the given structure both have Gaussian
distributions, and that the limit function has linear character. Furthermore, such a designing
code does not offer any algorithm for how to efficiently compute the first two probabilistic
moments for the functions R and E. On the other hand, an application of the secondorder reliability method (SORM) for practicing engineers is usually impossible because
it demands knowledge about a curvature of the limit state surface; this curvature cannot
be directly determined using the FEM software. This gave the authors a motivation
to look over mathematical works concerning probabilistic distance (divergence), which
could be applied to measure structural safety using some relative entropy in between the
distributions of R and E. It should be mentioned here that statistical divergence (or diversity)
has been studied in different areas of the applied sciences, particularly in biology [17],
econometrics [18], statistical physics [19] and even accidentally in civil engineering, where
it served in some seismic analyses [20].
Therefore, this paper aims to present an application of analytical and numerical probabilistic analysis in a reliability assessment of exemplary steels and popular civil engineering
structures. The Pratt steel truss girder has been selected to demonstrate such an approach
due to its popularity in academic analyses during civil engineering academic courses
and a huge number of the existing old steel bridges all around the world. A novelty in
this work is in the proposition of the alternative reliability index determination and also
in the comparison of the stochastic analytical and numerical approaches. The proposed
approach is based upon (i) polynomial response [1,21,22] determination via the series of
FEM experiments for varying design parameters and the least squares method (LSM) [23];
(ii) two various probabilistic numerical techniques, namely the Monte Carlo simulation as
well as the iterative generalized stochastic perturbation technique; and (iii) FORM [24] and
relative entropy-based reliability indices. The main goal of this work was achieved using
civil engineering-oriented FEM system Autodesk ROBOT (deterministic FEM series) and
computer algebra software MAPLE 2019.2 (LSM, probabilistic procedures and reliability indices determination). Future experiments with dynamic excitations of linear and nonlinear
large-scale structures, as well as a time-dependent reliability index of structures subjected
to corrosion and/or aging [25], was also considered.
2. Structural Analysis
The numerical experiments in this work were entirely focused on the simple Pratt truss
(Figure 1) schematically presented in Figure 2, whose height was adopted as h = 1.50 m,
whose number of segments equaled 12, and whose span l was treated as the input random
variable, having in turn triangular, uniform and also Gaussian probability distributions. The
random length was initially represented by the discrete set of values ranging from 15.00 up
to 21.00 m every 0.60 m to recover polynomial response functions via the series of traditional
FEM experiments. They connectextreme deformations and reduced stresses with input
uncertainty sources (e.g., parameter l in this study). Each truss FEM model in this series
was uploaded with the same constant load q redistributed throughout the upper chord
nodal points as the concentrated forces. Its characteristic value was set as qk = 15 kN/m,
while the design load was proposed for an illustration as qd = 20 kN/m—the adoption
Appl. Sci. 2022, 12, 2901
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and combination of the safety indices had a marginal influence on this study so it was
postponed here by arbitrary load values. All the cross-sections of structural components
were assumed as hot-finished square hollow steel profiles made of structural steel S235,
whose material parameters were assumed according to the design code Eurocode 3 [26];
structural parameters assigned to the individual elements have been presented in Table 1
below. The buckling lengths of the members making up the girder were assumed on
the basis of annex BB.1 to EN-1993-1-1: 2005 [26]. In-plane buckling of the chords was
assumed to be 1/12 of the length of the truss. Diagonals reduce the buckling length to
0.08 × l. In the case of out-of-plane buckling of the upper chord, the buckling length was
reduced by using every second purlin to support the transverse roof bracing to 0.16 × l. The
out-of-plane buckling coefficient for the lower chord was assumed with the assumption
of vertical bracing in the center of the spar. The buckling length of the diagonals was
reduced to 0.90 × l due to the degree of fastening of the bars in the chords, i.e., welded
joints. Let us note that the proposed cross-sections of structural elements were determined
based on initial FEM calculations, and designing procedures provided for the mean value
of the truss span l. This structure was initially designed by verification of both ultimate
and serviceability limit states (ULS and SLS) in all truss elements and also the bearing
capacity of the truss connections. Finally, the following profiles were proposed: (i) upper
chord—SHS 140 × 140 × 8, (ii) diagonals—SHS 90 × 90 × 8, and (iii) lower chord—SHS
120 × 120 × 8.
Figure 1. Road and railroad steel Pratt truss bridges (www.bridgehunter.com (26 January 2022)).
Figure 2. Static scheme of the given Pratt truss structure—Autodesk ROBOT 2021.
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Table 1. Structural parameters of steel structural elements in the given truss.
Parameter
Upper Chord
Diagonals
Lower Chord
Member length ly
Member length lz
Buckling length coefficient y
Buckling length coefficient z
0.90 × l
1.00 × l
0.08 × l
0.16 × l
1.00 × l
1.00 × l
0.90 × l
1.00 × l
0.90 × l
1.00 × l
0.08 × l
0.50 × l
The numerical experiments were completed assuming geometrical nonlinearities
and P-delta effect in the truss structural behavior, so that incremental Broyden–Fletcher–
Goldfarb–Shanno (BFGS) algorithm was engaged to determine the resulting deformations
and stresses. The FEM discretization was completed with the use of 49 linear two-noded
truss finite elements. The following additional parameters were used in the FEM analysis:
(i) load increment number equaled 5, (ii) maximum iteration number for one increment
equaled 40, (iii) increment length reduction number was set as 3, (iv) increment length
reduction factor was equal to 0.5, (v) the maximum number of line searches equaled 0,
(vi) control parameter for the line search method was as 0.5, (vii) the maximum number
of the BFGS corrections was automatically set as equal to 10, while (viii) relative tolerance
for the residual forces and displacements was predefined as 0.0001. Of course, all system
matrices were updated after each subdivision.
All the finite element method experiments were carried out in the civil engineering
system Autodesk ROBOT, where linear truss elements were selected, one finite element
was equivalent to a single structural element and each node had two degrees of freedom
(vertical and horizontal displacements). This structure was simply supported at both ends
in the lower chord, while the connections were all designed as perfectly non-deformable
(due to the welds). The results of numerical modeling in both deterministic and stochastic
contexts were contrasted with the analytical approach briefly presented below.
It is known that a deflection of the periodic and regular truss having two parallel
chords were determined using elementary knowledge following the rules of the strength
of materials and, particularly, fourth-order ordinary differential equations relevant to the
Euler–Bernoulli beam and its analogy to the truss structures with parallel upper and lower
chords. One may demonstrate that deflection of such a truss can be determined using the
following function u(x) including a distance 0 ≤ x ≤ l of the given point from its support
u( x ) =
x 3 x 4 ql 4 x
−2
+
,
24EJ l
l
l
(1)
where q is a characteristic value of the load, l is the theoretical span of the truss, E is the
Young modulus, J is the inertia moment of a system of the upper and lower chords. It
is very important that this model postpones a rigidity contribution of both columns and
diagonals during bending analysis of the truss. The center of gravity of the truss chords is
calculated as (see Figure 3)
A1 h
d=
,
(2)
A1 + A2
where A1 is the total cross-sectional area of the upper chord, and A2 is a lower chord
cross-sectional area. The resulting inertia moment of both truss chords according to the
Steiner theorem equals
I = I1 + A1 (h − d)2 + I2 + A2 d2 ,
(3)
where I1 is the moment of inertia of an upper chord, while I2 —inertia moment of the
lower chord.
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Figure 3. Symbols used for analytical calculations.
Having analytical formula for the deflection function and the given probability density
function of the truss span, one may derive the basic four probabilistic characteristics of
the deformation line u(x) of such a truss following classical definitions available from the
theory of probability [1]. It yields
E[u( x )] = 0.0417
qx 3
E [l ] + 3E[l ]σ2 (l ) − 2E[l ] x2 + x3 ,
EI
(4)
Var (u( x )) =
q2 x 2
0.001736σ2 (l ) E2 I 2 9E4 [l ] + 36E2 [l ]σ2 (l ) − 12E2 [l ] x2 + 15σ4 (l ) − 12σ2 (l ) x2 + 4x4
µ3 (u( x )) =
E [ l ] σ 4 ( l ) q3 x 3
E3 I 3
4
45σ (l ) + 48E2 [l ]σ2 (l ) − 24σ2 (l ) x2
0.001302
µ4 (u( x )) = 9.04224 · 10−6
+ 9E4 [l ] − 12E2 [l ] x2 + 4x4
,
,
(5)
(6)
σ 4 ( l ) q4 x 4
(3465 σ8 (l ) + 17280E2 [l ]σ6 (l ) − 2520x2 σ6 (l )+
E4 I 4
+11934E4 [l ]σ4 (l ) − 8136E2 [l ]σ4 (l ) x2 + 840σ4 (l ) x4 + 2160E6 [l ]σ2 (l ) − 3240E4 [l ]σ2 (l ) x2 +
+1440E2 [l ]σ2 (l ) x4 − 160σ2 (l ) x6 + 81E8 [l ] − 216E6 [l ] x2 + 216E4 [l ] x4 − 96E2 [l ] x6 + 16x8
(7)
where E[l] and σ(l) stand for the expectation and standard deviation of the truss girder
length. Such a characterization of probabilistic structural response is definitely wider
than that coming from numerical methods, because it is able to draw the diagrams of
expectations and variances throughout the girder length as the continuous function of their
parameters and, furthermore, this is exact from the probability theory point of view. The
engineers do not need to contrast these results with Monte Carlo simulation results and
the only error comes from the approximate deterministic model. The reliability index β
was determined for the numerical and analytical results of truss deflection (fourth-order
polynomial of the truss span l) and the numerical results of stresses in joints (third-order
polynomial of l appeared to be optimal here). The FORM reliability index was based upon
the limit function g defined as
g ( l ) = R ( l ) − E ( l ),
(8)
where R means general structural resistance, while E denotes the overall effect of external
actions. The reliability index β is with this notation defined as [16]
β(l ) =
E[ g(l )]
,
σ ( g(l ))
(9)
where E[g] is the expected value of the limit state function, σ(g) is its standard deviation.
It should be underlined that this definition is derived using the assumption that the limit
function has a linear form, which is not necessarily true in many engineering applications,
so the second-order reliability method (SORM) is frequently preferred. After analytical
transformations with the function g, the reliability index β takes the following form:
β(l ) =
E[ g(l )]
E[ R(l ) − E(l )]
E[ R(l )] − E[ E(l )]
E[ R(l )] − E[ E(l )]
=
= p
= p
.
σ ( g(l ))
σ ( R(l ) − E(l ))
Var ( R(l ) − E(l ))
Var ( R(l )) + Var ( E[l ])
(10)
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It is assumed, of course, that random functions R and E are uncorrelated here, which
reflects engineering practice very well. The expected value of admissible deformation in
the case for the SLS and ULS correspondingly equal
E[ R(l )] =
l
, E[ R(l )] = 235 MPa,
250
(11)
while the variance of admissible values equals in the SLS
Var ( R(l )) = σ2 (l ),
(12)
since R and l are connected by a linear transform. It is simply equal to 0 in the case of the
ULS analysis; Equation (12) holds true in the case of the random function E. Because of the
limitations of the FORM analysis, let us adopt the following proposition for the reliability
index estimation [27]:
+∞q
Z
0
p R ( x ) p E ( x ) dx,
(13)
β =
−∞
where pR (x), pE (x) define probability function associated with structural resistance and
probability function related to structural effort, respectively. It follows a more general idea
of the distance (divergence) in between two different probability distributions proposed
in [27]. One could alternatively use the Kullback–Leibler approach [28]; however, this
second distance has non-symmetric properties and may exhibit some unwanted properties.
It can be derived analytically that the new reliability index can be expressed for two
Gaussian distributions with the given expectations and standard deviations (µR , µE —
means of the structural resistance and σR , σE —their standard deviations) in the following,
relatively simple, algebraic form:
β0 =
2
1 ( E[ R] − E[ E])
σ ( R ) + σ2 ( E )
1
ln
+
.
4 σ2 ( R ) + σ2 ( E ) 2
2σ ( R)σ ( E)
(14)
Let us underline that the assumption about Gaussian distributions for the structural
resistance and effort should be justified from the engineering point of view and this is the
case of the basic structural demands included in Eurocode 0 [16].
3. Numerical Results for Structural Elements and Their Discussion
Numerical analysis was focused in turn on the following issues: (i) determination of
polynomial response functions of the structural effort and of the extreme deformation as
the functions of the truss span; (ii) determination of the basic probabilistic characteristics of
structural responses with the use of FEM series and of the analytical formulas; (iii) determination of the reliability indices using both analytics and numerical analysis, including the
new reliability index formula; and (iv) stochastic analysis of the most efforted structural
joint in the truss.
Three different probability distributions were selected for this analysis—triangular,
uniform and Gaussian—to study the Monte Carlo statistical approach and the iterative
generalized stochastic perturbation technique of the 10th order. The detailed analytical
derivations of the perturbation-based formulas for the first four central probabilistic moments could be found in [1] for the Gaussian PDF, whereas those relevant to triangular
and uniform PDFs are presented in [29]. The results of stochastic analysis have all been
collected for some range of the input uncertainty, i.e., α(l ) ∈ [0.00, 0.15], which has been
taken as relatively wide to study the convergence (and possible divergence) of all three
tested probabilistic numerical techniques.
The first results are presented in Figure 4 and they concern structural effort variations
in all groups of the structural elements determined as the functions of the truss span. A
numerical model was created in such a way that none of the designed steel profiles exceed
the limit effort (100%), even for the extremely large span of this structure. Figure 4 clearly
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documents that all these efforts almost linearly depend upon this span—the largest effort
was noticed within the welds, then, a slightly smaller one was observed for upper and
lower chord members, while the smallest was in the truss diagonals. One may conclude
that the numerical values of this effort ranged from about 65% up to 95% for the structural
joints (whereas l varies from 15.0 up to 21.0 m), whereas the effort for lower and upper
chord elements varied from 45% up to more than 85% at the same time. This enables us
to conclude that the ULS-based reliability index should be based upon the joints’ rational
designing, while the SLS, traditionally, upon this truss midspan deflection.
Figure 4. Structural effort variations of the Pratt truss structural components w.r.t. Pratt truss span.
Having verified the deterministic safety of the designed Pratt truss, the polynomial
response function was recovered via the series of FEM experiments with varying truss
span. Discretization of the span computational domain, the corresponding series of the
extreme truss deflections and third-order polynomial basis determined using the LSM
procedure are shown in Figure 5. It can be seen that LSM fitting corresponds almost
perfectly to the FEM data and that the approximating function is very smooth and regular.
It should be mentioned that extreme deformation corresponds to the last equilibrium
determined from the BFGS incremental path. Analytical representation was necessary for
further perturbation-based derivations and it was obtained on the basis of numerical results
delivered in Table 2 as
u(l ) = 0.0001522l 3 − 0.003769l 2 + 0.04322l − 0.17150.
(15)
It should be mentioned that this relatively low order of polynomial basis causes, that
higher than the fourth-order derivatives of the function u(l) simply vanish, so that the
Taylor series expansions inherent in the stochastic perturbation technique are very short
and contain a few terms only.
Probabilistic results concerning extreme deflection of the truss were further collected
(in Figures 6–9) and these are: expected value, coefficient of variation, skewness and kurtosis. They were collected for Monte Carlo analysis (abbreviated here as MC), perturbation
method (PM), separately for the FEM approach (NC) and following analytical expressions
(AC). Three different symmetric probability distributions were compared and these are triangular (TD), uniform (UD) and Gaussian (normal) distribution (ND). These distributions
Appl. Sci. 2022, 12, 2901
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have the same mean value and are all symmetric, while the ranges for TD and UD were
calculated from Gaussian distribution standard deviation with the use of the well-known
three-sigma rule. In this work, the values obtained from the 10th order iterative stochastic
perturbation method and the Monte Carlo Simulation were compared assuming a random
sample size equal to n = 105 .
Figure 5. The polynomial response function of the truss deflection depending upon its span.
Table 2. Truss deflection and its span interdependence—numerical results.
Length l [m]
15.00
15.60
16.20
16.80
17.40
18.00
18.60
19.20
19.80
20.40
21.00
Deflection u [cm]
1.42
1.63
1.86
2.13
2.41
2.73
3.08
3.46
3.88
4.34
4.83
Figure 6. Expected value E[u] in the SLS.
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Figure 7. Variance Var(u) in the SLS.
Figure 8. Skewness Skew(u) in the SLS.
A general conclusion which can be drawn from these data is that all probabilistic
characteristics monotonously increase together with an additional increase in the input
uncertainty level. This is quite an expected result, as is the fact that the first two moments
reach the largest values when the input distribution has a uniform PDF. The second general
conclusion is that the results of both probabilistic methods coincide almost perfectly for all
of the first four probabilistic characteristics, which is a very promising result in the context
of the relatively large input uncertainty level and the fact that a nonlinear BFGS solver was
used. It should be noticed at this point that a situation may be more complex for large-scale
engineering structures and the input CoV interval permitting such a perfect agreement may
become somewhat smaller, i.e., α(l ) ∈ [0.00, 0.10]; this should be verified through more
Appl. Sci. 2022, 12, 2901
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detailed FEM experiments. The next important information, which can be rather useful in
civil engineering practice, is that expectations computed via analytical formulas (Figure 6)
are remarkably smaller than those obtained from the FEM experiments. This may follow
the fact that analytics is available in the linear elastic regime only, whereas the FEM analysis
is conducted including geometrical nonlinearity; nevertheless, the first approach seems to
be insufficient in all these cases, where nonlinear effects may play a remarkable role during
exploitation time.
Figure 9. Kurtosis Kurt(u) in the SLS.
Higher order statistics such as skewness (Figure 8) and kurtosis (Figure 9) differ from 0,
so that the resulting distributions of extreme displacements are slightly different from the
Gaussian PDF, and this divergence increases while increasing the input uncertainty level.
It is noticeable that the largest values of both coefficients are obtained for the Gaussian
PDF of the truss span, slightly smaller values are obtained for the triangularly distributed
span and the smallest are obtained in the case of the uniform distribution. Skewness keeps
positive values for the entire variability interval of truss span uncertainty for all chosen
probability distributions, while kurtosis values are more complex. They are all positive for
the normal PDF, start from negative and tend towards positive while having a triangularly
distributed parameter l and are all negative when the uniform distribution is considered.
These higher-order statistics are usually postponed during the final reliability assessment;
however, these results demonstrate that the initial choice of PDF type remarkably affects the
resulting stochastic structural response. The first two probabilistic moments are contained
in Figures 6 and 7 and resulting from the Gaussian distribution of the truss span were used
to calculate final reliability values in the SLS state.
Having computed the probabilistic response of the Pratt truss, its reliability indices
were computed in the computer algebra software MAPLE 2019 according to Equations (9)
and (14), sequentially. This was also carried out for three different probability distributions:
triangular, uniform and Gaussian, to see an influence of the PDF choice on their final
values. Numerical values relevant to Equation (9) have been collected in Table 3, where an
influence of the input uncertainty of the truss span in the form of the coefficient of variance
α has been also included. Each cell of this table includes in turn the reliability index for
these three PDFs—the first one corresponds to the triangular probability function, the
middle to the uniform one and the lowest to the Gaussian distribution. The abbreviations
in this table heading are consistent with the notation of the previous figure, i.e., MCS-NC
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denotes the results corresponding to the Monte Carlo simulation method while using the
finite element method analysis. The data collected in this table confirm the well-known
trend of the reliability index, which usually exponentially decreases together with an
additional increase in the cumulated input uncertainty level. Monte Carlo simulation and
the iterative generalized stochastic perturbation technique both return practically the same
results, which, taking into account decisively smaller time and computer power effort of
this second technique, means that the perturbation-based SFEM is preferred in this case.
Table 3. Reliability indices β in the SLS via numerical and analytical methods based on the Monte
Carlo simulation method and the stochastic perturbation technique for the triangular, uniform and
Gaussian PDFs.
α [-]
0.025
0.050
0.075
0.100
0.125
0.150
MCS-NC
MCS-AC
PM-NC
PM-AC
37.93
37.40
38.11
18.95
18.66
19.05
12.61
12.40
12.69
9.44
9.26
9.50
7.53
7.36
7.59
6.26
6.09
6.31
38.28
37.83
38.43
19.12
18.88
19.21
12.73
12.55
12.79
9.53
9.37
9.58
7.61
7.46
7.66
6.32
6.17
6.37
37.93
37.40
38.11
18.95
18.66
19.05
12.61
12.40
12.69
9.44
9.26
9.50
7.53
7.36
7.59
6.26
6.09
6.31
38.28
37.83
38.43
19.12
18.88
19.21
12.73
12.55
12.79
9.53
9.37
9.58
7.61
7.46
7.65
6.32
6.17
6.37
As is expected, the lowest values of this index were returned for the uniform distribution of the truss span, slightly larger values were obtained for the triangular PDF of
this span and the extremely high reliability index was obtained in the case of the Gaussian
distribution. It is observable that an arbitrary assumption of the Gaussian character of
most structural parameters without a precise verification may lead to some inaccuracies or
even to unsafe designing of the engineering structures. A very interesting observation can
be made while comparing numerical and analytical analyses—these second series return
the results each time with no more than a few percent larger than the FEM-based analysis.
Nevertheless, this difference is almost negligible when the uncertainty level belongs to the
interval α ∈ [0.05, 0.10], which is highly expected in most of the statistical parameters.
Moreover, it is widely known that standard EN 1990 [16] defines the three reliability
classes RC1, RC2 and RC3, which are associated with the consequences classes CC1, CC2
and CC3, describing the possible consequences of a failure of the entire structure or just
its member. The table below (Table 4) contains the minimum values of the reliability
index β recommended by EN 1990 (Table B2) [16], which were recalled to determine the
admissibility level of input randomness for the given parameter of the Pratt truss span.
The results of this study are presented by inserting numerical values from Tables 3 and 4
into a single graph—see Figure 10 below.
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Table 4. Recommended minimum values for reliability index β.
Reliability Class
Minimum Values for β
1 Year Reference Period
50 Years Reference Period
5.2
4.7
4.2
4.3
3.8
3.3
RC3
RC2
RC1
Figure 10. The reliability index β in the SLS.
Truss girders are structures that are expected to last at least 50 years and the described
construction has been assigned to the third reliability class that allows resisting even if
the coefficient of variance of the truss span slightly exceeds the value of α > 0.150. An
intersection of the reliability curves delivered by numerical and analytical approaches
with the lines representing target values of the reliability index shows that this admissible
value equals α = 0.175. This value is enormously large for any measuring method for the
span, so one may conclude that the presented method generally has no limitations coming
from the engineering practice while investigating statistics in the truss spans. Taking into
account the well-known limitations of the FORM approach, the new reliability index has
been proposed following the relative entropy (probabilistic divergence) model proposed by
Bhattacharyya [27]. Its values computed using the first two probabilistic moments have
been collected above in Table 5, in the same manner as the results contained in Table 4.
Despite the different range of variability caused by the fact that Equation (9) contains a
linear form of the expectations of the functions R and E, one may see the same properties
and interrelations in between different methods and input distributions. This new approach
is more convenient in the engineering sense, as it measures a distance in between two
random distributions and no specific form of the limit function is required. Its value has
been scaled to compare with the existing designing algorithm and this has been performed
using the following semi-empirical formula:
β0
.
2
p
β00 =
(16)
Such a formula has been established by a comparison of the main components in
Equations (10) and (14) containing mean values of the structural response and its resis-
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tance. It is important to mention that this proposed scaling enables keeping the same
practical demands and values and to have a better mathematical and engineering justification of the entire procedure. It is evident after scaling that the new reliability index
(Table 6) is more demanding and does not allow for such a large input uncertainty interval;
nevertheless, the importance of the input PDF choice remains the same. Analytical and
numerical approaches here return almost the same result, so one could recommend inserting them directly into the engineering designing codes to give practicing engineers a fast
and widely available alternative to the variety of rather complex stochastic finite element
method implementations.
Table 5. A contrast of the estimated reliability indices β0 in the SLS by numerical and analytical
methods based upon the Monte Carlo simulation method and the stochastic perturbation technique
for the triangular, uniform and Gaussian probability distributions.
α [-]
0.025
0.050
0.075
0.100
0.125
0.150
MCS-NC
MCS-AC
PM-NC
PM-AC
11,426.46
11,422.07
11,428.21
716.04
714.92
716.46
143.47
142.94
143.67
47.21
46.87
47.34
20.92
20.64
21.02
11.46
11.19
11.55
11,579.79
11,575.61
11,581.44
725.84
724.76
726.23
145.57
145.05
145.75
48.00
47.65
48.12
21.34
21.05
21.44
11.74
11.46
11.85
11,426.45
11,421.98
11,427.93
716.04
714.91
716.42
143.47
142.94
143.65
47.21
46.86
47.33
20.92
20.64
21.01
11.46
11.19
11.55
11,579.77
11,575.53
11,581.19
725.84
724.75
726.20
145.57
145.05
145.74
48.00
47.65
48.11
21.34
21.05
21.44
11.74
11.46
11.84
Table 6. A contrast of the estimated reliability indices β” in the SLS by numerical and analytical
methods based upon the Monte Carlo simulation method and the stochastic perturbation technique
for the triangular, uniform and Gaussian probability distributions.
α [-]
0.025
0.050
0.075
0.100
0.125
0.150
MCS-NC
MCS-AC
PM-NC
PM-AC
53.45
53.44
53.45
13.38
13.37
13.38
5.99
5.98
5.99
3.44
3.42
3.44
2.29
2.27
2.29
1.69
1.67
1.70
53.80
53.79
53.81
13.47
13.46
13.47
6.03
6.02
6.04
3.46
3.45
3.47
2.31
2.29
2.32
1.71
1.69
1.72
53.45
53.44
53.45
13.38
13.37
13.38
5.99
5.98
5.99
3.44
3.42
3.44
2.29
2.27
2.29
1.69
1.67
1.70
53.80
53.79
53.81
13.47
13.46
13.47
6.03
6.02
6.04
3.46
3.45
3.47
2.31
2.29
2.32
1.71
1.69
1.72
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4. Stochastic Response of the Decisive Structural Connection and Its Discussion
Additionally, the load capacity of structural connection nodes, designed according to
the statements included in the designing code [30], was studied in the context of stochastic
structural response and reliability. This part of the numerical study was based on the
reduced von Mises stresses in their welds and some exemplary connections have been
schematically shown in Figure 11, whereas the nodes with extreme stresses accounted for
in further SFEM analysis have been marked in Figure 12 with the dashed line circles.
Figure 11. Axonometric view of the analyzed truss joints from the Autodesk ROBOT 2021.
Figure 12. Location of the most efforted truss joints.
Quite similarly to the SLS analysis in the previous section, one first needs to determine
the response function of the reduced stresses with respect to the truss span via the LSM
fitting on the basis of the results contained in Table 7. This is obtained in the form of the
third-order polynomial having the following analytical form:
σred = 0.007761l 3 − 0.421976l 2 + 23.5512l − 45.2738
(17)
Table 7. Extreme reduced stresses in the external weld in the truss column.
Length l [m]
15.00
15.60
16.20
16.80
17.40
18.00
18.60
19.20
19.80
20.40
21.00
Stress σ [MPa]
239.22
248.81
258.79
267.98
277.56
287.15
296.74
306.31
315.89
325.47
335.05
Furthermore, probabilistic characteristics of the structural response are determined
with the Monte Carlo simulation and the stochastic perturbation method; these are: expectations (Figure 13), coefficients of variance (Figure 14), skewness (Figure 15) and kurtosis
(Figure 16). This is conducted for three various probability distributions, but unfortunately,
any analytical method is available here. Figure 13 documents that the expected values
of the reduced stresses remarkably and nonlinearly decrease, contrary to the extreme deformations, while increasing input uncertainty. Furthermore, some small differences in
between the Monte Carlo simulation and the stochastic perturbation method can be noticed
for all PDFs, which additionally increase together with an increase in the input CoV. They
are quite remarkable in Figure 13, but their realistic range is smaller than one thousandth,
so they can be postponed in engineering practice. Variances of the reduced stresses quite
Appl. Sci. 2022, 12, 2901
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expectedly increase in Figure 13 while approaching the extreme values of the input α(l) and
now the differences between the statistical and perturbation approaches are invisible. The
largest resulting statistical scattering is obtained for the uniformly distributed truss span,
next, for the triangular PDF; and the smallest for the Gaussian distribution. This regularity
partially validates the entire methodology and, specifically, the efficiency of the iterative
generalized stochastic perturbation method for symmetric non-Gaussian distributions also.
Figure 13. Expected Value E in the ULS.
Figure 14. Variance Var in the ULS.
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Figure 15. Skewness Skew in the ULS.
Figure 16. Kurtosis Kurt in the ULS.
Skewness and kurtosis attached here for the completeness of probabilistic modeling
(Figures 15 and 16) are all very close to 0, so these reduced stresses could be used in reliability analysis of Gaussian with relatively small error, so that the first two moments could
be sufficient. Of course, this approximation is the most efficient when input uncertainty
has a Gaussian character. Now, analogously to Figure 14, two stochastic approaches return
the same values of both coefficients, which also confirms the applicability of the stochastic
perturbation method for higher statistics.
Finally, the reliability indices are contained in Table 8 and also compared in Figure 17
with the corresponding normative values [16]. Generally, one can observe that their values
are remarkably smaller than for the structural elements analyzed before, see cf. Table 6. This
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is consistent with traditional engineering practice, where the higher importance of structural
joints in the optimal designing of metal structures is considered. The highest reliability
indices occur when the input truss span is assumed to have the Gaussian PDF; then, slightly
smaller values are noticed while triangular distribution is assumed; whereas extremely
small indices are noticed for the uniformly distributed span. Furthermore, two probabilistic
methods return the same results correspondingly, which confirms the applicability of the
stochastic perturbation technique in reliability-oriented designing procedures of structural
joints. Contrary to the previous analysis, a range of admissible input statistical scattering
of the truss span is very small (α < 0.025); one can conclude that the truss span must be
measured very precisely and that any geometrical imperfection in this context may be
critical for the very optimally designed connections in the Pratt trusses.
Table 8. A contrast of the reliability indices β in the SLS by numerical and analytical methods based
upon the Monte Carlo simulation method and the stochastic perturbation technique for a various
probability distribution.
Distribution
Triangular
Uniform
Normal
α [-]
MCS-NC
PM-NC
MCS-NC
PM-NC
MCS-NC
PM-NC
0.025
0.050
0.075
0.100
0.125
0.150
5.79
2.9
1.93
1.45
1.15
0.96
5.79
2.9
1.93
1.45
1.15
0.96
4.75
2.37
1.58
1.18
0.94
0.78
4.75
2.37
1.58
1.18
0.94
0.78
6.34
3.17
2.11
1.58
1.27
1.05
6.33
3.16
2.11
1.58
1.26
1.05
Figure 17. Reliability index β in the ULS.
5. Conclusions
(1)
The numerical analysis delivered in this work proves that structural reliability of the
Pratt truss structures subjected to geometrical uncertainty can be efficiently modeled
using analytical formula describing its deflection line. This is due to a coincidence of
the first four probabilistic characteristics and reliability indices obtained via a Monte
Carlo simulation and the stochastic perturbation-based FEM analysis. This is a very
promising and important result because the deflection line for such a truss can be
easily implemented in any computer algebra system with probabilistic libraries, so
that some analytical expressions for the expectations and standard deviations could
Appl. Sci. 2022, 12, 2901
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(2)
(3)
be inserted into engineering design codes to perform reliability assessment, without a
rather complex implementation and time-consuming simulations of various SFEM
realizations. It would enable further applications of many correlated uncertainty
sources to have any desired probability density function. It can also be seen that this
approach can be extended with many existing analytical models in civil engineering
to develop stress formulas as well as deflection lines for some typical beams, plane
frames and other girders, to make analytical reliability analysis more popular. Let
us note that a difference in between numerical and analytical results in the case of
the reliability indices increases rather slowly together with input statistical scattering,
meaning it remains unbiased by the experimental statistics.
It was documented in this work that it is possible to use the relative entropy (probabilistic divergence) proposed by Bhattacharyya [27] in traditional civil engineering
reliability analysis, using the existing designing codes. Furthermore, it was demonstrated that this entropy may be efficiently rescaled to the FORM index as a half of this
entropy square root. This relatively simple formula may apply for the input coefficient
of variation not larger than α ≤ 0.15 while using either the Monte Carlo simulation or
the iterative generalized stochastic perturbation technique. This idea needs further
numerical experiments, not only in the case of steel structures, but also for other
branches and problems in engineering where input uncertainty may play a remarkable role. The new reliability index may be useful in the Stochastic Reliability-Based
Design Optimization (SRBDO) with triple perturbation-based, semi-analytical and
Monte Carlo simulation analysis of the first two probabilistic moments, analogously
to the study presented in [31].
It was shown that the admissible uncertainty level cannot be too large in the welded
steel structures because of the relatively high sensitivity of optimally designed connections of structural elements to the truss span. It was demonstrated that structural
effort in the range of 90% demands an input coefficient of variation α < 0.025. The
remaining open research question in this context is the possible correlation of the few
uncertainty sources inherent in the functions R and/or E. There are some engineering
examples showing that this correlation may remarkably decrease the overall structural
reliability index in some specific cases. Nevertheless, it should be included directly
into the final formulas such as in Equations (9) and (14) presented in this paper. The
numerical approaches shown above are capable of capturing and discussing this issue
as well. It would be very instructive to discuss these issues for structural stability
problems, which are of paramount importance in the area of steel structures [32].
Author Contributions: Conceptualization, M.K.; methodology, M.K.; software, R.B.; validation, M.K.;
formal analysis, M.K.; investigation, R.B. and M.K.; resources, R.B.; data curation, R.B.; writing—
original draft preparation, M.K. and R.B.; writing—review and editing, M.K.; visualization, R.B.;
supervision, M.K.; project administration, M.K.; funding acquisition, M.K. All authors have read and
agreed to the published version of the manuscript.
Funding: This paper has been written in the framework of the research grant OPUS no 2021/41/B/
ST8/02432 entitled “Probabilistic entropy in engineering computations” and sponsored by The
National Science Center in Poland.
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: Not applicable.
Conflicts of Interest: The authors declare no conflict of interest.
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