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- Herausgeber: Wiley-VCH
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
The book covers the application of numerical methods to reinforced concrete structures. To analyze reinforced concrete structures linear elastic theories are inadequate because of cracking, bond and the nonlinear and time dependent behavior of both concrete and reinforcement. These effects have to be considered for a realistic assessment of the behavior of reinforced concrete structures with respect to ultimate limit states and serviceability limit states. The book gives a compact review of finite element and other numerical methods. The key to these methods is through a proper description of material behavior. Thus, the book summarizes the essential material properties of concrete and reinforcement and their interaction through bond. These basics are applied to different structural types such as bars, beams, strut and tie models, plates, slabs and shells. This includes prestressing of structures, cracking, nonlinear stressstrain relations, creeping, shrinkage and temperature changes. Appropriate methods are developed for each structural type. Large displacement and dynamic problems are treated as well as short-term quasi-static problems and long-term transient problems like creep and shrinkage. Most problems are illustrated by examples which are solved by the program package ConFem, based on the freely available Python programming language. The ConFem source code together with the problem data is available under open source rules at concrete-fem.com. The author aims to demonstrate the potential and the limitations of numerical methods for simulation of reinforced concrete structures, addressing students, teachers, researchers and designing and checking engineers.
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Contents
Notations
Chapter 1: Finite Elements Overview
1.1 Modeling Basics
1.2 Discretization Outline
1.3 Elements
1.4 Material Behavior
1.5 Weak Equilibrium and Spatial Discretization
1.6 Numerical Integration and Solution Methods for Algebraic Systems
1.7 Convergence
Chapter 2: Uniaxial Structural Concrete Behavior
2.1 Scales and Short–Term Stress–Strain Behavior of Homogenized Concrete
2.2 Long-Term Behavior – Creep and Imposed Strains
2.3 Reinforcing Steel Stress–Strain Behavior
2.4 Bond between Concrete and Reinforcing Steel
2.5 The Smeared Crack Model
2.6 The Reinforced Tension Bar
2.7 Tension Stiffening of Reinforced Tension Bar
Chapter 3: Structural Beams and Frames
3.1 Cross-Sectional Behavior
3.2 Equilibrium of Beams
3.3 Finite Element Types for Plane Beams
3.4 System Building and Solution Methods
3.5 Further Aspects of Reinforced Concrete
3.6 Prestressing
3.7 Large Deformations and Second-Order Analysis
3.8 Dynamics of Beams
Chapter 4: Strut-and-Tie Models
4.1 Elastic Plate Solutions
4.2 Modeling
4.3 Solution Methods for Trusses
4.4 Rigid-Plastic Truss Models
4.5 More Application Aspects
Chapter 5: Multiaxial Concrete Material Behavior
5.1 Basics
5.2 Continuum Mechanics
5.3 Isotropy, Linearity, and Orthotropy
5.4 Nonlinear Material Behavior
5.5 Isotropic Plasticity
5.6 Isotropic Damage
5.7 Multiaxial Crack Modeling
5.8 The Microplane Model
5.9 Localization and Regularization
5.10 General Requirements for Material Laws
Chapter 6: Plates
6.1 Lower Bound Limit Analysis
6.2 Crack Modeling
6.3 Linear Stress–Strain Relations with Cracking
6.4 2D Modeling of Reinforcement and Bond
6.5 Embedded Reinforcement
Chapter 7: Slabs
7.1 A Placement
7.2 Cross-Sectional Behavior
7.3 Equilibrium of Slabs
7.4 Structural Slab Elements
7.5 System Building and Solution Methods
7.6 Lower Bound Limit Analysis
7.7 Kirchhoff Slabs with Nonlinear Material Behavior
Chapter 8: Shells
8.1 Approximation of Geometry and Displacements
8.2 Approximation of Deformations
8.3 Shell Stresses and Material Laws
8.4 System Building
8.5 Slabs and Beams as a Special Case
8.6 Locking
8.7 Reinforced Concrete Shells
Chapter 9: Randomness and Reliability
9.1 Basics of Uncertainty and Randomness
9.2 Failure Probability
9.3 Design and Safety Factors
Appendix A: Solution of Nonlinear Algebraic Equation Systems
Appendix B: Crack Width Estimation
Appendix C: Transformations of Coordinate Systems
Appendix D: Regression Analysis
Appendix E: Reliability with Multivariate Random Variables
Appendix F: Programs and Example Data
Bibliography
Index
Prof. Dr.-Ing. habil. Ulrich Häussler-CombeTechnische Universität DresdenInstitut für Massivbau01069 DresdenGermany
Cover: The photo shows a part of the façade of the Pinakothek der Moderne, Munich. The grid indicates the subdivision of a complex structure into small simple elements or finite elements, respectively.
Library of Congress Card No.:applied for
British Library Cataloguing-in-Publication DataA catalogue record for this book is available from the British Library.
Bibliographic information published by the Deutsche NationalbibliothekThe Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at <http://dnb.d-nb.de>.
© 2015 Wilhelm Ernst & Sohn, Verlag für Architektur und technische Wissenschaften GmbH & Co. KG, Rotherstraße 21,10245 Berlin, Germany
All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form by photoprinting, microfilm, or any other means nor transmitted or translated into a machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law.
Coverdesign: Sophie Bleifuß, BerlinTypesetting: le-tex Publishing Services GmbH, Leipzig
Print ISBN: 978-3-433-03054-7ePDF ISBN: 978-3-433-60362-8ePub ISBN: 978-3-433-60363-5eMob ISBN: 978-3-433-60364-2oBook ISBN: 978-3-433-60361-1
Preface
This book grew out of lectures the author gives at the Technische Universität Dresden. These lectures are entitled “Computational Methods for Reinforced Concrete Structures” and “Design of Reinforced Concrete Structures.” Reinforced concrete is a composite of concrete and reinforcement connected by bond. Bond is a key item for the behavior of the composite which utilizes compressive strength of concrete and tensile strength of reinforcement while leading to considerable multiple cracking. This makes reinforced concrete unique compared to other construction materials such as steel, wood, glass, masonry, plastic materials, fiber reinforced plastics, geomaterials, etc.
Numerical methods like the finite element method on the other hand disclose a way for a realistic computation of the behavior of structures. But the implementations generally present themselves as black boxes in the view of users. Input is fed in and the output has to be trusted. The assumptions and methods in between are not transparent. This book aims to establish transparency with special attention for the unique properties of reinforced concrete structures. Appropriate approaches will be discussed with their potentials and limitations while integrating them in the larger framework of computational mechancis and connecting aspects of numerical mathematics, mechanics, and reinforced concrete.
This is a wide field and the scope has to be limited. The focus will be on the behavior of whole structural elements and structures and not on local problems like tracking single cracks or mesoscale phenomena. Basics of multiaxial material laws for concrete will be treated but advanced theories for multiaxial concrete behavior are not a major subject of this book. Such theories are still a field of ongoing research which by far seems not to be exhausted up to date.
The book aims at advanced students of civil and mechanical engineering, academic teachers, designing and supervising engineers involved in complex problems of reinforced concrete, and researchers and software developers interested in the broad picture. Chapter 1 describes basics of modeling and discretization with finite element methods and solution methods for nonlinear problems insofar as is required for the particular methods applied to reinforced concrete structures. Chapter 2 treats uniaxial behavior of concrete and its combination with reinforcement while discussing mechanisms of bond and cracking. This leads to the model of the reinforced tension bar which provides the basic understanding of reinforced concrete mechanisms. Uniaxial behavior is also assumed for beams and frames under bending, normal forces and shear which is described in Chapter 3. Aspects of prestressing, dynamics and second-order effects are also treated in this chapter. Chapter 4 deals with strut-and-tie models whereby still a uniaxial material behavior is assumed. This chapter also refers to rigid plasticity and limit theorems.
Modeling of multiaxial material behavior within the framework of macroscopic contin-uum mechanics is treated in Chapter 5. The concepts of plasticity and damage are described with simple specifications for concrete. Multiaxial cracking is integrated within the model of continuous materials. Aspects of strain softening are treated leading to concepts of regular-ization to preserve the objectivity of discretizations. A bridge from microscopic behavior to macroscopic material modeling is given with a sketch of the microplane theory. Chapter 6 treats biaxial states of stress and strain as they arise with plates or deep beams. Reinforcement design is described based on linear elastic plate analysis and the lower bound limit theorem. While the former neglects kinematic compatibility, this is involved again with biaxial specifications of multiaxial stress–strain relations including crack modeling.
Slabs are described as the other type of plane surface structures in Chapter 7. But in contrast to plates their behavior is predominantly characterized by internal forces like bending moments. Thus, an adaption of reinforcement design based on linear elastic analysis and the lower bound limit theorem is developed. Kinematic compatibility is again brought into play with nonlinear moment–curvature relations. Shell structures are treated in Chapter 8. A continuum-based approach with kinematic constraints is followed to derive internal forces from multiaxial stress–strain relations suitable for reinforced cracked concrete. The analysis of surface structures is closed in this chapter with the plastic analysis of simple slabs based on the upper bound limit theorem. Chapter 9 gives an overview about uncertainty and in particular about the determination of the failure probability of structures and safety factor concepts. Finally, the appendix adds more details about particular items completing the core of numerical methods for reinforced concrete structures.
Most of the described methods are complemented with examples computed with a software package developed by the author and coworkers using the Python programming language.
Programs and example data should be available under
www.concrete-fem.com
. More details are given in Appendix F.
These programs exclusively use the methods described in this book. Programs and methods are open for discussion with the disclosure of the source code and should give a stimulation for alternatives and further developments.
Thanks are given to the publisher Ernst & Sohn, Berlin, and in particular to Mrs. Clau-dia Ozimek for the engagement in supporting this work. My education in civil engineering, and my professional and academic career were guided by my academic teacher Prof. Dr.-Ing. Dr.-Ing. E.h. Dr. techn. h.c. Josef Eibl, former head of the department of Concrete Structures at the Institute of Concrete Structures and Building Materials at the Technische Hochschule Karlsruhe (nowadays KIT – Karlsruhe Institute of Technology), to whom I express my gratitude. Further thanks are given to former or current coworkers Patrik Pröchtel, Jens Hartig, Mirko Kitzig, Tino Kühn, Joachim Finzel and Jörg Weselek for their specific contributions. I appreciate the inspiring and collaborative environment of the Institute of Concrete Structures at the Technische Unversität Dresden. It is my pleasure to teach and research at this institution. And I have to express my deep gratitude to my wife Caroline for her love and patience.
Ulrich Häussler-Combe
Dresden,inspring 2014
Notations
The same symbols may have different meanings in some cases. But the different meanings are used in different contexts and misunderstandings should not arise.
firstly used
General
•
T
transpose of vector or matrix •
Eq. (1.5)
•
−1
inverse of quadratic matrix •
Eq. (1.13)
δ
•
virtual variation of •, test function
Eq. (1.5)
δ
•
solution increment of • within an iteration of nonlinear equation solving
Eq. (1.70)
• transformed in (local) coordinate system
Eq. (5.15)
time derivative of •
Eq. (1.4)
Normal lowercase italics
a
s
reinforcement cross section per unit width
Eq. (7.70)
b
cross-section width
Section 3.1.2
b
w
crack-band width
Section 2.1
d
structural height
Section 7.6.2
e
element index
Section 1.3
f
strength condition
Eq. (5.42)
f
c
uniaxial compressive strength of concrete (unsigned)
Section 2.1
f
ct
uniaxial tensile strength of concrete
Section 2.1
f
t
uniaxial failure stress – reinforcement
Section 2.3
f
yk
uniaxial yield stress – reinforcement
Section 2.3
f
E
probability density function of random variable
E
Eq. (9.2)
g
f
specific crack energy per volume
Section 2.1
h
cross-section height
Section 3.1.2
m
x
, m
y
, m
xy
moments per unit width
Eq. (7.8)
n
total number of degrees of freedom in a discretized system
Section 1.2
n
E
total number of elements
Section 3.3.1
n
i
order of Gauss integration
Section 1.6
n
N
total number of nodes
Section 3.3.1
n
x
, n
y
, n
xy
normal forces per unit width
Eq. (7.8)
p
pressure
Eq. (5.8)
p
F
failure probability
Eq. (9.18)
distributed beam loads
Eq. (3.58)
r
local coordinate
Section 1.3
s
local coordinate
Section 1.3
s
bf
slip at residual bond strength
Section 2.4
s
b
max
slip at bond strength
Section 2.4
t
local coordinate
Section 1.3
t
time
Section 1.2
t
x
, t
y
, t
xy
couple force resultants per unit width
Eq. (7.67)
u
specific internal energy
Eq. (5.12)
υ
x
, v
y
shear forces per unit width
Eq. (7.8)
w
deflection
Eq. (1.56)
w
fictitious crack width
Eq. (2.4)
w
cr
critical crack width
Section 5.7.1
z
internal lever arm
Section 3.5.4
Bold lowercase roman
b
body forces
Section 1.2
f
internal nodal forces
Section 1.2
p
external nodal forces
Section 1.2
n
normal vector
Eq. (5.5)
t
surface traction
Section 1.2
t
c
crack traction
Eq. (5.123)
u
displacement field
Section 1.2
υ
nodal displacements
Section 1.2
w
c
fictitious crack width vector
Eq. (5.123)
Normal uppercase italics
A
surface
Section 1.2,
Eq. (1.5)
A
cross-sectional area of a bar or beam
Eq. (1.54)
A
s
cross-sectional area reinforcement
Example 2.4
A
t
surface with prescribed tractions
Section 1.2,
Eq. (1.5)
A
u
surface with prescribed displacements
Eq. (1.53)
C
material stiffness coefficient
Eq. (2.32)
C
T
tangential material stiffness coefficient
Eq. (2.34)
D
scalar damage variable
Eq. (5.106)
D
T
tangential material compliance coefficient
Eq. (5.160)
D
cT
tangential compliance coefficient of cracked element
Eq. (5.132)
D
cLT
tangential compliance coefficient of crack band
Eq. (5.132)
E
Young’s modulus
Eq. (1.43)
E
0
initial value of Young’s modulus
Eq. (2.13)
E
c
initial value of Young’s modulus of concrete
Section 2.1
E
s
initial Young’s modulus of steel
Section 2.3
E
T
tangential modulus
Eq. (2.2)
F
yield function
Eq. (5.64)
F
E
distribution function of random variable
E
Eq. (9.1)
G
shear modulus
Eq. (3.8)
G
flow function
Eq. (5.63)
G
f
specific crack energy per surface
Eq. (2.7)
I
1
first invariant of stress
Eq. (5.20)
J
determinantof Jacobian
Eq. (1.67)
J
2
,
J
3
second, third invariant of stress deviator
Eq. (5.20)
L
c
characteristic length of an element
Eq. (6.32)
L
e
length of bar or beam element
Section 1.3
M
bending moment
Section 3.1.2
N
normal force
Section 3.1.2
P
probability
Eq. (9.1)
T
natural period
Eq. (3.211)
V
shear force
Section 3.1.2
V
volume
Section 1.2,
Eq. (1.5)
Bold uppercase roman
B
matrix of spatial derivatives of shape functions
Section 1.2,
Eq. (1.2)
C
material stiffness matrix
Eq. (1.47)
C
T
tangential material stiffness matrix
Eq. (1.50)
D
material compliance matrix
Eq. (1.51)
D
T
tangential material compliance matrix
Eq. (1.51)
E
coordinate independent strain tensor
Eq. (8.15)
G
1
, G
2
, G
3
unit vectors of covariant system
Eq. (8.16)
G
1
, G
2
, G
3
unit vectors of contravariant system
Eq. (8.17)
I
unit matrix
Eq. (1.85)
J
Jacobian
Eq. (1.20)
K
stiffness matrix
Eq. (1.11)
K
e
element stiffness matrix
Eq. (1.61)
K
T
tangential stiffness matrix
Eq. (1.66)
K
Te
tangential element stiffness matrix
Eq. (1.65)
M
mass matrix
Eq. (1.60)
M
e
element mass matrix
Eq. (1.58)
N
matrix of shape functions
Section 1.2,
Eq. (1.1)
Q
vector/tensor rotation matrix
Eq. (5.15)
S
coordinate independent stress tensor
Eq. (8.24)
T
element rotation matrix
Eq. (3.109)
V
n
shell director
Section 8.1
V
α
,
V
β
unit vectors of local shell system
Eq. (8.2)
Normal lowercase Greek
α
tie inclination
Eq. (3.157)
α
E
,
α
R
sensitivity parameters
Eq. (9.13)
α
coefficient for several other purposes
β
shear retention factor
Eq. (5.137)
β
reliability index
Eq. (9.12)
ß
t
tension stiffening coefficient
Section 2.7
uniaxial strain
Section 1.4,
Eq. (1.43)
strain of a beam reference axis
Section 3.1.1,
Eq. (3.4)
1
,
2
,
3
principal strains
Section 5.2.3
ct
concrete strain at uniaxial tensile strength
Section 2.1
cu
concrete failure strain at uniaxial tension
Eq. (5.152)
c
1
concrete strain at uniaxial compressive strength (signed)
Section 2.1
cu
1
concrete failure strain at uniaxial compression (signed)
Section 2.1
I
imposed uniaxial strain
Section 2.2
V
volumetric strain
Eq. (5.102)
ϕ
cross-section rotation
Eq. (3.1)
ϕ
angle of external friction
Eq. (5.91)
φ
angle of orientation
Section 6.1,
Eq. (6.5)
φ
creep coefficient
Eq. (2.26)
φ
c
creep coefficient of concrete
Eq. (3.119)
γ
shear angle
Eq. (3.1)
γ
E
,
γ
R
partial safety factors
Eq. (9.44)
к
curvature of a beam reference axis
Section 3.1.1,
Eq. (3.4)
к
p
state variable for plasticity
Section 5.5.1
к
d
state variable for damage
Section 5.6
μ
E
mean of random variable
E
Section 9.1
v
Poisson’s ratio
Eq. (1.44)
v
coefficient of variation
Eq. (9.46)
θ
strut inclination
Eq. (3.148)
θ
deviatoric angle
Eq. (5.46)
ϑ
angle of internal friction
Eq. (5.89)
ρ
deviatoric length
Eq. (5.45)
ρ
s
reinforcement ratio
Eq. (6.8)
s
specific mass
Eq. (1.52)
σ
uniaxial stress
Section 1.4,
Eq. (1.43)
σ
1
,
σ
2
,
σ
3
principal stresses
Section 5.2.3
σ
E
standard deviation of random variable
E
Section 9.1
τ
bond stress
Section 2.4,
Eq. (2.44)
τ
time variable in time history
Section 2.2
τ
bf
residual bond strength
Section 2.4
τ
b
max
bond strength
Section 2.4
ω
circular natural frequency
Eq. (3.211)
ξ
hydrostatic length
Eq. (5.44)
Bold lowercase Greek
small strain
Section 1.2
generalized strain
Eq. (1.33)
p
plastic small strain
Eq. (5.61)
к
vector of internal state variables
Eq. (5.39)
σ
Cauchy stress
Section 1.2
σ
generalized stress
Eq. (1.34)
σ′
deviatoric part of Cauchy stress
Section 5.2.2
Normal uppercase Greek
Φ
standardized normal distribution function
Eq. (9.19)
Bold uppercase Greek
Σ
viscous stress surplus
Eq. (1.76)
Chapter 1
Finite Elements Overview
1.1 Modeling Basics
“There are no exact answers. Just bad ones, good ones and better ones. Engineering is the art of approximation.” Approximation is performed with models. We consider a reality of interest, e.g., a concrete beam. In a first view, it has properties such as dimensions, color, surface texture. From a view of structural analysis the latter ones are irrelevant. A more detailed inspection reveals a lot of more properties: composition, weight, strength, stiffness, temperatures, conductivities, capacities, and so on. From a structural point of view some of them are essential. We combine those essential properties to form a conceptual model. Whether a property is essential is obvious for some, but the valuation of others might be doubtful. We have to choose. By choosing properties our model becomes approximate compared to reality. Approximations are more or less accurate.
On one hand, we should reduce the number of properties of a model. Any reduction of properties will make a model less accurate. Nevertheless, it might remain a good model. On the other hand, an over-reduction of properties will make a model inaccurate and therefore useless. Maybe also properties are introduced which have no counterparts in the reality of interest. Conceptual modeling is the art of choosing properties. As all other arts it cannot be performed guided by strict rules.
The chosen properties have to be related to each other in quantitative manner. This leads to a mathematical model. In many cases, we have systems of differential equations relating variable properties or simply variables. After prescribing appropriate boundary and initial conditions an exact, unique solution should exist for variables depending on spatial coordinates and time. Thus, a particular variable forms a field. Such fields of variables are infinite as space and time are infinite.
As analytical solutions are not available in many cases, a discretization is performed to obtain approximate numerical solutions. Discretization reduces underlying infinite space and time into a finite number of supporting points in space and time and maps differential equations into algebraic equations relating a finite number of variables. This leads to a numerical model.
Figure 1.1: Modeling (a) Type of models following [83]. (b) Relations between model and reality.
A numerical model needs some completion as it has to be described by means of programming to form a computational model. Finally, programs yield solutions through processing by computers. The whole cycle is shown in Fig. 1.1. Sometimes it is appropriate to merge the sophisticated sequence of models into the model.
A final solution provided after computer processing is approximate compared to the exact solution of the underlying mathematical model. This is caused by discretization and round-off errors. Let us assume that we can minimize this mathematical approximation error in some sense and consider the final solution as a model solution. Nevertheless, the relation between the model solution and the underlying reality of interest is basically an issue. Both – model and reality of interest – share the same properties by definition or conceptual modeling, respectively. Let us also assume that the real data of properties can be objectively determined, e.g., by measurements.
Thus, real data of properties should be properly approximated by their computed model counterparts for a problem under consideration. The difference between model solution data and real data yields a modeling error. In order to distinguish between bad (inaccurate), good (accurate), and better model solutions, we have to choose a reference for the modeling error. This choice has to be done within a larger context, allows for discretion and again is not guided by strict rules like other arts. Furthermore, the reference may shift while getting better model solutions during testing.
A bad model solution may be caused by a bad model – bad choice of properties, poor relations of properties, insufficient discretization, programming errors – or by incorrect model parameters. Parameters are those properties which are assumed to be known in advance for a particular problem and are not object to a computation. Under the assumption of a good model, the model parameters can be corrected by a calibration. This is based upon appropriate problems from the reality of interest with the known real data. On one hand calibration minimizes the modeling error by adjusting of parameters. On the other hand, validation chooses other problems with known real data and assesses the modeling error without adjusting of parameters. Hopefully model solutions are still good.
Regarding reinforced concrete structures, calibrations usually involve the adaption of material parameters like strength and stiffness as part of material models. These parameters are chosen such that the behavior of material specimen observed in experiments is reproduced. A validation is usually performed with structural elements such as bars, beams, plates, and slabs. Computational results of structural models are compared with the corresponding experimental data.
This leads to basic peculiarities. Reproducible experiments performed with structural elements are of a small simplified format compared with complex unique buildings. Furthermore, repeated experimental tests with the same nominal parameters exhibit scattering results. Standardized benchmark tests carving out different aspects of reinforced concrete behavior are required. Actually a common agreement about such benchmark tests exists only in the first attempts. Regarding a particular problem a corresponding model has to be validated on a case-by-case strategy using adequate experimental investigations. Their choice again has no strict rules as the preceding arts.
Complex proceedings have been sketched hitherto outlining a model of modeling. Some benefit is desirable finally. Thus, a model which passed validations is usable for predictions. Structures created along such predictions hopefully prove their worth in the reality of interest.
This textbook covers the range of conceptual models, mathematical models, and numerical models with special attention to reinforced concrete structures. Notes regarding the computational model including available programs and example data are given in Appendix F. A major aspect of the following is modeling of ultimate limit states: states with maximum bearable loading or acceptable deformations and displacements in relation to failure. Another aspect is given with serviceability: Deformations and in some cases oscillations of structures have to be limited to allow their proper usage and fulfillment of intended services. Durability is a third important aspect for building structures: deterioration of materials through, e.g., corrosion, has to be controlled. This is strongly connected to cracking and crack width in the case of reinforced concrete structures. Both topics are also treated in the following.
1.2 Discretization Outline
The finite element method (FEM) is a predominant method to derive numerical models from mathematical models. Its basic theory is described in the remaining sections of this chapter insofar as it is needed for its application to different types of structures with reinforced concrete in the following chapters.
The underlying mathematical model is defined in one-, two-, or three-dimensional fields of space related to a body and one-dimensional space of time. A body undergoes deformations during time due to loading. We consider a simple example with a plate defined in 2D space, see Fig. 1.2. Loading is generally defined depending on time whereby time may be replaced by a loading factor in the case of quasistatic problems. Field variables depending on spatial coordinates and time are, e.g., given by the displacements.
Such fields are discretized by dividing space into
elements
which are connected by
nodes
, see
Fig. 1.3a
. Elements adjoin but do not overlap and fill out the space of the body under consideration.
Discretization basically means
interpolation
,, i.e., displacements within an element are interpolated using the values at nodes belonging to the particular element.
Figure 1.2: Model of a plate.
In the following this will be written as
(1.1)
with the displacements u depending on spatial coordinates and time, a matrix N of shape functions depending on spatial coordinates and a vector υ depending on time and collecting all displacements at nodes. The number of components of υ is n. It is two times the number of nodes in the case of the plate as the displacement u has components ux, uy. Generally some values of υ may be chosen such that the essential or displacement boundary conditions of the problem under consideration is fulfilled by the displacements interpolated by Eq. (1.1). This is assumed for the following.
Figure 1.3: (a) Elements and nodes (deformed). (b) Nodal quantities.
Strains are derived from displacements by differentiation with respect to spatial coordinates. In the following, this will be written as
(1.2)
with the strains depending on spatial coordinates and time, a matrix B of spatial derivatives of shape functions depending on spatial coordinates and the vector υ as has been used in Eq. (1.1). The first examples for Eqs. (1.1, 1.2) will be given in Section 1.3.
A field variable
u
is discretized with
Eqs. (1.1
,
1.2)
, i.e., the infinite field in space is reduced into a finite number
n
of variables in supporting spatial points or nodes which are collected in
υ
.
Thereby kinematic compatibility should be assured regarding interpolated displacements, i.e., generally spoken a coherence of displacements and deformations should be given.
Strains lead to stresses σ. A material law connects both. Material laws for solids are a science in itself. This textbook mainly covers their flavors for reinforced concrete structures. To begin with, such laws are abbreviated with
(1.3)
Beyond total values of stress and strain their small changes in time t have to be considered. They are measured with time derivatives
(1.4)
Nonlinear material behavior is mainly formulated as a relation between and . The first concepts about material laws are given in Section 1.4.
An equilibrium condition is the third basic element of structural analysis beneath kinematic compatibility and material laws. It is advantageously formulated as principle of virtual work leading to
(1.5)
for quasistatic cases with the volume V of the solid body of interest, its body forces b, its surface A, and its surface tractions t which are prescribed at a part At of the whole boundary A. Furthermore, virtual displacements δu and the corresponding virtual strains δ are introduced. They are arranged as vectors and δuT, δT indicate their transposition into row vectors to have a proper scalar product with σ, b, t which are also arranged as vectors. Fields of b and t are generally prescribed for a problem under consideration while the field of stresses σ remains to be determined. Surface tractions t constitute the natural or force boundary conditions.
Stresses
σ
and loadings
b
,
t
are in static equilibrium for the problem under consideration if
Eq. (1.5)
is fulfilled for arbitrary virtual displacements
δ
u
and the corresponding virtual strains
δ
.
Thereby, δu is zero at the part Au of the whole boundary A with prescribed displacement boundary conditions. Applying the displacement interpolation equation (1.1) to virtual displacements leads to
(1.6)
and using this with Eq. (1.5) to
(1.7)
with transpositions δυT, BT, NT of the vector δυ and the matrices B, N. As δυ is arbitrary a discretized condition of static equilibrium is derived in the form
(1.8)
with the vector f of internal nodal forces and the vector p of external nodal forces
(1.9)
Corresponding to the length of the vector υ the vectors f, p have n components.
Nonlinear stress–strain relations, i.e., physical nonlinearities, are always an issue for rein-forced concrete structures. It is a good practice in nonlinear simulation to start with a linearization to have a reference for the refinements of a conceptual model. Physical linearity is described with
(1.10)
with a constant material matrix C. Thus, using Eq. (1.2) internal forces f (Eq. (1.9)) can be formulated as
(1.11)
with a constant stiffness matrix K leading to
(1.12)
This allows for a direct determination of nodal displacements which is symbolically written as
(1.13)
Actually the solution is not determined with a matrix inversion but with more efficient techniques, e.g., Gauss triangularization. Stresses σ and strains follow with a solution υ given. A counterpart of physical linearity is geometric linearity:
Small displacements and geometric linearity are assumed throughout the following if not otherwise stated.
This was a fast track for the finite element method. The rough outline will be filled out in the following. Comprehensive descriptions covering all aspects are given in, e.g., [98], [99], [9], [3]. The special aspects of reinforced concrete structures are treated in [16], [44], [81].
1.3 Elements
Interpolation performed with finite elements will be described with more details in the following. We consider the mechanical behavior of material points within a body. A material point is identified by its spatial coordinates. It is convenient to use a different coordinate system simultaneously. First of all, the global Cartesian coordinate system, see Appendix C, which is shared by all material points of a body. Thus, a material point is identified by global Cartesian coordinates
(1.14)
in 3D space. In the following, we assume that the space occupied by the body has been divided into finite elements. Thus, a material point may alternatively be identified by the label I of the element it belongs to and its local coordinates
(1.15)
related to a particular local coordinate system belonging to the element e. A material point undergoes displacements. In the case of translations they are measured in the global Cartesian system by
(1.16)
Displacements in a general sense may also be measured by means of rotations
(1.17)
if we consider a material point embedded in some neighborhood of surrounding points. The indices indicate the respective reference axes of rotation.
Isoparametric interpolation will be used in the following. The general interpolation form (Eq. (1.1)) is particularized as
(1.18)
whereby the global coordinates of the corresponding material point are given by
(1.19)
The vector ve collects all nodal displacements of all nodes belonging to the element e and the vector xe all global nodal coordinates of that element. Isoparametric interpolation is characterized by the same interpolation for geometry and displacements with the same shape functions N(r). Global and local coordinates are related by the Jacobian
(1.20)
which may be up to a 3 × 3 matrix for 3D cases. Strains may be derived with displacements related to global coordinates through isoparametric interpolation. Their definition depends on the type of the structural problem. A general formulation
(1.21)
is used. Strains finally lead to stresses σ. Examples are given in the following.
All mentioned stresses and the corresponding strains are conjugate with respect to energy, i.e., the product corresponds to a rate of internal energy or a rate of specific internal energy. The concept of stresses may be generalized:
Depending on the type of structural element
σ
may stand for components of Cauchy stresses or for components of forces or for components of internal forces in a beam cross section, see Section 3.1.1.
Strains
are
generalized
correspondingly in order to lead to internal energy, e.g., including displacements in the case forces or curvature in the case of moments.
Another issue concerns continuity: For the four-node continuum element the interpolation is continuous between adjacent elements along their common boundary. One sided first derivatives of interpolation exist for each element along the boundary but are different for each element. Thus, the four-node continuum element has C0-continuity with these properties. Furthermore, the integrals for internal and external nodal forces (Eq. (1.9)) are evaluable. Other elements may require higher orders of continuity for nodal forces to be integrable.
Only a few element types were touched up to now. Further elements often used are 3D-continuum elements, 2D- and 3D-beam elements, shell elements and slab elements as a special case of shell elements. Furthermore, elements imposing constraints like contact conditions have become common in practice. For details see, e.g., [3]. Regarding the properties of reinforced concrete more details about 2D-beam elements including Bernoulli beams and Timoshenko beams are given in Section 3.3, about slabs in Section 7.4 and about shells in Chapter 8.
1.4 Material Behavior
From a mechanical point of view, material behavior is primarily focused on strains and stresses. The formal definitions of strains and stresses assume a homogeneous area of matter [64]. Regarding the virgin state of solids their behavior initially can be assumed as linear elastic in nearly all relevant cases. Furthermore, the behavior can be initially assumed as isotropic in many cases, i.e., the reaction of a material is the same in all directions. The concepts of isotropy and anisotropy are discussed in Section 5.3 with more details.
The following types of elasticity are listed exemplary:
Equations (1.43)–(1.45) are a special case of
(1.47)
with the constant material stiffness matrix C describing a linear material behavior. At the latest upon approaching material strength, the behavior becomes physically nonlinear. A simple case is given by the uniaxial elastoplastic law
(1.48)
and
(1.49)
In the case of nonlinear material equations at least an incremental form
(1.50)
should exist with the tangential material stiffness CT, which is no longer constant anymore but might depend on stress, strain, and internal state variables. On occasion the compliance is needed, as a counterpart of stiffness, i.e.,
(1.51)
1.5 Weak Equilibrium and Spatial Discretization
The preceding sections gave an introduction of (1) kinematic compatibility within the context of spatial discretization and of (2) material laws. The third cornerstone of structural mechanics is equilibrium which is formulated in a weak form as a principle of virtual work.
Boundary conditions have to be regarded in advance. Given a point on a boundary of a body, either a displacement boundary condition or a force boundary condition (zero force is also a condition) has to be prescribed for this point. Let us assume that displacements are prescribed with on surface part Au, tractions are prescribed with on surface part At while Au together with At contain the whole surface A but do not overlap. Thus, equilibrium is given by
(1.52)
under the conditions
(1.53)
and δu arbitrary otherwise. The meaning of the symbols is summarized as follows:
()
T
transpose of column vector () leading to row vector
u
field of displacement vector
ü
field of acceleration vector
δ
u
field of test functions or virtual displacement vector
δ
field of virtual strain vector corresponding to
δ
u
σ
field of stress vector
ϱ
specific mass
prescribed field of loads distributed in the body
prescribed field of tractions distributed over surface of the body
V
body volume
A
body surface
A
u
part of surface with prescribed displacements
A
t
part of surface with prescribed tractions
Formulation (1.52) covers structural dynamics and includes quasistatics as a special case. Concentrated loads are not explicitly included. For mathematically precise formulations also covering generalized variational principles see [96]. All listed parameters have to be considered as generalized. The following evaluations of are listed exemplary:
The principle of virtual work or weak integral forms of equilibrium conditions treat a body as a whole. Strong differential forms consider forces applied to infinitesimally small sections or differentials of a body and lead to differential equations. Both are equivalent from a mechanical point of view. This is exemplary demonstrated for beams in Section 3.2. Weak forms are the starting point for discretization with finite elements. This has the following steps regarding Eq. (1.52):
(1.61)
see Eqs. (1.58)1 and (1.21), with a constant element stiffness matrix Ke. Assembling leads to a system stiffness matrix K
(1.62)
and regarding Eq. (1.60) to
(1.63)
which is a system of linear ordinary differential equations of second order in time t.
To treat physical nonlinearities the system’s tangential stiffness is involved. The tangential stiffness matrix is needed for the solution of the nonlinear system and furthermore reveals characteristic properties, e.g., regarding stability properties. The tangential stiffness of an element is derived with
(1.64)
with
(1.65)
see Eqs. (1.58)1, (1.50), and (1.21), and a system tangential stiffness KT
(1.66)
Finally, the system (1.60) or (1.63) should be constrained with appropriate conditions regarding υ to prevent rigid body displacements.
1.6 Numerical Integration and Solution Methods for Algebraic Systems
The integral formulation of equilibrium conditions requires the evaluation of integrals as given by Eq. (1.58). The evaluation is performed element by element. The integration of a quad element, see Section 1.3, is exemplary discussed in the following. A general function f(x, y) indicates the integrand. The isoparametric quad element has a local coordinate system r, s with −1 ≤ r, s, ≤ 1, see Section 1.3. Thus, integration is performed by
(1.67)
with the determinant J of the Jacobian, see Eq. (1.37), and a thickness b. As closed analytical forms generally are not available for f(r, s) a numerical integration has to be performed
(1.68)
Integration accuracy increases with increasing integration order. On the other hand, numerical integration leads to a major contribution to computational costs.
Gauss integration generally is most efficient compared to other numerical integration schemes: an integration order ni gives exact results for polynoms of order 2ni + 1 disregarding roundoff errors, e.g., a uniaxial integration of order 1 with two sampling points exactly integrates a polynomial of the order 3. Alternative numerical integration schemes are given by schemes of Simpson, Newton–Cotes, Lobatto.
Table 1.1: Sampling points and weights for Gauss integration (15 digits shown).
n
i
ξ
i
η
i
0
0.0
2.0
1
±0.577350269189626
1.0
2
±0.77459 66692 41483
0.55555 55555 55556
0.0
0.88888 88888 88889
3
±0.86113 63115 94053
0.347854845137454
(1.69)
with a residualr, internal nodal forces f depending on displacements υ and external nodal loads p, which are assumed to be independent of υ. The general case is nonlinear dependence of f on υ. Thus, the solution of Eq. (1.69) has to be determined by an iteration with a sequence υ(0),…, υ(v). Regarding an arbitrary iteration step (v) we have r(υ(v)) ≠ 0 and seek for a correction δv. A linear Taylor expansion is used as a basic approach
(1.70)
with a tangential stiffness matrix, see also Eq. (1.65)
(1.71)
leading to the Newton–Raphson method
(1.72)
with (hopefully) an improved value υ(v+1). Iteration may stop if ||r(υ(v+1))|| ≪ 1 and ||δv|| ≪ 1 with a suitable norm || · || transforming a vector into a scalar. The method generally has a fast convergence but is relatively costly. The tangential stiffness matrix has to be computed in every iteration step (v) and a decomposition in order to solve (LU decomposition instead of inversion) has to be performed on it to determine δv. Alternative iteration methods use variants of the iteration matrix like the modified Newton–Raphson method or the BFGS method or other quasi-Newton methods [3, 8.4], [9, 6.3],[99, 7]. For more details, see Appendix A.
The procedure is illustrated in Fig. 1.4 and combined with integration according to Eq. (1.58) and assembling according to Eq. (1.59). The time t serves as a loading parameter in the quasistatic case. A scaling of time, i.e., multiplying time with a constant factor in each occurrence, does not have any influence upon the results.
This starts to become different with a transient analysis. A material behavior like creep, see Section 2.2, has to be regarded as transient. Such a behavior is modeled by incorporating viscosity [64, 6.4]. Thus, the incremental material law (Eq. (1.73)) is extended as
(1.76)
Figure 1.4: Flow of displacement-based nonlinear calculation.
with an additional term Σ depending on stress σ(t) and strain (t). In a similar way as done with Eq. (1.73) leading to Eq. (1.75) this is integrated with
(1.77)
Internal nodal forces are determined according to Eq. (1.58)1
(1.78)
(1.79)
The contributions may involve nonlinearities due to the dependence of CT,i+1, Σi+1 on strains and stresses. Equilibrium at a time ti+1 has the condition
(1.80)
according to Eqs. (1.69, 1.78). We apply the Newton–Raphson method (Eq. (1.72)) to solve this system of algebraic equations within in incrementally iterative scheme, see Eq. (1.72). An extended tangential stiffness, see Eq. (1.71), is given by
(1.81)
with the iteration counter (v) leading to an iteration scheme
(1.82)
The exact formulation of the extended tangential stiffness depends on the particular form of or Σ, respectively. In the case of time steps Δt being small is .
A particular case is given by the viscoelasticity of materials, see Section 2.2, leading to
(1.83)
with constant material terms V, W, see Eq. (2.27). Thus, and stress from Eq. (1.77) becomes
(1.84)
leading to
(1.85)
with the unit matrix I. Internal nodal forces according to Eq. (1.58)1 are given by
(1.86)
with Δυ as before and
(1.87)
The residual, see Eq. (1.80), is given by
(1.88)
leading to an iteration scheme
(1.89)
All quantities at time ti can assumed to be known within a time stepping scheme. A potential source of nonlinearity is still given by CT,i+1. Formulation (1.87) is used as solution method for Examples 2.2 and 3.3.
Real time t is also a key factor for a dynamic analysis regarding inertia. Based on Eq. (1.60) we have in analogy to Eq. (1.69)
(1.90)
A widespread approach for the temporal discretization of acceleration together with velocities is given in the Newmark method
(1.91)
(1.92)
with an auxiliary quantity
(1.93)
and the velocity
(1.94)
Finally, dynamic equilibrium equation (1.90) is applied for the time step i + 1 with the acceleration according to Eq. (1.92):
(1.95)
With the given parameters γ, β, Δt, a given previous state υi, i, t, given mass matrix M and load pi+1, Eq. (1.95) has to be solved for vi+i whereby the dependence of fi+1 on vi+1 is crucial and might be nonlinear.
We apply again the Newton–Raphson method (Eq. (1.72)). An extended tangential stiffness, see Eq. (1.71), is given by
(1.96)
leading to an iteration scheme
(1.97)
This includes the linear case with
(1.98)
and Eq. (1.97) simplifies to
(1.99)
with no iteration necessary [2, 9.2.4]. Numerical integration parameters γ, ß rule consistency and numerical stability of the method.
Stability and consistency are necessary to ensure that the error of the numerical method remains within some bounds for a finite time step length Δt. A choice is reasonable for the Newmark method to reach consistency and stability [2, 9.4].
This section completes the basic discussion of procedures as they are directly used to solve problems of reinforced concrete structures. The following last section of this chapter touches some theoretical background regarding the finite element method.
1.7 Convergence
The major contribution to the mathematical approximation error, see Section 1.1, is the discretization error arising from the difference between mathematical and numerical model, see Fig. 1.1. This difference should become smaller with a mesh refinement, i.e., the numerical model should converge with respect to the underlying mathematical model. Under the assumption of geometrical and physical linearity the convergence behavior of the finite element method can be analyzed theoretically. Quasistatic problems are considered in the following.
The following mathematical symbols are used in this section:
for all
∈
element of
⊂
subset of
it exists
∩
intersection
∪
union
Given a linear material law
(1.100)
the condition of weak, integral equilibrium equation (1.52), can be written as
(1.101)
Generalized strains , δ are derived from the generalized displacements u, δu by a differential operator depending on the type of the structural problem under consideration. The trial functions according to Eq. (1.18) and test functions according to Eq. (1.57) are assumed to belong to a Sobolev function space H (→ square integrable functions [2, 4.3.4]) defined over the body V and to fulfill the displacement boundary conditions.
Equation (1.101) can be written in a general form as
(1.102)
with a symmetric, bilinear operator a(·, ·), a further linear operator (f, ·), and v formally replacing δu. This has the following properties:
Due to Eq. (1.107)a(v, v) ≥ 0, i.e., a is a norm and may be physically interpreted as energy. It is twice the internal strain energy. It can be shown that the problem Eq. (1.102) – i.e., determine a function u ∈ H if such that Eq. (1.102) is fulfilled for all v ∈ H if − has a unique solution u, see, e.g., [2, 4.3]. This is the exact solution of the mathematical model, see Fig. 1.1.
Discretization uses trial and test functions uh, vh ∈ Hh of a subset Hh ⊂ H based upon the concept of meshes and interpolation with elements and nodes, see Section 1.3. To simplify the derivations, a uniform mesh of elements is assumed with a mesh size parameter h, e.g., a diameter or length of a generic element. For nonuniform meshes see [2, 4.3.5]. The approximate solutionuh ∈ Hh of Eq. (1.102) is determined by
(1.108)
The difference between approximate and exact solution gives the discretization error
(1.109)
The approximation uh, is known for Hh given, it can be determined according to the procedure described in Section 1.5. The error eh has to be estimated. The approximate solution has the following properties:
(1.110)
(1.111)
(1.112)
Combination of Eqs. (1.107), (1.112), and (1.106) leads to
(1.113)
where inf is infimum, the largest lower bound1. This is rewritten as
(1.114)
with
(1.115)
d is a “distance” of functions in Hh to the exact solution u, c depends on the structural problem type and the values of its parameters, but not on Hh.
Convergence
means
u
h
, →
u
or ||
u
–
u
h
||
1
→ 0 with mesh size
h
→ 0.
Convergence can be reached with an appropriate selection of function spaces Hh whereby reducing the distance d(u, Hh).
A more precise statement is possible using interpolation theory. This introduces the interpolant2ui ∈ Hh of the exact solution u. Complete polynomials3 of degree k are used for discretization and interpolation. Interpolation theory estimates the interpolation error with
(1.116)
with the mesh size h and a constant which is independent of h [2, (4.99)]. ||u||k+1 is the k+1- order Sobolev norm of the exact solution. On the other hand a relation infvh∈Hh ||u − vh||1 ≤ ||u − ui||1 must hold as ui ∈ Hh. Using this and Eqs. (1.114, 1.116) yields
(1.117)
The value c can be merged to c, which depends on the structural problem type and the values of its parameters, but not on h. A further merging of c and ||u||k+1 leads to the well-known formulation
(1.118)
whereby c depends on the structural problem type, the values of its parameters and the norm of the exact solution.
The following conditions for convergence can be derived [2, 4.3.2]:
