On the use of the Trigonometric Ritz method for general vibration analysis of rectangular Kirchhoff plates

A comprehensive study on the use of a set of trigonometric functions, originally proposed by Beslin and Nicolas [Journal of Sound and Vibration 1997;202:633–55], as admissible solutions in the Ritz method for general vibration analysis of rectangular orthotropic Kirchhoff plates is presented. The approach is denoted here as Trigonometric Ritz method (TRM). Since its introduction, application of TRM was limited to a very few plate problems. The aim of this work is to extend the potential of the method on predicting natural flexural frequencies of plates with various complicating factors, including in-plane loads, elastically restrained edges, rigid/elastic concentrated masses, intermediate line and point supports or their combinations. Computational efficiency, stability, convergence and accuracy of the method are discussed and supported by extensive analysis. TRM-based solutions are compared with many reference cases available in the literature obtained with other methods or Ritz functions. Numerical results indicate that TRM exhibits good to excellent accuracy for all cases considered. New solutions are also presented for future comparison purpose.     

A general finite strip for the static and dynamic analyses of folded plates

In the conventional semi-analytical finite strip analysis of folded plates, the boundary conditions and the intermediate support conditions must be satisfied a priori. The admissible functions used as the longitudinal part of the displacement functions are sometimes difficult to find, and they are valid for specific conditions only. In this paper, a general finite strip is developed for the static and vibration analyses of folded plate structures. The geometric constraints of the folded plates, such as the conditions at the end and intermediate supports, are modelled by very stiff translational and rotational springs as appropriate. The complete Fourier series including the constant term are chosen as the longitudinal approximating functions for each of the displacements. As these displacement functions are more general in nature and independent of one another, they are capable of giving more accurate solutions. The potential problem of ill-conditioned matrices is investigated and the appropriate choice of the very stiff springs is also suggested. The formulation is done in such a way to obtain a unified approach, taking full advantage of the power of modern computers. A few numerical examples are presented for comparison with numerical results from published solutions or solutions obtained from the finite element method. The results show that this kind of strips is versatile, efficient and accurate for the static and vibration analyses of folded plates.     

Static analysis of superelliptical clamped plates by Galerkin's method

In this study, clamped superelliptical plates under uniformly distributed surface load are statically analyzed. Linearly elastic, homogeneous, and isotropic material is considered. The classical thin plate model (Kirchhoff) is employed. The lack of contributions on the static behavior of this sort of plate shapes is the fundamental motivation of the current study. Galerkin's method is used to obtain solutions. The method is conducted for polynomial series at powers ranging from 2 to 8 in order to get converging solutions. Maximum deflections of the plates and mid-point moments are obtained and the results are arranged in tabular form. For purpose of understanding, the behavior trend of the structure with respect to the parameters, some of the solutions are organized in graphical form. The study is performed for a wide range of superelliptical plates. The results are also examined with respect to the parameters a/b ratios and n, which are the plate aspect ratio and the superelliptical power, respectively.     

The orthogonal meshless finite volume method for solving Euler–Bernoulli beam and thin plate problems

In this paper a new method untitled “orthogonal meshless finite volume method” (OMFVM) is developed for solving elastostatic problems in Euler–Bernoulli beam and thin plate. In this method, the weak formulation of a conservation law is discretized by restricting it to a discrete set of test functions. In contrast to the usual finite volume approach, the test functions are not taken as characteristic functions of the control volumes in a spatial grid, but are chosen from a Heaviside step function. The present approach eliminates the expensive process of directly differentiating the OMLS interpolations in the entire domain. This method was evaluated by applying the formulation to a variety of patch test and thin beam problems. The formulation successfully reproduced exact solutions. Numerical examples demonstrate the advantages of the present methods: (i) lower-order polynomial basis can be used in the OMLS interpolations; (ii) smaller support sizes can be used in the OMFVM approach; and (iii) higher accuracies and computational efficiencies are obtained.     

最小二乘法解弹性薄板强度——正弦级数与多项式试函数

本文推导了正弦级数与多项式的梁函数,它可以满足梁的简支、固定、自由等各种边界条件。弹性弯曲薄板在各种边界条件下,用最小二乘法极值原理可以解算待定常数。由于它满足边界条件,因此是加权残数法内部法的一种试函数。     

Finite strip method for the free vibration and buckling analysis of plates with abrupt changes in thickness and complex support conditions

The free vibration problem of a stepped plate supported on non-homogeneous Winkler elastic foundation with elastically mounted masses is formulated based on Hamilton's principle. The stepped plate is modelled by finite strip method. To overcome the problem of excessive continuity of common beam vibration functions at the location of abrupt change of plate thickness, a set of C¹ continuous functions have been chosen as the longitudinal interpolation functions in the finite strip analysis. The C¹ continuous functions are obtained by augmenting the relevant beam vibration modes with piecewise cubic polynomials. As these displacement functions are built up from beam vibration modes with appropriate corrections, they possess both the advantages of fast convergence of harmonic functions as well as the appropriate order of continuity. The method is further extended to the buckling analysis of rectangular stepped plates. Numerical results also show that the method is versatile, efficient and accurate.     

Benchmark symplectic solutions for bending of corner-supported rectangular thin plates

This work presents the first ever analytical solutions for bending of a rectangular, thin plate supported only at its four corners. This breakthrough analysis employs a new symplectic elasticity approach that extends beyond the limitation of the classical plate bending methods such as Timoshenko's method, Navier method, Levy method and the polynomial approximation analysis of Lee and Ballesteros (Int J Mech Sci 1960;2:206). The classical methods are, in fact, special cases of this symplectic approach in the real eigenvalue regime for wavenumber with at least one pair of opposite sides of plates simply supported. For plate problems that do not fall into this category, the classical methods fail to yield any analytical solutions, but the symplectic approach does because in these cases the plate bending problems enter the complex eigenvalue regime for wavenumber. Another distinctive feature of this new approach is its necessity to pose an eigenvalue problem even for plate bending. In short, this innovative approach establishes the relationship between eigenvalue problem and bending. The novelty of this approach lies in the use of the Hamiltonian principle in a symplectic geometry space to derive a Hamiltonian system and a full state vector. The free boundaries with corner supports are dealt with using the variational principle. Analytical bending modes are then derived by expansion of eigenfunctions. The solutions are compared with other known (approximate) results and numerical finite element solutions but some of the results are not in agreement. Because the analytical bending moment and shear force solutions thus derived fulfill all natural and geometric boundary conditions, it leaves ample room for authentication of the benchmarks in the future. In addition, the twisting moment at the corners satisfies the condition for static bending equilibrium, in which the finite element solutions fail.     

Uncertainty propagation in dynamics of composite plates: A semi-analytical non-sampling-based approach

In this study, the influences of spatially varying stochastic properties on free vibration analysis of composite plates were investigated via development of a new approach named the deterministic-stochastic Galerkin-based semi-analytical method. The material properties including tensile modulus, shear modulus, and density of the plate were assumed to be spatially varying and uncertain. Gaussian fields with first-order Markov kernels were utilized to define the aforementioned material properties. The stochastic fields were decomposed via application of the Karhunen-Loeve theorem. A first-order shear deformation theory was assumed, following which the displacement field was defined using admissible trigonometric modes to derive the potential and kinetic energies. The stochastic equations of motion of the plate were obtained using the variational principle. The deterministic-stochastic Galerkin-based method was utilized to find the probability space of natural frequencies, and the corresponding mode shapes of the plate were determined using a polynomial chaos approach. The proposed method significantly reduced the size of the mathematical models of the structure, which is very useful for enhancing the computational efficiency of stochastic simulations. The methodology was verified using a stochastic finite element method and the available results in literature. The sensitivity of natural frequencies and corresponding mode shapes due to the uncertainty of material properties was investigated, and the results indicated that the higher-order modes are more sensitive to uncertainty propagation in spatially varying properties.     

半离散法分析薄壁柱的畸变屈曲

采用广义梁理论(GBT)的耦合差分方程解决了半离散法分析薄壁柱的畸变屈曲问题。作者近期发表的两篇文章对类似GBT的新型半离散分析方法进行了阐述。对横截面进行离散分析,寻找沿梁变化的解析解。采用新方法,利用齐次和非齐次方程建立确定梁全部变形的一般GBT方程并求解,从而使GBT的(复杂)变形方程变形为可降阶的微分方程。提出的半离散方法在广义梁理论(GBT)基础上增加了用于柱的失稳分析和失稳形态识别的几何刚度因素。通过势能的变化并在梁理论中引入约束条件,对初始应力下建立的GBT齐次微分方程进行修正,以分析其变形特性。对简支梁梁端施加横向位移和轴力,建立GBT初始应力方程,通过该方程寻求失稳的解决方法。根据已知的边界条件,利用三角函数关系式和求解特征值的方法求解这些可降阶的微分方程,使得屈曲形态和相关特征值与分叉荷载因素相符。因此,无需通过模态分解,可由耦合的GBT方程直接求得屈曲形态的解析解。通过实例分析了柱的整体失稳、屈曲变形和局部纵弯失稳,以及如何将新方法用于描述特征曲线和弹性屈曲曲线。将该方法的分析结果与ABAQUS、GBTUL和CUFSM软件的分析结果进行对比,验证了该方法的正确性。     

Stability criteria for rectangular plates subjected to intermediate and end inplane loads

This paper is concerned with the elastic buckling of rectangular plates subjected to intermediate and end uniaxial inplane loads, whose direction is parallel to two simply supported edges. The aforementioned buckling problem is solved by decomposing the plate into two subplates at the location where the intermediate uniaxial load acts. Each subplate buckling problem is solved exactly using the Levy approach and the two solutions brought together by matching the continuity equations at the separated interface. It is worth noting that there are five possible solutions for each subplate and consequently there are 25 combinations of solutions to be considered. For different boundary conditions, the buckling solutions comprise of different combinations. For each boundary condition, the correct solution combination depends on the ratio of the intermediate load to the end load. The exact stability criteria, presented both in tabulated and in graphical forms, should be useful for engineers designing walls or plates that have to support intermediate floors/loads.     

双参数地基板的无单元法

将无单元方法用于双参数地基上弹性板的计算分析,给出双参数地基上自由边界弹性板的虚功方程;基于滑动最小二乘(MLS)插值函数的特征,选择二次多项式基函数和指数型权函数,满足板弯曲问题对插值函数C1连续性和完备性的要求;根据所得虚功方程推出双参数地基板的无单元方法基本方程,并给出其刚度矩阵的特征。最后通过双参数地基板算例,说明了该方法的正确性和可行性。     

Vibration of general triangular composite plates with elastically restrained edges

Triangular fibre reinforced composite plates are important structural elements in modern engineering structures. In this paper a computationally efficient and accurate numerical model is presented for the study of free vibration behaviour of anisotropic triangular plates with edges elastically restrained against rotation and translation. The approach developed is based on the Rayleigh–Ritz method and the use of non-orthogonal right triangular co-ordinates. The deflection of the plate is approximated by a set of beam characteristic orthogonal polynomials generated using the Gram–Schmidt procedure. Several examples are solved and some results which correspond to particular cases are compared with existing values in the literature. New results are also presented for single layer composite plates with different fibre orientations and combinations of boundary conditions. For some plates, mode shapes of free vibration are also shown. Selected new transverse vibration mode shapes are presented to illustrate the effects of boundary constraints, aspect ratio and fibre orientation.The method can be applied to a wide range of elastic restraint conditions, any aspect ratio and for higher modes. The effect of the fibre orientation on the natural frequencies for plates with these restraint conditions are also considered.     

双参数弹性地基厚板分析的边界元法

以弹性地基上Ｒｅｉｓｓｎｅｒ板为研究对象，地基采用双参数模型，把地基效应归并到厚板的弯曲微分方程中。利用「４」导出的双参数地基上弯曲问题的基本解，从虚功原理出发，依据在胡海昌的中厚板弯曲理论，推导出三个广义位移表示的边界积分方程。适用于任意边界条件，任意形状及任意荷载的薄板及中厚板的弯曲问题。     

Buckling of centerline-stiffened plates subjected to uniaxial eccentric compression

A solution for the elastic buckling of flat rectangular plates with centerline boundary conditions subjected to non-uniform in-plane axial compression is presented. The loaded edges are simply supported, the non-loaded edges are free, and the centerline is simply supported with a variable rotational stiffness. The Galerkin method is used to establish an eigenvalue problem and a series solution for plate buckling coefficients is obtained by using combined trigonometric and polynomial functions that satisfy the boundary conditions. It is demonstrated that the formulation approaches the classical solution of a plate with a clamped edge as the variable rotational stiffness is increased. The variation of buckling coefficient with aspect ratio is presented for various stress gradient ratios. The coupling between plate aspect ratio, centerline rotational stiffness, and gradient of applied compressive stress is illustrated and discussed. The solution is applicable to stiffened plates and I-shaped beams that are subjected to biaxial bending or combined flexure and torsion, and is important to estimate the reduction in elastic buckling capacity due to stress gradient.     

弹性地基上厚板的计算分析

以弹性地基上Reissner板为研究对象,地基采用双参数模型,把地基效应归并到厚板的弯曲微分方程中。利用导出的双参数地基上厚板弯曲问题的基本解,从虚功原理出发,依据胡海昌的中厚板弯曲理论,推导出三个广义位移表示的边界积分方程。适用于任意边界条件、任意形状及任意荷载的薄板及中厚板的弯曲问题。文中给出了算例,并对各种参数变化进行了分析,获得了满意的结果。     

体外预应力碳纤维布加固钢-混凝土组合梁的试验研究

本文主要研究体外预应力碳纤维板加固钢-混凝土组合结构简支梁桥的力学性能。从顶杆间引入碳纤维板预应力筋,既能对主梁施加预应力,又能在主梁底端提供一个向上的顶升力,减小原结构的挠度。因此,通过顶杆间引入碳纤维板预应力筋,可以提高加固施工的速度和效率。通过缩尺模型静力试验,研究了碳纤维布用量、预应力筋尺寸、顶杆数量等参数对组合梁刚度和承载力的影响。在张拉组合梁的弯曲试验中,通过增加预应力水平和CFRP板数量,有效提高了组合梁截面的弯曲惯性矩,对组合梁屈服承载力和极限弯曲承载力的加固效果提升明显,并且加固试件仍具有良好的韧性。对于第三个组合梁,使用15%预应力水平和3 mm厚度的CFRP板进行加固时,其极限承载能力提高了47.9%,下翼缘的屈服荷载提高了39.8%,弹性阶段的刚度提高了21.66%。     

Mechanical and thermal post-buckling analysis of FGM rectangular plates with various supported boundaries resting on nonlinear elastic foundations

Mechanical and thermal post-buckling analysis is presented for FGM rectangular plates resting on nonlinear elastic foundations using the concept of physical neutral surface and high-order shear deformation theory, and investigations on post-buckling behavior of FGM rectangular plates with two opposite simply supported edges and other two opposite clamped edges are also new. Approximate solutions of FGM rectangular plates are given out using multi-term Ritz method, and influences played by different supported boundaries, foundation stiffnesses, thermal environmental conditions and volume fraction index are discussed in detail. It is worth noting that the effect of nonlinear elastic foundation is small at the pre-buckling and initial post-buckling state and is significant with increasing deflection at the deep post-buckling state. Especially, comparisons of post-buckling for FGM rectangular plates resting on nonlinear elastic foundations with movable simply supported edge subjected to compression acting on the geometric middle surface and the physical neutral surface are innovative, and may be helpful to clarify typical mistakes in literature.     

荷载横向变位下箱梁顶板与底板的剪滞效应分析

不同于以往荷载作用于箱梁的肋板处时剪力滞效应的研究,考虑当荷载不作用在箱梁的肋板处时顶板与底板位移及剪力滞效应的差异,对顶板与底板分设不同的纵向位移差函数,采用二次抛物线作为箱梁翼缘板的纵向位移沿梁宽分布函数,通过能量变分法,研究荷载在顶板横向变位下箱梁顶板与底板的位移,应力及剪力滞效应的变化,根据对于简支箱梁受均布荷载作用下跨中剪力滞效应的计算结果,随着作用在顶板的对称荷载从对称中心向肋板处移动,顶板将经历一个产生负剪力滞效应到无剪力滞效应,再到产生正剪力滞效应的过程。而与此同时底板的剪力滞效应规律受荷载的移动的影响很小,始终保持正剪力滞效应。计算结果与有限元数值模拟结果吻合较好,验证了理论的可靠性,同时也说明当荷载不作用在箱梁的肋板处时,对顶板与底板分设不同的纵向位移差函数的考虑是正确的。     

Analysis of functionally graded plates by an efficient finite element method with node-based strain smoothing

This paper presents an improved finite element approach in which a node-based strain smoothing is merged into shear-locking-free triangular plate elements. The formulation uses only linear approximations and its implementation into finite element programs is quite simple and efficient. The method is then applied for static, free vibration and mechanical/thermal buckling problems of functionally graded material (FGM) plates. In the FGM plates, the material properties are assumed to vary across the thickness direction by a simple power rule of the volume fractions of the constituents. The behavior of FGM plates under mechanical and thermal loads is numerically analyzed in detail through a list of benchmark problems. The numerical results show high reliability and accuracy of the present method compared with other published solutions in the literature.     

A new meshless approach for bending analysis of thin plates with arbitrary shapes and boundary conditions

An efficient and meshfree approach is proposed for the bending analysis of thin plates. The approach is based on the choice of a set of interior points, for each of which a basis function can be defined. Plate deflection is then approximated as the linear combination of those basis functions. Unlike traditional meshless methods, present basis functions are defined in the whole domain and satisfy the governing differential equation for plate. Therefore, no domain integration is needed, while the unknown coefficients of deflection expression could be determined through boundary conditions by using a collocation point method. Both efficiency and accuracy of the approach are shown through numerical results of plates with arbitrary shapes and boundary conditions under various loads.