On the Flexure of a Saint-Venant Cylinder

The flexure of a Saint-Venant cylinder is defined from a kinematic point of view. A complete solution for the field equations is provided and the gauge choices to define the center of shear and rotation are made consequently. This seems to eliminate some ambiguities present in the literature. This revised version was published online in August 2006 with corrections to the Cover Date.     

重新认识横力弯曲梁的切应力强度效应

通过多层叠合梁的横力弯曲问题分析, 证明弯曲切应力不仅直接导致多层叠合梁的脱层, 而且这种脱层还会进一步引发弯曲正应力的急剧增加. 这种由弯曲切应力引发的连锁效应和潜在危害, 理应引起同学们的足够重视.     

基于剪切变形的矩形梁剪力滞求解方法

Timoshenko梁通过假设截面的剪切刚度和附加平均剪切转角变形的方式来近似修正初等梁中未考虑剪切变形能的问题,这与梁剪应力沿梁高变化的实际不符。本文基于材料力学剪应力计算式和相应的剪切变形理论,从剪切变形与梁的位移关系入手,导出矩形梁考虑剪切变形时的纵向位移沿梁高方向的函数关系式,证明该位移可分解为纯弯曲引起的位移和剪力引起的剪力滞翘曲位移之和。应用剪力滞广义坐标与广义力的概念,基于能量变分原理得到等截面梁剪力滞控制微分方程组及其通解形式。对均布荷载作用下矩形简支梁的算例分析表明,本文算法与弹性力学精确解对比,两者的应力和挠度剪力滞系数求解结果非常接近,本文算法有足够的精度,且比弹性力学简单。     

A nonlinear planar beam formulation with stretch and shear deformations under end forces and moments

A new nonlinear planar beam formulation with stretch and shear deformations is developed in this work to study equilibria of a beam under arbitrary end forces and moments. The slope angle and stretch strain of the centroid line, and shear strain of cross-sections, are chosen as dependent variables in this formulation, and end forces and moments can be either prescribed or resultant forces and moments due to constraints. Static equations of equilibria are derived from the principle of virtual work, which consist of one second-order ordinary differential equation and two algebraic equations. These equations are discretized using the finite difference method, and equilibria of the beam can be accurately calculated. For practical, geometrically nonlinear beam problems, stretch and shear strains are usually small, and a good approximate solution of the equations can be derived from the solution of the corresponding Euler–Bernoulli beam problem. The bending deformation of the beam is the only important one in a slender beam, and stretch and shear strains can be derived from it, which give a theoretical validation of the accuracy and applicability of the nonlinear Euler–Bernoulli beam formulation. Relations between end forces and moments and relative displacements of two ends of the beam can be easily calculated. This formulation is powerful in the study of buckling of beams with various boundary conditions under compression, and can be used to calculate post-buckling equilibria of beams. Higher-order buckling modes of a long slender beam that have complex configurations are also studied using this formulation.     

A new higher-order shear deformation theory and refined beam element of composite laminates

A new higher-order shear deformation theory based on global-local superposition technique is developed. The theory satisfies the free surface conditions and the geometric and stress continuity conditions at interfaces. The global displacement components are of the Reddy theory and local components are of the internal first to third-order terms in each layer. A two-node beam element based on this theory is proposed. The solutions are compared with 3D-elasticity solutions. Numerical results show that present beam element has higher computational efficiency and higher accuracy.The project supported by the National Natural Science Foundation of China (10172023)     

Higher-order shear beam theories and enriched continuum

The buckling of higher-order shear beam-columns is studied in the light of enriched continuum. We show the equivalence between the enriched kinematics of usual higher-order shear beam theories with the nonlocal and gradient nature of the associated constitutive law. These equivalences are useful for a hierarchical classification of usual beam theories comprising Euler-Bernoulli beam theory, Timoshenko and third-order shear beam theories. A consistent variationnally presentation is derived for all generic theories, leading to meaningful buckling solutions. It is shown that Timoshenko or some other higher-order shear theories can be considered as nonlocal or gradient Euler-Bernoulli theories. The buckling problem of a third-order shear beam-column is analytically studied and treated in the framework of gradient elasticity Timoshenko theory. Some different gradient elasticity Timoshenko models are presented at the end of the paper with available buckling solutions for repetitive structures and microstructured beams.     

Inelastic axial-flexure–shear coupling in a mixed formulation beam finite element

In this paper a beam element that accounts for inelastic axial-flexure–shear coupling is presented. The mathematical model is derived from a three-field variational form. The finite element approximation for the beam uses shape functions for section forces that satisfy equilibrium and discontinuous section deformations along the beam. No approximation for the beam displacement field is necessary in the formulation. The coupling of the section forces is achieved through the numerical integration of an inelastic multi-axial material model over the cross-section. The proposed element is free from shear-locking. Examples confirm the accuracy and numerical robustness of the proposed element and showcase the interaction between axial force, shear, and bending moment.     

Upper and lower bounds for the shear stress in the Saint-Venant theory of flexure

Upper and lower bounds are derived for the shear stress as it is determined by Saint-Venant's theory of flexure, and used to establish the asymptotic character of the classical Strength of Materials formula in the limit of vanishing thickness.

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Résumé On dérive des limites supérieures et inférieures des contraintes tangentielles suivant la théorie de la flexion de Saint-Venant, que l'on utilise aux fins d'établir le caractère asymptotique de la formule de la Résistance des Matériaux dans le cas limite d'une épaisseur extrêmement petite.

     

各向异性非均质梁截面特性计算的映射法

Ｇｉａｖｏｔｏ建立了确定各向异性梁截面特性和翘曲函数的二维有限元法。在此基础上，本文建立了一种映射法。在利用Ｇｉａｖｏｔｏ方法计算得某一具体截面的特性和翘曲函数后，与该截面具有相同形状不同尺寸的任意截面的特性和翘曲函数可用该截面的相应量通过显式确定，使计算大为简化。     

Variational formulation of the static Levinson beam theory

In this communication, we provide a consistent variational formulation for the static Levinson beam theory. First, the beam equations according to the vectorial formulation by Levinson are reviewed briefly. By applying the Clapeyron's theorem, it is found that the stresses on the lateral end surfaces of the beam are an integral part of the theory. The variational formulation is carried out by employing the principle of virtual displacements. As a novel contribution, the formulation includes the external virtual work done by the stresses on the end surfaces of the beam. This external virtual work contributes to the boundary conditions in such a way that artificial end effects do not appear in the theory. The obtained beam equations are the same as the vectorially derived Levinson equations. Finally, the exact Levinson beam finite element is developed.     

CONTACT PROBLEMS AND DUAL VARIATIONAL INEQUALITY OF 2－D ELASTOPLASTIC BEAM THEORY

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Saint-Venant Decay Rates for an Isotropic Inhomogeneous Linearly Elastic Solid in Anti-Plane Shear

The purpose of this research is to investigate the effects of material inhomogeneity on the decay of Saint-Venant end effects in linear isotropic elasticity. This question is addressed within the context of anti-plane shear deformations of an inhomogeneous isotropic elastic solid. The mathematical issues involve the effects of spatial inhomogeneity on the decay rates of solutions to Dirichlet or Neumann boundary-value problems for a second-order linear elliptic partial differential equation with variable coefficients on a semi-infinite strip. The elastic coefficients are assumed to be smooth functions of the transverse coordinate. The estimated rate of exponential decay with distance from the loaded end (a lower bound for the exact rate of decay) is characterized in terms of the smallest positive eigenvalue of a Sturm–Liouville problem with variable coefficients. Analytic lower bounds for this eigenvalue are used to obtain the desired estimated decay rates. Numerical techniques are also employed to assess the accuracy of the analytic results. A related eigenvalue optimization question is discussed and its implications for the issue of material tailoring is addressed. The results of this paper are applicable to continuously inhomogeneous materials and, in particular, to functionally graded materials. This revised version was published online in August 2006 with corrections to the Cover Date.     

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对于连续截面直梁平面横力弯曲情形, 分析了剪切变形引起横截面翘曲的影响. 基于位移模式假定, 用材料力学方法得到相应的正应力与切应力表达式, 并以均布力作用的矩形截面简支梁为例说明截面翘曲对于应力的影响. 该分析方法及其结果适用于材料力学教学.     

On the cross section of minimum stress concentration in the Saint-Venant theory of torsion

A lower bound is derived for the maximum stress in the torsion of cylindrical solids of simply-connected cross section. This bound, which is expressed in terms of the applied torque and the cross-sectional area, is isoperimetric in that it coincides with the maximum stress when the cross section is circular. It confirms the notion that given the applied torque and the area of the cross section, the least maximum stress occurs when the section is circular. Related isoperimetric upper bounds are derived for the minimum value of the stress at boundary points.     

On the behaviour of gelled fibre suspensions in steady shear

The shear rheological properties of suspensions of gelled agar fibres in a low viscosity Newtonian matrix fluid were investigated. Two classes of fibres, low aspect ratio fibres and high aspect ratio fibres with an aspect ratio of the order of 10 and 100 respectively were included in the investigations. For all fibre phase volumes investigated, from as low as 0.01 upwards, the flow curves are characterised by an apparent yield stress followed by shear-thinning which was independent of the fibre aspect ratio. Based on our analysis of the flow curves, we conclude that the high aspect ratio fibres behave like flexible threads in contrast to the low aspect ratio fibres whose high shear relative viscosity is successfully described by a relation for long rigid rods. These findings are supported by flow visualisation using an optical shearing stage coupled to a light microscope.     

Modeling and bifurcation analysis of the centre rigid-body mounted on an external Timoshenko beam

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On the use of Laplace transform inversion for reconstruction of relaxation spectra

We discuss the use of Fourier transforms to construct approximate inversion formulae for the Laplace transform and apply this technique to recover the relaxation spectrum from “data” for the relaxation modulus. We show that regularization by a Gaussian, as proposed by Davies and Anderssen [A.R. Davies, R.S. Anderssen, Sampling localization in determining the relaxation spectrum, J. Non-Newt. Fluid Mech. 73 (1997) 163–179] yields reasonable results even for data with significant noise. We also show that, in principle, other choices of regularization allow the relaxation spectrum to be reconstructed from data which are taken in any interval of time or frequency, however short and wherever located. We construct formulae which would do this, based on polynomial approximation of a function in an exponentially weighted space. This algorithm, however, turns out not to be practical, and we elucidate the reasons for that.     

关于"平面弹性悬臂梁剪切挠度问题"的进一步研究

对于狭矩形截面的平面弹性悬臂短梁,给出了悬臂端受切向外力或一边受均匀载荷作用时精 确的有限元结果和DQ (differential quadrature, 微分求积法) 法结果作为比较依据;基于铁摩辛柯梁理论并采用Gowper导出的剪切系数,导出了与弹性力 学解精度相当的结果.