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- Sprache: Englisch
Beton ist aufgrund seiner Vorteile der mit Abstand meistverwendete Baustoff: er ist formbar, preiswert und überall verfügbar. Kombiniert mit Bewehrung bietet dies eine immense Bandbreite an Eigenschaften und kann für eine Vielzahl von Zwecken angepasst werden. Damit ist Beton der Baustoff des 20. Jahrhunderts. Um der Baustoff des 21. Jahrhunderts zu sein, muss seine Nachhaltigkeit in den Fokus rücken. Bewehrte Betonkonstruktionen müssen mit geringerem Materialaufwand konstruiert werden, wobei ihr Tragfähigkeitspotential optimal ausgeschöpft werden muss. Computergestützte Methoden wie die Finite-Elemente-Methode (FEM) bieten wesentliche Werkzeuge, um das Ziel zu erreichen. In Kombination mit experimenteller Validierung ermöglichen sie ein tieferes Verständnis der Tragmechanismen. Im Vergleich zu herkömmlichen Ansätzen kann eine realistischere Abschätzung der Grenzzustände der Tragfähigkeit und der Gebrauchstauglichkeit erreicht werden. Dies ermöglicht eine deutlich verbesserte Ausnutzung der Baustoffe. Damit eröffnet sich auch ein weiterer Horizont für innovative Tragwerksentwürfe. Anspruchsvolle numerische Rechenverfahren werden aber in der Regel als "Black Boxes" bereitgestellt. Daten werden eingegeben, die Ausgaben ungeprüft übernommen, aber das Verständnis für die dazwischenliegenden Schritte ist oft rudimentär. Dies birgt die Gefahr von Fehlinterpretationen, um nicht zu sagen ungültigen Ergebnissen im Vergleich zu den getroffenen Problemdefinitionen. Das Risiko ist insbesondere bei nichtlinearen Problemen hoch. Bewehrter Beton weist als Verbundmaterial in seinen Grenzzuständen ein nichtlineares Verhalten auf, verursacht durch Verbund und nichtlineare Eigenschaften seiner Bestandteile. Seine Rissbildung ist ein reguläres Verhalten. In diesem Buch werden die Mechanismen des bewehrten Betons unter dem Blickwinkel numerischer Methoden aufgezeigt. So sollen auch "Black Boxes" transparent werden. Das Buch beschreibt entsprechende Methoden für Balken, Scheiben, Platten und Schalen im Rahmen von Quasi-Statik und Dynamik. Betonkriechen, Temperatureinwirkungen, Vorspannung, große Verformungen werden beispielhaft behandelt. Weiterhin werden aktuelle Materialmodelle für Beton dargestellt. Dabei werden sowohl die Möglichkeiten als auch die Fallstricke numerischer Methoden aufgezeigt. Die Theorie wird durch eine Vielzahl von Beispielen veranschaulicht. Die meisten von ihnen werden mit dem in Python implementierten und unter Open-Source-Bedingungen verfügbaren Softwarepaket ConFem durchgeführt.
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Table of Contents
Cover
Title Page
Copyright
Preface
List of Examples
Notation
1 Introduction
Why Read This Book?
Topics of the Book
How to Read This Book
2 Finite Elements Overview
2.1 Modelling Basics
2.2 Discretisation Outline
2.3 Elements
2.4 Material Behaviour
2.5 Weak Equilibrium
2.6 Spatial Discretisation
2.7 Numerical Integration
2.8 Equation Solution Methods
2.9 Discretisation Errors
3 Uniaxial Reinforced Concrete Behaviour
3.1 Uniaxial Stress–Strain Behaviour of Concrete
3.2 Long–Term Behaviour – Creep and Imposed Strains
3.3 Reinforcing Steel Stress–Strain Behaviour
3.4 Bond between Concrete and Reinforcement
3.5 Smeared Crack Model
3.6 Reinforced Tension Bar
3.7 Tension Stiffening of Reinforced Bars
4 Structural Beams and Frames
4.1 Cross-Sectional Behaviour
4.2 Equilibrium of Beams
4.3 Finite Elements for Plane Beams
4.4 System Building and Solution
4.5 Creep of Concrete
4.6 Temperature and Shrinkage
4.7 Tension Stiffening
4.8 Prestressing
4.9 Large Displacements – Second-Order Analysis
4.10 Dynamics
5 Strut-and-Tie Models
5.1 Elastic Plate Solutions
5.2 Strut-and-Tie Modelling
5.3 Solution Methods for Trusses
5.4 Rigid Plastic Truss Models
5.5 Application Aspects
6 Multi-Axial Concrete Behaviour
6.1 Basics
6.2 Continuum Mechanics
6.3 Isotropy, Linearity, and Orthotropy
6.4 Nonlinear Material Behaviour
6.5 Elasto-Plasticity
6.6 Damage
6.7 Damaged Elasto-Plasticity
6.8 The Microplane Model
6.9 General Requirements for Material Laws
7 Crack Modelling and Regularisation
7.1 Basic Concepts of Crack Modelling
7.2 Mesh Dependency
7.3 Regularisation
7.4 Multi-Axial Smeared Crack Model
7.5 Gradient Methods
7.6 Overview of Discrete Crack Modelling
7.7 The Strong Discontinuity Approach
8 Plates
8.1 Lower Bound Limit State Analysis
8.2 Cracked Concrete Modelling
8.3 Reinforcement and Bond
8.4 Integrated Reinforcement
8.5 Embedded Reinforcement with a Flexible Bond
9 Slabs
9.1 Classification
9.2 Cross-Sectional Behaviour
9.3 Equilibrium of Slabs
9.4 Reinforced Concrete Cross-Sections
9.5 Slab Elements
9.6 System Building and Solution Methods
9.7 Lower Bound Limit State Analysis
9.8 Nonlinear Kirchhoff Slabs
9.9 Upper Bound Limit State Analysis
10 Shells
10.1 Geometry and Displacements
10.2 Deformations
10.3 Shell Stresses and Material Laws
10.4 System Building
10.5 Slabs and Beams as a Special Case
10.6 Locking
10.7 Reinforced Concrete Shells
11 Randomness and Reliability
11.1 Uncertainty and Randomness
11.2 Failure Probability
11.3 Design and Safety Factors
12 Concluding Remarks
Appendix A: Solution Methods
A.1 Nonlinear Algebraic Equations
A.2 Transient Analysis
A.3 Stiffness for Linear Concrete Compression
A.4 The Arc Length Method
Appendix B: Material Stability
Appendix C: Crack Width Estimation
Appendix D: Transformations of Coordinate Systems
Appendix E: Regression Analysis
References
Index
Wiley End User License Agreement
List of Tables
Chapter 2
Table 2.1 Sampling points and weights for Gauss integration (15 digits shown).
Chapter 3
Table 3.1 Material parameters Example 3.4.
Chapter 6
Table 6.1 Material parameters of Example 6.2.
Table 6.2 Example 6.4. Material parameters
Chapter 7
Table 7.1 Discretisation refinement of Example 7.1.
Table 7.2 Material parameters of Example 7.6.
Chapter 8
Table 8.1 Parameters of Example 8.3.
Chapter 9
Table 9.1 Sampling points and weights for triangular numerical integration.
Table 9.2 Example 9.5. Reinforcement material parameters and derived quantities.
Chapter 11
Table 11.1 Parameters of Example 11.1.
Table 11.2 Parameters of Example 11.3.
List of Illustrations
Chapter 1
Figure 1.1 (a) Stuttgart television tower, from Kleinmanns and Weber (2009), pho...
Figure 1.2 (a) Ganter bridge, from Billington (2014), photography: Nicolas Janbe...
Figure 1.3 (a) Office building: Züblin-Haus, from Bachmann et al. (2021). (b) Hi...
Figure 1.4 (a) Power plant, RWE, Niederaußem, Germany, from Krätzig et al. (2007...
Figure 1.5 (a) FEM and reinforced concrete bases. (b) Uniaxial structures.
Figure 1.6 (a) Multi-axial concrete and its implications. (b) Multi-axial struct...
Chapter 2
Figure 2.1 Modelling. (a) Type of models following Schwer (2007). (b) Relations ...
Figure 2.2 Model of a plate.
Figure 2.3 (a) Elements and nodes (deformed). (b) Nodal quantities.
Figure 2.4 Newton–Raphson method.
Figure 2.5 Flow of displacementbased nonlinear calculation.
Chapter 3
Figure 3.1 Uniaxial compressive stress–strain curve of concrete.
Figure 3.2 Simplified model for force transfer in the composition of aggregates ...
Figure 3.3 (a) Cracking in mesoscale with the process zone. (b) Uniaxial tensile...
Figure 3.4 Scheme of homogenised localisation within a crack band in a tension b...
Figure 3.5 Example 3.1. Scheme of system and discretisation.
Figure 3.6 Example 3.1. (a) Reaction force displacement curve. (b) Strains along...
Figure 3.7 Approximation of stresses with the fictitious crack width.
Figure 3.8 Uniaxial strain depending on time for a material with creep.
Figure 3.9 (a) Kelvin–Voigt and Maxwell elements. (b) Chain and series.
Figure 3.10 Example 3.2. Time dependencies. (a) Strain. (b) Stress.
Figure 3.11 Reinforcing steel. (a) Uniaxial stress–strain behaviour. (b) Cyclic ...
Figure 3.12 (a) Basic bond set-up. (b) Main bond mechanism.
Figure 3.13 (a) Schematic bond equilibrium. (b) Typical bond law.
Figure 3.14 The smeared crack concept.
Figure 3.15 Linearised cohesive crack model for the 1D smeared crack (see Figure...
Figure 3.16 Scheme of a reinforced tension bar model.
Figure 3.17 Example 3.4. Reaction force displacement curve.
Figure 3.18 Example 3.4. (a) Concrete stresses. (b) Rebar stresses.
Figure 3.19 Example 3.4. (a) Bond stresses. (b) Displacements in the final state...
Figure 3.20 (a) Cracks and stresses. (b) Tension stiffening model.
Chapter 4
Figure 4.1 Kinematics of a plane beam.
Figure 4.2 Reinforced concrete cross-section. (a) Geometry. (b) Internal forces.
Figure 4.3 Reinforced concrete cross-section with linear compressive concrete.
Figure 4.4 Example 4.1. Moment-curvature curves.
Figure 4.5 Equilibrium of an infinitesimal beam element.
Figure 4.6 Beam orientation in 2D space.
Figure 4.7 Example 4.2. (a) System. (b) Load factor lf depending on mid-span def...
Figure 4.8 Example 4.2. Final loading state. (a) Deflection of reference axis (s...
Figure 4.9 Example 4.2. Final loading state. (a) Reinforcement strain. (b) Upper...
Figure 4.10 Shear stiffness. (a) Struts. (b) Ties.
Figure 4.11 Example 4.3. (a) Mid-span deflection during time. (b) Concrete and r...
Figure 4.12 Example 4.4. Bending moments. (a) Linear elastic. (b) RC (different ...
Figure 4.13 Example 4.4. RC. (a) Curvature. (b) Strain of the reference axis.
Figure 4.14 Crack pattern of RC beam with constant moment.
Figure 4.15 Example 4.5. (a) Moment. (b) Normal force.
Figure 4.16 (a) Redirection forces from prestressing. (b) Internal forces with p...
Figure 4.17 Example 4.6. (a) System. (b) Mid-span load–deflection curve.
Figure 4.18 Example 4.6. Final stage. (a) RC bending moment Mc. (b) Prestressing...
Figure 4.19 Equilibrium of beam section in the deformed configuration.
Figure 4.20 Cantilever column.
Figure 4.21 Example 4.8. (a) System. (b) Vertical load-horizontal displacement c...
Figure 4.22 Example 4.8. (a) Moments along column for different loading factors....
Figure 4.23 Example 4.9. (a) System and loading. (b) Linear elastic mid-span def...
Figure 4.24 Example 4.9. Linear elastic. (a) Moments along beam until the first ...
Figure 4.25 Example 4.9. RC mid-span deflection with time.
Figure 4.26 Example 4.9. RC. (a) Moments along beam until first maximum displace...
Chapter 5
Figure 5.1 The deep beam system.
Figure 5.2 Deep beam principal stresses.
Figure 5.3 (a) Example truss system. (b) Compression field as a strut-and-tie mo...
Figure 5.4 Truss. (a) Member strain. (b) Member force and nodal forces.
Figure 5.5 Truss system types.
Figure 5.6 Example 5.2. (a) Member stresses [MN∕m2]. (b) Proposed reinforcement ...
Figure 5.7 Corbel example 5.3. (a) System. (b) Strut-and-tie model.
Figure 5.8 Example 5.3. (a) Load displacement curve. (b) Member stresses [MN∕m2]...
Figure 5.9 Nodes. (a) Compression. (b) Compression–tension. (c) Reinforcement re...
Chapter 6
Figure 6.1 (a) Concrete at mesoscale. (b) Microcracking.
Figure 6.2 (a) Body in reference and deformed configuration. (b) Infinitesimal s...
Figure 6.3 (a) Rotation of coordinate system. (b) Principal stress space.
Figure 6.4 (a) Hydrostatic length and deviatoric plane. (b) Deviatoric length an...
Figure 6.5 Triaxial cell.
Figure 6.6 Strength surfaces. (a) General view direction. (b) Pressure axis view...
Figure 6.7 (a) Biaxial strength. (b) Stress paths.
Figure 6.8 Nonlinear material classification.
Figure 6.9 Surfaces of Mohr–Coulomb and Drucker–Prager yield functions in princi...
Figure 6.10 Mohr circle for the Mohr–Coulomb yield type.
Figure 6.11 Intersections of Mohr–Coulomb, Drucker–Prager, and Willam–Warnke sur...
Figure 6.12 Damage variable D depending on equivalent strain κd.
Figure 6.13 Example 6.2. (a) Uniaxial stress–strain curves. (b) Loading, unloadi...
Figure 6.14 Microplane. (a) Interaction layers (Bažant et al. 2000), and (b) Uni...
Figure 6.15 (a) V-D-split. (b) Microplanes by triangularisation of the unit sphe...
Figure 6.16 Example 6.4. (a) Uniaxial stress–strain curves. (b) Damage on microp...
Chapter 7
Figure 7.1 (a) Fracture modes. (b) Material failure types.
Figure 7.2 Cohesive crack model with fictitious crack.
Figure 7.3 (a) Model for a softening bar. (b) Material model for a softening bar...
Figure 7.4 Load-displacement relations for a softening bar.
Figure 7.5 Example 7.1. (a) System. (b) Load displacement curves.
Figure 7.6 Example 7.2. (a) Scaled tensile uniaxial stress relations. (b) Scalin...
Figure 7.7 Example 7.2. Load–displacement curves.
Figure 7.8 Example 7.2. Principal stresses on a deformed structure (scaling fact...
Figure 7.9 Non-local uniaxial strain.
Figure 7.10 Crack band (2D).
Figure 7.11 Linearised cohesive crack model with crack energy Gf.
Figure 7.12 (a) Phase field regularisation (Miehe et al. 2010). (b) Reference ba...
Figure 7.13 Example 7.5. Load displacement curves.
Figure 7.14 Example 7.5. (a) Strains ϵ along the bar. (b) Phase fields d along t...
Figure 7.15 Kinematics of the discontinuous part Eq. (7.117).
Figure 7.16 Normal components of traction–separation relations with unloading an...
Figure 7.17 Example 7.6. Load displacement curves.
Figure 7.18 Example 7.6. Principal stresses on deformed structure (scaling facto...
Figure 7.19 Example 7.6. Principal stresses on deformed structure (scaling facto...
Figure 7.20 Example 7.6. Load displacement curves.
Chapter 8
Figure 8.1 Mohr circle.
Figure 8.2 (a) sin φc cos φc and solution range of Eq. (8.10). (b) Mohr circles ...
Figure 8.3 Strength square for biaxial concrete strength.
Figure 8.4 Example 8.1. (a) Characteristic stress points with required reinforce...
Figure 8.5 Cohesive crack model with loading, unloading, and re-loading stages.
Figure 8.6 Bond (a) Rigid. (b) Flexible.
Figure 8.7 Overlay of elements.
Figure 8.8 Example 8.2. (a) Discretisation. (b) Load–displacement curve.
Figure 8.9 Example 8.2. Principal stresses on the deformed structure (scale 30)....
Figure 8.10 Example 8.2. Comparison of load–displacement behaviour.
Figure 8.11 Example 8.2. Principal stresses of concrete for the microplane model...
Figure 8.12 Reinforcement embeddings. (a) 2D. (b) 3D.
Figure 8.13 Embedded reinforcement. (a) Spatial discretisation. (b) Bond discret...
Figure 8.14 Example 8.3. (a) System. (b) Bond law between fibre and concrete con...
Figure 8.15 Example 8.3. (a) Discretisation. (b) Load–displacement curve.
Figure 8.16 Example 8.3. Near maximum load (displacements scaled by 5). (a) Cont...
Figure 8.17 Example 8.4. (a) System. (b) Load-displacement curve.
Figure 8.18 Example 8.4. Point A (Figure 8.17b). Principal stress field.
Figure 8.19 Example 8.4. Point A (Figure 8.17b). Rebar stresses [MN∕m2].
Figure 8.20 Example 8.4. Point B (Figure 8.17b). Principal stress field.
Figure 8.21 Example 8.4. Point B (Figure 8.17b). Rebar stresses [MN∕m2].
Figure 8.22 Example 8.4. Deformed mesh (scaled by 50).
Chapter 9
Figure 9.1 (a) Structural types. (b) Coordinate system for slabs.
Figure 9.2 Stresses at the slab element.
Figure 9.3 Slab equilibrium.
Figure 9.4 Layer model.
Figure 9.5 Triangular element and area coordinates.
Figure 9.6 Example 9.1. (a) System. (b) Discretisation and principal moments.
Figure 9.7 Example 9.1. (a) Deflections [m]. (b) Boundary support reactions [MN]...
Figure 9.8 Slab reinforcement with positive bending.
Figure 9.9 Example 9.2. Sum asx + asy of reinforcement [cm2∕m] (a) Bottom side. ...
Figure 9.10 Example 9.2. Required reinforcement of selected points [cm2∕m].
Figure 9.11 The mechanism of shear.
Figure 9.12 Example 9.3. (a) Computed shear forces vx, vy. (b) Sum |v1| + |v2|[M...
Figure 9.13 Example 9.4. (a) Discretisation and principal moments. (b) Deflectio...
Figure 9.14 Concrete bending with tensile and compressive reinforcement.
Figure 9.15 Example 9.5. (a) Discretisation. (b) Load deflection–behaviour.
Figure 9.16 Example 9.5. Principal moments; for magnitudes of values, see Table ...
Figure 9.17 Simple slab with yield lines.
Chapter 10
Figure 10.1 Shell element (Dvorkin and Bathe 1984). (a) Geometry. (b) Local coor...
Figure 10.2 Slab element as a special case of a shell element.
Figure 10.3 Example 10.1. Quarter slab discretisation.
Figure 10.4 Example 10.2. (a) Load displacement curve (simulation result ×4). (b...
Figure 10.5 Example 10.2. Quarter slab in final state. (a) Principal moments. (b...
Figure 10.6 Example 10.2. Load– deflection behaviour for material models (simula...
Chapter 11
Figure 11.1 (a) Normal distribution. (b) Correlation.
Figure 11.2 Uniaxial random fields with five samples each.
Figure 11.3 (a) Actions and their extreme values. (b) Joint probability density ...
Figure 11.4 Nonlinear limit state functions and linearisation.
Figure 11.5 Example 11.2. (a) Normalised histograms. (b) Sampling around mean.
Figure 11.6 Example 11.2. (a) Importance sampling. (b) Importance sampling with ...
Figure 11.7 (a) Two-span beam. (b) Abstract failure model for the two-span beam.
Figure 11.8 (a) Safety margin. (b) Design value and characteristic value.
Appendix A
Figure A.1 NR cycle iteration examples. (a) Curvature change. (b) Two dof system...
Figure A.2 (a) Modified Newton–Raphson method (b) Secant method.
Appendix B
Figure B.1 Discontinuity curve in 2D.
Appendix C
Figure C.1 (a) Strains at cracked cross-section. (b) Equilibrium with bond stres...
Figure C.2 Crack states. (a) Single cracks. (b) Stabilised cracks.
Appendix D
Figure D.1 Plane coordinate transformation.
Guide
Cover
Table of Contents
Title Page
Copyright
Preface
List of Examples
Notation
Begin Reading
Appendix A: Solution Methods
Appendix B: Material Stability
Appendix C: Crack Width Estimation
Appendix D: Transformations of Coordinate Systems
Appendix E: Regression Analysis
References
Index
End User License Agreement
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Computational Structural Concrete
Theory and Applications
Ulrich Häussler-Combe
Second enlarged and improved Edition
Author
Univ.-Prof. Dr.-Ing. habil. Ulrich Häussler-Combe
Technische Universität Dresden Faculty of Civil Engineering Institute of Concrete Structures 01069 DresdenGermany
Cover and copyright:
Cut Concrete Structure (archive Ernst & Sohn GmbH);
Interaction Layers (With the use of a figure in “Microplane model m4 for concrete. I. Formulation with work conjugate deviatoric stress, II: Algorithm and calibration” by Zdenek P. Bažant et al., Journal of Engineering Mechanics 126 (2000), pp. 944–980, ASCE.).
Photo editing:
Petra Franke, Ernst & Sohn GmbH
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Preface
This book grew out of lectures that the author gave at the Technische Universität Dresden. These lectures were 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 behaviour of the composite, which utilises the compressive strength of concrete and the tensile strength of reinforcement while allowing for controlled crack formation. This makes reinforced concrete unique compared to other construction materials such as steel, wood, glass, masonry, plastic materials, fibre reinforced plastics, geomaterials, etc. The theory and use of reinforced concrete in structures falls in the area of structural concrete.
Numerical methods like the finite element method, on the other hand, basically allow for a realistic computation of the behaviour of all types of structures. But the implementations are generally presented as black boxes in the view of the 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 provide transparency with special attention being paid to the unique properties of reinforced concrete structures. Corresponding methods are described with their potentials and limitations while integrating them into the larger framework of computational mechanics connected to reinforced concrete. This is aimed at advanced students of civil and mechanical engineering, academic teachers, designing and supervising engineers involved in complex problems of structural concrete, and researchers and software developers interested in the broader picture. Most of the methods described are complemented with examples computed with a PyTHON software package developed by the author and coworkers. Program package and example data should be available at https://www.concrete-fem.com. The package exclusively uses the methods described in this book. It is open for discussion with the disclosure of the source code and should give stimulation for alternatives and further developments.
This book represents a fundamental revision of the book CompUTATIONAL MeTHOds FOR ReINFORCed CONCReTe STRUCTURes, which was published in 2014. In particular, the chapter on multi-axial concrete material laws was expanded, and the topics of crack formation and the regularisation of material laws with strain softening were dealt with in a separate chapter. Thanks are given to the publisher Ernst & Sohn, Berlin, and in particular to Mrs Claudia 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 Eibl1), 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). Further thanks are given to former coworkers Patrik Pröchtel, Jens Hartig, Mirko Kitzig, Tino Kühn, Joachim Finzel, Tilo Senckpiel-Peters, Daniel Karl, Ahmad Chihadeh, Ammar Siddig Ali Babiker, Evmorfia Panteki, and Alaleh Sehni for their specific contributions. I deeply appreciate the inspiring and collaborative environment of the Institute of Concrete Structures at the Technische Unversität Dresden, which is directed by Prof. Dr.-Ing. Dr.-Ing. E.h. Manfred Curbach. It was 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.
Dresden, Spring 2022
Ulrich Häussler-Combe
1)
He passed away in 2018.
List of Examples
3.1 Tension bar with localisation
3.2 Tension bar with creep and imposed strains
3.3 Simple uniaxial smeared crack model
3.4 Reinforced concrete tension bar
4.1 Moment–curvature relations for given normal forces
4.2 Simple reinforced concrete (RC) beam
4.3 Creep deformations of RC beam
4.4 Effect of temperature actions on an RC beam
4.5 Effect of tension stiffening on an RC beam with external and temperature loading
4.6 Prestressed RC beam
4.7 Stability limit of cantilever column
4.8 Ultimate limit for RC cantilever column
4.9 Beam under impact load
5.1 Continuous interpolation of stress fields with the quad element
5.2 Deep beam with strut-and-tie model
5.3 Corbel with an elasto-plastic strut-and-tie model
6.1 Mises elasto-plasticity for uniaxial behaviour
6.2 Uniaxial stress–strain relations with Hsieh–Ting–Chen damage
6.3 Stability of Hsieh–Ting–Chen uniaxial damage
6.4 Microplane uniaxial stress–strain relations with de Vree damage
7.1 Plain concrete plate with notch
7.2 Plain concrete plate with notch and crack band regularisation
7.3 2D smeared crack model with elasticity
7.4 Gradient damage formulation for the uniaxial tension bar
7.5 Phase field formulation for the uniaxial tension bar
7.6 Plain concrete plate with notch and SDA crack modelling
8.1 Reinforcement design for a deep beam with a limit state analysis
8.2 Simulation of cracked reinforced deep beam
8.3 Simulation of a single fibre connecting a dissected continuum
8.4 Reinforced concrete plate with fiexible bond
9.1 Linear elastic slab with opening and free edges
9.2 Reinforcement design for a slab with opening and free edges with a limit state analysis
9.3 Computation of shear forces and shear design
9.4 Elasto-plastic slab with opening and free edges
9.5 Simple RC slab under concentrated loading
9.6 Simple RC slab with yield line method and distributed loading
9.7 Simple RC slab with yield line method and concentrated loading
10.1 Convergence study for linear simple slab
10.2 Simple RC slab with interaction of normal forces and bending
11.1 Analytical failure probability of cantilever column
11.2 Approximate failure probability of cantilever column with Monte Carlo integration
11.3 Simple partial safety factor derivation
Notation
The same symbols may have different meanings in some cases. But the different meanings are used in different contexts, and misunderstandings should not arise.
General
firstly used
∙
T
transpose of vector or matrix ∙
Eq. (2.5)
∙
−1
inverse of quadratic matrix ∙
Eq. (2.13)
δ∙
virtual variation of ∙,testfunction
Eq.(2.5)
δ∙
solution increment of ∙ within iterations
Eq. (2.75)
∙transformed in (local) coordinate system
Eq. (6.14)
time derivative of ∙
Eq. (2.4)
∙
e
∙ related to single finite element
Eq. (2.18)
Normal lowercase italics
a
s
reinforcement cross-section per unit width
Eq. (9.61)
b
cross-section width
Eq. (4.9)
b
w
crack band width
Eq. (3.6)
d
cross-section effective height
Eq. (9.67)
e
element index
Eq. (2.18)
f
strength condition
Eq. (6.48)
f
c
uniaxial compressive strength of concrete (unsigned)
Eq. (3.2)
f
ct
uniaxial tensile strength of concrete
Eq. (3.4)
f
t
uniaxial failure stress of reinforcement
Eq. (3.41)
f
y
uniaxial yield stress of reinforcement
Eq. (2.48)
f
E
,
f
R
probability density functions of random variables
E
,
R
Eqs. (11.2)
,
(11.3)
g
f
specific crack energy per unit volume
Eq. (3.7)
ℎ
cross-section geometric height
Eq. (4.10)
m
x
,
m
y
,
m
xy
moments per unit width
Eq. (9.7)
n
total number of degrees of freedom in a discretised system
Eq. (2.70)
n
E
total number of elements
Section 4.3
n
i
order of Gauss integration
Eq. (2.69)
n
N
total number of nodes
Section 4.3
n
x
,
n
y
,
n
xy
normal forces per unit width
Eq. (9.7)
p
pressure
Eq. (6.8)
p
f
failure probability
Eq. (11.19)
loading distributed along beam
Eq. (4.49)
r
,
s
,
t
local spatial coordinates
Eq. (2.15)
S
slip
Section 3.4
S
bf
slip at residual bond strength
Section 3.4
S
b
max
slip at bond strength
Section 3.4
t
clock time or loading time
Eq. (2.4)
t
x
,
t
y
,
t
xy
couple force resultants per unit width
Eq. (9.58)
U
i
i
-th displacement component
Eq. (6.1)
υ
x
,
υ
y
shear forces per unit width
Eq. (9.7)
ω
deflection
Eq. (2.56)
ω
fictitious crack width
Eq. (3.5)
ω
cr
critical crack width
Eq. (3.9)
x
,
y
,
z
global spatial coordinates
Eq. (2.14)
compression zone height
Eqs. (4.29)
,
(9.66)
internal lever arm
Eqs. (4.115)
,
(9.58)
Bold lowercase roman
b
body forces
Eq. (2.5)
f
internal nodal forces
Eq. (2.9)
p
external nodal forces
Eq. (2.9)
n
normal vector
Eq. (6.5)
s
slip
Eq. (8.53)
t
surface tractions
Eq. (2.5)
t
b
bond force
Eq. (8.54)
t
cL
crack traction in local system
Eq. (7.3)
t
c
crack traction in global system
Eq. (7.133)
u
displacement field
Eq. (2.1)
υ
nodal displacement vector
Eq. (2.1)
υ
e
nodal displacement vector related to a single element
Eq. (2.18)
w
cL
fictitious crack width in local system
Eq. (7.2)
w
c
fictitious crack width in global system
Eq. (7.133)
Normal uppercase italics
A
cross-sectional area of a bar or beam
Eq. (2.54)
A
s
cross-sectional area reinforcement
Section 3.6
A
t
part of surface with prescribed tractions
Eq. (2.5)
A
u
part of surface with prescribed displacements
Eq. (2.53)
C
material stiffness coefficient
Eq. (3.35)
C
T
tangential material stiffness coefficient
Eq. (3.37)
D
scalar damage variable
Eq. (6.105)
E
Young’s modulus
Eq. (2.43)
E
0
initial Young’s modulus
Eq. (3.16)
E
c
initial Young’s modulus of concrete
Eq. (3.1)
E
s
initial Young’s modulus of steel
Eq. (3.41)
E
T
tangential hardening material stiffness coefficient
Eq. (3.41)
F
yield function
Eq. (6.64)
F
damage function
Eq. (6.108)
F
E
distribution function of random variable
E
Eq. (11.1)
G
shear modulus
Eq. (4.8)
G
flow potential
Eq. (6.63)
G
f
specific crack energy per surface
Eq. (3.8)
I
1
first invariant of stress
Eq. (6.19)
J
determinant of Jacobian matrix
Eq. (2.37)
J
2
,
J
3
second, third invariant of stress deviator
Eq. (6.19)
K
slab bending stiffness
Eq. (9.12)
L
c
characteristic length of an element
Eq. (7.18)
L
e
length of bar or beam element
Eq. (2.23)
M
bending moment
Eq. (4.9)
N
normal force
Eq. (4.9)
P
probability
Eq. (11.1)
T
natural period
Eq. (4.209)
V
shear force
Eq. (4.9)
V
volume
Eq. (2.5)
Bold uppercase roman
B
matrix of spatial derivatives of trial functions
Eq. (2.2)
C
material stiffness matrix
Eq. (2.47)
C
T
tangential material stiffness matrix
Eq. (2.50)
C
cLT
tangential local crack stiffness matrix
Eq. (7.9)
D
material compliance matrix
Eq. (2.51)
D
T
tangential material compliance matrix
Eq. (2.51)
D
cT
tangential crack band compliance matrix
Eq. (7.38)
D
cLT
tangential local crack compliance matrix
Eq. (7.9)
E
isotropic linear elastic material stiffness matrix
Eq. (6.23)
G
1
,
G
2
,
G
3
unit vectors of covariant system
Eq. (10.16)
G
1
,
G
2
,
G
3
unit vectors of contravariant system
Eq. (10.17)
I
unit matrix
Eq. (6.100)
J
Jacobian matrix
Eq. (2.20)
K
stiffness matrix
Eq. (2.11)
K
T
tangential stiffness matrix
Eq. (2.67)
M
mass matrix
Eq. (2.61)
N
matrix of trial functions
Eq. (2.1)
Q
stress / strain rotation matrix
Eq. (6.14)
T
element rotation matrix
Eq. (4.105)
V
n
shell director
Eq. (10.2)
V
α
,
V
β
unit vectors of local shell system
Eqs. (10.2)
,
(10.3)
Normal lowercase Greek
α
for several purposes in a local context
α
E
,
α
R
sensitivity parameters
Eq. (11.14)
α
s
shear retention factor
Eq. (7.7)
β
for several purposes in a local context
β
t
tension stiffening coefficient
Eq. (3.65)
ϵ
uniaxial strain
Eq. (2.43)
ϵ
strain of a beam reference axis
Eq. (4.4)
ϵ
1
,
ϵ
2
,
ϵ
3
principal strains
Section 6.2.3
ϵ
ct
concrete strain at uniaxial tensile strength
Figure 3.3
ϵ
cu
concrete failure strain at uniaxial tension
Figure 3.3
ϵ
c
1
concrete strain at uniaxial compressive strength (signed)
Eq. (3.1)
ϵ
cu
1
concrete failure strain at uniaxial compression (signed)
Eq. (3.1)
ϵ
I
imposed uniaxial strain
Eq. (3.35)
ϵ
V
volumetric strain
Eq. (6.101)
ϕ
cross-section rotation
Eq. (4.1)
ϕ
angle of external friction
Eq. (6.90)
φ
for several purposes in a local context
φ
c
orientation of concrete principal compression
Eq. (8.5)
φ
s
orientation of reinforcement
Eq. (8.6)
γ
shear angle
Eq. (4.1)
γ
E
,
γ
R
partial safety factors
Eqs. (11.58)
,
(11.59)
κ
curvature
Eq. (4.4)
κ
p
internal state variable for plasticity
Eq. (6.64)
κ
d
internal state variable for damage
Eq. (6.107)
μ
R
,
μ
E
means of random variables
R
and
E
Eqs ((11.3)
,
(11.6)
)
υ
Poisson’s ratio
Eq. (2.44)
υ
R
,
υ
E
coefficients of variation
Eq. (11.60)
θ
Lode angle
Eq. (6.45)
ϑ
angle of internal friction
Eq. (6.88)
ρ
deviatoric length
Eq. (6.44)
ρ
s
reinforcement ratio
Eq. (8.8)
ϱ
s
specific mass
Eq. (2.52)
σ
uniaxial stress
Eq. (2.43)
σ
1
,
σ
2
,
σ
3
principal stresses
Section 6.2.3
σ
R
,
σ
E
standard deviations of random variables
R
,
E
Eqs. (11.3)
,
(11.6)
τ
bond stress
Eq. (3.47)
τ
for several purposes in a local context
τ
bf
residual bond strength
Figure 3.13
τ
b
max
bond strength
Figure 3.13
ω
circular natural frequency
Eq. (4.209)
ω
related crack width
Eq. (7.5)
ξ
hydrostatic length
Eq. (6.43)
Bold lowercase Greek
ϵ
small strain
Eq. (2.2)
ϵ
generalised strain
Eq. (2.33)
ϵ
p
small plastic strain
Eq. (6.61)
κ
vector of internal state variables
Eq. (6.38)
σ
Cauchy stress
Eq. (2.3)
σ
generalised stress
Eq. (2.34)
σ
′
deviatoric part of Cauchy stress
Eq. (6.9)
Uppercase Greek
Φ
standardised normal distribution function
Eq. (11.20)
Σ
stress extension
Eq. (2.82)
1Introduction
Why Read This Book?
Concrete is by far the most used building material in the world. Concrete can be given arbitrary forms, its basic constituents are available everywhere, its processing is basically simple, and it is inexpensive. Furthermore, concrete can be customised to fulfil special requirements – e.g. high strength, resistance in rough environments, impermeability, ductility – through adjustment of binder, aggregates, fibres, and additives. Its major characteristic from a mechanical point of view is given by a relatively high compressive strength but a low tensile strength. Thus, it is reinforced with bars, wire mats, fabrics of steel, carbon, glass, and more, which leads to an immense variety of composite building materials.
With this we see architectural landmark buildings like the television tower in Stuttgart, Germany, the first of this type designed and engineered by Fritz Leonhardt and built in 1956, Figure 1.1a, the Palazzetto dello Sport in Rome, Italy, a coliseum for the Olympic games 1960 built in 1956 and engineered by Pier Lucri Nervi, Figure 1.1b, the Ganter bridge within the access road to the Simplon pass in the Swiss Alps built in 1980 and designed and engineered by Christian Menn, Figure 1.2a, and the National Veterans Memorial and Museum, Columbus, Ohio, USA, built in 2018 and engineered by Knippers Helbig, Figure 1.2b, to mention only a few.
Figure 1.1 (a) Stuttgart television tower, from Kleinmanns and Weber (2009), photography: Landesmedienzentrum Baden-Württemberg: Albrecht Brugger. (b) Palazetto dello Sport, from Ehmann and Pfeffer (1999).
Figure 1.2 (a) Ganter bridge, from Billington (2014), photography: Nicolas Janberg. (b) National Veterans Memorial and Museum, from Helbig et al. (2020), photography: Knippers Helbig Stuttgart – New York – Berlin.
Figure 1.3 (a) Office building: Züblin-Haus, from Bachmann et al. (2021). (b) High-speed railway viaduct over the valley Unstruttal, Germany, photomontage, from Schenkel et al. (2009).
A countless number of concrete buildings contribute to everyday life; for example, office buildings, Figure 1.3a (Züblin headquarters, Stuttgart, Germany; precast concrete with steel–glass atrium), railway bridges, Figure 1.3b (Unstruttal viaduct, Thuringia, Germany), power plants, Figure 1.4a (RWE, Niederaußem, Germany), station concourses, Figure 1.4b (Stuttgart 21, Germany; final state visualisation, still under construction). This demonstrates some visible contributions of the application of concrete. Indispensable infrastructures providing freshwater, drainage, and wastewater processing, waste disposal processing in general, generation and provision of electricity, support of transport via vehicles, trains, ships, and airplanes are generally hidden from immediate visibility. To sum it up, today’s civilisation would be unthinkable without concrete as a building material.
It can be stated that reinforced concrete is the building material of the twentieth century. But will it also be the building material of the twenty-first century?
Presumably yes, due to its advantages listed above. But sustainability has to become a predominant topic also for reinforced concrete besides bearing capacity, usability, and durability. Production of cement – the predominant binder for concrete – causes a high output of CO2 due to its energy consumption on the one hand and chemical conversion processes on the other hand. The same also applies to reinforcing steel whereupon its contribution to reinforced concrete is relatively small measured by weight ratio. Construction waste makes up the largest proportion of the total amount of waste. What is the conclusion?
Figure 1.4 (a) Power plant, RWE, Niederaußem, Germany, from Krätzig et al. (2007), photography: RWE. (b) Underground station concourse, Stuttgart 21, from Bechmann et al. (2019), visualisation: Ingenhoven Architekten, Düsseldorf.
We have to use less concrete and fewer reinforcement materials and at the same time achieve a higher quality of building components.
Structural design plays a key role to reach this goal. We should gain a better understanding of load carrying mechanisms of building components in order to fully utilise load bearing potentials and to optimise structural forms and materials. There is still a lot of room for improvement in this regard.
Computational methods are an extremely important tool for this. Numerical simulation in combination with experimental investigations allows for a comprehensive understanding of the deformation behaviour, force flow, and failure mechanisms of building components. This permits weak points to be identified and eliminated in a targeted manner. New concepts may be initiated, and a simulation-based rapid prototyping may be performed for initial assessments of new innovative structural forms and materials. On the basis of the knowledge gained from this, the design and elaboration of components in building practice can be carried out more efficiently and with higher quality using computational methods.
Topics of the Book
Such methods are generally demanding with respect to methodology, implementation, and application. This is especially true for nonlinear problems as are typical for structural concrete. Computational methods for nonlinear structural analysis offer a wide range of capabilities. But they are made available to users as black boxes. This hides the fact that numerical methods usually have application limits. If these are not observed, the results become questionable. Often, this is not obvious to users providing input for black boxes and accepting output without hesitation. This motivates the goals and contents of this textbook about computational methods – in particular, the finite element method (FEM) – for reinforced concrete (RC):
Survey of the key aspects of the FEM.
Understanding of basic mechanisms of RC regarding interaction of concrete and reinforcement through bond.
Specifics of FEM regarding structural elements like RC-beams, plates, slabs, and shells.
Essential characteristics of the multi-axial mechanical behaviour of concrete.
Pitfalls related to FEM treating structural concrete and in particular the failure behaviour.
Knowing these issues, the black boxes should become more transparent, and their results should be better comprehensible. The finite element method is the preferred method also for the computation of reinforced concrete structures due to its versatility and adaptability.
Chapter 2 gives an overview of modelling in general and summarises items of FEM as far as is required for its application to reinforced concrete structures. Chapter 3 describes basic mechanisms of structural concrete, which relies on the interaction of concrete and reinforcement by continuous transfer of forces through bond. This is restricted to uniaxial behaviour in a first approach to point out essential properties and describes the mechanisms of the reinforced uniaxial tension bar as prototype of structural concrete. In Chapter 4, this is extended to reinforced concrete beams and frames, which are characterised by bending that may be superimposed with normal forces whereby still basing on uniaxial behaviour of materials. This also includes first aspects of creep, temperature, and shrinkage. Furthermore, prestressing of beams is treated, which is an important technology to extend the application range of reinforced concrete. The chapter closes with the analysis of large displacements and dynamics, exemplarily in each case with their application to beams. A first extension of bending of beams to high beams and plates is given in Chapter 5 with strut-and-tie models, which utilise the uniaxial behaviour of concrete and reinforcement for a design of plane structures with in-plane loading. Furthermore, limit theorems of plasticity – which are an important basis for design in structural concrete – are exemplarily developed within this context. Chapter 6 treats multi-axial concrete behaviour as extension of the uniaxial approach applied in the foregoing chapters. Multi-axial material concrete models are the basis for the structural models for plates, slabs, and shells treated in the following chapters. Basic topics of continuum mechanics are described in as far as they are necessary to understand multiaxial nonlinear stress–strain and failure behaviour of concrete. Material models like elasto-plasticity, damage, and microplane are applied with respect to concrete modelling. A major item regarding material modelling occurs with strain softening – increasing strains with decreasing stresses – which requires a regularisation to reach reliable numerical solution. A further major item concerns the cracking of concrete, which separates parts of a continuum into a discontinuum. This couples discretisation issues with material modelling and is described in Chapter 7. Chapter 8 treats design and simulation of reinforced concrete plates with high beams as a special but common case. In this respect, the design is considered separately, as it may be based on linear solutions for plate stresses utilizing a limit theorem of plasticity. On the other hand, simulation considers nonlinear stress–strain relations additionally leading to solutions for the deformation behaviour. Reinforced concrete slabs, which are treated in Chapter 9, extend uniaxial 1D-bending of beams into biaxial 2D-bending. As before with plates, aspects of design and simulation may be separated in an analogous manner. The most general approach for structural analysis is given with shells, which combine in-plane actions of plates and transverse actions of slabs whereby extending fiat geometries to folded or curved geometries. Shells require complex mechanical models, which is exemplarily treated in Chapter 10 together with the application to reinforced concrete. Chapter 11 treats first aspects of randomness, which is a major topic regarding structural concrete behaviour. Deterministic models – however sophisticated they may be – always give a more or less restricted view of the real world. First notions of an extended view are given in this chapter. Finally, a number of topics are treated in the appendices insofar they are reasonable for better understanding of the main text but might disturb the line of concise arguing therein.
How to Read This Book
The treatment of the above combines methods of mechanics, structural analysis, and applied mathematics. This recourse should be self-explanatory and conclusive to a large degree, so that a study of accompanying literature is generally not required. In doing so, essential lines of development are worked out on the one hand, but on the other hand, the available concepts and methods cannot be described with all details. Furthermore, not every problem addressed is provided with a comprehensive solution. The book is intended to encourage the reader to deepen and explore such topics independently.
Nevertheless, the book involves a large volume. Proposals for shorter tracks are given in the following thereby also enlightening the structure of the book content and the relations between sections. Major groups are characterised as
FEM and reinforced concrete bases, see
Figure 1.5a
.
Uniaxial structures, see
Figure 1.5b
.
Multi-axial concrete and its implications for numerical methods, see
Figure 1.6a
.
Multi-axial structures such as plates, slabs, and shells, see
Figure 1.6b
.
This includes a short track (left column) and branches (right column) for each of these. Chapter 11.1 Randomness and Reliability falls out of this scheme. Nevertheless, basic knowledge of stochastics related to reinforced concrete is considered necessary.
Many topics are illustrated with examples. Most of them are computational and are processed with the PyTHON 3.6 program package CONFem. A few are performed with stand-alone PyTHON scripts or are short, illustrating theoretical derivations. Environments to perform PyTHON are freely available on the internet for all common platforms.
Figure 1.5 (a) FEM and reinforced concrete bases. (b) Uniaxial structures.
Figure 1.6 (a) Multi-axial concrete and its implications. (b) Multi-axial structures.
All Python sources for ConFem, a basic documentation, example input data, and reference result data are available at https://www.concrete-fem.com under open-source conditions.
Thus, all book examples should be reproducible by the reader. But the CONFem project is not finished and may be subject to continuous development. The user should see it as an inspiring challenge to master this tool. The interplay of theory, implementation, and application – possibly with overcoming resistance – ultimately leads to a deeper understanding of numerical methods, structural concrete, and their dependencies.
