INVESTIGATION OF POST CRACKING BEHAVIOUR OF REINFORCED CONCRETE GIRDER BRIDGE

Abstract

The methods of analysis and design of highway bridges have undergone a vast improvements throughout the world in the last 2-3 decades. These developments have been necessitated due to large increase in traffic, introduction of heavy vehicles and increased aesthetics awareness for bridge forms. The easy accessibility to large memory digital computers has made the acceptability of these refined methods by the profession more readily. The current methods of analysis are mainly based on the following approaches: (i) Orthotropic plate theory, (ii) Grillage analogy method; (iii) Finite strip method; and (iv) Finite element method. Generally the members of a bridge super structure experience significant shear deformations which can not be considered in the first two approaches. Further, these approaches are based on classical theory of elasticity and thus, handicapped to predict the response of the structure in post cracking stages. In the finite strip approach, the structure is idealized as composed of interconnected longitudinal finite strips and the solution is obtained by resorting to harmonic analysis. This method also has the limitations of being a linear analysis technique applicable to bridges with a uniform cross section throughout the span. Because of the versatility of the finite element method in obtaining solutions to complex structural and continuum mechanics problems, many attempts have been made to extend its applicability to the nonlinear analysis of reinforced concrete (r.c. ) structures. The basic prerequisites for the post cracking analysis of r.c. girder bridges, using finite element method, are the adoption of an appropriate element that can suitably idealize the three dimensional character of the structure and accurately predict the deformational characteristics. Further a realistic material constitutive relationship need to be employed in order to take cognizance of material nonlinearities arising due to cracking of concrete, spread of plasticity and yielding of steel in the inelastic stages of loading. Reinforcement details , which may take complicated form, should also be accurately idealized. Most efforts in the nonlinear analysis of bridges have been limited to predict their deformational characteristics and the prediction of failure loads. vii From practical design point of view, designers of reinforced concrete girder bridges are primarily concerned with the transverse distribution of the wheel loads amongst the main girders, and for this, they are resorting to simplified methods. The recent trend in the design of r.c. girder bridges is also to replace the traditional working stress method, which is still in use in many countries including India, by a more rational method based on limit state approach. The spread of cracking and the degradation of material stiffness due to the nonlinearity of concrete and yielding of steel lead to redistribution of loads amongst the girders in the post cracking stages so that the prediction of the limiting load and failure mechanism of these structures becomes rather a complex problem. There is thus a need to undertake a rigorous analysis of r.c. girder bridges to investigate their complete behavior including the load redistribution amongst the main girders in the post cracking stages up to failure. There is also a need to predict the failure mechanism and the overloading capacity available in the conventionally designed bridges In the present study, an attempt had been made to develop a 3-D finite element code to investigate the true behaviour of r.c. girder bridges under monotonically increasing load up to failure. The study includes prediction of the failure load and failure mechanism, distribution of the load amongst the main girders during different stages of loading, variation of load distribution along the span of the main girders in the elastic stage and near failure load, and variation of the support reactions in the pre and post cracking stages. The effect of number of cross beams and their locations on the overall response of the bridge has also been studied. Several important aspects such as material nonlinearity, cracking and crushing of concrete, yielding of steel, tension stiffening and aggregate interlock of cracked concrete etc. have been considered for a correct appraisal of the behaviour of r.c. multi T-beam bridges. A general shear deformable 9 noded shell element, with 6 degree of freedom at each node, has been implemented using layered concept. This element has been used to represent the deck slab and the webs of the main girders and cross beams. The problem of spurious zero energy mode viii resulting from using full reduced integration rule to control the shear and membrane locking problems, is treated by using artificial stabilization based on the projection operator technique. The stabilization coefficients have been efficiently evaluated to take into account the stiffness degradation during different stages of loading, throughout the thickness of the shell element at each Gauss point, such that the spurious zero energy modes are properly restrained. An accurate and flexible attainment in the idealization of the reinforcement details and prestressing tendons, has been achieved by implementing three different models; these are; (i) Smeared layers to represent the deck slab mesh reinforcement and the web steel of the main girders and cross beams, (ii) Discrete bar 3-noded elements to represent the steel that lies in the mid surface of the shell element and connected to the nodes of the shell whenever possible and (iii) Embedded model to idealize the steel bar of complicated geometry that is randomly located within the domain of the concrete shell element. The nonlinear behaviour of concrete material has been idealized by employing an elasto plastic strain hardening model. A yield function has been used to define the initial and subsequent yield surfaces, and the same function has been adopted to check the crushing condition of the concrete. Crushing of concrete is assumed to takes place when an equivalent uniaxial plastic strain exceeds a prescribed value. For this a uniaxial stress strain relationship has been used to relate the multi-dimensional stress strain relation into an equivalent uniaxial stress strain relation from which the plasticity hardening modulus is evaluated. An associated flow rule to relate the plastic strain increment to stress increment has been adopted, and isotropic hardening which describes the motion of subsequent yield surfaces, has been considered. Smeared crack model is incorporated in the model allows concrete to crack, fully or partially, in one or two orthogonal directions, taking into consideration the tension stiffening and the aggregate interlock phenomena. If in a certain layer, concrete is cracked, the crack direction is assumed to remain fixed at that location throughout the solution. i Based on the above premises, a procedure for non-linear analysis of multi-girder r.c. and prestressed concrete bridges has been formulated, ix and the proposed formulation coded in FORTRAN. The developed software can handle different types of loading, namely; the self weight, external point and uniformly distributed live loads and internal prestressing force. Different combinations of these loads can be considered with different sequences of application as per the requirement. The restarting capability has also been facilitated in the developed program. In the incremental nonlinear analysis technique, both normal and modified iterative Newton Raphson methods have been implemented, using the active column solver, with the provision to switch from one method to the other in different load steps during the solution. Several numerical examples have been solved to demonstrate the validity of the proposed model. These examples include linear analysis of a simple cantilever beam, a cantilever slab and a cylindrical shell to test the validity of the proposed stabilization technique. Nonlinear analysis of a simply supported r. c. beam and a T-beam, simply supported r.c. slabs and prestressed slabs have been carried out for validation of the inelastic analysis procedures. Although relatively course meshes have been adopted in the idealization of the above test structures, the analytical solutions compare fairly well with the experimental and exact solution findings. To high-light the importance of the study undertaken, the proposed formulation has been used to investigate thoroughly two bridge structures, as described below. Firstly the developed computer program has been used to predict the overall behaviour of a 21 m span conventionally designed r.c. T-beam bridge having 3 main girders. The study indicates that the load distribution amongst the main girders of the bridge is significantly changed at different stages of loading, and these changes depend mainly upon the stiffness degradations of the bridge members due to cracking of concrete and yielding of steel. The predicted load distribution factors show a significant variations along the span in both the elastic and post cracking stages depending upon the number of cross beams and their locations, position of the load, boundary conditions at the end supports, and the failure mechanism of the main members. The use of 5 cross beams shows an excellent improvement in the response of the bridge over that of 2,3 and 4 cross beam bridges in many respects, such as failure load, deformation behaviour , and the load distribution amongst the main girders. The applicability of the developed model to predict the nonlinear response of prestressed concrete girder bridges, with a perfect bond assumed between steel and concrete, and the long term effects in the prestress losses neglected, has been then demonstrated by analyzing a prestressed concrete T-beam girder bridge as a second example. This has similar plan geometry as that of the r.c. bridge with 5 cross beams. In the nonlinear analysis, the dead load and prestressing forces have been applied instantaneously in the first load step and subsequently the external load has been incrementally applied up to failure. An important conclusions have been withdrawn from the analytically predicted results. These are mainly concerned with the distribution of the load in the pre and post cracking stages amongst the main girders, overloading capacity available in the conventionally design bridges, effects of number of cross beams on the overall behaviour.