Lateral post-buckling analysis of beams

Summary The nonlinear lateral buckling response of perfect stocky beams in the vicinity of the critical bifurcational state is discussed. Attention is focused on the initial post-buckling response. This depends on the nature of the critical branching point which is explored by using a nonlinear (lateral) bending-curvature relationship. It is found that the plane of loading (associated with the major moment of inertia) looses its stability through a stable symmetric bifurcation point. Hence, the above beams are not sensitive to imperfections, exhibiting post-buckling strength. However, the post-buckling equilibrium path is quite shallow, so that the load-carrying capacity of such beams above the critical state is rather limited. The analysis is supplemented by illustrative examples forI-beams for which the effect of various parameters on the initial postbuckling path is also discussed.

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Analyse des lateralen nachkritischen Knickverhaltens von Balken

Übersicht Das nichtlineare Knickverhalten von perfekten gedrungenen Balken in der Nachbarschaft des kritischen Verzweigungspunktes wird untersucht, wobei sich die Aufmerksamkeit insbesondere auf den Beginn der nachkritischen Phase richtet. Diese hängt vom Charakter des kritischen Verzweigungspunktes ab, der seinerseits mit Hilfe einer nichtlinearen (lateralen) Biegemoment-und Krümmungs-Beziehung analysiert wird. Es zeigt sich, daß die Stabilität in der Belastungsebene-verbunden mit dem maximalen Flächenmoment zweiter Ordnung-über einen stabilen symmetrischen Verzweigungspunkt verlorengeht. Daher sind solche Balken unempfindlch gegenüber Imperfektionen, sie zeigen nachkritische Belastbarkeit. Der nachkritische Gleichgewichtspfad ist jedoch sehr flach, so daß die Tragfähigkeit dieser Balken oberhalb des kritischen Zustandes recht begrenzt ist. Die Untersuchung ist durch illustrative Beispiele für I-Balken untermauert, für welche der Einfluß der verschiedenen Parameter auf den anfänglichen nachkritischen Pfad herausgearbeitet wird.

     

纵横载荷作用下功能梯度梁的弯曲和过屈曲

基于一阶非线性梁理论和物理中面概念,导出了纵横向载荷作用下功能梯度材料(FGM)梁非线性弯曲和过屈曲问题的控制方程,并获得了该问题的精确解;据此解研究了梯度材料性质、外载荷、横向剪切变形以及边界条件等因素对功能梯度材料梁非线性力学行为的影响,分析中假设功能梯度材料性质只沿梁厚度方向,并按成分含量的幂指数函数形式变化。结果表明:纵横载荷共同作用下,功能梯度梁的弯曲构形将有无限多个;随着梯度指数的增大,梁的变形减小,临界载荷升高;随着长高比的增大,横向剪切变形的影响减小。     

Thermal post-buckling and nonlinear vibration analysis of FGM beams based on physical neutral surface and high order shear deformation theory

Thermal post-buckling and nonlinear vibration behaviors of FGM beams are analyzed by using concept of physical neutral surface, von Kármán strain-displacement relationships and high order shear deformation theory. Material properties are assumed to be temperature dependent and vary along the thickness. The prominent character of physical neutral surface higher-order shear deformation beam theory is that stretching-bending couplings are eliminated in constitutive equations, and governing equations have the similar forms as homogeneous isotropic beams. Approximate solutions are given out by Ritz method, and influences played by different supported boundaries, thermal environmental conditions and volume fraction index are discussed in detail.     

On the elastic stability of beams under unilateral constraints

Summary We analyze the problem of the axially loaded elastic beam on a one-sided elastic foundation. The existence of the critical load is proved and its expression is given.

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Sommario Per il problema della trave elastica su fondazione elastica unilaterale assialmente caricata si prova l'esistenza del carico critico e se ne dà l'espressione.

     

Nonlinear free vibration and post-buckling analysis of functionally graded beams on nonlinear elastic foundation

In this study, simple analytical expressions are presented for large amplitude free vibration and post-buckling analysis of functionally graded beams rest on nonlinear elastic foundation subjected to axial force. Euler–Bernoulli assumptions together with Von Karman’s strain–displacement relation are employed to derive the governing partial differential equation of motion. Furthermore, the elastic foundation contains shearing layer and cubic nonlinearity. He’s variational method is employed to obtain the approximate closed form solution of the nonlinear governing equation. Comparison between results of the present work and those available in literature shows the accuracy of this method. Some new results for the nonlinear natural frequencies and buckling load of the FG beams such as the effect of vibration amplitude, elastic coefficients of foundation, axial force, and material inhomogenity are presented for future references.     

On shear lag effects in orthotropic composite beams

Based upon the linearised theories of the bending and stretching of thin plates, an analysis is presented for the interaction between non-prismatic beams and an orthotropic concrete plate. It is shown that an exponential representation for the steel beam profiles provides a suitable basis for studying interaction in continuous non-prismatic beams, and for deducing suitable effective widths of slab for design purposes. The influence of “elastic” shear connection modulus is studied, as well as the effect of the varying flexural rigidity of the steel beams. The dependence of interaction on shear connection modulus in continuous beams is demonstrated through deflexion and slip characteristics, and so also is the dependence of interaction on the severity of the variation of flexural rigidity. The solution can be specialised to the limiting case of prismatic steel beams and a concrete slab and also to the solutions of rectangular plates with certain edge conditions.     

On the post-buckling analysis of plastic structures under impulsive loading

An analytical method is suggested to analyze the plastic post-buckling behavior under impulsive loading. The fundamental equation of motion of a cylindrical shell is taken as an example to explain the main concept and procedure. The axial and the radial displacements are decoupled by an approximate scheme, so that only one non-linear equation for the radial buckling displacement is to be solved. By expanding it in terms of an amplitude measure as a time variable, we may get the post-buckling behavior in the form of a series solution. The post-buckling behavior of a rectangular plate used as a special case of cylindrical shell is discussed.     

THERMAL POST-BUCKLING OF FUNCTIONALLY GRADED MATERIAL TIMOSHENKO BEAMS

Analysis of thermal post-buckling of FGM (Functionally Graded Material) Timoshenko beams subjected to transversely non-uniform temperature rise is presented. By accurately considering the axial extension and transverse shear deformation in the sense of theory of Timoshenko beam, geometrical nonlinear governing equations including seven basic unknown functions for functionally graded beams subjected to mechanical and thermal loads were formulated. In the analysis, it was assumed that the material properties of the beam vary continuously as a power function of the thickness coordinate. By using a shooting method, the obtained nonlinear boundary value problem was numerically solved and thermal buckling and post-buckling response of transversely non-uniformly heated FGM Timoshenko beams with fixed-fixed edges were obtained. Characteristic curves of the buckling deformation of the beam varying with thermal load and the power law index are plotted. The effects of material gradient property on the buckling deformation and critical temperature of beam were discussed in details. The results show that there exists the tension-bend coupling deformation in the uniformly heated beam because of the transversely non-uniform characteristic of materials.     

Dynamic characteristics of multi-span spinning beams with elastic constraints under an axial compressive force

A theoretical model for the multi-span spinning beams with elastic constraints under an axial compressive force is proposed. The displacement and bending angle functions are represented through an improved Fourier series, which ensures the continuity of the derivative at the boundary and enhances the convergence. The exact characteristic equations of the multi-span spinning beams with elastic constraints under an axial compressive force are derived by the Lagrange equation. The efficiency and ac...     

考虑弯扭形变的薄壁箱形梁的非线性后屈曲

在大位移和扭转的前提下,通过一中等弯曲扭转的位移场描述了薄壁箱形梁在偏心载荷作用下的静稳定性问题.该非线性公式可用于分析简支薄壁箱形梁在不同载荷作用下的屈曲和后屈曲行为.采用伽辽金方法将非线性微分系统离散,并通过牛顿-拉普森增量迭代法求解得代数方程组.数值计算结果表明,当前屈曲位移不可忽略时,经典的横向屈曲预测是保守的...     

计及剪切影响的RC梁的非线性分析

为提高钢筋混凝土RC梁的计算效率和精度,提出了一种基于梁截面弯矩-曲率关系的宏观有限元方法,可用于各种跨高比RC梁的材料非线性分析。首先假定了混凝土和钢筋的非线性应力-应变关系,然后引入经过修正的Rodriguez截面模型,根据边界顶点把截面划分成若干梯形单元,利用quasi-Newton法求解由两个变量耦合而成的截面非线性平衡方程,由此建立RC截面的弯矩-曲率关系。在此基础上利用Timoshenko梁弯曲理论建立考虑横向剪切变形影响的RC梁的有限元分析模型。通过对试验梁的分析对比验证了所提出的分析方法的适用性。     

Thermal post-buckling analysis of imperfect laminated plates using a higher-order shear deformation theory

Thermal post-buckling analysis is presented for a simply supported, composite laminated plate subjected to uniform or non-uniform tent-like temperature loading. The initial geometrical imperfection of the plate is taken into account. The formulations are based on the Reddy's higher-order shear deformation plate theory, and include thermal effects. The analysis uses a mixed Galerkin-perturbation technique to determine thermal buckling loads and post-buckling equilibrium paths. Numerical examples cover the performances of perfect and imperfect, antisymmetrically angle-ply and symmetrically cross-ply laminated plates. The effects played by transverse shear deformation, thermal load ratio, plate aspect ratio, total number of plies, fiber orientation and initial geometrical imperfections are studied. Typical results are presented in dimensionless graphical form.     

Initial thermo-mechanical post-buckling of beams with temperature-dependent physical properties

The objective of this work is to analyze the elastic buckling and initial post-buckling behavior of slender beams subjected to uniform heating. The beams are assumed to be double-hinged with fixed ends, preventing thermal expansion. Consequently, destabilizing compressive forces arise that may lead to beam buckling. When the temperature is further increased, the beam experiences finite displacements, with the result that the analysis is geometrically non-linear. The modulus of elasticity and the thermal induced strain, key material properties for this problem, are temperature-dependent. Thus, the coefficients of the governing equations are not constant. This suggests the physical non-linearity of the mathematical model. Hence, the analysis is geometrically and physically non-linear. The analysis is sensitive to the beam initial temperature, as the thermal strain is a function of the initial and final temperatures. The material is considered to be linear elastic, and consequently viscoelastic and plastic effects are not taken into account. Furthermore, the beam cross-section properties are assumed to be constant, which is consistent with the small strain formulation. A perturbation method is applied to the governing non-linear differential equations so that the initial post-buckling behavior may be analytically determined when temperatures above the critical temperature are applied to the beam. To illustrate the application of the formulation we present a case study for the aluminum 7075-T6 alloy, a material commonly used in aerospace and naval industries. Nonetheless, it is expected similar behavior for other metallic materials. The curves that define the variation of the modulus of elasticity, the thermal strain and the yield stress with temperature are considered in our analysis. The change in length, reaction forces at the supports and geometric configurations are obtained as a function of temperature and the beam slenderness ratio. The critical buckling loads and temperatures and the initial post-buckling analysis are also calculated in the context of the temperature-independent physical properties. Our results emphasize the importance of modeling the material's non-linearity if accuracy is required. However, from a practical application point of view results are acceptable if temperature-independent physical properties are employed, especially for large slenderness ratios.     

Failure analysis of unidirectional fiber composites under combined axial tension and shear

A method of analysis is proposed for investigating the mechanics of failure of unidirectional, fiber composite materials subjected to axial tension and shear. The mechanisms of failure are assumed to result from the interaction of the applied shear stress and local matrix shear stress concentrations which arise as a result of the scattered fiber breaks that occur throughout the material. Two modes of failure are identified. One is associated with the unstable growth of shear failure regions in the matrix. The other is primarily a tensile failure mode which is influenced by the applied shear. The analysis predicts that a variety of shear-tensile stress failure surfaces are possible, depending on material properties. The results suggest that radical changes in the shape of failure surfaces can occur as the result of environmental effects. This has significant implications for reliability.     

Imperfection sensitivity of the post-buckling behavior of higher-order shear deformable functionally graded plates

This paper investigates the sensitivity of the post-buckling behavior of shear deformable functionally graded plates to initial geometrical imperfections in general modes. A generic imperfection function that takes the form of the product of trigonometric and hyperbolic functions is used to model various possible initial geometrical imperfections such as sine type, local type, and global type imperfections. The formulations are based on Reddy’s higher-order shear deformation plate theory and von Karman-type geometric nonlinearity. A semi-analytical method that makes use of the one-dimensional differential quadrature method, the Galerkin technique, and an iteration process is used to obtain the post-buckling equilibrium paths of plates with various boundary conditions that are subjected to edge compressive loading together with a uniform temperature change. Special attention is given to the effects of imperfection parameters, which include half-wave number, amplitude, and location, on the post-buckling response of plates. Numerical results presented in graphical form for zirconia/aluminum (ZrO₂/Al) graded plates reveal that the post-buckling behavior is very sensitive to the L2-mode local type imperfection. The influences of the volume fraction index, edge compression, temperature change, boundary condition, side-to-thickness ratio and plate aspect ratio are also discussed.     

An analysis of the shear failure of rigid-linear hardening beams under impulsive loading

A theoretical rigid-plastic analysis for the dynamic shear failure of beams under impulsive loading is presented when using a travelling plastic shear hinge model which takes into account material strain hardening. The maximum dynamic shear strain and shear strain-rate can be predicted in addition to the permanent transverse deflections and other parameters. The conditions for the three modes of shear failure, i.e., excess deflection failure, excess shear strain failure and adiabatic shear failure are analyzed. The special case of an infinitesimally small plastic zone is discussed and compared with Nonaka's solution for a rigid, perfectly plastic material. The results can also be generalized to examine the dynamic response of fibre-reinforced beams.     

轴向力作用任意不连续梁自由振动分析的广义函数解

摘要：首先运用分布理论建立了轴向力作用下含多个不连续点的欧拉梁的自由振动的统一微分方程。不连续点的影响由广义函数（Dirac delta函数）引入梁的振动方程。微分方程运用Laplace变换方法求解;与传统方法不同的是,本文方法适用于含任意类型的不连续点和多种不连续点组合情况的梁,求得的模态函数为整个不连续梁的一般解。由于模态函数的统一化以及连续条件的退化,特征值的求解得到了极大的简化。最后,以轴向力作用下多跨梁—弹簧质量块系统模型为例验证了本文方法的正确性与有效性。