Shear Lag Theory Of T Beam And Box Girder With Wide Flange Based On Symplectic Elasticity

At the end of the 60's to the early 70s in the 20th century,four instability or damage accidents to the steel beam took place in Austria, the United Kingdom, Australia and Germany. The subsequent analysis results showed that the serious problems existed in the calculation of the four bridges, one of which was that shear lag effect had not been considered in the design. Therefore, it led to excessive stress concentration and made the instability or partial destruction happen in the structure, which attracted wide attention of scholars in the world. In recent years, a large number of thin-walled box girder bridges, T structure, rigid frame, cable-stayed bridges, especially some bridges with larger ratio of width to span and width to height have been constructed in domestic. However, there are no specific provisions on the shear lag effect of box girder in current bridge design specifications. Therefore, the problem is neglected in general engineering design, which makes the number of horizontal cracks increase. In recent decades, many scholars at home and abroad have been working on this study, many new ideas and theories about shear lag have been put forward through the aspects of analytical theory, numerical method, model tests, etc. and a number of research results having been gained.Although numerical analysis method possesses the main position in current shear lag analysis, it would not reduce the significance of analytical method. The reasons are as follows: First,the theoretical basis for numerical method can not detached from numerical method. Secondly, analytical method is the preferred method for the engineers because of its simplicity and shortcut. Elastic theory solution is the effective means to solve simple mechanical model, which based on classical elastic theory. And more precise answer can be obtained if using this method. However, the solution of"Elasticity mechanics"gives priority to semi inverse method, which relies on the specific issues and lack of universality. Semi inverse method can often find some solutions but can not prove all solutions, which makes the reader trouble in how to piecing so that the problem can be solved. Semi inverse method is used due to the complexity of equations. The analytical method has always been used within the scope of one kind of variables. The stress method or the displacement method or other methods are used to eliminate unknown function. And a high-order partial differential equations is obtained, which needs to be solved. From the aspect of mathematical system,solving one kind of variables belongs to the Lagrange system method. Therefore, it will lead to the appearance of higher-order partial differential equations inevitably, so that separation of variables,eigenfunction expansion method and other effective methods of mathematical physics can not be implemented. As a result, the semi-inverse method has failed to break for a long time. The reason is that there is no thorough change in the solving system and the complexity of analysis and calculation makes the application having been restricted in practical engineering. Therefore, the elasticity theory solution can only solve quite a few problems.Symplectic system is introduced to the theory of elasticity mechanics by professor Zhong Wanxie. And then symplectic system of elasticity is established and developed so that many effective mathematics and physics methods such as separation of variables, conjugated symplectic orthogonal and symplectic eigenfunction expansion method etc. are made to be implemented, which contributes to the creation the symplectic system of elasticity mechanics. Based on symplectic solution system of elastic mechanics, reasonable simplification is made for T beam and box girder. Then the analysis on the shear lag in the bridge structure is carried out in the paper. The main study works and conclusions are as follows:1,The paper proposes the application of the symplectic elasticity solution system in solving of shear lag problem exists in bridge structures for the first time, which provides a new method for solving shear lag problem and promote the use of symplectic elasticity solution in solving practical problems at the same time. This method is based on the symplectic solution system of elasticity mechanics and can obtain more precise solutions. Because of the change in the solution system of elasticity mechanics, the analysis and calculation formula obtained is very simple and easy to be used in practical problems.2,The flange slab of cantilever and simple-supported T beam with wide flange both are simplified into plane stress plate when using symplectic elasticity in analyzing shear lag problem. The equations set are established according to the deformation in the combining site and static equilibrium conditions and separately deduce the Saint-Venant analytical solutions and exact solutions for the wing flange of cantilever and simple-supported T beam under uniform and linear shear loads and parabola loads. Also, the closed polynomial exact expression and Saint-Venant expression for shear lag coefficient. The comparison results between the exact solutions and finite element solution for cantilever and simple-supported T beam under different load is coincidence with each other. And a slight error at the end is because that non-zero eigenvalue is taken relatively small.3,The influence of self-equilibrium force system at the end is covered by Saint-Venant principle. The influence mentioned above is partial and will decay rapidly with distance. And the decay solution is just expressed by non-zero solution. Therefore, the exact solutions are composed of Saint-Venant solutions and non-zero solutions, which make the exact solutions and Saint-Venant solutions entirely different and provide the condition for even more effective analysis and understand the shear lag problem.4,The method in the paper and finite element method are used to analyze the wing flange of cantilever T beam in bending state. The results are nearly the same. When cantilever T beam is subjected to different loads, not only the positive shear lag happens, but also negative shear lag happens. The positive shear lag and negative shear lag both are due to the different deform of the points in one section. The constrains on sides are the intrinsic factors which result in negative shear lag, and the load types out of surfaces are the extrinsic factors which result in negative shear lag. The exact solutions of shear lag in the paper detach the effects of constrains of both sides, the bounders of two sides and loads on shear lag. The reasons of negative lag and positive shear lag are easily to be understood.5,Influence of material parameter Poisson Ratio is considered in expressions of shear lag coefficient. Generally, the small can not affect much more on the shear lag coefficient. But the influence can not be ignored at the larger .6,Effective flange widths of T-shaped solutions by this paper are compared with British and German norms. It is shown that solutions by this paper agree well with norms in the case of small width span ratio. Effective width formula given by this paper can be used as simple calculation formula at Initial Stages of design.7,The closed polynomial expressions for shear lag coefficient and effective width are given at the condition of Saint-Venant analytical solutions to full-span Uniform Load on flange slab of box girder. Availability of formula are demonstrated by comparing results by this paper and finite element method.8,Longitudinal displacement differential functions for shear lag of flange slab are rationally established based on symplectic solution to T-shaped beams and box girders. System total potential energy is constructed relied on strain-displacement relation. According to energy-variational principle, shear lag control differential equations and corresponding boundary conditions are deduced and general closed solutions are obtained. Results show that longitudinal displacement differential functions in this paper are more reasonable.9,The matrix perturbation method is brought out for the modes analysis during the bridge design. This can avoid that once the structure parameter is modified, generalized eigenvalue has to be solved again, which is useful for the calculation of shear lag in mode analysis.