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Stiffness-oriented numerical model for non-linear reinforced concrete beam
systems
Thesis · February 2011
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SILESIAN UNIVERSITY OF TECHNOLOGY
Faculty of Civil Engineering
master thesis
Stiffness-oriented numerical model
for non-linear reinforced concrete
beam systems
Author:
SEng Agnieszka KNOPPIK–WRÓBEL
KBI-A CIS, year 2010/2011
February 22, 2011
Supervisor:
PhD SEng Grzegorz WANDZIK
Contents
1 Introduction
1.1 Objective of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Range of problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Scope of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
5
5
6
2 Models for analysis of reinforced concrete beam elements
2.1 Material non-linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Models based on theory of elasticity . . . . . . . . . . . . . . . . . .
2.1.2 Models based on theory of plasticity . . . . . . . . . . . . . . . . .
2.1.3 Rheological models . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Material models in standards . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Concrete models . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Steel models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Geometrical non-linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
9
10
10
11
15
15
17
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3 Refinement of linear-elastic analysis results
3.1 Analysis of reinforced concrete members beyond elastic phase . . . . . . . .
3.1.1 Linear-elastic vs. non-linear analysis . . . . . . . . . . . . . . . . .
3.1.2 Plastic properties of reinforced concrete . . . . . . . . . . . . . . . .
3.2 Linear-elastic analysis with moment redistribution . . . . . . . . . . . . . .
3.3 Plastic analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Limit equilibrium method . . . . . . . . . . . . . . . . . . . . . . .
3.4 Non-linear analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
19
19
20
23
25
27
28
4 Stiffness degradation in R/C flexural members
4.1 Stiffness of reinforced concrete flexural members . . . . . . . . . . . . . . .
4.2 Bending stiffness of cross-section . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Material behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.2 Cross-section behaviour . . . . . . . . . . . . . . . . . . . . . . . .
4.2.3 Creep and shrinkage effects . . . . . . . . . . . . . . . . . . . . . .
4.2.4 Tension stiffening effect . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.5 Mean moment–curvature relationship . . . . . . . . . . . . . . . . .
4.3 Bending stiffness of member . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Influence of cracks . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.2 Influence of reinforcement . . . . . . . . . . . . . . . . . . . . . . .
31
31
32
32
36
39
39
40
40
40
42
3
4
CONTENTS
5 Numerical model
5.1 Static scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.2 Discretisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Material model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Cross-section model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.1 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.2 Bending moment . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.3 Moment–curvature relationship . . . . . . . . . . . . . . . . . . . .
5.4 Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.1 Stiffness of cross-section . . . . . . . . . . . . . . . . . . . . . . . .
5.4.2 Stiffness of segment . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5 Static analysis with FEM . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.1 Static calculations . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.2 Bending moment distribution . . . . . . . . . . . . . . . . . . . . .
43
44
44
45
45
46
46
47
50
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51
53
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57
6 Summary
6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3 Abstract (in Polish) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
65
65
67
67
Chapter 1
Introduction
1.1
Objective of thesis
The objective of this thesis is to derive a numerical model for designing of flexural reinforced
concrete beams taking into consideration a non-linear behaviour of reinforced concrete
and redistribution of internal forces as a result of stiffness degradation of the elements due
to crack formation at flexure, thus providing a unified algorithm for static calculations
and dimensioning.
1.2
Range of problems
The use of the linear-elastic analysis in determination of internal forces distributions
in reinforced concrete structures has a virtue of simplicity and allows of results from a
series of analyses to be combined using the principle of superposition. Popularity and
long-lasting tradition of application of the approach is undeniable. The linear-elastic
approach eliminates problems in numerical analysis, such as incremental calculations,
in which the stiffness of the loaded element is locally modified along with the changing
distribution of internal forces, iterative calculations connected with mutual correlation
between amount and distribution of reinforcement and distribution of bending moments
and, as a result, problems with convergence of computational process.
The assumption of the linear-elastic behaviour is reasonable at low levels of loading but
becomes invalid at higher loads due to cracking and development of plastic deformations.
Once an element cracks, the behaviour becomes non-linear, so application of the elastic
theory provides unreliable values of internal forces and has a series of disadvantages.
In the limit states analysis complex material models are used while the static analysis is
based on a very primitive model. The assumptions are inconsistent [20]: the load-bearing
capacity of the section MRd is determined according to the ultimate limit state of flexure
(section failure criterion) but it is compared with the values of internal forces (MSd )
determined for a completely different state of work, even before the level of cracking
moment is reached. The results of the static analysis do not consider the influence of the
amount and distribution of reinforcement or the capacity use of section. Concrete cracking
is considered to be a disadvantage requiring reinforcement and diminishing structural
tightness and durability. The positive effects of cracking – reduction and redistribution of
cross-sectional forces – are either neglected or poorly simulated [30]. In static calculations
of elements subjected to non-mechanical loads (thermal loads, fire) in the design process
“corrected”, reduced values of temperatures, safety factors or moduli of elasticity are
5
6
CHAPTER 1. INTRODUCTION
introduced. This frequently results in uneconomic and unsafe structures, and should be
improved by a broader application of stiffness-oriented design procedures.
The notion of the moment redistribution in cracked elements is useful for practical
design as it allows of some flexibility in the arrangement of reinforcement. Bending moments
may be transferred into the less congested areas or standard reinforcement layouts may
be applied where small differences occur in the bending moment distributions for a series
of beams, thus avoiding the need to detail each beam separately. In addition, economy
can be achieved when moment redistribution is applied to different load combinations,
resulting in a smaller bending moment envelope which still satisfies equilibrium.
Moment redistribution in beams has traditionally been considered as an ultimate limit
state (ULS) phenomenon, but experiments [31] prove that a significant portion of this
redistribution will almost always occur at the serviceability limit state (SLS) because
of the mismatch between the flexural stiffness assumed when calculating moments for
the ULS and those actually occurring at the SLS due to variations in the reinforcement
layout along the member and the influence of cracking. That is why the ULS and SLS
assumptions and procedures should be conjugated and permissible level of redistribution
should be defined.
The analysis of the plastic properties of reinforced concrete structures has been performed throughout the years in stages [21]. In the first stage changes of the internal
forces under increasing loads were determined experimentally, especially under loads close
to the load-bearing capacity of the structure. In the second stage the limit equilibrium
method was adopted, primarily to the steel structures. The notion of the fact that the
plastic hinge in a reinforced concrete structure differs from the plastic hinge in a steel
one was a motive for the next, third stage, covering development of general methods for
calculation of structures according to the theory of plasticity. The fourth stage, which has
not yet been finished, is characterised by development of simple design methods, useful in
everyday practice, and finally the last stage, still under development, is connected with
incorporation of numerical calculations thanks to the possibility of the use of computers.
The linear-elastic FE analysis is already world-wide accepted but for the reasons
mentioned before there arises an urgent need for implementation of the non-linear analysis
into design procedures. This results from the use of more and more complex material
models, close-to-reality modelling of multi-phenomena processes or advanced analysis of
whole complex models instead of their subdivisions. Nevertheless, the nature of the nonlinear FE analysis makes it much more difficult to provide the same level of automatisation
as for the linear one. The factors which prevent wide acceptability of the non-linear FE
analysis procedures are connected with [24]: no full identification of constitutive properties
of concrete, no generally accepter material law available to model concrete behaviour, high
costs and required experience.
1.3
Scope of thesis
Given the range of problems stated in Sec. 1.2, the thesis challenges a range of issues
aimed at deriving of a close-to-reality numerical model of reinforced concrete structures
on the example of continuous flexural beams.
The theoretical background is a concise overview of the Polish and foreign literature
connected with the topic of non-linear concrete mechanics as well as international standards,
primarily the European Standard EN 1992 – Eurocode 2 with detailed useful information
from the CEB-FIP Model Code 1990 and final draft of Model Code 2010, the Polish
1.3. SCOPE OF THESIS
7
Standard PN-B-03264 (the author’s previous national standard), the British Standard BS
8110 and the American Standard ACI 318 1 .
The practical part covers a creative analysis of the solutions proposed in the literature
and description of the numerical model derived in cooperation with the supervisor of
this thesis, PhD Grzegorz Wandzik, proposed on the basis of the referred theoretical
background and personal experience in the topic.
Chapter 2 presents an overview of the reasons of non-linearity in the analysis of
concrete structures. In this chapter the two general aspects of non-linear mechanics are
described: non-linear material models and behaviour of structures considering deformations
and their influence on the distribution of internal forces.
Chapter 3 presents the methods of refinement of the linear-elastic analysis result
proposed in the literature, taking into consideration plastic properties of reinforced concrete
and the consequent mechanism of stiffness degradation and redistribution of internal forces.
Chapter 4 introduces the theory of stiffness degradation in a flexural reinforced
concrete member. The idea of a cross-section model is presented and stiffness a segment
is defined. The influences of rheological phenomena occurring in steel and concrete are
considered.
Chapter 5 is a description of a numerical model for calculations of continuous reinforced
concrete beams derived in co-operation with the supervisor of this thesis. This chapter
provides a description of the algorithm along with a concise overview and evaluation of
computer methods which can be applied in the approach aimed in this thesis.
Chapter 6 summarises the work done in this thesis and evaluates the applicability of
the presented model together with the future prospects.
1
Model Code is a generalised theoretical basis for codes while the Eurocode 2 is the currently valid
standard. Each time a reference to the standard is made, the reader should understand the EC2, unless
stated otherwise.
Chapter 2
Models for analysis of reinforced
concrete beam elements
Two main reasons of non-linearity in engineering structures can be distinguished:
1. material (physical) if the material has a non-linear σ– characteristic,
2. geometrical (kinematic), when initial and final configuration of the system is considered or as a result of initial deformations.
Two characteristics are important for good understanding of these phenomena [21]: a
material behaviour and a cross-section behaviour. The first one defines the relationship
between stress and strain (σ–) for a given material while the other represents the
relationship between the internal force in the section and its deformation (moment and
curvature of the deformation line M –κ).
The structure can be analysed in a number of ways, depending on the initial assumption
of a designer. One must remember that no model can perfectly represent the real behaviour
of the structure and that the assumptions of the model are made to provide optimal
simplicity and accuracy of results for satisfaction of the design requirements. The choice
of the method depends on the assumed behaviour of the material and the possible
consideration of the effects of deformation on the action effects (second order effects). This
thesis presents the issues connected with implementation of some variants of the non-linear
analysis to reinforced concrete beam systems (considering 1-dimensional stress state).
2.1
Material non-linearity
There exists a great number of models representing the stress–strain behaviour of engineering materials. These models can be divided into groups according to the exhibited
properties they have in common. This might be the phenomena they represent, level and
time of loading, properties of occurring strain or whether the behaviour of the material at
compression and tension is the same or different. This section presents basic models for
structural analysis and dimensioning of reinforced concrete elements.
Concrete exhibits a complex structural response with various non-linearities: a nonlinear stress–strain behaviour, tensile cracking and compression crushing material failures,
and creep strains. This non-linearities together with non-linearities introduced by reinforcing steel should be taken into account [24].
9
10 CHAPTER 2. MODELS FOR ANALYSIS OF REINFORCED CONCRETE BEAM ELEMENTS
2.1.1
Models based on theory of elasticity
This group constitutes of a number of elastic models, from simple linear elasticity, through
various types of non-linear elasticities, to non-linear incremental elasticity. Such models
are characterised with the strain being dependent only on the actual state of stress, not
on the history of loading. Strains are totally reversible and the material behaves in the
same manner at loading, unloading and reloading.
Assumption of the linear elasticity signifies that the relationships between the components of stress and strain are linear (Fig. 2.1a1 , Fig. 2.1b). It is valid for infinitesimal
strains or small deformations, and for stress states that do not produce yielding.
(a) linear-elastic model
(b) perfectly-rigid model
Figure 2.1: Models based on theory of elasticity
The linear elasticity is the simplest example of the more general non-linear theory
of elasticity, convenient for the materials in elements which undergo large deformations,
and includes such material models as the Cauchy model, hyperelastic or Green model and
hypoelastic model. Nevertheless, these models are good representation for materials such
as polymers, and are rather useless with respect to reinforced concrete.
2.1.2
Models based on theory of plasticity
(a) elastic–perfectly-plastic model
(b) rigid–perfectly-plastic model
Figure 2.2: Models based on theory of plasticity
1
cu and fc signify the ultimate compressive strain and strength while tu and ft the ultimate tensile
strain and strength, respectively.
11
2.1. MATERIAL NON-LINEARITY
(a) plastic phase with hardening
(b) plastic phase with softening
Figure 2.3: Elasto-plastic models with hardening and softening
Operation of that group of models can be divided into initial elastic phase (usually
linear or rigid) and non-elastic phase, where relationship between stress and strain can be
presented in an incremental manner (plastic flow). Here the strain depends not only on
the actual stress state but also on the loading history. Though, such models exhibit both
reversible and irreversible strains and unloading goes on the path other than initial loading.
The idealised stress–strain diagram in Fig. 2.2a2 is referred to as the elastic-perfectly
plastic behaviour. Plastic deformations are often significantly larger than in the elastic
phase so the model can be further simplified to the rigid-perfectly plastic (Fig. 2.2b).
Plastic phase can be treated in more sophisticated way as plastic with hardening
(Fig. 2.3a), if the progressive increase of yield stress after yielding needs to be taken into
account (such as in steel), or softening (Fig. 2.3b) in which stress decreases with increasing
strain (i.e. in materials experiencing brittle damage due to cracks such as concrete). Linear
hardening and linear softening in plastic phase is a convenient approximation. However, a
more realistic description of hardening in metals can be achieved with non-linear hardening.
On the other hand, softening in quasi-brittle materials such as concrete can be modelled
in a bilinear or exponential manner [6].
2.1.3
Rheological models
In some materials like concrete, steel, soil or polymers a phenomenon of flow can be
observed even under very small stresses. The phenomenon is observed to demonstrate
the features of viscous flow and the material models can be identified as rheological or
rate-dependent. For these materials the elasto–plastic, let alone linear-elastic models are
highly inaccurate, so the viscous effects must be incorporated.
The viscous effects can be considered in the material model in different manners, by
assuming that they [8]:
• are connected only with elastic strains (viscoelastic and viscoelasto-plastic models),
• are connected only with plastic strains (elasto-viscoplastic models),
• occur in the whole range of material work (viscoelasto-viscoplastic models).
2
In the diagram the symbols el and pl signify the elastic and plastic component of strain. Their
meaning is only qualitative.
12 CHAPTER 2. MODELS FOR ANALYSIS OF REINFORCED CONCRETE BEAM ELEMENTS
(a) Hooke’s element
(b) Saint-Venant’s element
(c) Newton’s element
Figure 2.4: Basic elements for decription of rate-dependent materials
The physical characteristics of the rheological materials can be described as a sequence
of basic elements:
1. Perfectly elastic – represented by the Hooke’s spring element (Fig. 2.4a) with
stiffness E. The response of the spring to stress σ is instantaneous and the recovery
after release of the stress is instantaneous and complete. The stress σ applied is
proportional to the deformation , and the proportionality constant is the modulus
of elasticity E:
σ = E · .
2. Perfectly plastic – represented by the Saint-Venant’s sliding frictional element
(Fig. 2.4b) in which the friction force acts against the exerted tensile force P .
When P reaches its limit value P = Plim , the element undergoes permanent and
irreversible elongation. The behaviour of the model can be expressed as:
σ = σy
where σy is the yield limit.
3. Perfectly viscous – represented by the Newton’s dash-pot element (Fig. 2.4c) characterised by a viscosity η. There is no instantaneous response and no recovery takes
place. Stress rate is proportional to the rate of deformation, and the proportionality
constant is the viscosity η:
d
.
σ=η·
dt
The viscoelastic or viscoplastic behaviour is comprised of elastic and/or plastic and
viscous components modelled as a linear combination of springs, dash-pots and sliding
frictional elements. Viscotic behaviour in concrete is connected with creep (increase of
deformation under constant load) while in steel – with relaxation (decrease of stress under
constant deformation). The following rheological models can be distinguished [8].
The Maxwell model is a combination of perfectly elastic and perfectly viscous elements
combined in series (Fig. 2.5a). For a description of creep in concrete the model assumes
that the response under constant load is the sum of elastic and viscous response (Fig. 2.5b):
!
σ
σ
1
1
= 1 + 2 = + = σ ·
+
.
E
η
E η
13
2.1. MATERIAL NON-LINEARITY
(a) scheme
(b) creep in concrete
(c) relaxation in steel
Figure 2.5: Viscoelastic model (Maxwell model)
To represent the relaxation in steel, the stress change in time under constant deformation
is analysed (Fig. 2.5c). An exponential decay can be observed. A relaxation time τ is
defined as a time after which the initial stress is reduced by 1/e. In the generalised Maxwell
model3 a set of spring–dash-pot Maxwell elements are used to represent that the relaxation
occurs at a distribution of times.
(a) scheme
(b) creep in concrete
Figure 2.6: Viscoelastic model (Kelvin–Voight model)
The Kelvin–Voight model is used for description of creep. In the model, the elastic
and viscous elements are combined in parallel (Fig. 2.6a). Subjected to constant stress
(σ1 + σ2 = const.), it demonstrates an exponential increase of strain with relaxation time
σ
τ asymptotically approaching the steady-state value max = :
E
σ(t) = E · (t) + η ·
d
.
dt
After removal of the load also an exponential decay of strain (reversible) can be observed
with the same relaxation time (Fig. 2.6b).
The Standard Linear Solid model (Fig. 2.7a) is a combination of the Maxwell model
and the Kelvin–Voight model. Under constant load total deformation tot is a sum of
three types of deformations (Fig. 2.7b): spontaneous elastic deformation 1 , delayed elastic
deformation (reversible creep) 2 and irreversible creep (flow) 3 .
The elasto-viscoplastic model in a general form (Fig. 2.8a) consists of a spring, a
dash-pot and a sliding frictional element connected in parallel, i.e. the spring element of
3
also known as the Maxwell–Wiechert model
14 CHAPTER 2. MODELS FOR ANALYSIS OF REINFORCED CONCRETE BEAM ELEMENTS
(a) scheme
(b) creep in concrete
Figure 2.7: Viscoelastic model (Standard Linear Solid model)
(a) general elasto-viscoplastic
(b) rigid-viscoplastic
Figure 2.8: Elasto-viscoplastic models
the Kelvin–Voight model replaced by the sliding frictional element acting as a rigid body.
This model is used for representation of creep in concrete. A specific type of that model is
the Bingham model (Fig. 2.8b) without a spring element (a rigid-viscoplastic model). In
that model the strain is irreversible afrer reaching the yield stress (yield limit) σy .
Figure 2.9: Viscoelasto-plastic model
Figure 2.10: Viscoelasto-viscoplastic model
The viscoelasto-plastic model (Fig. 2.9) is a modification of viscoelastic model in
which a plastic phase is considered. The model might be regarded as development of
the Kelvin–Voight model in which a frictional sliding element is added to simulate the
stress–strain relationship after the yield limit is reached.
Further improvement into the elasto-viscoplastic and viscoelasco-plastic models is
2.2. MATERIAL MODELS IN STANDARDS
15
consideration of viscous effects both in elastic and plastic phase. This leads to viscoelastoviscoplastic models. Despite their best representation of rheological properties of concrete,
such models are not widely presented because of their complexity. Nevertheless, development of computer methods should increase their applicability. Two basic viscoelastoviscoplastic formulations are given by Perzyna and Duvaut–Lions. In Fig. 2.10 the
viscoelasto-viscoplastic model according to Perzyna formulation is presented.
2.2
Material models in standards
2.2.1
Concrete models
Concrete in compression
Concrete is a brittle material as it has good properties in compression but fractures abruptly
when its tensile strength is reached as a result of decohesion. The linear elastic model is
used extensively in structural analysis and engineering design of concrete. Major design
codes such as the Model Code, BS 8110, EC2 and ACI 318, as well as PN-B-03264 endorse
this model.
However, in the linear-elastic analysis the plastic properties of concrete in compression,
which may in reality increase the load-bearing capacity of concrete member, are neglected.
Therefore, the standards enable application of non-linear material models, taking these
favourable properties into account. The models are based on Model Code and incorporated
into other European standards. In this section they are presented acc. to EC2.
For the structural analysis of elements under short-term loading the following stress–
strain relationship is proposed, presented in Fig. 2.11a and described by the formula:
kη − η 2
σc
=
fcm
1 + (k − 2)η
(2.1)
where:
σc – compressive stress,
fcm – mean compressive strength,
c
η =
,
c1
c – compressive strain,
c1 – compressive strain at reaching maximum compressive strength fcm ,
cu1 – ultimate compressive strain4 ,
c1
.
k = 1.05Ecm
fcm
This model is an example of the (elasto-)plastic model with softening. The linear-elastic
behaviour can be assumed for stresses not greater than 0.4fcm , above which the behaviour
becomes non-linear. When the material (compressive) strength fcm is reached, the element
can still transfer load until the ultimate strain ecu1 is reached which signifies failure.
Dimensioning of sections can be based on one of two proposed models: parabola–
rectangle model, presented in Fig. 2.11b, and the bi-linear model, presented in Fig. 2.11c.
4
For normal concretes cu1 = cu2 = cu3 = 3.5h, c2 = 2.0h, c3 = 1.75h, n = 2.0 [36].
16 CHAPTER 2. MODELS FOR ANALYSIS OF REINFORCED CONCRETE BEAM ELEMENTS
(a) model for static analysis
(b) parabola–rectangle model
(c) bi-linear model
Figure 2.11: Standard models for concrete in compression [36]
The parabola–rectangle model is given by the formula:

n

fcd 1 − 1 − c
for 0 < c < c2
c2
σc = 
f
for c2 < c < cu2
cd
and the bi-linear model by the formula:

f c

cd
c3
σc =

fcd
where:
fck , fcd
c2 , c3
cu2 , cu3
n
for 0 < c < c3
(2.2)
(2.3)
for c3 < c < cu3
– characteristic and design compressive strength,
– compressive strain at reaching maximum compressive strength fcd ,
– ultimate compressive strain,
– exponent.
Concrete in tension
Tensile failure of concrete is always a discrete phenomenon. Therefore, to describe the
tensile behaviour a stress–strain diagram (Fig. 2.12a) should be used for uncracked concrete
and a stress–crack opening diagram (Fig. 2.12b) should be used for cracked section [35].
(a) σ– diagram for uncracked concrete
(b) σ–w diagram for cracked concrete
Figure 2.12: Standard models for concrete in tension [35]
For uncracked concrete a bilinear stress-strain relationship is given in equations:



Eci ct







σct = 
ctu − ct 



f

1
−
0.1

ctm


0.9f

ctm 


−

ctu
Eci
for σct ≤ 0.9fctm
for 0.9fctm < σct ≤ fctm
(2.4)
17
2.2. MATERIAL MODELS IN STANDARDS
where:
Eci
fctm
σct
ct
ctu
– tangent modulus of elasticity,
– mean tensile strength,
– tensile stress,
– tensile strain,
– ultimate tensile strain5 .
For cracked section a bilinear stress-crack opening relationship is given in equations:

w


fctm 1 − 0.8
w1
σct = 
w

fctm 0.25 − 0.05
w1
for w ≤ w1
(2.5)
for w1 < w ≤ wc
where:
w
w1
wc
GF
– crack opening,
= GF /fctm , crack opening for σct = 0.2fctm ,
= 5GF /fctm , crack opening for σct = 0,
– fracture energy.
2.2.2
Steel models
Plastic material models are also ideal for representation of the properties of steel, namely
ductility, i.e. the ability to deform plastically without rupture, specifically under tensile
stresses. This is endorsed by the standards where two models are proposed: for hot-rolled
steel (Fig. 2.13a) and cold-worked steel (Fig. 2.13b).
(a) hot-rolled steel
(b) cold-worked steel
(c) idealised and design model
Figure 2.13: Standard models for steel [36]
In the hot-rolled–steel model the yield stress is explicit as the yielding of that type of
steel can be easily defined (visible yield plateau). The value of the yield stress is referred
to as the yield strength fyk of steel. The behaviour of the cold-worked steel is much more
difficult to describe. As the boundary between elastic and plastic behaviour is not that
visible, the assumptions for the model are more implicit. The yield limit is determined as
the f0.2k proof stress, i.e. the stress after reaching 0.2% strain. The tensile strain uk 6 is
related to reaching the tensile strength ft 7 of steel, after which steel begins to flow until
rupture. For design, simplified models presented in Fig. 2.13c are proposed.
5
ctu = 0.15h [35]
uk ≥ 2.5% for steel A, 5.0% for steel B and 7.5% for steel C
7
ft = kfyk or ft = kf0.2k , where k is coefficient dependent on the class of steel [36]
6
18 CHAPTER 2. MODELS FOR ANALYSIS OF REINFORCED CONCRETE BEAM ELEMENTS
2.3
Geometrical non-linearity
Long elements subjected to a large compressive force are on the verge of buckling, as under
such a load the lateral stiffness of the element reduces significantly and a small lateral
load may cause the element to buckle. Consideration of the influence of the secondary
effects caused by reduction of that “geometrical” stiffness is referred to as P –∆ analysis,
since additional moments arising in the element are the product of compressive force P
and displacement ∆. The problem is marginal for the issues analysed in this thesis and is
introduced juts for the notion of the reader.
Assuming the displacements in the structure to be small, the linear buckling theory
can be applied to determine the value of critical force Pcr (which leads to buckling of
compressed element). Buckling of the beam elements is assumed to be caused only by
bending moment exerted on that element; influence of transverse forces and shortening of
the element’s axis is neglected.
The value of the critical force is derived from a differential equation of deformation line.
It occurs that the solution of that equation is not exact, i.e. there is an infinite number
of critical forces and buckling modes that satisfy that equation. Buckling occurs for the
smallest value of the critical force for the first mode of buckling (described by sine/cosine
functions). The value of critical force is therefore:
Pcr =
where:
EI
lw
µ
π 2 EI
lw2
(2.6)
– bending stiffness of section,
– buckling length of compressed bar; lw = µl,
– buckling coefficient dependent on the support conditions of the element.
If the displacements in the structure are large, the curvature of buckled elements should
be described by a precise non-linear differential equation. Hence, the P (∆) relationship
becomes exact and signifies that after reaching the critical force any increase of compressive
force leads to extensive deflections and catastrophic increase of normal stresses.
The assumption that the “axial” force is subjected to the element along its axis and
that the element is straight is just an approximation. In reality, the elements have an initial
geometrical imperfection, so as a result the subjected force acts eccentrically which leads
to increase of bending moments exerted on this element. Nevertheless, that phenomenon
has no qualitative influence of the structure, merely defines the direction in which the
element will buckle and provides the exact P (∆) relationship.
In the presence of transverse forces the critical force decreases, so the element is more
prone to buckling under the axial compression. The shortening of the element’s axis may
have additional influence on the reduction of the critical force, especially in short elements
with a high elasticity limit.
Chapter 3
Refinement of linear-elastic analysis
results
3.1
Analysis of reinforced concrete members beyond
elastic phase
3.1.1
Linear-elastic vs. non-linear analysis
Under a given load bending moments as well as normal and shear forces take given values in
every point of the structure, depending on the static scheme and material properties of the
structure. As calculation of the structure (or cross-section) determination of distribution
and values of the internal forces in the structure, usually presented in a form of graphs,
with the assumed properties under given internal or external load should be understood.
Thus, distribution of forces and moments is a known state which can be determined
according to different theories of mechanics. Computation according to the theory of
elasticity is based on the assumption that the relationship between stresses and strains in
every cross-section of the structure remains linear until the moment of failure while the
theory of plasticity assumes that the stress–strain relationship is non-linear in a whole
range.
The linear-elastic analysis is the most frequently used for analysis of structures. However,
distribution of internal forces determined in such a way differs from the real distribution
because it poorly resembles the real behaviour and random characteristics of the materials.
Hence, the non-linear analysis is the only consistent way to verify the safety of the
structure satisfying both equilibrium and compatibility. The Model Code specifies that
“non-linear analysis is a realistic description of the physical behaviour and therefore a
method completely consistent with the assumptions used for the local verification and
member design” and “it should be used as a reference for other more simplified approaches”.
The following idealisations of structural behaviour are proposed:
1. linear-elastic behaviour,
2. linear-elastic behaviour with limited redistribution (Sec. 3.2),
3. plastic behaviour (Sec. 3.3),
4. non-linear behaviour (Sec. 3.4).
19
20
CHAPTER 3. REFINEMENT OF LINEAR-ELASTIC ANALYSIS RESULTS
Eurocode 2 specifies that linear analysis of elements based on the theory of elasticity may
be used for both the serviceability and ultimate limit states, having assumed that:
• cross-sections are uncracked,
• stress–strain relationships are linear,
• mean value of elastic modulus is considered.
For thermal deformation, settlement and shrinkage effects at the ultimate limit state,
a reduced stiffness corresponding to the cracked sections, neglecting tension stiffening but
including the effects of creep, may be assumed. For the serviceability limit state a gradual
evolution of cracking should be considered.
As it will be presented in the following sections, these assumptions are very simplified
and allow for structural analysis of only a limited range of cases. Therefore, the more
complex idealisations should be introduced in the design.
3.1.2
Plastic properties of reinforced concrete
In a rationally reinforced concrete beam reinforcement is located in the tensile zone while
concrete works in compression in all loading stages. When the tensile strength of concrete
is exceeded at the tensiled side, concrete is cracked and the work of reinforcing steel
becomes more intensive taking the overall tensile stress. As a result, particular parts of
the beam are in different stress stages, called phases, and the stiffness of the element is
reduced and no longer constant. In general, three phases of reinforced concrete member
work can be distinguished [13]:
1. phase I before the formation of the first crack,
2. phase II when section is cracked in tensile zone and steel takes all tensile forces,
3. phase III when equilibrium of forces is reached at the moment of failure.
Figure 3.1: Real work of flexural reinforced concrete beam [13]
Under the change of load in the most-loaded parts of the structure plastification of
concrete or steel occurs. Plastified parts of the structure work in a different manner
than before plastification. Incremental applied load, greater than the load producing first
yielding, is assumed to produce inelastic rotation at the yielded section, but no change to
the applied moment, so incremental moments are developed at sections other than the
initially yielded section. Static scheme and initial distribution of internal forces is changed.
3.1. ANALYSIS OF REINFORCED CONCRETE MEMBERS BEYOND ELASTIC PHASE
21
Under increasing load in the structure with a new static scheme a new distribution of
internal forces is formed in the parts which are still in elastic or partially-plastic phase.
Redistribution of forces and moments can be observed. Implicit in the current use of the
moment redistribution is the assumption that sections possess sufficient ductility for the
requisite plastic deformations to occur.
The parts in which plastification of sections is observed (i.e. where plastic deformations
occurred in concrete or steel) are called plastification zones. Plastic deformations from the
whole plastification zone are concentrated in a plastic hinge. Three types of hinges can be
distinguished [21]:
1. real hinge, not capable of transferring any moments (Fig. 3.2a),
2. partial plastic hinge in which rotation occurs and which can transfer the increment
of bending moment up to failure (Fig. 3.2b),
3. full plastic hinge which can undergo limited or unlimited plastic deformations but
cannot transfer the increment of bending moment over a given value of moment Mu
(Fig. 3.2c).
(a) real hinge
(b) partial plastic hinge
(c) full plastic hinge
Figure 3.2: Types of hinges [21]
A place in the structure where the plastic hinge can be formed or failure can occur is
called a critical section. In a statically-indeterminate structure there are usually several
critical sections – their number is generally greater than the level of statical indetermination
of the structure. Location of the critical sections can be estimated according to the elastic
theory – they usually occur in the places of the extreme bending moments.
The classical theory of reinforced concrete focuses on phase I and II. Dimensioning of
flexural members is based on the comparison of stresses in steel and concrete with their
allowable values. Deflections are calculated on the basis of Hooke’s law, both in phase I
and II, but for different values of modulus of elasticity. According to that theory, transition
between these two phases is instantaneous, which is not true – formation of a single crack
determines treatment of the member as in phase II while in the reality most parts which
are not yet cracks still work in phase I.
The theory of ultimate limit state focuses only on the critical sections neglecting
neighbouring sections in which the load-bearing capacity is not fully used, in such a
way that limit state (phase III) occurs almost independently of the previous stage, as
if a rigid, undeformable member would instantaneously undergo local failure. It is not
true in reality. In an arbitrarily supported beam under loading all of the phases can be
observed simultaneously, as presented in Fig. 3.1. Formation of a plastic hinge (phase
III) is demonstrated by widening and deepening of cracks, and excess deformations of the
member, and there is a transition between phase I and II and then between phase II and III.
The influence of stiffness degradation in a flexural element is crucial for adequate
representation of the behaviour of that element under incremental load. There is a mutual
relationship between the stiffness, the load and the deformation, which can be defined by
the equality on the basis of Bernoulli hypothesis:
22
CHAPTER 3. REFINEMENT OF LINEAR-ELASTIC ANALYSIS RESULTS
κ=
where:
v(x)
φ(x)
κ
M
B
dφ
d2 v
M
=
=
dx
d x2
B
(3.1)
– function of deformation line,
– function of rotation of the cross-section,
– function of curvature of deformation line,
– bending moment in cross-section,
– stiffness of the cross-section.
Thus, the deformation function can be found by the solution of such a second-order
differential equation. The stiffness law has to be known, which is not that difficult in
case of statically determinate elements but becomes highly analytically complicated in
continuous ones, as there is a mutual dependency between bending moment and stiffness
distributions.
(a) real [21], [6]
(b) approximated [20], [6]
Figure 3.3: M –κ relationship for flexural reinforced concrete elements
Figure 3.3a shows the relationship between moment Mx and curvature κ as a result
of stiffness degradation, which in statically-indeterminate elements decides about the
actual distribution of internal forces. Analysing that diagram it is visible that for small
values of load (section in phase I) the value of stiffness is constant and so the curvature is
proportional to the value of bending moment:
κel =
M
.
B0
(3.2)
Nevertheless, along with the increase of bending moment the value of the moment
M0 is reached for which this relationship is no longer linear. The increment of bending
moment is connected with displacements greater than for M < M0 , as the tensiled concrete
undergoes plastic deformations until the first crack is formed and the stiffness decreases.
After cracking, the compressed concrete continues to act linearly-elastic but it finally starts
to deform plastically, too. For that reason, the increase of curvature becomes faster and
is no longer linear when concrete is in plastic phase (κtot = κel + κpl ). Further increase
of load leads to the bending moment reaching its ultimate value Mu . Depending on the
proposed model, beyond this load limit the section undergoing further deformations can be
assumed to either sustain moments to transfer them into other sections, until the ultimate
curvature κu is reached.
3.2. LINEAR-ELASTIC ANALYSIS WITH MOMENT REDISTRIBUTION
23
The failure of an engineering structure falls into one of two simple categories: material
failure or structural instability. If the section fails due to exceeding its resistance this may
happen either as a result of tension (excess deformation or rupture of reinforcement) or
compression (crushing of concrete). When calculations are based on the plastic hinges
method, global (local) failure occurs if the structure with nth level of statical indetermination looses its global (local) stability as a result of formation of n + 1 plastic hinges
(turning into a mechanism). The load at which a mechanism forms in any span is called
the limit load in that span.
Precise modelling of the behaviour of a structure is significant in a process of development of new structural solutions, with a tendency to minimise the weight of structure and
economically use the materials. It is undeniable that, although we have precise methods
for determination of deformation properties of steel and concrete, distribution of internal
forces is determined based on the assumption of ideally-elastic cooperation of steel and
concrete, neglecting the phenomenon that plastification of both materials influences vast
areas beyond critical sections of the member.
The method based on plastic deformations seems to be rational, especially in case of
economical dimensioning, as it takes into account:
• plastic properties of concrete,
• magnitude of bending moment leading to failure of the section (phase III), against
which the structure is protected with safety factors > 1,
• assumption of simultaneous destruction of steel and concrete (condition of allowable
percentage of reinforcement).
Algorithms for dimensioning according to the theory of plasticity are based on experimental
data, considering deformability of concrete and steel, changes of neutral axis location,
magnitude of fracture moment and deflection as well as general properties of concrete such
as fire resistance, corrosion of steel and concrete, resistance to cracking, water permeability,
resistance to high temperatures, etc.
The approaches to refine the linear-elastic analysis results presented in the following
sections are presented in a form proposed by Eurocode 2.
3.2
Linear-elastic analysis with moment redistribution
The plastic behaviour of reinforced concrete at the ultimate limit state affects the distribution of moments in a structure. To allow of this, the moments derived from the elastic
analysis my be redistributed based on the assumption that plastic hinges have formed
at the sections with the largest moments. EC2 allows of such an adjustment of bending
moment with the ratio of the redistributed and initial bending moment diagrams being an
indicator of the amount of redistribution which has occurred. The percentage of moment
redistribution δ[%] at a section along a beam is calculated as follows:
Mred
· 100%
(3.3)
Mel
where Mred signifies the moment after redistribution and Mel the moment before redistribution, calculated for linear state.
δ=
24
CHAPTER 3. REFINEMENT OF LINEAR-ELASTIC ANALYSIS RESULTS
The initial elastic bending moment diagram thus forms the baseline for the redistribution
calculation and any assumptions or approximations made in its determination will directly
affect the level of redistribution calculated using the above expression.
It must be noted that in the static analysis based on the linear-elastic analysis with
limited redistribution the elastic–plastic σ– relationships are used, so the name “linear”
becomes a bit ambiguous as the method is non-linear in a mathematical sense.
The linear analysis with limited redistribution may be applied to the analysis of
structural members only for the verification of ULS and only for continuous beams and
slabs predominantly subjected to flexure and with the ratio of the lengths of adjacent
spans in the range of 0.5 to 2. Formation of plastic hinges requires relatively large rotations
with yielding of the tension reinforcement. Design codes achieve this by specifying rules
which ensure that the tension steel must have yielded, explicitly in the case of ACI 318
(which specifies a minimum reinforcement strain of 7500 microstrains) and implicitly in
the case of PN-B, BS 8110 and EC2 (which link percentage redistribution to neutral axis
depth) [31].
To ensure large strains in the tension steel, EC2 restricts the depth of the neutral axis:
xu
for fck ≤ 50MPa,
δ ≥ k1 + k2
d
xu
δ ≥ k3 + k4
for fck > 50MPa,
d
δ ≥ k5
if reinforcement type B or C is used,
δ ≥ k6
if reinforcement type A is used1 .
where:
xu
d
– the depth of the neutral axis at the ultimate limit state after redistribution
– the effective depth of the section.
0.0014
The recommended values of ki are : k1 = 0.44, k2 = 1.25 0.6 +
, k3 = 0.54,
cu2
0.0014
k4 = 1.25 0.6 +
, k5 = 0.7 and k6 = 0.8. Therefore, the limitation in the method
cu2
due to the limitation of the rotation in the plastic hinge results in the requirement that
the maximum allowable moment redistribution cannot exceed ±30% [20]. Redistribution
should not be carried out in circumstances where the rotation capacity cannot be defined
with confidence.
The BS 8110 imposes a minimum neutral axis depth of 0.11d (where d – effective
depth of beam) which, acc. to [31], has the effect of restricting reinforcement strains to
a maximum of 28000 microstrains when making the usual assumption of linear strain
distribution across the section. In reality, this value is largely meaningless since gross
yield of the reinforcement will have occurred by the time this neutral axis depth has
been reached, leading to strains greatly in excess of this nominal value. Therefore, the
reinforcement will be able to develop the required strain and the failure of section will be
caused by crushing of the compressed concrete.
The ACI Code provides two methods for determination of allowable redistribution [16].
In the first method the negative moments at the supports can be changed by the value:
2
1
2
A stands for low ductility steels while B and C for high ductility steels
In PN-B k1 = 0.44, k2 = 1.25, k5 = 0.7, k6 = 0.8. High strength concretes are not considered.
25
3.3. PLASTIC ANALYSIS
ρ − ρ0
δ ≤ 20 1 −
ρb
!
(3.4)
for ρ − ρ0 ≤ 0.5ρb , where:
ρ
ρ0
ρb
– ratio of tension reinforcement,
– ratio of compression reinforcement,
– reinforcement ratio producing balanced strain condition3 .
Alternatively, the negative moments can be changed by no more than 1000t %4 , with
a maximum of 20%, provided that t ≥ 7.5h at the section where moments are being
reduced.
The method of the moment redistribution presented in standards is a great simplification
of the real process of redistribution. Only a fact of transferring bending moments from the
most loaded to the less loaded cross-sections of the flexural element is considered without
a detailed analysis of the process itself. For that reason it is obvious that the proposal is
provided in a very implicit and uncontrollable way.
3.3
Plastic analysis
Methods based on plastic analysis shall only be used for check at the ULS. The effects of
previous applications of loading may generally be ignored and a monotonic increase of the
intensity of actions may be assumed.
The plastic analysis without any direct check of the rotation capacity may be used if
the ductility of the critical sections is sufficient for the envisaged mechanism to be formed.
The ductility condition is assumed to be satisfied if [36]:
xu
≤ 0.25 for
d
concrete strength classes at most C50/60, and ≤ 0.15 for concrete strength classes
at least C55/67,
1. the area of tension reinforcement is limited such that at any section
2. reinforcing steel is either class B or C5 ,
3. the ratio of the moments at intermediate supports to the moments in the span shall
be between 0.5 and 2.
If the rotation capacity has to be controlled, the simplified procedure is proposed
based on the rotation capacity of beam zones over a length of approximately 1.2 times the
depth of the section (Fig. 3.4). It is assumed that these zones undergo plastic deformation
(formation of yield hinges is observed) under the relevant combination of actions. Thus,
the plastic hinge in reinforced concrete is not a single-section phenomena, but occurs as a
result of large curvatures in the section and in its neighbourhood due to concentration of
deformations [9].
The verification of the plastic rotation in the ultimate limit state is considered to be
fulfilled if it is shown that under the relevant action the calculated rotation θs is less than
or equal to the allowable plastic rotation θpl,d .
The rotation θs should be determined on the basis of the design values for actions and
materials. In the simplified procedure, the allowable plastic rotation may be determined
3
i.e. when tension reinforcement reaches the strain corresponding to its specified yield strength fy as
26
CHAPTER 3. REFINEMENT OF LINEAR-ELASTIC ANALYSIS RESULTS
Figure 3.4: Plastic rotation θs of reinforced concrete sections in continuous beams [36]
by multiplying the basic value of the allowable rotation by a correction factor kλ that
depends on the shear slenderness, according to the formula:
s
kλ =
λ
3
(3.5)
where λ is the ratio of the distance between point of zero and maximum moment after
redistribution and effective depth d. As a simplification λ may be calculated for the
concordant design values of the bending moment and shear, as:
λ=
MSd
.
VSd · d
(3.6)
Figure 3.5: Allowable plastic rotation θpl,d of reinforced concrete sections [36]
The recommended values of θpl,d for steel classes B and C (the use of class A steel is
not recommended for plastic analysis) and concrete strength classes less than or equal to
concrete in compression reaches its assumed ultimate strain (3h).
4
t signifies the tensile strain in the reinforcement
5
In PN-B requirements for steel are expressed as “high ductility steel”, which actually coincides with
EC2 steel class B and C
27
3.3. PLASTIC ANALYSIS
C50/60 and C90/105 are given in Fig. 3.5. The values for concrete strength classes C55/67
to C90/105 may be interpolated accordingly. The values apply for the shear slenderness
λ = 3.0. For different values of the shear slenderness the θpl,d should be multiplied by kλ .
It must be noted that the concept of the rotational capacity of a reinforced concrete
section is not well-examined [9]. There is a number of factors to be controlled with respect
to their influence on the plastic properties of reinforced concrete, such as: scale (dimensions
of the element), length of segment in which the concentration of deformations occurs,
compression reinforcement, cyclic loads as well as normal and transverse forces.
3.3.1
Limit equilibrium method
The plastic analysis should be based either on the lower bound (static) method or on
the upper bound (kinematic) method. This can be presented on the simple example of
the limit equilibrium method. The idea of the limit equilibrium method, contrary to the
redistribution method in which the history of loading is important, is to find the limit load.
The distribution of the internal forces is determined based on the assumption that the
cross-section of the structure behaves elastically up to some given value of stresses (which
generates the ultimate moment Mu ), above which the full plastic hinge is formed (refer
to Fig. 3.2c) which is no longer able to carry any increments of stress but can undergo
unlimited deformations (Fig. 3.3b). The remaining cross-sections in which the value of
acting moment M < Mu behave in a linear-elastic manner.
The limit equilibrium method is acc. to [26] a simplification of the linear-elastic
analysis with redistribution. As it does not satisfy requirements of EC2, it should be –
as a consequence – forbidden. Nevertheless, under certain precautions, the method has a
number of practically approved advantages, such as:
1. economy is reinforcement design, especially visible with the increasing
q
ratio;
g+q
2. unification of reinforcement layout in mid-span and over supports (with a little
additional reinforcement of the edge span);
3. simplification of calculations.
Kinematic method
The kinematic method is referred to as the upper bound method as is provides the upper
boundary for the searched limit load.
In the kinematic approach the limit load is determined by comparison of a virtual work
of load and a virtual work in plastic hinges under an exerted bending moment (a detailed
scheme of the procedure is presented in Fig. 3.6 acc. to [20]).
Figure 3.6: Determination of limit load Qu with kinematic method [20]
28
CHAPTER 3. REFINEMENT OF LINEAR-ELASTIC ANALYSIS RESULTS
Under the incremental force Q two hinges are formed along with the displacement of
the point 1. From the equality of the virtual works of the external forces and bending
moments in the hinges, the value of the limit force can be derived as:
1
1
+ MBu .
(3.7)
bc
c
This solution is very easy, but the location of plastic hinges has to be known prior to
calculations of the limit load, which is not obvious in all cases.
Qu = M1u
Static method
The static method is referred to as the lower bound method as is provides the lower
boundary for the searched limit load.
In the static approach the analysis of transferring of the incremental load q to the
continuous beam is performed (a detailed scheme of the procedure is presented in Fig. 3.7
acc. to [20]).
The proportionally increased load q eventually reaches the value q1 which leads to the
internal support bending moment MBq1 reaching the ultimate value MBu . This results in
the plastic hinge formation in that cross-section. Hence, the static scheme of the structure
is modified and the additional load transferred to that model would be analysed for the
new scheme.
Along with the further increase of the load from q1 to q2 , the ultimate moment
MCu = MCq1 + MCq2 is reached over the next support, another plastic hinge if formed
and the static scheme is again modified. The process of the load increase resulting in
the formation of the subsequent plastic hinges can be repeated as long as the number of
the hinges does not exceed the level of static indetermination of the structure. In that
case for a 3-span beam with 4 supports and consequently 4 unknown support reactions,
5 hinges can occur under the load q3 in the structure until it turns into a mechanism.
qu = q1 + q2 + q3 is the limit load for that structure.
3.4
Non-linear analysis
The EC2 allows of application of non-linear methods of analysis for both ULS and
SLS, provided that equilibrium and compatibility are satisfied and an adequate non-linear
behaviour for materials is assumed. The analysis may be first or second order. Nevertheless,
no consistent design procedure is provided except for some design requirements which
must be satisfied:
• At the ultimate limit state, the ability of local critical sections to withstand any
inelastic deformations implied by the analysis shall be checked, taking appropriate
account of uncertainties.
• For structures predominantly subjected to static loads the effects of previous applications of loading may generally be ignored and a monotonic increase of the intensity
of the actions may be assumed.
• The use of material characteristics which represent the stiffness in a realistic way
but take account of the uncertainties of failure shall be used when using non-linear
analysis. Only these design formats which are valid within the restricted fields of
application shall be used.
3.4. NON-LINEAR ANALYSIS
29
Figure 3.7: Determination of limit load qu with static method [20]
• For slender structures, in which second order effects cannot be ignored:
– equilibrium and resistance shall be verified in the deformed state;
– deformations shall be calculated taking into account the relevant effects of
cracking, non-linear material properties and creep;
– where relevant, analysis shall include the effect of flexibility of adjacent members
and foundations (soil-structure interaction);
– the structural behaviour shall be considered in the direction in which deforma-
30
CHAPTER 3. REFINEMENT OF LINEAR-ELASTIC ANALYSIS RESULTS
tions can occur and biaxial bending shall be taken into account when necessary;
– uncertainties in geometry and position of axial loads shall be taken into account
as additional first order effects based on geometric imperfections.
Having understood the real behaviour of the flexural reinforced concrete members and
based on the methods proposed by the standards along with the requirements for non-linear
analysis, the model can be derived.
Chapter 4
Stiffness degradation in R/C flexural
members
4.1
Stiffness of reinforced concrete flexural members
The stiffness of a reinforced concrete beam decreases as the magnitude of the load increases.
Analysis and mathematical interpretation of this phenomenon is complicated for such a
complex material as reinforced concrete. Classic plastic properties have influence only on
the compressive side of the section. On the other side, cracks in concrete, cooperation of
tensiled concrete with reinforcing steel and non-uniform stress state in steel in cracks and
between them can be observed. Because of the complexity of these processes considering
all of the factors in a form of a multi-parameter cause analysis becomes difficult.
It is very important for an engineering practice to formulate a theory describing the
real work of statically-indeterminate elements in which there is a significant influence of
stiffness degradation in the process of flexure. In the analysis of a reinforced concrete
beam considering the changing stiffness at the length of the beam, classic approach is
insufficient. As it was stated in Sec. 3.1.2, in the beam under heavy loading all of the
phases of the work can be distinguished simultaneously. Therefore, the stiffness of the
beam is different at the supports and in the mid-span. Moreover, the stiffness, apart
from differing at the length of the beam, degrades in a given cross-section along with the
loading being increased. Hence, it is basically a function of two parameters [13]: location
x
ξ = and load P . However, there is a number of phenomena influencing the stiffness,
l
such as cooperation of steel and concrete at crack due to the bond or the effects of creep
and shrinkage regardless of the applied load.
The notion of stiffness degradation allows of solution of two groups of problems:
calculation of displacements and analysis of the work of statically-indeterminate beams.
In determination of deflection (as a consequence of action of bending moment) three main
approaches can be distinguished [20]:
1. consideration of the local degradation of stiffness in the cracked cross-section and
the neighbouring cross-sections,
2. averaging of the stiffnesses of cracked and uncracked sections,
3. averaging of the stiffness for the whole element.
It was experimentally proved that the process of deflection of the beam under the load
is continuous [13], and so the (1) approach is used very rarely [20]. The effects of the
31
32
CHAPTER 4. STIFFNESS DEGRADATION IN R/C FLEXURAL MEMBERS
cracks in the tensile zone, even if instantaneous, are compensated on one side by elastic
properties of steel and on the other – in the compressive zone – by elasto–plastic properties
of concrete. A drop in stiffness caused by formation of the crack occurs in a very small
part of the beam and the process is performed together with the increment of the moment,
which results in almost unnoticeable discontinuity. That is why the cross-section behaviour
M –κ can be represented by a smooth curve (as it was shown in Fig. 3.3a). Of course, for
such a non-uniform material as concrete, this theory to be true must be based on the
statistical data (mean values). Only in that sense concrete can be treated as continuum
and, consequently, the material and cross-sectional characteristics can be described by
analytical functions (for which the differential and integral calculus can be applied)[13].
To derive the model for a reinforced concrete flexural member based on the theory of
stiffness degradation, the following assumptions have to be made [13]:
• a specified reinforced concrete material and cross-section model,
• stiffness change both along the length of the beam and with the increase of load,
• validity of the theory in statistical sense,
• validity of the Hooke’s law in the initial phase of the beam’s work,
• limit state of failure occurring for stresses in steel reaching yield strength and in
compressed concrete compressive strength at flexure, equivalent to formation of
plastic hinge1 ,
• formation of cracks connected with cracking moment being equivalent to the second
ultimate limit state of the beam,
• curvature of the beam being related to the central axis of the beam (line along the
centres of gravity of the sections) and in physical meaning not coinciding with the
neutral axis (line along the zero-stresses in the sections), with neutral axis being a
theoretical (not material) axis.
4.2
Bending stiffness of cross-section
4.2.1
Material behaviour
Figure 4.1 presents the behaviour of the cross-section of the flexural reinforced concrete
member in different phases of work.
Phase I - uncracked section
For a reinforced concrete section in phase I a perfect bond between steel and concrete
can be assumed. The section, called a transformed section, is composed of concrete
section and n-time reinforcement section. The area of the transformed section Atrans for a
doubly-reinforced section:
Atrans = bh + n(As1 + As2 )
(4.1)
1
Note that in all proposed elasto–plastic reinforced concrete models failure is not a direct result of
exceeding material strength but reaching the ultimate compressive strain leading to crushing of compressed
concrete fibers.
4.2. BENDING STIFFNESS OF CROSS-SECTION
33
Figure 4.1: Phases of work of reinforced concrete member. Stresses [13]
where:
b, h – width and height of section,
As1 – reinforcement in tensile zone,
As2 – reinforcement in compressive zone.
Phase Ia (Fig. 4.1a). Under small loads stresses are linear, which means they are still
within the range of relative proportionality. Elastic modulus of concrete is constant so
it does not influence the change of stresses. The use of reinforcing steel is very small,
however, the neutral axis is located a bit lower than the centre of gravity of the section.
The depth of the compressive zone x for a rectangular cross-section:
bh2
+ nAs1 d + nAs2 a2
xI = 2
Atrans
where:
d
a1
a2
(4.2)
– effective height of beam, d = h − a1 ,
– distance of centroid of reinforcement in tensile zone from tensiled edge,
– distance of centroid of reinforcement in compressive zone from compressed edge.
The moment of inertia II :
II =
b(h − xI )3
bx3I
+
+ nAs2 (xI − a2 )2 + nAs1 (d − xI )2 .
3
3
(4.3)
The static moment of reinforcement about the centroid of the section SI :
SI = As1 z1,I + As2 z2,I
(4.4)
where z1,I and z2,I denote location of the reinforcement As1 and As2 with respect to the
central line of the section2 .
For a singly-reinforced section the component As2 is omitted.
Phase Ib (Fig. 4.1b). Increase of load, thus increase of bending moment leads to non-linear
stress distribution in the tensile zone. In the compressive zone, where stresses are still
2
Central line of the section (line along the center of gravity) does not necessarily coincide with a neutral
axis of the section for a given geometry and level of loading.
34
CHAPTER 4. STIFFNESS DEGRADATION IN R/C FLEXURAL MEMBERS
within proportionality range, their distribution remains linear. The use of reinforcement
increases but is still insubstantial. The neutral axis is raised. Determination of the
equilibrium equations for phase Ib is much more difficult than in phase Ia as distribution
of stresses is no longer linear.
Phase I last until the tensile strength of concrete is reached. The respective bending
moment reaches the value of the cracking moment Mcr . The cracking moment can be
determined under the assumption that a crack is formed when the tensile stress in concrete
reaches the mean tensile strength fctm and can be calculated as [9]:
Mcr = fctm Wc
(4.5)
where the bending index Wc is usually calculated for the concrete section (i.e. without
reinforcement), although it is allowed to use the transformed section characteristics. There
used to be a practice to account for plastic properties of concrete in tension, which led to
increase of the value of the cracking moment, but it no longer functions.
Phase II - cracked section
Phase IIa (Fig. 4.1c). Fracture of the reinforced concrete section (formation of cracks)
signifies transition between phase I and II. Initially, cracks are short and small but along
with increase of the bending moment they get wider and lengthen towards the compressed
side. Formation of the first crack is a random phenomenon, thus its location is also random
and unpredictable. Only after the first crack occurs, a pattern in formation of subsequent
cracks can be observed.
In the compressive zone stresses remain linear. Only a small part of concrete section
works in tension – the gross tensile stresses are transferred to reinforcing steel. The range
of phase IIa depends on the class of concrete and for very strong ones lasts almost up to
the moment of failure.
For singly-reinforced section:
• the depth of the compressive zone (from equilibrium of static moments):
s
xII =
• the moment of inertia:

nAs1 
2bd
1+
− 1 ,
b
nAs1
bx3II
III =
+ nAs1 (d − xII )2 ,
3
(4.6)
(4.7)
• the static moment of reinforcement about center of gravity of the section
SII = As1 z1,II .
(4.8)
v

u
n(As1 + As2 ) u
2b(A
d
+
A
a
)
s1
s2
2
t1 +
xII =
− 1 ,
(4.9)
For doubly-reinforced section:
• the depth of the compressive zone:
b
n(As1 + As2 )2
35
4.2. BENDING STIFFNESS OF CROSS-SECTION
• the moment of inertia:
III =
bx3II
+ nAs1 (d − xII )2 + nAs2 (xII − a2 )2 ,
3
(4.10)
• the static moment of reinforcement about center of gravity of the section
SII = As1 z1,II + As2 z2,II .
(4.11)
Phase IIb (Fig. 4.1d). Along with further increase of the load, the tensiled concrete
undergoes heavy cracking while in the compressive zone significant plastification of concrete
can be observed. Stress distribution becomes totally non-linear and the Hooke’s law no
longer applies. Therefore, similarly as in phase Ib, determination of that distribution is
very difficult.
Phase III - ultimate limit state
Figure 4.1e presents the cross-section in the ultimate limit state of flexure (when bending
moment reaches its maximum value). Distribution of stresses is non-linear. In compressive
zone its shape is close to the 2nd or 3rd order curve. In practice a non-linear diagram of
stress in concrete – as a result of chosen diagram for dimensioning (refer to Fig. 2.11b) –
can be substituted with [18]:
1. the rectangular–parabolic stress block (Fig. 4.2b),
2. the equivalent rectangular stress block (Fig. 4.2c).
(a) cross-section
(b) rectangular–parabolic
(c) equivalent rectangular
Figure 4.2: Simplified diagrams for distribution of stresses in ULS [18]
In the classical reinforced concrete design it is assumed that the structural failure
occurs when the ultimate limit state is reached in one section (i.e. when the bending
moment induced exceeds the maximum allowable moment). Actually, the load-bearing
capacity of the section is not yet depleted when the material strength is reached – the
section can still undergo some limited deformations (strains). When yielding of steel occurs
(i.e. when σs = fyd ), a plastic hinge is formed and the section posseses some rotational
capacity to sustain loads and transfer them into the less congested areas. Not until the
number of plastic hinges exceeds the degree of statical indetermination, does the failure
occur as a result of the global stability loss. Hence, the load-bearing capacity is not a
property of a single cross-section but of a structure as a whole.
36
4.2.2
CHAPTER 4. STIFFNESS DEGRADATION IN R/C FLEXURAL MEMBERS
Cross-section behaviour
Models for a cross-section behaviour (M –κ relationship) are collectively presented in
Fig. 4.3 acc. to [20].
(a) model 1
(b) model 2
(c) model 3
(d) model 4
(e) model 5
(f) model 6
(g) model 7
(h) model 8
(i) model 9
Figure 4.3: Models for cross-section behaviour [20]
Figure 4.3a presents the linear-elastic model for the flexural element. The constant
stiffness is a product of the modulus of elasticity Ec and the moment of inertia of the
section Ic , i.e. B = Ec Ic . Depending on the standard, the value of the modulus of elasticity
is assumed to be Ec = Ec0 (line 1), Ec = 0.85Ecm (line 2) or Ec = 0.625Ecm (line 3),
while the moment of inertia is derived for the uncracked transformed section Ic = II . The
model presented in Fig. 4.3b is a modification of the previous model taking into account
occurrence of the cracks. Phase I is neglected in that model.
The models in Fig. 4.3c and Fig. 4.3d are the first models considering the stiffness
degradation in a flexural reinforced concrete element in different phases of work. The first
one (Fig. 4.3c), given by Muraszow and incorporated by old Russian standards, assumes a
stiffness jump at crack formation. This model was applied initially in Polish Standard as
37
4.2. BENDING STIFFNESS OF CROSS-SECTION
well, but in the revision published in 1976 it was replaced by the one in Fig. 4.3d, where a
transition zone between the cracked and uncracked section under the load in the vicinity
of the cracking moment is more smooth.
The influence of stiffness degradation is well approximated by the model in Fig. 4.3e
(line 1), especially for the loads M ≤ 0.9Mu . The value of strain derived for the uncracked
section is increased by the value of strain caused by crack formation. The behaviour of the
section in phase II can be treated as either linear or non-linear. The ultimate limit state
is determined for the modified version of that model in which the failure is assumed to
occur for the value of (Mu , κu ) (line 2). However, that far-from-reality assumption leads to
serious design errors, which in contrary is not a problem for a solution developed by Levi
presented in Fig. 4.3f. The best representation in a form of a broken lines was derived by
Macchi for CEB and is shown in Fig. 4.3g.
The last group of models, presented in Fig. 4.3h, assumes a continuous stiffness
degradation. The most popular are the models proposed by Kuczyński in [13]. These
models satisfy the assumptions stated in Sec. 4.1 and coincide with the experimental
results (Fig. 4.3i shows the experimental data in which discrete values are approximated
with a parabolic function).
The bending stiffness B with the initial value of B0 is degraded in the whole range of
work of the flexural element proportionally to the ratio of the active bending moment Mi
and ultimate bending moment Mu in the section under the load qi . This proportionality
can be either direct (linear) or power. In the first case the stiffness degradation is referred
to as mutation φ and defined by the formula:
Mi
1−φ
,
Mu
B = B0
(4.12)
while in the other it is referred to as mutation ψ and defined by the formula:
"
Mi
B = B0 1 −
Mu
ψ #
.
(4.13)
The values φ and ψ are empirical coefficients so in order to achieve a satisfactory approximation, these values must be indicated empirically. The approximation accuracy is limited
to the value Mi /Mu < 0.85, but the analyses prove that the mutation φ can be used up to
Mi /Mu = 1 [20].
For a given state of stress in the section, the curvature of deflection line at this section
is defined as [21]:
s − c
κ=
(4.14)
d
where:
c – maximum compressive strain (of outermost concrete fibre),
s – maximum tensile strain (of tensile reinforcement),
d – effective depth of cross-section.
Phase I - uncracked section
The cross-section works within an elastic range and concrete cooperates in transferring
tensile stresses (M < Mcr ). The maximum compressive and tensile strains are equal to
(Fig. 4.4a):
38
CHAPTER 4. STIFFNESS DEGRADATION IN R/C FLEXURAL MEMBERS
Figure 4.4: Phases of work of reinforced concrete member. Strains [18]
c = −
σc
M xI
=−
,
Ec
Ec II
(4.15)
s = −
σs
M (d − xI )
=−
,
Ec
Ec II
(4.16)
M
Ec II
(4.17)
and the curvature:
κI =
where:
Ec
II
– elastic modulus of concrete,
– moment of inertia for uncracked section, acc. to Eq. 4.3.
Phase II - cracked section
The curvature of the cracked section (Fig. 4.4b) can be derived analogically to the curvature
of uncracked section, neglecting the cracked tensile part of concrete section, for the moment
of inertia of the cracked section III acc. to Eq. 4.7 or Eq. 4.10 [18]:
κII =
M
.
Ec III
(4.18)
Phase III - ultimate limit state
The ultimate value of the curvature κu can be easily determined assuming the ultimate
value of c = cu and x = xlim (Fig. 4.4c), for which:
cu
κu = −
.
(4.19)
xlim
For normal concretes the ultimate value of compressive strain should be taken as 3.5h. The
ultimate value of the location of the neutral axis should be limited according to the balanced
design recommendation, i.e. in a way that provides simultaneous depletion of concrete
and reinforcement resistance (concrete and tension steel reach their ultimate strains at the
same time). When no distribution is considered this value is xlim = xbal = 0.45d, and is
smaller if moments are redistributed [18].
4.2. BENDING STIFFNESS OF CROSS-SECTION
4.2.3
39
Creep and shrinkage effects
The works focused on derivation of a good model for the behaviour of the flexural reinforced
concrete elements result in creation of newer and better models. Nevertheless, so far
there is no model which, with sufficient accuracy, resembles all of cases of the element’s
work, let alone under the long-term loading. It is essential to prepare a model taking into
consideration the influence of rheological effects, which at the moment are modelled with:
• change of moduli of elasticity of steel and concrete,
• application of multiplication factors increasing the final value of deflection calculated
for short-term loading,
• application of additional multiplication factors increasing the initial value of deflection
to account for creep and independently addition of the value of deflection due to
shrinkage.
Creep
The effect of creep results in increasing deflections with time and should be included in
calculations by using an effective modulus of elasticity Ec,ef f [18]:
Ec,ef f =
Ecm
1 + φ(∞, t0 )
(4.20)
where:
φ(∞, t0 ) – creep coefficient equal to the ratio of creep strain to initial elastic strain3 ,
Ecm
– mean elastic modulus of concrete.
Shrinkage
The effect of shrinkage of the concrete is a change of curvature by κcs and consequently
the deflection of the member; it can be calculated according to the formula [18]:
κcs =
Es cs S
Ec I
(4.21)
where:
cs
Es
S
I
– free shrinkage strain4 ,
– modulus of elasticity of steel,
– static moment of reinforcement about the centroid of the section (S = SI or SII ),
– moment of inertia of the section (I = II or III ).
4.2.4
Tension stiffening effect
In a cracked cross-section all tensile forces are balanced by the steel only. However,
between adjacent cracks, tensile forces are transmitted from the steel to the surrounding
concrete by bond forces. The contribution of the concrete may be considered to increase
3
4
For detailed procedure of determination of φ(∞, t0 ) refer to [36].
For detailed procedure of determination of cs refer to [36].
40
CHAPTER 4. STIFFNESS DEGRADATION IN R/C FLEXURAL MEMBERS
the stiffness of the tensile reinforcement. This effect is called the tension stiffening effect.
If the tension stiffening effect is neglected, the stiffness of a reinforced concrete bar or a
structural member is underestimated.
The influence of the tension stiffening effect can be introduced with a coefficient [18]:
ξ(M ) =


0
if M < Mcr
Mcr

1 − β
M
if M ≥ Mcr
(4.22)
where:
β
Mcr
M
– load duration factor; β = 1 for short-term load, β = 0.5 for sustained loads,
– cracking moment,
– design bending moment for calculation of curvature and deflection.
4.2.5
Mean moment–curvature relationship
The mean curvature κmean should be based on both cracked and uncracked sections, taking
into consideration the effects of creep, shrinkage and cooperation of steel and concrete at
crack in a form of tension stiffening effect. The modified values of curvatures κI and κII
allowing for creep and shrinkage effects are given by formulas:
κI =
M − Es cs SI
,
Ec,ef f II
(4.23)
κII =
M − Es cs SII
.
Ec,ef f III
(4.24)
An average value of curvature κm can be obtained using a formula [18]:
κmean = (1 − ξ)κI + ξκII
(4.25)
where:
κI
κII
ξ
– curvature for uncracked case, considering creep and shrinkage, acc. to Eq. 4.23,
– curvature for cracked case, considering creep and shrinkage, acc. to Eq. 4.24,
– coefficient allowing for tension stiffening effect.
4.3
Bending stiffness of member
4.3.1
Influence of cracks
Distribution of stiffness
The most popular is an assumption that the stiffness of a member is constant along its
length (Fig. 4.5b). Providing the considerations of this thesis it becomes obvious that such
an assumption is far from reality and more sophisticated models should be introduced.
The influence of cracks may be treated as a local phenomenon, thus locally influencing
the stiffness of the member (Fig. 4.5c). It can be assumed that beyond the cracked area
sections behave as in phase I, while the cracked parts are considered as either parts of
limited length, working in phase II or as points where additional elastic rotations occur.
41
4.3. BENDING STIFFNESS OF MEMBER
The continuous stiffness degradation acc. to [13] is presented in Fig. 4.5d. This is
quite a good approximation, except that in the zone of minimum moments there should
be a segment of constant stiffness (which results from the analysis of the experimentally
obtained M –κ relationship presented in Fig. 4.3i). This problem is solved by modification
proposed in [20] presented in Fig. 4.5e.
(a) cracked flexural beam
(b) constant stiffness
(c) local stiffness degradation
(d) continuous stiffness degradation
(e) modified continuous stiffness degradation
Figure 4.5: Stiffness of flexural beam along the element [20]
Cracking pattern
During the state of crack formation one crack after another occurs decreasing the stiffness of
the member. When cracks appear, single cracks play an important role. In this state some
parts of the area between cracks remain in phase I. After the crack formation is finished,
the mean spacing between cracks where bonded reinforcement is fixed at reasonably close
centres within the tension zone (spacing ≤ 5(c + φ/2)) can be taken as [36]:
sr,max = k3 c + k1 k2 k4
φ
ρp,ef f
(4.26)
42
where:
c
φ
k1
k2
k3
k4
ρp,ef f
CHAPTER 4. STIFFNESS DEGRADATION IN R/C FLEXURAL MEMBERS
– concrete cover,
– diamater of reinforcement,
– coefficient dependent on bonding properties; k1 = 0.8 for ribbed bars,
– coefficient dependent on strain distribution; k2 = 0.5 for bending,
– coefficient; recommended value k3 = 3.4,
– coefficient; recommended value k3 = 0.425,
– effective reinforcement ratio, given by formula:
As
ρp,ef f =
Ac,ef f
where:
As
Ac,ef f
(4.27)
– area of reinforcement,
– effective area, for rectangular section Ac,ef f = hef f b, where:
!
h−x
hef f ≤ min 2.5(h − d);
.
(4.28)
3
Where spacing of the bonded reinforcement exceeds 5(c + φ/2) or where there is no bonded
reinforcement within the tension zone, the maximum crack spacing may be assumed as:
sr,max = 1.3(h − x)
(4.29)
where:
h – height of section,
x – depth of neutral axis.
In the analysis with numerical methods it can be assumed that cracks are either a fuzzy
or concentrated phenomenon [9].
4.3.2
Influence of reinforcement
In the analysis, the flexural stiffness EI is used to determine the values of the internal
forces in statically indeterminate systems. Since the reinforcement is not know until the
end of the design process, in static calculations the concrete section is most frequently
used. The use provides simplicity but also has implications for the moment redistribution.
It is intuitive that distribution of bending moments in a statically indeterminate system
is different if initially a constant flexural stiffness is assumed and when a local degradation
of stiffness is considered where reinforcement is applied, even though all sections along
the member are behaving in a linearly-elastic fashion. Therefore, it can be said that the
so-called relative not the absolute values of EI determine the moment distribution and
that the stiffness changing along the member influences not only the ULS but also the
SLS (when reinforcement is behaving elastically) [31].
The redistribution resulting from the mismatch between the assumed and actual stiffness
values can be termed the elastic redistribution while the post-yield redistribution (after
the reinforcement yields) can be referred to as the plastic redistribution [31]. The plastic
redistribution is widely recognised and used. Nevertheless, the total redistribution in ULS
design is actually a sum of elastic and plastic redistribution, as the elastic redistribution
will occur even if it is not anticipated.
In order to take into account the actual stiffness distribution of the reinforcement, the
reinforcement layout has to be known prior to undertaking the analysis. Such an approach
is possible only with iterative numerical methods.
Chapter 5
Numerical model
There are two main criteria that should be satisfied by the model of a physical phenomenon
[24]: the model should be as simple as possible, but reproduce the important characteristics
consistent with experimental results and should be theoretically sound and computationally
stable, so that reliable analysis results are obtained.
A distinction has to be made between a continuous and a discrete approach [22]. A
discrete approach is such a situation in which a finite number of components can be
extracted while a continuous approach is defined using the mathematical functions of an
infinitesimal which leads to differential equations or equivalent statements implying an
infinite number of elements.
Discrete problems can be solved easily, even for situations with a great number of
elements, which is not so obvious in continuous problems. That is why a discretisation
of continuous problems can be applied where it is inefficient or impossible to solve those
problems exactly by mathematical manipulation. A design performed for a non-linear
constitutive material model assumed in this thesis requires discretisation and numerical
approach.
Derivation of a numerical simulation of a physical phenomenon is an interdisciplinary
task. In case of this thesis a structural engineer and a software developer must be engaged
in the process, which is schematically preseted in Fig. 5.1.
Figure 5.1: Process of numerical simulation of physical phenomenon
This thesis is oriented towards the first and the second phase of the model derivation
process. If a problem in the model can be solved in a number of ways, their advantages
and disadvantages are analysed and the applicability evaluated. No choice is made as the
optimisation of the computational performance is not the aim of this thesis.
43
44
CHAPTER 5. NUMERICAL MODEL
5.1
Static scheme
5.1.1
Geometry
As an input data an arbitrary continuous beam is chosen with known geometry and
reinforcement layout. An exemplary beam is shown in Fig. 5.2.
Figure 5.2: Exemplary continuous beam
Such a beam can be represented as a bar structure in which each span is a bar of a
given effective length Lef f and defined geometrical and material properties. It is assumed
that each bar spans between two nodes and that the bar is supported in the nodal points
only. The static scheme of the beam is shown in Fig. 5.3.
Figure 5.3: Static scheme of the beam
According to Sec. 4.3.2, location and amount of reinforcement has the influence on the
stiffness of cross-sections and consequently on the distribution of internal forces. Therefore,
reinforcement is assumed from the beginning of the computational process and as a result
four types of sections are distinguished. Definition of sections is provided in Fig. 5.4. For
a beam with a greater number of spans and different reinforcement layouts the amount of
types of section may be greater.
Figure 5.4: Material and geometrical characteristics
5.2. MATERIAL MODEL
5.1.2
45
Discretisation
To perform the finite element analysis, the beam has to be discretised, i.e. divided into a
finite number of sub-elements. In this model the beam is divided into the sub-elements,
referred to as segments, of the same length l and in such a way that each ith segment
includes cross-sections of one type only; the type of the ith segment is denoted j = t(i).
Thus, one can conclude that the geometry of the beam and the reinforcement layout
should be defined with the precision not greater than the precision of the definition of the
segments, i.e. if the length of the segment is assumed to be l, then the length of the beam
and the length and location of reinforcement has to be a multiplication of l (Lef f = n · l).
Figure 5.5 shows the static scheme of the beam after discretisation.
Figure 5.5: Static scheme of the beam after discretisation
The segment has the same properties (in a statical sense) as a bar: it has a given
length, a start and an end node, and defined geometrical and material characteristics. At
the beginning of the computational process each ith segment has the stiffness of the j th
type of cross-section it includes, determined for the the phase I (Bt(i) = Bj = EII,j ).
5.2
Material model
The material model for concrete is chosen to be composed of the MC’s models: the model
presented in Fig. 2.11a and described in detail in Sec. 2.2.1, to represent the reinforced
concrete σ– relationship in compression and the model presented in Fig. 2.12a and
described in detail in Sec. 2.2.1 for tension, for stresses ≤ fctm , beyond which cracking is
assumed and cooperation of concrete in section is neglected. The input concrete material
model is shown in Fig. 5.6a.
The material model for steel in chosen to be an EC2 model presented in Fig. 2.13c and
described in detail in Sec. 2.2.2. The steel is considered to be an isotropic material, i.e. its
characteristic in tension and compression is the same. The input steel material model is
shown in Fig. 5.6b.
The strengths of concrete can be taken as characteristic values fck , fctk , design values
fcd , fctd or mean values fcm , fctm , depending on the nature of the problem being solved.
The ultimate strains for characteristic, designs and mean values are cu = 3.5h and
ctu = 0.15h for compression and tension, respectively.
The strengths of steel can be also taken as either characteristic (ftk = fck and fyk ) or
design (ftd = fcd and fyk ) values. The ultimate strain cu = ctu = u depends on the class
of steel, and for characteristic values is equal to 25h, 50h and 75h for class A, B and C,
respectively.
Note that an arbitrarily complex material model can be used without depriving the
model of its computational performance.
46
CHAPTER 5. NUMERICAL MODEL
(a) for concrete
(b) for steel
Figure 5.6: Chosen σ– relationship
5.3
Cross-section model
The cross-section model (M –κ relationship) is derived separately for each type of crosssection. To find one point of the M –κ diagram a numerical procedure of finding the
strain distribution in the section is performed. It must be noted that there is a possibility
that each section can be subjected to either a positive or negative bending moment, i.e.
compression can occur in either top or bottom fibers1 .
5.3.1
Curvature
The procedure is carried out in increasing steps for a constant increment of curvature
∆κ. A geometrical interpretation of the curvature is a tangent of the inclination angle
α of the strain line (a) representing the strain distribution as a function of a vertical
location a ∈ [0, h]. The strain line inclination angle must be assumed as α ∈ [−90◦ , 90◦ ] to
determine M –κ values within both positive and negative range of M –κ set (Fig. 5.7).
(a) κ > 0
(b) κ < 0
Figure 5.7: Geometrical interpretation of curvature
For the assumed κk the location of a strain line is determined in iterations in such a
way that there is pure bending in the section (N = 0). The bisection method is directly
applicable, because the strain distribution in a section is a monotonic function with a root
within the range of interest. It is also the simplest and the most stable iterative method.
Other iterative methods include linear interpolation methods (the regula falsi method and
the secant method) or the Newton–Raphson method 2 .
1
Conventionally, a positive bending moment denotes tension of bottom fibres while negative their
compression.
2
also known as Newton method or tangent method
47
5.3. CROSS-SECTION MODEL
The group of linear interpolation methods is similar to bisection method, but although
they all assure convergence of the solution, the former reach it faster. i.e. in a smaller
number of iterations, thanks to introduction of the further knowledge of the function.
Choosing between the linear interpolation and the Newton method it is advised to think
what is easier to calculate for a given function – its value or the value of its derivative.
Because the function to analyse is the function of axial force N (a) (as formulated in
Eq. 5.3) and taking into account the problems of convergence in the Newton method, the
linear interpolation methods, or at least the bisection method, seem to be better solution
in this case.
The procedure of finding the strain line is presented on the example of the bisection
method. Two parameters are required to determine the strain distribution in the section.
For a known curvature κ the values of strains along the height of the section can be
determined assuming either the value of strain at one edge or the location of the neutral
axis x ((x) = 0). In the first case, when the value of strain is assumed – let it be the
maximum strain on top edge tmax – the strain distribution function has a form:
(a) = κa + tmax .
(5.1)
In the other case, this function has a form:
(a) = κ(a + x).
(5.2)
In each approach addition and multiplication is needed to find one value of strain,
so for a defined finite number of strains m to be calculated both approaches require the
same amount of operations, i.e. 2m operations. As the strain distribution is needed to
determine the stress distribution, the choice of the approach is based on the convenience
it will provide in the next steps of the process. However, both the values of the outermost
strains and the location of the neutral axis are necessary for further computations. The
choice can be arbitrary, so let us choose that the procedure will be presented for the second
approach.
In the first step, the curvature κ1 = ∆κ is assumed. The boundaries for bisection are
the cases when x = 0 and x = h, for which the resultant forces have the opposite signs.
These two boundary forces can be denoted N |x=0 = N01 and N |x=h = Nh1 . N0k and Nhk
always have the opposite signs. Thus, in the first iteration the neutral axis is assumed to
lie in the middle of the section, i.e. x11 = h2 . The top and the bottom strains are equal
(t,11 = b,11 ). The resultant axial force N11 is calculated. If N11 6= 0 then it is either a
compressive or tensile force. The bisection procedure must continue within the range of (0,
h
) if N01 · N11 < 0 or ( h2 , h) if Nh1 · N11 < 0, and then again and again, until in the F th
2
iteration there is an x1F found for which N1F = 0. The x1F is the location of the neutral
axis for the first step. A graphical representation of that procedure is presented in Fig. 5.8.
The process is preformed for the next steps: κ2 = κ1 + ∆κ, κ3 = κ2 + ∆κ, ...,
κk = κk−1 +∆κ = k∆κ, until after K steps, for κK the ultimate value of strain (compressive
or tensile) is reached on one edge, which refers to reaching the ultimate value of curvature
κu . The following steps of the process are schematically depicted in Fig. 5.9.
5.3.2
Bending moment
The values of the internal forces acting in the section are computed with integration
of the function of stress distribution in the section σ((a)) corresponding to this strain
distribution acc. to the chosen concrete and steel material models. Although the chosen
48
CHAPTER 5. NUMERICAL MODEL
(a) curvature κ1
(c) first iteration
(b) boundary forces N01 and Nh1
(d) second iteration
(e) last iteration
Figure 5.8: Determination of strain distribution in cross-section with bisection method
(a) first step
(b) intermediate step
(c) last step
Figure 5.9: Distribution of strains in cross-section for different levels of loading
material models are defined with the analytically differentiable functions, application
of numerical integration is chosen because of its flexibility – a material model of any
complexity can be used in further modifications of the program, including models obtained
in laboratory tests (presented in a form of set of measurements).
For a known strain distribution function in the f th iteration of the k th step, kf (a), a
stress distribution σkf (kf (a)) can be derived based on the assumed material model. If the
stress in concrete is represented in a form of a density of load as a function of location,
i.e. b · σc,kf (kf (a)) and the stress in ith of Ns groups of reinforcing steel of the area Asi
and location asi as a resultant force σsi,kf (kf (asi )) · Asi , then the axial force acting in the
section can be calculated from the formula:
Nkf = b
Z h
0
σc,kf (kf (a)) d a +
Ns
X
σsi,kf (kf (asi ))Asi
(5.3)
i=1
and the corresponding bending moment from the formula:
Mkf = b
Z h
0
(xkf − a)σc,kf (kf (a)) d a +
Ns
X
(xkf − a)σsi,kf (kf (asi ))Asi .
(5.4)
i=1
Analytic integration can be efficiently replaced with numerical integration in which
instead of summation of the infinitesimal increments, an approximate method of the
49
5.3. CROSS-SECTION MODEL
finite summation is performed. The values of stresses are derived for a finite number m
of strains at the height of the section at constant distances ∆a. Precision of numerical
integration depends on the number of sub-intervals (value of m/length of ∆a) and method
of σ((a)) function approximation in subintervals. The most popular methods of function
approximation are the Rectangular Rule (interpolation with a 0th order polynomial) and
the Trapezium Rule (interpolation with a 1st order polynomial).
Let us divide the height of the section into m intervals of width ∆a and denote each ith
interval as [ai , ai+1 ]. In the Rectangular Rule one value of the stress function is required
in each interval, lat us say σ((ac,i )), such that ai ≤ ac,i ≤ ai+1 . It is equivalent to
the assumption that ∀a ∈ (ai , ai+1 ) σ((a)) ≃ σ(ac,i ). The function can be now easily
integrated, as:
Z h
σ((a)) ≃ ∆a
0
m
X
σ((aci )).
(5.5)
i=1
The ac,i can be any value from [ai , ai+1 ] interval. If ac,i = ai , then it is a left or lower
Riemman sum, if ac,i = ai+1 it is a right or upper Riemman sum and if ac,i = 12 (ai + ai+1 ) it
is a middle Riemman sum. The value of ac,i should be chosen in such a way that provides
optimum performance (maximum possible precision at minimum possible computational
effort).
In the Trapezium Rule two values of stress function are needed: σ((ai )) and σ((ai+1 )).
Then the stress function is interpolated with a linear function and the integral can be
calculated as:
Z h
σ((a)) ≃ ∆a
0
m
X
σ((ai )) + σ((ai+1 ))
2
i=1
.
(5.6)
Of course, the function can be approximated with a polynomial of any desired order or
any other easily integrable function, however, their use in numerical integration becomes
more and more questionable along with the increasing complexity of the approximating/interpolating function. Moreover, linearisation of the function for very small intervals
provides the greatest flexibility in the choice of material model as an input data without
the necessity of adjustment of the approximation method to best suite the material model
function.
The process of numerical integration of the stress distribution function is presented for
the middle Riemman sum. The axial force is calculated in each f th iteration of each k th
step according to the formula:
Nkf = b
m
X
σc,kf kf
i=1
1
i−
∆a
2
∆a −
Ns
X
σsi,kf (kf (asi ))Asi .
(5.7)
i=1
The corresponding bending moment, calculated after the last iteration of the k th step and
denoted as Mk , is calculated from the formula:
Mk = b
m X
i=1
xk − i −
1
∆a σc,k k
2
i−
1
∆a
2
∆a +
Ns
X
(xk − asi )σsi,k (k (asi ))Asi .
i=1
(5.8)
The operation of finding of the internal forces with numerical integration is presented
in Fig. 5.10 for an exemplary stress distribution.
50
CHAPTER 5. NUMERICAL MODEL
(a) cross-section
(b) strains (a)
(c) stresses σ((a))
(d) forces M and N
Figure 5.10: Determination of internal forces in section with numerical integration
5.3.3
Moment–curvature relationship
A set of pairs of Mj,k –κj,k values create a M –κ relationship – the cross-section model for
the j th type of section. The M –κ diagrams are prepared in both positive and negative
range of bending moments (curvatures), but it must be remembered that the M –κ diagram
is symmetrical (with respect to the origin of coordinate system) only for symmetrical
cross-sections. An exemplary M –κ diagram obtained in that method is presented in
Fig. 5.11.
Figure 5.11: Exemplary numerically-determined M –κ diagram
Mu denotes the ultimate bending moment – the maximum moment which the section can
transfer. There are two independent Mu values for each section: Mu,pos which signifies the
moment resistance of the section against positive bending moment and Mu,neg the moment
resistance against negative bending moment. Mu values for a given type of sections can be
determined from the Mj,k –κj,k set according to the conditions that Mu,pos,j,k = max{Mj,k }
and Mu,neg,j,k = min{Mj,k }.
It should be noted that such a solution has one drawback – a constant increment
of curvature does not necessarily result in a constant increment of bending moment, so
the M –κ pairs cannot be prepared for all desired values of bending moments (at least
5.4. STIFFNESS
51
not that easily). This will result is a certain approximation error in static calculations
in which the values of bending moments, not curvatures, are needed. Nevertheless, the
simplicity of determination of the bending moment value with a known curvature instead
of doing otherwise was decided to be more computationally convenient, having in mind
considerations from Sec. 5.4.1.
5.4
Stiffness
5.4.1
Stiffness of cross-section
The basis of this thesis is the notion that the stiffness of the section is a function of bending
moment acting in this section, i.e. B = B(M ). In a mathematical sense, the value of the
stiffness of the section Br after the rth loading step with the bending moment Mr is the
first derivative of M (κ) function, i.e. Br = M 0 (κr ). In a geometrical sense it is tangent of
the inclination angle β of the line tangent to the M –κ function in a given (κr ,Mr ) point,
as presented graphically in Fig. 5.12.
Figure 5.12: Geometrical interpretation of section’s stiffness
For the purpose of this model, one should be able to determine the value of the stiffness
for every stress state occurring in every segment of the analysed member. However, the
M –κ function derived in the procedure described in Sec. 5.3 is only a finite set of M –κ
pairs. Therefore, some approximation has to be made in the stiffness determination
process.
Determination of the cross-section stiffness is a two-step process. First, for a given
bending moment Mr acting in the cross-section the corresponding curvature κr must be
found, so that (κr , Mr ) point is known, and then a derivative of M (κ) in that point, i.e.
M 0 (κr ) must be determined as the stiffness B(Mr ) = Br .
The easiest solution is preparation of a very “dense” diagram, i.e. M and κ values
are determined for a very “small” increment of curvature ∆κ. The terms “dense” and
“small” should be understood in the context of the precision of results: if the maximum
difference between two adjacent values of bending moments in the M –κ set is smaller than
the precision of the result values of bending moments obtained in a static analysis, then
such a solution is satisfactory. Then the value of active bending moment Mr is rounded to
the nearest value Mk ∈ {Mk }, for which κk is known. The rounding is proceeded in such a
way that the values of Mk1 , Mk2 adjacent to Mr are found, i.e. Mk1 = max Mk : Mk < Mr
and Mk2 = min Mk : Mk > Mr , and Mr is taken as Mr = Mk1 if |Mr − Mk1 | < |Mr − Mk2 |
or Mr = Mk2 otherwise.
52
CHAPTER 5. NUMERICAL MODEL
The value of derivative can be then determined with one of the finite difference methods,
which enable to replace derivative expressions with approximately equivalent difference
quotients. The basis for this group of methods is the assumption that the domain of the
function is uniformly discretised – it this case that the spacing between the subsequent κk
is equal (∆κ = const.). The finite difference may be of any order – the order determines
the number of points (or (κk , Mk ) pairs) to be used to calculate the value of the derivative.
The most commonly considered are the forward, backward and central differences. For
a (κk , Mk ) pair the forward difference requires the knowledge of the (κk+1 , Mk+1 ) pair
such that κk+1 > κk and κk+1 − κk = ∆κ, the backward difference the knowledge of the
(κk−1 , Mk−1 ) pair such that κk−1 < κk and κk − κk−1 = ∆κ, while the central difference
existence of two pairs, (κk− 1 , Mk− 1 ) and (κk+ 1 , Mk+ 1 ), such that κk− 1 < κk < κk+ 1 and
2
2
2
2
2
2
κk+ 1 − κk− 1 = ∆κ. The central difference to be used for {Mk , κk } obtained in this model
2
2
must be calculated for (κk−1 , Mk−1 ) and (κk+1 , Mk+1 ), such that κk−1 < κk < κk+1 and
κk+1 − κk−1 = 2∆κ.
Provided that Mk is the value of bending moment from {Mk } nearest to Mr , the
stiffness Br is calculated with a forward, backward and central finite difference method as:
Br = M 0 (κk ) ≃
Mk+1 − Mk
,
∆κ
(5.9)
Br = M 0 (κk ) ≃
Mk − Mk−1
,
∆κ
(5.10)
Mk+1 − Mk−1
.
(5.11)
2∆κ
In a geometrical sense the stiffness Br determined with this operation is a tangent of the
inclination angle of the line crossing points {(κk+1 , Mk+1 ), (κk , Mk )}, {(κk−1 , Mk−1 ), (κk , Mk )}
and {(κk+1 , Mk+1 ), (κk−1 , Mk−1 )} for the forward (Fig. 5.13a), backward (Fig. 5.13b) and
central (Fig. 5.13c) finite difference method, respectively.
Br = M 0 (κk ) ≃
(a) forward difference
(b) backward difference
(c) central difference
Figure 5.13: Stiffness of section after rth step with first-order finite difference methods
A different idea is an operation of curve fitting – an approximation of the set with an
analytically differentiable function, preferably a polynomial. The operation of approximation with a differentiable function enables to proceed with analytic computations, because
differentiation of polynomials is not only analytically possible but also very simple. The
approximation can be performed for the whole set at once or segmentally.
5.4. STIFFNESS
53
The first method of curve fitting is interpolation with a polynomial. The interpolation
methods assume that the interpolating function – let us denote it as W (κ) – for a chosen
set of values of the interpolated function – in this case M (κ) – has the same values, i.e. as
the the M (κ) function is a set of points, {Mk , κk }, interpolation with a polynomial W (κ)
is an operation of finding such a polynomial that ∀k W (κk ) = Mk .
If the {Mk , κk } has K elements, then the W (κ) will be a polynomial of the K − 1 order.
It must be noted, that the higher the order of the polynomial (the greater the number of
the points), the less stable becomes the interpolating function3 . Moreover, as the value of
κr for a given Mr must be known, the procedure of root finding of the W (κ) polynomial
has to be performed. This is analytically possible only for polynomials of at most 4th
order. For higher order polynomials the iterative methods for non-linear equations root
finding must be used. Hence, a choice of interpolation is a compromise between precision
and computational convenience.
Another solution can be derivation of an approximating function. In that sense,
approximation is a standard optimisation problem of minimalisation of the error function,
or bluntly speaking, minimalisation of the differences between the corresponding values of
the approximated (M (κ)) and the approximating (A(κ)) function. The approximation
method is deprived of the Runge’s phenomenon problem.
The process of interpolation/approximation can be also performed locally, for a defined
neighbourhood of the desired Mr value.
The method which can be considered a half-way solution between the interpolation
and approximation of the whole set is spline interpolation, a form of interpolation where
the interpolant is a special type of piecewise polynomial called a spline. The spline is
composed of a set of polynomials, preferably of a low degree, defined over the subintervals
of the interpolated function, in such a way that the polynomials have the same values in
the joining points, called knots. The proper use of splines minimises the risk of polynomial
oscillation and preserves the characteristic of the interpolated function.
For the purpose of this model it is demanded that the spline was a smooth curve of
the smoothness at least C 1 , i.e. the spline function and its derivative are continuous. It is
possible to achieve using at least a quadratic spline (the spline of the 2nd order). However,
to assure stability of the spline, the special type of spline called a natural cubic spline
would be more convenient.
The operation of differentiation of the spline is very similar to that used in typical
interpolation and approximation. The only additional condition, which may be regarded
as a drawback of this method, is a need of checking between which knots is the desired
value of Mr to indicate which segment of the spline to consider.
5.4.2
Stiffness of segment
The stiffness determined in the process described in Sec. 5.4.1 is the stiffness of a single
cross-section. It was stated in Sec. 5.1 that initially each segment is given the stiffness of
the cross-section type j it includes as a mean stiffness, i.e. Bmean,i = Bt(i),0 = Bj,0 , and
this stiffness is changing directly according to the proper cross-section model. As long as
there is no crack in the segment, it is valid. However, once the crack is formed, i.e. the
value of the active bending moment Mr = Mcr , the stiffnesses of particular cross-sections
in a segment differ. The stiffness of the cracked cross-section is degraded from BI towards
BII . Thus, the mean stiffness of the segment must be some average of the stiffnesses of
3
The problem of oscillation of the interpolating polynomial is known as the Runge’s phenomenon.
54
CHAPTER 5. NUMERICAL MODEL
the cracked and uncracked sections. The tension stiffening effect coefficient ξ, given by
Eq. 4.22, can be used, and the mean stiffness after the rth loading step Bmean,t(i),r can be
determined as:
Bmean,t(i),r = (1 − ξi,r )Bt(i),r
(5.12)
where Bt(i),r is the stiffness determined from the Mj –κj diagram and ξi,r is the tension
stiffening effect coefficient for the ith segment for the bending moment Mi,r obtained after
the rth loading step.
With an increasing load, the stiffness will be decreasing till, for the ultimate moment
Mu , it will degrade to 0 (M 0 (κMu ) = const.) It is when the plastic hinge is formed. It
must be remembered that once the stiffness of a segment Bt(i),r is reduced after the rth
loading step, it cannot be “regained” if the value of the bending moment is reduced in
that segment in the following steps, nor can be the (1 − ξi,r ) coefficient. This fact can be
mathematically defined as:
ξi,r = max (ξ(Mi,r ); ξi,r−1 )
Bt(i),r = max 0; min Bt(i),r ; Bt(i),r−1
5.5
(5.13)
.
(5.14)
Static analysis with FEM
In is not possible to determine the values of internal forces in a statically-indeterminate
structure directly with simple equilibrium conditions. A method must be used which takes
into consideration stiffnesses of all structural components. A displacement method is a
very good choice. In a matrix formulation, the method can be easily implemented to be
used in computer calculations. The numerical version of this method is referred to as the
displacement finite element method.
Displacement finite element method is a method of finding a displacement function
of the structure with defined stiffness of the components and external loading pattern.
Computational procedure is a two-phase process. In the first phase the structure is
descretised, i.e. divided into the finite number of bars, each bar having two nodes and
the bars joining in the nodes. In case of the exemplary beam analysed in this chapter, as
shown in Fig. 5.5, these bars are referred to as segments.
In the second phase a global stiffness matrix K is formed as a result of definition
of relationship between generalised forces (in a form of a matrix of nodal loads P ) and
generalised displacements of nodes (in a form of a nodal displacements matrix u), which in
a linear-elastic approach leads to a matrix formulation of the system of linear equations:
Ku = P.
(5.15)
In the non-linear analysis the elements of the stiffness matrix K are no longer constant
as they are functions of displacements u (stiffness is a function of bending moment, while
bending moment is a function of displacement), and the resulting system is a system of
non-linear equations of the form:
K(u)u = P.
(5.16)
5.5. STATIC ANALYSIS WITH FEM
55
Solution of such a system of non-linear equations must be found in steps and with iterative
search in each step. Thus, the methods of numerical solution of non-linear problems are
ofter referred to as incremental–iterative methods. The approximated solution can be
also found in a process of linearisation of the problem: the global stiffness matrix can
be calculated in steps, for very small increments. In each step it can be then assumed
that the problem is no longer non-linear and the stiffness is independent on the deflection.
Precision of such a solution increases (in comparison to iterative method) as the value of
increment decreases. The incremental approach is presented in this section.
5.5.1
Static calculations
Computations are proceeded with an incremental method. A uniformly distributed load q
is applied to the whole member in constant increments ∆q. Initially, the beam has a static
scheme as shown in Fig. 5.5 and is loaded with a load q1 = ∆q (Fig. 5.14). The value of
the bending moment is determined in each node, and distribution of bending moments
is interpolated. As the decisive bending moment for each segment, Mi , the maximum
bending moment in this segment is understood.
Figure 5.14: Static calculations. Step 1
After this step the stiffness of each segment Bt(i),0 is modified according to the relevant
Mj –κj relationship and the procedure is being repeated for the next increments q2 = ∆q,
q3 = ∆q, ..., qr = ∆q, each time for corrected stiffness value Bt(i),r−1 = B(Mi,r−1 ). The
total value of bending moment for each segment i after the rth loading step is the sum of
P
all bending moments obtained in all the previous steps, i.e. Mtot,r = rm=1 Mm .
Finally, in the Rth step, the total bending moment Mtot,R in one node reaches the
ultimate value Mu , which is equivalent to depletion of the load-bearing capacity of this
section and formation of a plastic hinge (Fig. 5.15). These R steps are the “first cycle”.
As it is known, formation of the plastic hinge does not signify failure of the member,
because although the section can no longer transfer additional bending moments, the
moments can be redistributed and the section can undergo further deformations as long
as they do not exceed their ultimate values. In the model, a static scheme of the beam
is changed (a structural hinge is introduced in the place of the plastic hinge), and the
beam can undergo another loading “cycle”. The “correcting” bending moments Mdif f
are introduced in the section where plastic hinge is formed to substitute the process of
moment capacity decrease in section acc. to the material model assumed (Fig. 5.16). The
value of the additional moments is equal to the difference between the value of the bending
moment capacity of the section before and after the given loading step.
56
CHAPTER 5. NUMERICAL MODEL
Figure 5.15: Static calculations. Formation of first plastic hinge
Figure 5.16: Static calculations. Step R1 + 1
57
5.5. STATIC ANALYSIS WITH FEM
The process continues until the number of structural hinges exceeds the degree of
statical indetermination of the beam. Each “cycle” c terminates in the step Rc up to the
P c
final step RC of the final cycle C. The final value of the load qtot,RC = R
m=1 qm = Rc ∆q,
leading to the formation of the last plastic hinge, is the ultimate load qu for the beam
(Fig. 5.17).
Figure 5.17: Static calculations. Formation of the last plastic hinge
Formation of the second plastic hinge is equivalent to the global loss of stability and
failure of the member. Hence, the total number of cycles C = 2, R2 is the last step of the
whole procedure and the load qtot,R2 = R2 ∆q is the ultimate load for the exemplary beam.
5.5.2
Bending moment distribution
Solution of the exemplary beam with a finite element method is based on the Euler–
Bernoulli partial differential equation for bending with application of Galerkin’s method of
weighted residuals to develop the finite element formulation and the corresponding matrix
equations [1].
A deformation function (deformation line) can be derived with a finite element method
from a displacement function which assigns displacements to the nodes. In the displacement
method a displacement function is a (1xDOF ) vector, where DOF is a global number of
degrees of freedom (number of nodes x degrees of freedom/node). It can be calculated
after transformation of Eq. 5.15 as:
u = K −1 P.
(5.17)
The deformation of a beam must have continuous slope and continuous deflection at
any two neighbouring elements. In case of the analysed beam any two neighbouring beam
elements (segments) have common deflection vi and slope φi at the shared nodal point i.
Based on the Bernoulli hypothesis of plane sections it is concluded that slope is the first
58
CHAPTER 5. NUMERICAL MODEL
derivative of deflection in terms of location along the beam x, φ = dd xv . As there are four
nodal variables in each beam element (4 degrees of freedom), the deformation line can be
described with a 3rd order polynomial function:
v(x) = c0 + c1 x + c2 x2 + c3 x3
(5.18)
The notion of this fact is crucial for derivation of bending moment function.
Firstly, the stiffness matrices Ki are prepared separately for each segment. A segment
with stiffness Bj which spans over two nodes – the first node being denoted as P and the
second as K – and with 2 degrees of freedom in each node (vertical displacement v[m]
and rotation φ[rad]) has a total number of 4 degrees of freedom. 4 values: forces VP and
VK and moments MP and MK are found for 4 cases: unit vertical displacement of node
P and K (vP = 1 and vK = 1) and unit rotation of node P and K (φP = 1 and φK = 1)
with all the remaining displacements taken as 0. This is possible with application of the
Hermitian shape functions, the functions which are derived from the deformation function
given by Eq. 5.18 for the appropriate boundary conditions, i.e. vP = 1, φP = 1, vK = 1
and φK = 1, respectively, and have the form:
3x2 2x3
+ 3
l2
l
2
2x
x3
H2 (x) = x −
+ 2
l
l
3x2 2x3
H3 (x) = 2 − 3
l
l
2
x
x3
H4 (x) = − + 2
l
l
H1 (x) = 1 −
(5.19)
(5.20)
(5.21)
(5.22)
The analysed sub-cases with corresponding shape functions and resultant V and M forces
are shown in Fig. 5.18.
The resulting deformation function within one element is a linear combination of shape
functions, decribed by the formula:
v(x) = H1 (x)vP + H2 (x)φP + H3 (x)vK + H4 (x)φK
(5.23)
In such a way the deformation function can be derived for the whole length of the element
when exact values of displacements are introduced.
The calculated values of the forces are the second derivatives of the shape functions for
the assumed stiffness and create the stiffness matrix of the segment Ki , of the form:
"
Ki =
kP,P,i kP,K,i
kK,P,i kK,K,i
#
(5.24)
where kP,P,i , kP,K,i , kK,P,i and kK,K,i are the stiffness sub-matrices which represent the
reaction of the unit-displacement force applied in the node P on node P , in the node P
on node K, in the node K on node P and in the node K on node K, respectively, of the
forms [19]:
59
5.5. STATIC ANALYSIS WITH FEM
(a) unit vertical displacement of P node for H1 (x)
(b) unit vertical displacement of K node for H3 (x)
(c) unit rotation of P node for H2 (x)
(d) unit rotation of K node for H4 (x)
Figure 5.18: Determination of elements of stiffness matrix of segment Ki acc. to [19]
60
CHAPTER 5. NUMERICAL MODEL
12Bt(i),r

l3



kP,P,i = 


kP,K,i = 
VP,i |vP =1
VP,i |φP =1
MP,i |vP =1 MP,i |φP =1
VP,i |vK =1
VP,i |φK =1


kK,P,i = 

MP,i |vK =1 MP,i |φK =1
VK,i |vP =1
VK,i |φP =1
MK,i |vP =1 MK,i |φP =1


=



kK,K,i = 
VK,i |vK =1
VK,i |φK =1
MK,i |vK =1 MK,i |φK =1
6Bt(i),r 

l2

(5.25)



6Bt(i),r 4Bt(i),r
− 2
l
l

12Bt(i),r
6Bt(i),r 
− 2
 −

l3
l




=


6Bt(i),r
l2
2Bt(i),r
l
(5.26)





T
 = kP,K,i
(5.27)
12Bt(i),r

l3

6Bt(i),r 

l2

6Bt(i),r
l2
l


−


=


(5.28)
.

4Bt(i),r 
The complete stiffness matrix of the element is a (4x4) matrix.
Formation of a global stiffness matrix K is carried out in a process of aggregation. The
aggregation is a process of addition of the matrices of segments, Ki , into the appropriate
places of the global matrix K. Aggregation ensures equal displacements of the nodes
belonging to more than one segment.
For a continuous beam, being an example in this model, a Ki is a stiffness matrix for a
segment i with the first node P = i and the second node K = i + 1, and not more than
two elements joining in one node. The sub-matrices kP,P,i , kP,K,i , kK,P,i and kK,K,i have
the same form for all segments, with only the value of stiffness Bt(i),r changing between
segments and in loading steps. The beam has 11 segments and 12 nodes, with 2 degrees
of freedom in each node, so the total number of degrees of freedom is DOF = 24. The
resultant global stiffness matrix K can be presented in a (24x24) matrix form, with all
values of forces corresponding to all the degrees of freedom, or in a simplified way in a
(12x12) matrix form with sub-matrices corresponding to the subsequent nodes, in a form:

k1,2,1
k2,2,1 + k2,2,2
k3,2,2
..
.
0
k2,3,2
k3,3,2 + k3,3,3
..
.
0
0
k3,4,3
..
.
0
0
0
0
0
0
0
k1,1

 k2,1

 0

=
 ...
k1,2
k2,2
k3,2
..
.
0
k2,3
k3,3
..
.
0
0
k3,4
..
.
...
...
...
..
.
0
0
0
0
0
0
0
k11,10
0
k1,1,1

 k2,1,1

 0

K=
 ...


 0



 0
0
0
0
0
0
0
...
...
...
..
.

0
0
0





=

0


k11,12,11 
k10,11,10
k11,10,10 k11,11,10 + k11,11,11
0
k12,11,11
k12,12,11








0


k11,12 
k10,11
k11,11
k12,11 k12,12
0
0
0
(5.29)
61
5.5. STATIC ANALYSIS WITH FEM
where ka,b,c is a sub-matrix representing the reaction of the forces acting in the node a on
the node b, calculated for segment c, or in a graphical form in Fig. 5.19a.
(a) global stiffness matrix K
(b) vector of nodal loads P
Figure 5.19: Graphical presentation of matrices formation in aggregation process
It must be emphasised that for a continuous beam only the entries of the matrix laying
on its diagonal (hatched entries) are being modified in the aggregation process and that
the resultant matrix is a band matrix.
Figure 5.20: Determination of elements of matrix of nodal loads Pi
The matrix of loads represents the loads to which the beam is subjected in a form of
generalised nodal loads, which are statically equivalent to the external load acting on the
beam. It is a (1xDOF ) vector. For each segment a pair of transverse forces and bending
moments is obtained.
For a single segment subjected to the uniformly distributed load qr the resultant nodal
loads are presented in Fig. 5.20. The resultant vector of nodal loads has a form:
62
CHAPTER 5. NUMERICAL MODEL

VP,i |qr




 MP,i |
"
#

qr
pP,i

Pi =
=

pK,i
 VK,i |

qr


MK,i |qr





qr l

=

2




qr l 2
−
12
qr l
2
qr l 2
12
T
.
(5.30)
For a general arbitrary loading pattern q(x) such a segment would produce a load
vector of a form:
H1 (x)
 H (x) 


Pi =
q(x)  2
 d x.
 H3 (x) 
0
H4 (x)


Z l
(5.31)
The aggregation process can be carried out to form a load vector for the whole beam
and boundary conditions can be introduced for the nodes where supports are defined. The
known shear force and or bending moment can be included in the system force vector;
otherwise, they remain unknown. The resultant load vector has a form:

V1,1
 M

1,1

 V2,1 + V2,2

 M +M
2,1
2,2

 .
.
P =
 .













 V11,10 + V11,11 


 M


11,10 + M11,11 


 V12,11

(5.32)
M12,11
where Va,c is an entry representing the equivalent nodal force in the node a calculated
for segment c subjected to the load qr , or in a graphical form in Fig. 5.19b. It must be
noted that for the range of work where κr > κ|Mu additional bending moments Mdif f are
introduced in the node where plastic hinge was formed, to reflect the softening nature of
concrete.
The last step is introduction of boundary conditions, i.e. conditions of support.
Introduction of boundary conditions assures zero displacements in the points where
supports are defined. There is a number of ways to achieve this goal.
One of them is limiting or depriving the node of the possibility of displacement –
depending on the support a vertical displacement or rotation – by introducing a very big
force responsible for causing this displacement, e.g. multiplying the value by 1020 . In case
of the exemplary beam, the hinge supports are introduced in nodes 1, 7 and 12. This is
equivalent to the condition that v1 = 0, v7 = 0 and v12 = 0. Therefore the vertical forces
responsible for causing this displacements must be increased to the “computer’s infinity”.
In the global stiffness matrix only the ki,i entries are modified to account for the direction
of dicplacement being blocked. In that case the vertical forces causing unit displacements
of the supported nodes in the k1,1 , k7,7 and k12,12 matrices are modified as follows:
63
5.5. STATIC ANALYSIS WITH FEM

VP,1 |v1 =1 · 1020
k1,1 = kP,P,1 = 
VP,1 |φ1 =1

MP,1 |v1 =1
 
k7,7 = kK,K,6 + kP,P,7 = 

MP,1 |φ1 =1

(5.33)


VK,6 |v7 =1 + VP,7 |v7 =1 · 1020
VK,6 |φ7 =1 + VP,7 |φ7 =1
MK,6 |v7 =1 + MP,7 |v7 =1
MK,6 |φ7 =0 + MP,7 |φ7 =1




(5.34)

k12,12 = kK,K,12 = 
VK,12 |v1 =1 · 1020
VK,12 |φ1 =1

MK,12 |v1 =1
MK,12 |φ1 =1


.
(5.35)
In case of the nodal loads vector these boundary conditions must signify that there is no
possibility of bending moment transfer in the external nodes where hinge supports are
introduced, i.e. M1 = 0 and M12 = 0.
Another solution, coming directly from the classic displacement method, is to neglect
the displacements which values are known. In the FEM it means crossing-out columns and
rows which corresond to zero displacement. In the analysed case the rows and columns in
K, u and P matrices corresponding to vertical displacements in nodes 1, 7 and 12 should
be crossed out – in 24x24 presentation these are columns no. 1, 13 and 23 in K matrix
and rows no. 1, 13 and 23 in all matrices.
Finally, the u vector can be calculated. The u vector returns a finite sequence of
displacements of a form:
v1


 φ1

 v2



=
 φ2

 .

 ..



u1

 u2
u=
 ..
 .
u12













 v12 
(5.36)
φ12
where ua is a vector of displacements for a node a which includes a vertical displacement
va and a rotation φa in the node, when the complete matrix is analysed (then it should be
v1 = v7 = v12 = 0), or in a condensed form, with only unknown displacements provided.
In computer calculations the K matrix is not actually inversed (it is computationally
uneconomiacal). Solution of the matrix form of the system of equations is proceeded with
one of the methods of solutiton of linear equations systems, e.g. Gaussian elimination,
LU decomposition, Hausholder or Gram-Schmidt orthonormalisation process, Cholesky
decomposition (as K is a symmetrical, positive-definite matrix) or with simple-type
iterative methods (Chebyshev method).
The values of internal forces along the whole memeber can be obtained by summation
of forces resulting from the displacements of the nodes and forces resulting from external
loads to which the memeber is subjected, in any cross-section. This must be performed
for a local case, so for each segment. The values if internal forces in an every node i must
64
CHAPTER 5. NUMERICAL MODEL
be equal regardless of the choice of segment i (for which node i = P ) or i − 1 (for which
node i = K).
For a single element the forces resulting from the displacements of the nodes are the
sums of the products of multiplication of the forces resulting from the unit displacements
of the nodes and displacement of the corresponding nodes obtained from Eq. 5.17 in a
form of the vector u, given by Eq. 5.36, and the forces resulting from the action of the
external load.
In case of a segment, as analysed in this thesis, these forces are obtained by addition
of five cases: four representing the unit-displacement forces, presented in Fig. 5.18 and the
fifth being a solution of the substitute static scheme of the segment under the uniformly
distributed load qr , which is presented in Fig. 5.20. For segments in which a plastic hinge
was formed, the additional bending moments Mdif f must be considered in calculations.
The border values of bending moments in the segment i with nodes i and i + 1 can be
calculated acc. to the superposition rule, for the nodes i and i + 1, respectively:
Mi = MP |vP =1 vi + MP |vK =1 vi+1 + MP |φP =1 φi + MP |φK =1 φi+1 + MP |qr
(5.37)
Mi+1 = MK |vP =1 vi + MK |vK =1 vi+1 + MK |φP =1 φi + + MK |φK =1 φi+1 + MK |qr
(5.38)
The same can be done with shear forces:
Vi = VP |vP =1 vi + VP |vK =1 vi+1 + VP |φP =1 φi + VP |φK =1 φi+1 + VP |qr
(5.39)
Vi+1 = VK |vP =1 vi + VK |vK =1 vi+1 + VK |φP =1 φi + + VK |φK =1 φi+1 + VK |qr .
(5.40)
It must be noted that since the deformation function is given by a differentiable function
and since internal forces are derivatives of the deformation line, the resultant distribution
of internal forces can be obtained as a continuous, not a discrete solution.
Chapter 6
Summary
6.1
Conclusions
The process of design of a structure requires a series of idealisations: idealisation of
geometry, idealisation of materials and idealisation of loads. Hence, this process is actually
performed on a model. The model should be as simple as possible, but reproduce the
important characteristics consistent with experimental results and should be theoretically
sound and numerically stable, so that reliable analysis results are obtained. In the
structural analysis of reinforced concrete members four main approaches are proposed by
the standards: linear-elastic analysis, linear-elastic analysis with limited redistribution,
plastic analysis and non-linear analysis.
In the linear-elastic analysis the values of internal forces are derived based the assumption that the material has a linear-elastic characteristic. This simplification is not much far
from reality, because the values of bending moments MSd obtained with such an approach
are usually beyond the level of cracking moment Mcr , so in the range where the material
is in phase I and its behaviour is very close to linear. However, this values are compared
to the cross-sectional moment resistance MRd determined from the ultimate limit state of
flexure, as for the cross-section being in the phase III, so for a completely different stage of
work. This inconsistence is the reason for uneconomic design.
The notion of the plastic properties of reinforced concrete, so the ability of concrete to
undergo plastic deformations without rupture, provides additional load-bearing capacity,
the so-called plastic reserve, which – if applied reasonably – allows of designing of lighter,
more economical elements. It also extends the understanding of the behaviour of reinforced
concrete in the plastic phase, when a cracked element is subjected to irreversible strains.
This notion is the basis for the non-linear methods of analysis. The linear-elastic analysis
with limited redistribution is the simplest method of refinement of linear-elastic analysis
results. Bending moments obtained in the linear-elastic static analysis are redistributed
within a limited range from the most loaded cross-sections to other, less congested areas.
The method is simple, but diifcult to control and thus questionable.
The standards provide a designer with information about the plastic properties of
concrete. The notions of plastic hinge formation and rotational capacity of the plastified
regions are introduced. Moreover, it is emphasised that the reinforced concrete behaviour is
in reality totally non-linear due to stiffness degradation as a result of material characteristics,
crack formation and damage of internal structure, which finally leads to depletion of loadbearing capacity and fracture.
65
66
CHAPTER 6. SUMMARY
Nevertheless, modelling of the non-linear behaviour of reinforced concrete is not an
easy task. The phenomena occurring in reinforced concrete members, although repeatably
observed in laboratory tests and in full-scale objects, are very difficult in mathematical
description. Thus, analytic computations are impossible and, as a consequence, numerical
analysis is required.
Derivation of a numerical simulation of the physical phenomena is an interdisciplinary
task. The process requires co-operation between the scientist/engineer and the software
developer. A numerical model, derived on the basis of theoretical background of the
physical phenomenon and creative application of mathematics and numerical methods,
can be implemented and optimised for performance. The results obtained with such a
numerical analysis can be then validated with experimental results and the applicability
of the model can be evaluated by the comparison to the currently used methods.
The objective of this thesis, which was to derive a stiffness-oriented numerical model
for non-linear analysis of flexural reinforced concrete beams, was achieved. The literature
review was carried out to prepare a theoretical background of the phenomenon of stiffness
degradation in flexural reinforced concrete elements as a result of plastic behaviour of
concrete and crack formation at flexure. The currently used methods of static analysis and
dimensioning of reinforced concrete structures were investigated. The basic requirements
given by standards, mainly Model Code and Eurocode 2, were cited.
The numerical model was derived in cooperation with the supervisor of this thesis
on the basis of the referred theoretical background and personal experience in the topic.
The procedure of calculations was presented in a very detail, mathematical or geometrical
interpretations of physical phenomena were introduced, supported with schematic graphical
representations. Wherever noticed possible, simplifications and patterns resulting from
the specification of the design task were emphasised. The aim of these actions was to
generalise the physical problem in a way that enables the software developer, who is not
necessary familiar with the analysed physical problem, to implement the model.
Each time a step in the procedure could be solved in more than one way, the advantages
and disadvantages of the possible methods were analysed and the applicability evaluated.
Although it is believed that the software developer would have great experience in this
field, such a pre-analysis was performed to show the possible problems or – in contrary –
convenient solutions resulting from the nature of the design task.
The whole procedure of numerical analysis is presented on a chosen example. A very
simple – from a mechanical point of view – case was chosen because the objective of this
model is qualitative and quantitative assesment of redistribution process. It is believed
that a more complicated case could make validation of the model more difficult and bias
the results of the analysis.
The objective of this thesis was hence two-fold. The main aim was connected with the
necessity of derivation of a unified design algorithm for non-linear analysis of reinforced
concrete. The other, minor aim, was to take the floor in the topic of the future prospects
of engineering. The accessibility of fast computing machines and variety of computer
programming environments and numerical methods provides excellent possibilities for
solution of engineering task that used to be impossible to solve. Nevertheless, beside
the necessity of understanding the engineering problem there arises an urgent need of
programming skills.
Therefore, there are two choices for the engineer on these crossroads: either to learn
programming or to establish co-operation with software developer. The second choice has
always been a privilege of big commercial companies, leaving the first option the only
6.2. ACKNOWLEDGMENTS
67
choice for a common engineer or scientist. Have the fact been acknowledged that each
expert is the best in their domain, both parties would gain, let alone the benefits for
engineering itself. Thus, the actions should be undertaken which aim at triggering such
co-operation at all levels of engineering/scientific work.
The author and the supervisor of this thesis hope that it will be possible to successfully
implement the derived numerical model. It would be then possible to evaluate the process
of redistribution of internal forces in reinforced concrete elements and define its real
magnitude and nature. The results obtained in the numerical analysis could be compared
to the currently used simplified methods. Probably a more efficient and more general
algorithm for concrete members design and reinforcement could be derived.
6.2
Acknowledgments
To the supervisor of this thesis, PhD SEng Grzegorz Wandzik, for co-operation, guidance,
knowledge and time spent helping me complete this thesis.
To my husband, Michał Wróbel, for engagement, creativity, encouragement and support.
6.3
Abstract (in Polish)
Proces projektowania konstrukcji, w tym konstrukcji żelbetowych, wymaga szeregu idealizacji, stąd też proces ten przeprowadza się właściwie na modelu. Model ten powinien być
możliwe prosty, ale w sposób powtarzalny odwzorowywać najważniejsze charakterystyki
zgodnie z wynikami badań oraz powinien być teoretycznie poprawny i numerycznie stabilny,
aby uzyskane dzięki niemu wyniki pokrywały się z rzeczywistymi obserwacjami. Normy
proponują cztery grupy metod analizy elementów żelbetowych: analizę liniowo-sprężystą,
analizę liniowo-sprężystą z ograniczoną redystrybucją, analizę plastyczną oraz analizę
nieliniową.
W analizie liniowo-sprężystej wartości sił wewnętrznych otrzymuje się przy założeniu
liniowo-sprężystej charakterystyki materiału. Uproszczenie to nie jest bardzo dalekie od
rzeczywistości, ponieważ n.p. wartości momentów zginających MSd otrzymanych przy
takim podejściu są zwykle poniżej poziomu momentu rysującego Mcr , a więc w zakresie,
gdzie element znajduje się w fazie I, a jego zachowanie jest bliskie liniowemu. Jednakże
wartości te w metodzie stanów granicznych porównywane są z nośnością przekroju na
zginanie MRd , wyznaczoną dla stanu granicznego nośności na złamanie, jak dla przekroju w
fazie III, więc w zupełnie innej fazie pracy. Ta niespójność jest powodem nieekonomicznego
projektowania.
Wiedza na temat plastycznych właściwości betonu zapewnia dodatkową nośność, zwaną
rezerwą plastyczną, która – wykorzystywana rozsądnie – pozwala na projektowanie lżejszych, bardziej oszczędnych elementów. Wiedza ta jest podstawą nieliniowych metod
analizy konstrukcji. Analiza liniowo-sprężysta z ograniczoną redystrybucją jest najprostszą
z metod urzeczywistniania wyników analizy liniowo-sprężystej z uwzględnieniem plastycznych właściwości betonu. Momenty zginające uzyskane w liniowej analizie statycznej
są redystrybułowane w ograniczonym zakresie z najbardziej obciążonych przekrojów w
mniej wytężone obszary. Metoda ta jest prosta, ale trudna do kontrolowanie, przez co
kontrowersyjna.
Normy dostarczają projektantowi informacje dotyczące plastycznych właściwości betonu. Wprowadzone są informacje na temat powstawania przegubów plastycznych oraz
68
CHAPTER 6. SUMMARY
zdolności do obrotu obszarów uplastycznionych. Co więcej, podkreśla się, iż zachowanie
żelbetu w rzeczywistości jest całkowicie nieliniowe z powodu spadku sztywności jako efektu
właściwości materiałowych, zarysowania oraz zniszczenia wewnętrznej struktury, co w
konsekwencji prowadzi do wyczerpania nośności i zniszczenia.
Niemniej jednak modelowanie nieliniowego zachowania żelbetu nie jest prostym zadaniem.
Zjawiska zachodzące w elementach żelbetowych, choć w sposób powtarzalny obserwowane
w badaniach laboratoryjnych jak i w pełno wymiarowych obiektach, są trudne w opisie
matematycznym. Obliczenia analityczne są niemożliwe, stąd konieczne jest wykorzystanie
numerycznych metod analizy.
Stworzenie numerycznej symulacji zjawisk fizycznych jest zadaniem interdyscyplinarnym,
wymagającym współpracy pomiędzy naukowcem/inżynierem a programistą. Model numeryczny, stworzony na postawie teoretycznego podłoża oraz kreatywnego zastosowania
matematyki i metod numerycznych, może zostać zaimplementowany. Wyniki uzyskane
dzięki takiej analizie numerycznej mogą następnie zostać porównane z obserwacjami oraz
wynikami badań, a przydatność modelu oceniona w odniesieniu do obecnie istniejących,
uproszczonych metod analizy.
Celem tej pracy było stworzenie modelu numerycznego do nieliniowej analizy żelbetowych belek ciągłych przy uwzględnieniu degradacji ich sztywności. Cel ten został
osiągnięty. Przeprowadzono przegląd literaturowy w celu przygotowania teoretycznego
podłoża dla zagadnienia degradacji sztywności w zginanych elementach żelbetowych zakładając sprężysto–plastyczne zachowanie żelbetu. Zbadano obecnie stosowane metody
analizy konstrukcji oraz wymiarowania. Zacytowano również najważniejsze wymagania normowe, głównie CEB-FIP Model Code oraz Eurokod 2, dotyczące analizy i wymiarowania
elementów żelbetowych.
Model numeryczny został opracowany przy współpracy z kierującym tę pracę, dr
inż. Grzegorzem Wandzikiem, na podstawie przytoczonego przeglądu literaturowego oraz
własnego doświadczenia w tym zakresie. Szczegółowo opisano algorytm obliczeń, wprowadzono matematyczne i geometryczne interpretacje poszczególnych zjawisk fizycznych oraz
przedstawiono ich graficzne prezentacje. Gdziekolwiek było to możliwe, podkreślono uproszczenia oraz zależności wynikające z charakteru analizowanego zagadnienia. Celem
takiego działania była generalizacja analizowanych zjawisk, aby umożliwić programiści,
który niekoniecznie jest biegły w omawianym zagadnieniu, implementację modelu.
Ilekroć dany krok w procedurze można byłoby wykonać na kilka sposobów, przydatność
każdego rozwiązania została oceniona na podstawie przytoczonych zalet i wad. Taka
wstępna analiza pozwoliła na zwrócenie uwagi na możliwe problemu lub, przeciwnie,
korzyści wynikające z zastosowanie danego rozwiązania do tego konkretnego zagadnienia.
Procedura analizy numerycznej została przedstawiona na prostym przykładzie, ponieważ
celem modelu była ilościowa i jakościowa ocena procesu redystrybucji. Wprowadzenie
bardziej skomplikowanego przypadku mogłoby w znaczny sposób utrudnić ocenę wyników
analizy. Niemniej jednak za pomocą tego modelu w łatwy sposób można analizować
dowolne układy prętowe, o dowolnych charakterystykach geometrycznych i materiałowych
oraz pod dowolnym obciążeniem.
Autor i kierujący tą pracą wierzą, iż będzie możliwe zaimplementowanie tego modelu
numerycznego. Umożliwiłoby to jego weryfikację, a następnie ocenę badanego zjawiska.
Docelowo rozwój tego modelu mogłoby zaowocować powstaniem zaawansowanego oprogramowania do bezpieczniejszego i bardziej ekonomicznego projektowania oraz zbrojenia
elementów żelbetowych.
List of Figures
2.1 Models based on theory of elasticity . . . . . . . . . . . . . . . . . . . . . .
2.2 Models based on theory of plasticity . . . . . . . . . . . . . . . . . . . . . .
2.3 Elasto-plastic models with hardening and softening . . . . . . . . . . . . .
2.4 Basic elements for decription of rate-dependent materials . . . . . . . . . .
2.5 Viscoelastic model (Maxwell model) . . . . . . . . . . . . . . . . . . . . . .
2.6 Viscoelastic model (Kelvin–Voight model) . . . . . . . . . . . . . . . . . .
2.7 Viscoelastic model (Standard Linear Solid model) . . . . . . . . . . . . . .
2.8 Elasto-viscoplastic models . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.9 Viscoelasto-plastic model . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.10 Viscoelasto-viscoplastic model . . . . . . . . . . . . . . . . . . . . . . . . .
2.11 Standard models for concrete in compression [36] . . . . . . . . . . . . . .
2.12 Standard models for concrete in tension [35] . . . . . . . . . . . . . . . . .
2.13 Standard models for steel [36] . . . . . . . . . . . . . . . . . . . . . . . . .
10
10
11
12
13
13
14
14
14
14
16
16
17
3.1
3.2
3.3
3.4
3.5
3.6
3.7
Real work of flexural reinforced concrete beam [13] . . . . . . . . . . . . .
Types of hinges [21] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
M –κ relationship for flexural reinforced concrete elements . . . . . . . . .
Plastic rotation θs of reinforced concrete sections in continuous beams [36]
Allowable plastic rotation θpl,d of reinforced concrete sections [36] . . . . .
Determination of limit load Qu with kinematic method [20] . . . . . . . . .
Determination of limit load qu with static method [20] . . . . . . . . . . . .
20
21
22
26
26
27
29
4.1
4.2
4.3
4.4
4.5
Phases of work of reinforced concrete member. Stresses [13] . . . . . . . . .
Simplified diagrams for distribution of stresses in ULS [18] . . . . . . . . .
Models for cross-section behaviour [20] . . . . . . . . . . . . . . . . . . . .
Phases of work of reinforced concrete member. Strains [18] . . . . . . . . .
Stiffness of flexural beam along the element [20] . . . . . . . . . . . . . . .
33
35
36
38
41
5.1 Process of numerical simulation of physical phenomenon . . . . . . . . . .
5.2 Exemplary continuous beam . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Static scheme of the beam . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4 Material and geometrical characteristics . . . . . . . . . . . . . . . . . . .
5.5 Static scheme of the beam after discretisation . . . . . . . . . . . . . . . .
5.6 Chosen σ– relationship . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.7 Geometrical interpretation of curvature . . . . . . . . . . . . . . . . . . . .
5.8 Determination of strain distribution in cross-section with bisection method
5.9 Distribution of strains in cross-section for different levels of loading . . . .
5.10 Determination of internal forces in section with numerical integration . . .
5.11 Exemplary numerically-determined M –κ diagram . . . . . . . . . . . . . .
43
44
44
44
45
46
46
48
48
50
50
69
70
LIST OF FIGURES
5.12 Geometrical interpretation of section’s stiffness . . . . . . . . . . . . . . . .
5.13 Stiffness of section after rth step with first-order finite difference methods .
5.14 Static calculations. Step 1 . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.15 Static calculations. Formation of first plastic hinge . . . . . . . . . . . . .
5.16 Static calculations. Step R1 + 1 . . . . . . . . . . . . . . . . . . . . . . . .
5.17 Static calculations. Formation of the last plastic hinge . . . . . . . . . . .
5.18 Determination of elements of stiffness matrix of segment Ki acc. to [19] . .
5.19 Graphical presentation of matrices formation in aggregation process . . . .
5.20 Determination of elements of matrix of nodal loads Pi . . . . . . . . . . . .
51
52
55
56
56
57
59
61
61
Bibliography
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PWN, Warszawa (2003)
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odkształceń plastycznych. Wydawnictwo Arkady, Warszawa (1971)
[22] Zienkiewicz O.C., Taylor R.L.: The Finite Element Method. Vol. 1: The Basis, 5th
ed. Butterworth–Heinemann, Oxford (2000)
[23] Zienkiewicz O.C., Taylor R.L.: The Finite Element Method. Vol. 2: Solid Mechanics,
5th ed. Butterworth–Heinemann, Oxford (2000)
Articles
[24] Bathe K.J., Walczak J., Welch A., Mistry N.: Nonlinear analysis of concrete structures.
Computers & Structures vol. 32, no. 3/4, p. 563–590 (1989)
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[26] Dąbrowski K.: Kilka uwag na temat metody plastycznego wyrównania momentów
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[29] Noakowski P.: Continuous Theory for the Determination of Crack Width under
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Standards
[32] ACI 318:2002 Building Code Requirements for Structural Concrete
[33] BS 8110:1997 Structural Use of Concrete Part 1: Code of Practice for Design and
Construction
[34] CEB-FIP Model Code 1990
[35] CEB-FIP Model Code 2010 (final draft)
[36] EN 1992-1-1:2008 Eurocode 2: Design of Concrete Structures. Part 1-1: General
Rules and Rules for Buildings
[37] PN-B-03264:2002: Konstrukcje betonowe, żelbetowe i sprężone. Obliczenia statyczne i
projektowanie
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