A modified finite segment method for thin-walled single-cell box girders with shear lag

According to the principle of minimum potential energy and energy differential methods and assuming the shear lag warp displacement function with consideration of prestressed steel, a set of differential equations are deduced with different boundary conditions in considering moment, torsion, shear lag coupling. And the numerical solution of the differential equations is gained by Galerlein method. In addition, explores the influence on shear lag with prestressed force or not, dead loading, live loading and prestressed forces of the prestessed steels the areas of them basing on the model experiment and finite element model. Comparisons of results for typical numerical examples are given to verify the accuracy and applicability of the present method.

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.

Chapter 8 - Basic Equations and Solution Procedure

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

Nonlinear electromechanical bending of bi-modular piezoelectric laminated beams

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

Nonaxisymmetric free vibrations of a spherically isotropic spherical shell embedded in an elastic medium

SUMMARYIn this study, effects of applying higher order axial displacement distribution to solve a continuum model for static analysis of the combined system of framed tube, shear core and outrigger–belt truss system in high‐rise buildings are investigated. Framed tube system is modeled using an orthotropic box beam analogy approach and the interaction between shear core and outrigger–belt truss system on framed tube is modeled with rotating spring placed at outrigger–belt truss location. The axial displacement distributions in web and flange panels along structure's height are proposed to be fifth‐order and fourth‐order polynomials, respectively. Analytic analyses are carried out on the basis of the principle of minimum potential energy. A detailed work is carried out through two numerical examples and the accuracy of proposed approximation functions is compared with previous work and finite element analysis. Results demonstrate that the proposed parametric stress distribution and displacement functions are more accurate than previously proposed functions in comparison with the finite element solution. Copyright © 2012 John Wiley & Sons, Ltd.

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.

A refined plate theory for functionally graded carbon nanotube-reinforced plates with piezoelectric actuator

The aim of this research is to analyze buckling and bending behavior of a sandwich Reddy beam with porous core and composite face sheets reinforced by boron nitride nanotubes (BNNTs) and shape memory alloy (SMA) wires resting on Vlasov\'s foundation. To this end, first, displacement field\'s equations are written based on the higher-order shear deformation theory (HSDT). And also, to model the SMA wire properties, constitutive equation of Brinson is used. Then, by utilizing the principle of minimum potential energy, the governing equations are derived and also, Navier\'s analytical solution is applied to solve the governing equations of the sandwich beam. The effect of some important parameters such as SMA temperature, the volume fraction of SMA, the coefficient of porosity, different patterns of BNNTs and porous distributions on the behavior of buckling and bending of the sandwich beam are investigated. The obtained results show that when SMA wires are in martensite phase, the maximum deflection of the sandwich beam decreases and the critical buckling load increases significantly. Furthermore, the porosity coefficient plays an important role in the maximum deflection and the critical buckling load. It is concluded that increasing porosity coefficient, regardless of porous distribution, leads to an increase in the critical buckling load and a decrease in the maximum deflection of the sandwich beam.

Static bending of a twisted beam subjected to distributed transverse loads is established by using the Timoshenko beam theory and applying the principle of minimum potential energy. The closed-form solution for the static deflection of cantilever twisted beam is then derived. The effects of the twist angle, thickness ratio and shear deformation on the static bending characteristics of twist beams are investigated and discussed.

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.

The conventional methods used to analyze the shear lag effect in simply supported box girders assume that the neutral axis of the section coincides with the centroid, which does not strictly satisfy the axial equilibrium condition. To address this problem, this study proposes an analysis method in which three independent functions for the shear lag are employed to define the different shear lag strengths of the top slab, the bottom slab, and the cantilever slab. To fulfill the axial equilibrium condition of the box girder and to automatically locate the neutral axis position, the longitudinal displacement of the web is introduced. The shear deformation of the box girder is also considered. The governing differential equations and corresponding boundary conditions for displacement variables such as deflection and rotation of the box girder are derived through the application of the principle of virtual work. The differential equations are solved by utilizing the boundary conditions to obtain the analytical expressions of the shear lag function, longitudinal displacement of the web, rotation, deflection, and neutral axis position. Furthermore, after performing the finite element analysis, the effectiveness of the proposed method is verified by comparing the results with those obtained from conventional methods and finite element analysis. Furthermore, the influence of the axial equilibrium condition is quantified on axial force and stress difference ratios under three methods. Finally, extensive parametric analysis is carried out to investigate the effect of different parameter ratios on the ratios of the stress difference of the flanges. The results show that when the axial equilibrium condition is not considered, the axial stresses in the upper flange of the simply supported box girder are underestimated, especially at the intersection of the top, cantilever slab, and web, and the axial stresses in the lower flange are overestimated. As a result, the method in this study is able to calculate the axial stresses and deflections on simply supported box girders more accurately.