The Shear-Lag Effect on Curved Box Girder Bridge Considering Prestress and Initial Curvature

In this paper, the differential equations of box beams are established based on the principle of minimum potential energy and the variational method. The elastic stiffness matrix and geometric stiffness matrix considering shear lag and compression-flexure effects are induced in this paper. And a finite element program is developed. Then the influence of compression-flexure effects to shear lag effect of box beam is analyzed.

The shear lag effect in high-rise framed tube structures is a significant issue when considering temperature variations. This effect can result in uneven axial forces in the columns, where the axial force in the middle part of the structure is smaller than in the peripheral parts. Such a phenomenon can lead to material waste or structural unsafety if the structure is not properly designed. This study proposes a two-parameter displacement function that accounts for the difference in amplitude between the two sides of a flange framed tube. A governing equation is established to describe the shear lag effect under temperature variations. The principle of minimum potential energy is used to analyze the shear lag effect in framed tube structures subjected to thermal loading. To analyze the shear lag effect, the framed tube is idealized as an equivalent continuum solid web tube structure. An analytical solution is derived based on the proposed governing equation. Furthermore, several examples are provided to illustrate the validity of the proposed theory. The study reveals that temperature variations significantly affect the shear lag effect in framed tube structures, leading to non-uniform axial force distributions that depend on the magnitude and distribution of the thermal load. The proposed analytical model, which is validated through finite element analysis, offers a simplified yet accurate method for predicting temperature-induced stresses and deformations.

Nonlinear electromechanical bending of bi-modular piezoelectric laminated beams

This paper presents a modified finite segment method for shear-lag analysis of thin-walled box girders. The method is based on the hypothesis that the warping displacement of a flange plate is governed by a third-order parabolic curve. Based on the principle of minimum potential energy, the governing differential equations with two unknown displacement functions are derived. The homogeneous solutions of the differential equations are used to represent the displacement functions of the finite element. By mean of the variational principle, the element stiffness equation is developed and the normal longitudinal stresses taking into account the shear lag are obtained. To verify the accuracy of the proposed method, three experiments are conducted so that the present results are compared with those obtained by the model tests and other methods.

Chapter 8 - Basic Equations and Solution Procedure

With the displacement field taken as the only fundamental unknown field in a mixed-boundary-value problem for linear elastostatics, the principle of minimum potential energy asserts that a potential energy functional, which is defined as the difference between the free energy of the body and the work done by the prescribed surface tractions and the body forces --- assumes a smaller value for the actual solution of the mixed problem than for any other kinematically admissible displacement field which satisfies the displacement boundary condition. This principle provides a weak or variational method for solving mixed boundary-value-problems of elastostatics. In particular, instead of solving the governing Navier form of the partial differential equations of equilibrium, one can search for a displacement field such that the first variation of the potential energy functional vanishes. A similar principle of minimum complementary energy, which is phrased in terms of statically admissible stress fields which satisfy the equilibrium equation and the traction boundary condition, is also discussed. The principles of minimum potential energy and minimum complementary energy can also be applied to derive specialized principles which are particularly well-suited to solving structural problems; in this context the celebrated theorems of Castigliano are discussed.

In order to accurately calculate the cross‐sectional stresses and deflections of the improved composite box girder with corrugated steel webs, the control differential equations and boundary conditions are established based on the energy variational principle, taking into account the effects of corrugated steel web properties, shear lag, and shear deformation. A cantilever composite box girder is used as a numerical example. The influence of width span ratio and corrugated steel web bending angle on the shear lag and fold effect is analyzed. The results show that the analytical solution is in good agreement with the finite element solution. A shear lag effect and a negative shear lag effect exist in the cantilever composite box girder under uniform load. The fold effect under concentrated load has a greater influence on the mechanical properties of the wing plate than under uniform load. With the increase of the width span ratio, the effects of both shear lag and fold effect increases. With the increase of the corrugated steel web bending angle, the fold effect is obviously enhanced, but the change of the bending angle has little effect on the shear lag effect.

The solution of elastostatic problems and the principles of minimum potential energy, minimum complementary work, under fuzzy boundary conditions

Shear lag effect of composite girders in cable-stayed bridges under dead loads

Stocky thin- or thick-walled beams: Theory and analysis

A strip element method is presented to determine bending solutions of orthotropic plates. In this method, the plate is discretized into strip elements in one direction. By using the principle of minimum potential energy to a strip element, a set of fourth-order ordinary differential equations with respect to the other direction are derived. By assembling the corresponding equations of adjacent strip elements, differential equations for the whole plate are obtained. These equations can be solved analytically by using the Fourier transform method. The deflection distribution is determined by using the boundary conditions. Numerical examples for an orthotropic plate are presented and it is shown that the strip element method is an efficient and accurate numerical method for the plate bending analysis.

As few multi-tower single-box multi-cell cable-stayed bridges with corrugated steel webs have been built, analysis is mostly achieved by combining single-girder model, beam grillage model and solid model in support of the design. However, such analysis methods usually suffer from major limitations in terms of the engineering applications: single-girder model fails to account for spatial effect such as shear lag effect of the box girder and the relevant girder width and eccentric load coefficient; owing to the approximation in the principle equivalence, the plane grillage model cannot accurately capture shear stress distribution and local stress state in both top and bottom flange of composite box girder; and solid model is difficult to be practically combined with the overall calculation. The usual width method fails to provide a uniform and accurate effective length (and the codes fail to provide a unified design approach at those circumstance) considering different shear lag effects resulting from dead load, prestress and cable tension in the construction. Therefore, a novel spatial grid model has been developed to account for shear lag effect. The theoretical principle of the proposed spatial grid model has been elaborated along with the relevant illustrations of modeling parameters of composite box girder with corrugated steel webs. Then typical transverse and longitudinal shear lag coefficient distribution pattern at the side-span and mid-span key cross sections have been analyzed and summarized to provide reference for similar bridges. The effectiveness and accuracy of spatial grid analysis methods has been finally validated through a practical cable-stayed bridge.

On the basis of the three-dimensional theory of elasticity, a graded finite element method capable of modeling static and dynamic behaviors of elliptical plates made of functionally graded materials (FGMs) subjected to uniform pressure is developed. In the present paper, two different material properties distributions are considered. For the dynamic analysis, the effective through-the-thickness continuous material properties distribution of the FGM (which is assumed to be composed of ceramic and metallic constituents) is determined based on Mori–Tanaka homogenization technique. The three-dimensional graded finite element formulation is derived based on the principle of minimum potential energy and Rayleigh Ritz method. To solve the time-dependent equations, Newmark’s direct integration method is employed. To present the efficiency of the present work, several numerical examples are included. Since similar results are not available in the literature, results of the present formulations are verified by comparing them with available ones of a homogenous elliptical plate.

Mechanical adjustment and prediction of metal-composite reconfigurable tubes

We present a method for verifying the contact stress of elastic bodies. An explicit formulation of the total contact force, a fraction function with the numerator as a linear function and the denominator as a quadratic convex function, is derived by using the surrogate model of quadratic optimization for the contact problem. Then a bound formulation is obtained for the sum of the nodal contact forces, which is an explicit formulation of matrices of the finite element model, derived by maximizing the fraction function under the constraint that the sum of the normalized nodal contact forces is one. The bound is solved with the problem dimensions being only the number of contact nodes or node pairs, which are much smaller than the dimension for the original problem. Next, a scheme for constructing an upper bound on the contact stress is proposed that uses the bound on the sum of the nodal contact forces obtained on a fine finite element mesh and the nodal contact forces obtained on a coarse finite element mesh. Finally, the proposed method is verified through an example to demonstrate its feasibility and robustness. Keywords-contact stress; bound; surrogate; verification; finite elements I. INTRODUCTION Based on the principle of minimum potential energy, an elastic contact problem is essentially an optimization problem with the contact condition as a constraint. In the Karush-Kuhn- Tucker (KKT) conditions for these optimization problems, the Lagrange multipliers that represent the nodal forces are in an unbounded positive space. It is fortunate that when aggregating the constraints with a so-called surrogate constraint (1,2), we obtain an explicit formulation of the sum of the nodal contact forces, and the variable — the normalized nodal contact force — is only constrained in a bounded simplex. Therefore, it is possible to find an upper bound on the sum of nodal contact forces, and when this is done, we can obtain a bound on the contact stress with the information provided by the normalized nodal contact forces solved with a coarse mesh. For construction of the bounded feasible region, we have to solve an optimization problem with an objective being a fraction function. We first prove that the objective is pseudo concave in a neighborhood of the optimum, and then, with a suitable initial solution, optimization methods can be used to solve the fractional programming problem to obtain the optimum value. In this paper, we construct a bound that is explicitly formulated in terms of matrices regarding the finite element model, by maximizing the fraction function with a larger constraint field.