The mathematical modelling of concrete constitutive relationships
The available experimental evidence demonstrates the extreme nonlinear material behaviour of reinforced concrete structures. These nonlinear effects are attributed to the collective behaviour of the constituent materials in addition to factors such as cracking, crushing, aggregate interlock, creep, shrinkage, bond slip and rate of loading. Analytical methods have been improved in the past two decades as a result of the availibility of more powerful computers. It is, therefore, feasible to model these nonlinear features in order to conduct an analysis of the behaviour of reinforced concrete structures. The present research is concerned with some of these nonlinear effects. These include the formulation of a constitutive model for the three-dimensional stress-strain relationships of concrete and the mathematical modelling of cracked and crushed concrete. The proposed models have been implemented into a finite element system for the analysis of reinforced and pre-stressed concrete structures. Chapter One is a general introduction to structural nonlinearities and the finite element method. The structure of the thesis is also outlined. Chapter Two reviews available theoretical approaches used for the formulation of the concrete behaviour and assesses their relative advantages. The theory of plasticity is discussed in greater depth as it forms the foundation of the work in Chapter Three. A three-dimensional concrete yield surface is developed in Chapter Three. This yield surface is used in the theory of hardening plasticity to establish the incremental constitutive relationships for concrete. Furthermore, this model is extended to represent the strain-softening effect in concrete. The hardening and softening rule which has been developed is based on experimental results obtained from the literature. The results of the proposed model are compared with these experimental data. The cracking and crushing of concrete have been studied in Chapter Four. A rough crack model is developed for concrete and crack stress-displacement relationships due to aggregate interlock are formulated. A mathematical model is proposed for the effect of dowel forces in cracked reinforced concrete structures. The effect of bond stress between a steel bar and concrete has been introduced by a tension-stiffening factor and suitable formulations has been proposed. The results from the crack related models have also been compared with experimental data from the literature. Finally, stiffness matrices for cracked plain and reinforced concrete have been developed using a smeared crack approach. The concrete constitutive model and the crack model developed in Chapters Three and Four have been implemented into a finite element program for the numerical analyses given in Chapter Five. This implementation has been carried out for plane stress and axisymmetric solid stress problems. A reinforced concrete beam and a prestressed concrete reactor vessel have been analysed and the results compared with experimental data. Finally Chapter Six presents the overall conclusions and recommendations for further research.