Theoretical Study And Application Of Shear-Lag And Temperature Effect In Box-Girder Bridges

With the development of box girders towards simple single-cell box section with long cantilever slab and large web-space, the shear lag effect in box girders is increasingly paid close attention. In recent years, many scholars devoted themselves to the study of shear lag effect and certain achievements were made. However, there are still some problems worth further studying, especially in the shear lag effect of curved and skew box girders. Thermal effect is thought to be one of the main reasons causing long-span prestressed concrete box girder bridges crack, but the study for complex structures e. g. skew box girders is rare at present. In this paper, the shear lag effect and thermal effect in box girders are studied systematically. The main work and achievements are as follows.The theoretical system of the shear lag analysis is established by the energy variation method aiming at box girders with general single-cell trapezoid sections. The generalized moment corresponding to shear lag displacement is firstly defined and is called the shear lag moment. Its magnitude and distribution are studied for cantilever, simply supported, continuous and non-uniform box girders respectively. It is shown that the shear lag moment and bending moment are two close internal forces under vertical loads. On the basis of the two defined geometrical properties of cross section called the moment and product of inertias for shear lag, the general formula of stress is presented through the internal forces and geometrical properties. It is the extension of the flexural stress formula in mechanics of materials. Model test confirms the formula.One-dimensional finite element method is presented to analyze the shear lag effect of box girders. The homogeneous solution of the differential equation for shear lag is chosen as the element displacement function. According to the definition of the elements in stiffness matrix, their expressions with shear lag are derived taking the constants of integration in displacement function as middle variables. Based on the present stiffness matrix and the principle of virtual work, the vectors of equivalent joint loads are further derived for the vertical uniform and concentrated loads. The finite element program BOXSL is worked out and is used to calculate a model of linear non-uniform cantilever box girder and one of two-span continuous box girder. The calculated result is in good agreement with the test one.Based on the theory of thin-walled curved members, the flexural-torsional differential equations and boundary conditions with shear lag are established by the energy variation method for thin-walled curved box girders considering the effect of the secondary shear deformation on warping in restraint torsion. The Galerkin's method is applied to solve the differential equations. If the generalized warping displacement in restraint torsion is substituted for the first order of twist angle, the differential equations turn into the ones in the existing references in which the secondary shear deformation is ignored. If the curvature radius is infinitely great, they turn into the ones of straight box girders. If the shear lag is ignored and the warping displacement is substituted for the first order of twist angle, they change into the Vlasov equations. Therefore, they are more universal. Model test validates the present theoretical analysis.The generalized conforming element for thin plate bending and the membrane element for plane stress are combined to obtain a plane shell element to analyze the shear lag effect in continuous skew box girders. On the bases of the defined shear lag coefficient for skew box girders, Two-span continuous skew box girders with different degrees of skew are analyzed to investigate the longitudinal and transversal distributions of shear lag effect under concentrated and uniform loads, and are compared with that of right box girders. The results show that a skew continuous box girder under uniform load has more remarkable negative shear lag behavior than a right box girder. Remarkable coefficients occur at the inner support section of continuous skew box girders. The difference of coefficients at webs in support section is also great. Degree of skew produces great influence on them but little on the coefficients at centers of sections.Three independent generalized displacements are employed in analyzing the shear lag effect of T-beams with wide flanges. The differential equations and boundary conditions are established where the Timoshenko shear deformation is considered. The analytical method is conducive to refining deflection calculation, but the shear lag and the Timoshenko shear effects are independent. In order to consider the influence of web shear deformation on the shear lag of flanges, the twin-warping displacement function method is presented in which U₁(x) and U₂(x) are employed for flange and web respectively. The related differential equations and boundary conditions are established by the energy variation method.Under the relative changes of temperature in flange and web, beams with wide flanges still have shear lag behavior. An analytic method is presented which is based on the mechanics of elasticity. The flange and web plates are considered to be in a state of plane stress. The deformation compatibility and equilibrium at the connection of flange and web plates are used to establish additional equations to determine the constants of integration in stress function. The analysis formulas of thermal stress in flange and web are derived. Through the stress formulas of flanges, the non-uniform distribution of thermal stress is fully reflected and the shear lag is automatically taken into account. Numerical example confirms the method. It shows that the series converges fast and so the method is efficient.The general formulas of solar thermal stress for continuous box girder bridges under thermal gradient are derived. The deformation curvatures produced by the self-restraint and redundant constraint are considered simultaneously. According to the modes of solar thermal gradients provided in the specifications for bridges, a practical method is proposed to calculate the primary and secondary internal forces and thermal stresses. The distribution of thermal stresses in a non-uniform continuous box girder bridge is analyzed systematically. Research shows that in design of a non-uniform continuous box girder bridge much attention should be paid to checking the fracture resistance at mid-span normal sections and inclined ones near internal supports.The simplified mechanical model for skew box girders is established. The calculation method of solar thermal effect and secondary internal forces in single span and multiple-span continuous skew box girders is presented. The influence of skew angle and ratio of bending and torsion stiffness on the thermal effect is studied in detail. A large number of curve diagrams drawn show the influence regularities. Research shows that secondary internal forces occur in both single span and multiple-span continuous skew box girders under solar thermal gradient. The secondary bending and torsion moments are especially great. Skew angle and ratio of bending and torsion stiffness significantly affect the secondary internal forces of skew box girders. For smaller ratio of bending and torsion stiffness, its variation affects more remarkably the secondary internal forces. Skew angle affects more remarkably the secondary torsion moment than other internal forces. Within the usual range of skew angles, tensile stresses at bottom and compression stresses at top in the middle span of continuous skew box girders decrease as the skew angle increases, and increase as the ratio of bending and torsion stiffness increases. Large camber displacement exists in the side spans, and it decreases with the increase of skew angle, while increases with the increase of the ratio of bending and torsion stiffness. Deflection in the middle of central span changes from downwards to upwards with the increase of skew angle, and the critical skew angle increases with the increase of the ratio of bending and torsion stiffness.