Two-dimensional thermoelastic analysis of beams with variable thickness subjected to thermo-mechanical loads

Two-dimensional thermoelastic analysis for simply supported beams with variable thickness and subjected to thermo-mechanical loads is investigated. An approximate analytical method is proposed. Firstly, the heat conduction equation is analytically solved to obtain the temperature distributions for two kinds of boundary conditions at the beam ends, which are the harmonic series with unknown coefficients. Then the two-dimensional equilibrium differential equations are analytically solved to obtain the displacement component series with unknown coefficients and the stress component series is obtained. The unknown coefficients in the temperature series and the stress component series are approximately determined by using the upper surface and lower surface conditions of the beam. With the proposed procedure, the solutions satisfy the governing differential equations, the loading conditions, and the simply supported end conditions. The proposed solution method shows a good convergence and the results agree well with those obtained from the commercial finite element software ANSYS. Several examples are used to demonstrate the effectiveness of the proposed solution method. The simultaneous effects of temperature change and applied mechanical load on the behavior of the beam are examined.     

一端固支一端简支变厚度梁的弹性力学解

研究了一端固支另一端简支连续变厚度梁在静力荷载作用下的应力和位移分布.通过引入单位脉冲函数和Dirae函数,将固支边等效为简支边与未知水平反力的叠加,利用平面应力问题的基本方程,导出满足控制微分方程及左右两端边界条件的位移函数的一般解,对上下表面的边界方程作Fourier级数展开,结合固支边位移为O的条件确定待定系数,得到的解是高精度的.数值结果与商业有限元软件ANSYS进行了比较,显示出很好的精度.     

Two-dimensional analysis of simply supported piezoelectric beams with variable thickness

A two-dimensional analysis is presented for piezoelectric beam with variable thickness which is simply supported and grounded along its two ends. According to the governing equations of plane stress problems, the displacement solutions, which exactly satisfy the governing differential equations and the simply-supported boundary conditions at two ends of the beam, are derived. The unknown coefficients in the solution are then determined by using the Fourier sinusoidal series expansion to the boundary equations on the upper and lower surfaces of the beams. The present solutions show a good convergence and the numerical results are presented and compared with those available in the literature. The method could be applied to control engineering and other projects with highly accurate demand on stress and displacement analysis such as the design of micro-mechanical apparatuses.     

两对边简支中间有任意多个单向弹性线支矩形板横向振动的一个解析解法*

本文给出了两对边简支另两对边任意支承的中间有任意多个单向弹性线支矩形板横向振动的一个新的解析解法、将弹性线支反力看作是作用于板上的待求外力,求得了含有来知的弹性线支反力的两对边简支矩形板的运动方程的解析解;利用弹性线支反力与板横向位移之间的线性关系导出频率方程;频率方程及振型函数的表述均与已有方法不同.     

Analytical solutions of refined plate theory for bending,buckling and vibration analyses of thick plates

Analytical solutions for bending, buckling, and vibration analyses of thick rectangular plates with various boundary conditions are presented using two variable refined plate theory. The theory accounts for parabolic variation of transverse shear stress through the thickness of the plate without using shear correction factor. In addition, it contains only two unknowns and has strong similarities with the classical plate theory in many aspects such as equations of motion, boundary conditions, and stress resultant expressions. Equations of motion are derived from Hamilton’s principle. Closed-form solutions of deflection, buckling load, and natural frequency are obtained for rectangular plates with two opposite edges simply supported and the other two edges having arbitrary boundary conditions. Comparison studies are presented to verify the validity of present solutions. It is found that the deflection, stress, buckling load, and natural frequency obtained by the present theory match well with those obtained by the first-order and third-order shear deformation theories.     

A two-variable first-order shear deformation theory considering in-plane rotation for bending,buckling and free vibration analyses of isotropic plates

This paper presents a two-variable first-order shear deformation theory considering in-plane rotation for bending, buckling and free vibration analyses of isotropic plates. In recent studies, a simple first-order shear deformation theory (S-FSDT) was developed and extended. It has only two variables by separating the deflection into bending and shear parts while the conventional first-order shear deformation theory (FSDT) has three variables. However, the S-FSDT provides incorrect predictions for the transverse shear forces on the insides and the twisting moments at the boundaries except simply supported plates since it does not consider in-plane rotation. The present theory also has two variables but considers in-plane rotation such that it is able to correctly predict the responses of plates with any boundary conditions. Analytical solutions are obtained for rectangular plates with two opposite edges that are simply supported, with the other edges having arbitrary boundary conditions. Numerical results of deflections, stress resultants, buckling loads and natural frequencies are presented with the FSDT, the S-FSDT and the present theory. Comparative studies demonstrate the effects of in-plane rotation and the accuracy of the present theory in predicting the bending, buckling and free vibration responses of isotropic plates.     

四边简支对称正交层合矩形板的非线性弯曲问题

本文在von Kármán型板理论的基础上,采用双重Fourier级数方法,研究了对称正交层合矩形板在简支边条件下,承受任意分布横向载荷和面内载荷联合作用的非线性弯曲问题,得到了满足控制方程和边界条件的解.     

简支夹层矩形板的非线性弯曲

本文应用变分法导出了具有软夹心的夹层矩形板的非线性弯曲理论的基本方程和边界条件.然后,使用摄动法研究了均布横向载荷作作用下简支夹层矩形板的非线性弯曲问题,得到了相当精确的解析解.     

Levy-type solution for free vibration analysis of orthotropic plates based on two variable refined plate theory

Closed-form solutions for free vibration analysis of orthotropic plates are obtained in this paper based on two variable refined plate theory. The theory, which has strong similarity with classical plate theory in many aspects, accounts for a quadratic variation of the transverse shear strains across the thickness, and satisfies the zero traction boundary conditions on the top and bottom surfaces of the plate without using shear correction factors. Equations of motion are derived from the Hamilton’s principle. The closed-form solutions of rectangular plates with two opposite edges simply supported and the other two edges having arbitrary boundary conditions are obtained by applying the state space approach to the Levy-type solution. Comparison studies are performed to verify the validity of the present results. The effects of boundary condition, and variations of modulus ratio, aspect ratio, and thickness ratio on the natural frequency of orthotropic plates are investigated and discussed in detail.     

Levy-type solution for buckling analysis of thick functionally graded rectangular plates based on the higher-order shear deformation plate theory

In this article, an analytical approach for buckling analysis of thick functionally graded rectangular plates is presented. The equilibrium and stability equations are derived according to the higher-order shear deformation plate theory. Introducing an analytical method, the coupled governing stability equations of functionally graded plate are converted into two uncoupled partial differential equations in terms of transverse displacement and a new function, called boundary layer function. Using Levy-type solution these equations are solved for the functionally graded rectangular plate with two opposite edges simply supported under different types of loading conditions. The excellent accuracy of the present analytical solution is confirmed by making some comparisons of the present results with those available in the literature. Furthermore, the effects of power of functionally graded material, plate thickness, aspect ratio, loading types and boundary conditions on the critical buckling load of the functionally graded rectangular plate are studied and discussed in details. The critical buckling loads of thick functionally graded rectangular plates with various boundary conditions are reported for the first time and can be used as benchmark.     

On fourth-order boundary value problem with singular data

We consider a simply supported beam with restoring and external forces given as a sum of a continuous function and a Dirac delta distribution. We present sufficient conditions on these data in order to guarantee a unique positive or negative solution, respectively.     

Minimum-weight design of elastic sandwich beams with deflection constraints

In this paper, minimum-weight design of an elastic sandwich beam with a prescribed deflection constraint at a given point is investigated. The analysis is based on geometrical considerations using then-dimensional space of discretized specific bending stiffness. Since the present method of analysis is different from the method based on the calculus of variations, the conditions of piecewise continuity and differentiability on specific bending stiffness can be relaxed. Necessary and sufficient conditions for optimality are derived for both statically determinate and statically indeterminate beams. Beams subject to a single loading and beams subject to multiple loadings are analyzed. The degree to which the optimality condition renders the solution unique is discussed. To illustrate the method of solution, two examples are presented for minimum-weight designs under dual loading of a simply supported beam and a beam built in at both ends. The present analysis is also extended to the following problems: (a) optimal design of a beam built in at both ends with piecewise specific stiffness and a prescribed deflection constraint and (b) minimum-cost design of a sandwich beam with prescribed deflection constraints.The results presented in this paper were obtained in the course of research supported partly by the US Army Research Office, Durham, North Carolina, Research Grant No. DA-ARO-31-G1008, and partly by the Office of Naval Research, Contract No. N00014-67-A-0109-0003, Task No. NR 064-496. The authors wish to express their thanks to Professor H. Halkin for pointing out the applicability of optimal control theory to the present problem and to Professor W. Prager for his valuable suggestions.     

Highly accurate algorithms endowing with boundary functions for solving a nonlinear beam equation involving an integral term

In this paper, the problem of a nonlinear beam equation involving an integral term of the deformation energy, which is unknown before the solution, under different boundary conditions with simply supported, 2‐end fixed, and cantilevered is investigated. We transform the governing equation into an integral equation and then solve it by using the sinusoidal functions, which are chosen both as the test functions and the bases of numerical solution. Because of the orthogonality of the sinusoidal functions, we can find the expansion coefficients of the numerical solution that are given in closed form by using the Drazin inversion formula. Furthermore, we introduce the concept of fourth‐order and fifth‐order boundary functions in the solution bases, which can greatly raise the accuracy over 4 orders than that using the partial boundary functions. The iterative algorithms converge very fast to find the highly accurate numerical solutions of the nonlinear beam equation, which are confirmed by 6 numerical examples.     

均布荷载作用下压电材料简支梁的解析解

采用逆解法求解了均布荷载作用下压电材料简支梁的解析解。首先给出应力函数和电位移函数的多项式表达式,进而根据相容方程以及应力和电位移、位移和电势的边界条件,求得了同时考虑材料弹性参数、密度参数和压电参数呈梯度变化时,简支梁在均布荷载作用下的解析解。作为特例还得到了常体力以及材料参数为常数时的解答。并对结果进行了讨论。     

Analytical analysis of free vibration of non-uniform and non-homogenous beams: Asymptotic perturbation approach

This paper applies the asymptotic perturbation approach (APA) to obtain a simple analytical expression for the free vibration analysis of non-uniform and non-homogenous beams with different boundary conditions. A linear governing equation of non-uniform and non-homogeneous beams is obtained based on the Euler–Bernoulli beam theory. The perturbative theory is employed to derive an asymptotic solution of the natural frequency of the beam. Finally, numerical solutions based on the analytical method are illustrated, where the effect of a variable width ratio on the natural frequency is analyzed. To verify the accuracy of the present method, two examples, piezoelectric laminated trapezoidal beam and axially functionally graded tapered beam, are presented. The results are compared with those results obtained from the finite element method (FEM) simulation and the published literature, respectively, and a good agreement is observed for lower-order beam frequencies.     

一类指定应力问题的变分原理与应用

针对有限元分析中对应力或内力有指定条件的问题,引入非弹性应变作为实现指定应力条件的附加未知量,在小变形条件下描述了指定应力条件应当满足的弹性力学控制方程;以位移和未知非弹性应变作为独立变量,建立了具有指定应力条件问题的势能变分原理和虚功方程;以位移、弹性应变、未知非弹性应变和应力为独立变量,建立了一个含四类变量的广义变...     

Analytical solution of piezolaminated rectangular plates with arbitrary clamped/simply-supported boundary conditions under thermo-electro-mechanical loadings

This paper extends an analytical method for static analysis of general cross-ply piezolaminated rectangular plates with any combination of clamped/simply-supported boundary conditions under uncoupled thermo-electro-mechanical loadings. This method is based on the novel superposition method and the first-order shear deformation theory (FSDT). The FSDT enables this expanded method to consider the effect of shear deformation of the plate. The process of applying electrical and thermal resultants causes some advantages due to its simplicity and less computational process. In this analysis displacement components are written in terms of unknown force and moment resultants. Using Fourier series for displacement components, mechanical, thermal, and/or electrical stress resultants, the complex governing differential equations of the plate are reduced to a set of linear algebraic equations with non-trivial solution. The obtained equations may be solved analytically to determine the unknown stress resultants. Several examples are proposed, and their obtained numerical results are compared with those available in the literature to verify the convergence, high accuracy, and the capability of the present method to analyze the static behavior of piezolaminated plates. It is found that there is high agreement between the present results with those obtained by other investigators.     

Elasticity solutions for functionally graded rectangular plates with two opposite edges simply supported

England (2006) [13] proposed a novel method to study the bending of isotropic functionally graded plates subject to transverse biharmonic loads. His method is extended here to functionally graded plates with materials characterizing transverse isotropy. Using the complex variable method, the governing equations of three plate displacements appearing in the expansions of displacement field are formulated based on the three-dimensional theory of elasticity for a transverse load satisfying the biharmonic equation. The solution may be expressed in terms of four analytic functions of the complex variable, in which the unknown constants can be determined from the boundary conditions similar to that in the classical plate theory. The elasticity solutions of an FGM rectangular plate with opposite edges simply supported under 12 types of biharmonic polynomial loads are derived as appropriate sums of the general and particular solutions of the governing equations. A comparison of the present results for a uniform load with existing solutions is made and good agreement is observed. The influence of boundary conditions, material inhomogeneity, and thickness to length ratio on the plate deflection and stresses for the load x²yq are studied numerically.     

Large deformation and stability of a shear-deformable beam with a variable length under a uniformly distributed force

A beam held by two spatially fixed supports, may slide relative to these as soon as external loads are imposed. In particular, the possibly large deformation of a shear-deformable beam under a uniformly distributed, transverse force is investigated, which is clamped at its left side, while it may slide horizontally through another clamping device at its right side. Consequently, the material point of the beam that is currently located at the latter and the length of the portion of the reference configuration situated in between the two supports depend on the external forces and therefore are not known in advance. In order to obtain approximate solutions, a finite element scheme is utilized, in which a coordinate transformation is introduced, by which the difficulties of non-material boundary conditions and the unknown length of the beam are circumvented elegantly. It turns out that no static equilibrium is found, if the external forces are increased beyond a critical value, since the beam would slide out infinitely in that case. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

层合板一个新的高阶理论解析解

本文以复合材料的Reddy高阶理论为基础,引进一个位移函数Φ,将原来求解的微分方程组转化为一个高阶微分方程,得到了四边简支情况下的Navier型解,和一对边简支另一对边任意情况下的Levy型解.文中列举了算例进行比较,其数值结果和文献上已有结果相吻合,表明本文采用的解法是可靠的.Reddy高阶理论未知数不多,但精度比一阶剪切变形理论要好,计算时无需用剪切修正系数,计算较为简单.