**Nonlinear Finite Element Analysis of Composite and Reinforced Concrete Beams**

**Contents **

Preface vii

1 Introduction 1

1.1 General introduction 1

1.2 Scopes and structure 5

References 6

2 Finite element analysis of beams 9

2.1 Beam theories 9

2.2 Finite element analysis of beams 13

2.3 Nonlinear finite element analysis of beams 18

References 23

3 Finite element analysis of composite beams 29

3.1 Introduction 29

3.2 A one-dimensional two-node **composite** beam element 30

3.3 Finite element equations and analysis procedures 34

3.4 Finite element analysis of homogeneous isotropic beams 35

3.5 Finite element analysis of composite beams 41

References 45

4 Finite element analysis of reinforced concrete beams 47

4.1 Introduction 47

4.2 A composite beam element for reinforced concrete beams 48

4.3 Material models 52

4.4 Nonlinear finite element analysis procedures 55

4.5 Finite element analysis of reinforced concrete beams 57

References 62

5 Finite element analysis of reinforced concrete beams with bond–slip 65

5.1 Introduction 65

5.2 A composite beam element for reinforced concrete beams with bond–slip 66

5.3 Material models 71

5.4 Nonlinear finite element analysis procedures 77

5.5 Finite element analysis of reinforced concrete beams

with bond-slip 77

References 80

6 Finite element analysis of reinforced concrete beams at elevated temperatures 83

6.1 Introduction 83

6.2 A composite beam element for reinforced concrete beams at elevated temperatures 84

6.3 Temperature-dependent material models 89

6.4 Nonlinear finite element analysis procedures 94

6.5 Finite element analysis of reinforced concrete beams at elevated temperatures 95

References 99

7 Finite element analysis of FRP-strengthened reinforced

concrete beams under static and cyclic loads 101

7.1 Introduction 101

7.2 Finite elements in numerical models 102

7.3 Material models 106

7.4 Bond–slip model 108

7.5 Material behaviours of concrete, steel, and FRP

under cyclic load 109

7.6 Loading conditions 112

7.7 Finite element analysis of FRP-strengthened RC beams 112

References 117

Appendix A: List of notations 119

Appendix B: Gaussian integration 123

Appendix C: Temperature-dependent material properties of concrete 125

Appendix D: Temperature-dependent material properties of steel 129

Appendix E: Temperature-dependent material properties of FRP 133

Appendix F: Finite element code for composite beam element:

Linear analysis 135

Appendix G: Finite element code for composite beam element:

Nonlinear analysis 149

Appendix H: Finite element code for composite beam element:

Nonlinear analysis with bond–slip 169

Appendix I: Finite element code for composite beam element:

Nonlinear analysis with temperature effect 189

Appendix J: User subroutine for concrete under cyclic load 227

Appendix K: User subroutine for steel under cyclic load 233

Appendix L: User subroutine for FRP under cyclic load 243

Index 245

**Preface **

Finite element method has been one of the most popular, powerful, and robust numerical modelling techniques for finding approximate solutions to practical engineering problems. In recent decades, with the knowledge advancement in science and engineering, composite materials, such as fibre-reinforced polymers (FRP), have been increasingly used for innovative design and retrofitting/repairing of structural components to enhance durability, sustainability, and resilience of infrastructures. Effective analysis and reliable prediction of the performance of composite structures especially those under extreme loading conditions, such as cyclic, fire, impact, and blast loads, is not an easy task, but it is essential and crucial for the structural design. Thus, there is a high demand of further development and advancement of numerical modelling methods and techniques.

This book provides comprehensive finite element analysis (FEA) procedures and modelling techniques for composite beams with a focus on the FRP-reinforced/ strengthened concrete beams. In particular, the critical issues encountered in the structural analysis of composite FRP-reinforced concrete beams have been addressed by using the composite beam elements developed by the authors. Additionally, numerical techniques for establishing effective finite element model using the commercial FEA software ANSYS are introduced. The FEA source codes are also provided in this book. The readers would benefit from the ideas for the development of new elements, the modelling techniques applied in the analysis of composite beamlike structural components, and the source codes for FEA of composite beams. With the growing demand for reliable, accurate, and highly efficient numerical predictions, I believe this book forms a useful and valuable complement to many other excellent books already published in the field, and I wish the book can provide a valuable reference for researchers and engineers.

I have been very fortunate to have received excellent mentoring in the field of computational mechanics from world-leading high-profile scholars, particularly my PhD supervisor Professor Y.K. Cheung who has made pioneering contributions to the finite element method. My understanding and appreciation of the finite element method have been largely inspired by him, and I have been working on the finite element development and computer-aided numerical simulation of composite materials and structures in the past 20 years. I was also fortunate to have my first PhD student, a very bright and talented girl, Dr. Xiaoshan Lin, who is the first author of this book, to work with me on the FEA of composite beams. The majority part of this book is from her PhD research. The contribution from the third author Prabin Pathak, who was also my research student, is acknowledged.