Computational procedures for a matrix analysis of the stability and vibration of thin flat-walled structures in compression

This paper is concerned with computational problems arising in the application of a previously published matrix analysis of the stability and vibration of structures consisting entirely of a series of thin flat rectangular plates connected together along longitudinal edges. The theory requires that the conditions at the ends of the structure permit a mode which varies sinusoidally in the longitudinal direction, and it is then possible to consider each individual flat plate as one element having four degrees of freedom at each of its two longitudinal edges. The corresponding stiffness matrices are essentially exact within the spirit of thin plate theory. The main computational problem is that of testing whether or not, for a specified wavelength of buckle, any chosen value of the compressive stress is less than or greater than the lowest buckling stress of the structure. This problem is discussed in detail, and a systematic test procedure is derived. Examples of panels with integral unflanged stiffeners, bonded Z-section stiffeners, bonded top-hat stiffeners and corrugated-core sandwich panels are discussed.     

Buckling and vibration of anisotropic or isotropic plate assemblies under combined loadings

This paper describes the underlying theory, and a general-purpose computer program, VIPASA, for determining the critical buckling stresses or natural frequencies of vibration of thin prismatic structures, consisting of a series of plates rigidly connected together along longitudinal edges. Each plate may be either isotropic or anisotropic and may carry a basic stress system consisting of longitudinal and transverse direct stress combined with shear. The structure is assumed to be subjected to a “dead load” system which does not cause buckling; in addition a “live load” system, defined in magnitude by a single load factor, may be applied and the value of the load factor at buckling is determined. Alternatively the natural frequencies of vibration of the structure when subjected to the dead load system are determined. Any number of critical load factors or natural frequencies can be obtained. The theory is based upon the assumption that all modes are sinusoidal, in the sense that all three components of displacement vary sinusoidally along any longitudinal line, but phase differences are incorporated to allow for the effects of anisotropy and shear. Apart from this assumption no further approximations are made other than those inherent in thin plate theory.     

Buckling and vibration of any prismatic assembly of shear and compression loaded anisotropic plates with an arbitrary supporting structure

The VIPASA computer program accurately treats buckling and vibration of prismatic plate assemblies with a response that varies sinusoidally in the longitudinal direction. In-plane shear loading of component plates produces skewed mode shapes that do not conform to desired support conditions, and this has placed a limitation on the general applicability of VIPASA.This problem is overcome in the present paper by coupling the VIPASA stiffness matrices for different wavelength responses through the method of Lagrangian Multipliers. Supports at arbitrary locations, including support provided by any elastic structure, are included in the theory. Examples illustrate the accuracy and convergence of the method and some of the principal features of the solution. The complete generality and capability of VIPASA have been retained in a computer program VICON that permits constraints and a supporting structure consisting of any number of transverse beam-columns.     

A post-buckling analysis for isotropic prismatic plate assemblies under axial compression

This paper presents a post-buckling analysis for prismatic plate assemblies made of isotropic materials. The structures are assumed to consist of a series of long flat strips rigidly connected together at their edges, subjected to longitudinal in-plane compressive load. The buckling load and corresponding buckling mode of the structure are first obtained as the results of transcendental eigenvalue problems, which arise when exact solutions to the member differential equations are used to form the stiffness matrix of the plate assemblies. The other post-buckling field functions are also obtained analytically as exact solutions to the member differential equations. Results for the load end-shortening and load–deflection relationships for long prismatic plate assembly examples are obtained and compared with results obtained by other authors.     

Buckling and vibration of stiffened plates subjected to partial edge loading

The buckling and vibration characteristics of stiffened plates subjected to in-plane partial and concentrated edge loadings are studied using finite element method. The initial stresses are obtained considering the pre-buckling conditions. Buckling loads and vibration frequencies are determined for different plate aspect ratios, edge conditions and different partial non-uniform edge loading cases. The non-uniform loading may also be caused due to the supports on the edges. The analysis presented determines the stresses all over the region for different kinds of loading and edge conditions. In the structural modelling, the plate and the stiffeners are treated as separate elements where the compatibility between these two types of elements is maintained. The vibration characteristics are discussed and the results are compared with those available in the literature. Buckling results show that the stiffened plate is less susceptible to buckling for position of loading near the supported edges and near the position of stiffeners as well.     

Thermal buckling of functionally graded rectangular plates subjected to partial heating

We present the thermal buckling analysis of functionally graded rectangular plates subjected to partial heating in a plane and uniform temperature rise through its thickness. The plate is simply supported for out-of-plane deformation and perfectly clamped for in-plane deformation. It is assumed that the functionally graded material properties such as the coefficient of linear thermal expansion and Young's modulus are changed individually in the thickness direction of the plate with the power law, while Poisson's ratio is assumed to be constant. Analytical developments consist of two stages. First, the nonuniform in-plane resultant forces are determined by solving a plane thermoelastic problem. Then the critical buckling temperatures of the plates with the predetermined resultant forces are calculated as the generalized eigenvalue problem which is constructed by using the Galerkin method. Finally, the effects of material inhomogeneity, aspect ratio, and heated region on the critical buckling temperatures are examined.     

Refined theory for vibrations and buckling of laminated isotropic annular plates

Hamilton's variational principle is used for the derivation of equations of transversally isotropic laminated annular plates motion. Nonlinear strain—displacements relations are considered. Linearized vibration and buckling equations are obtained for the annular plates uniformly compressed in the radial direction. The effects of transverse shear and rotational inertia are included. A closed form solution is given for the mode shapes in terms of Bessel, power and trigonometric functions. The eigenvalue equations are derived for natural frequencies and buckling loads of annular and circular plates elastically restrained against rotation along edges. Classical-type plate theory results are obtained then by letting the transverse shear stiffness go to infinity and rotational inertia go to zero. Numerical examples are presented by tables and figures for 2- and 3-layered plates with various geometrical and physical parameters. The transverse shear, rotational inertia and boundary conditions effects are discussed.     

Analytical buckling solutions for mindlin plates involving free edges

This paper investigates the buckling behaviour of rectangular Mindlin plates having two parallel edges simply supported, one edge free and the remaining edge free, simply supported or clamped. The proper boundary conditions at free edges subjected to in-plane loads have been examined. The buckling analysis is performed by applying the concept of state space to the Levy-type solution method to obtain the closed-form critical loads from the governing differential equations. The results, where possible, are compared with existing solutions to verify the validity of the solution method. The differences between buckling factors obtained with the appropriate and inappropriate free edge conditions are reported. Several design charts representing the essential features of the critical load characteristics of rectangular plates with two opposite edges simply supported at least one free edge are obtained. The critical loads can be determined from the design charts without difficulty.     

Buckling of unsymmetrical, cross-ply laminates with matrix cracks

The effect of matrix cracks on the buckling of unsymmetrical, cross-ply laminates is investigated. The cracks are modelled as aligned slit cracks across the ply thickness and transverse to the laminate plane. The distribution of cracks is assumed to be statistically homogeneous corresponding to an average crack density. The results are given for antisymmetrically- and unsymmetrically-laminated plates with a given density of matrix cracks. The effect of reduction in the laminate stiffness is studied by plotting the ratio of the buckling loads of cracked and uncracked laminates against the aspect ratio, crack density and relative layer thicknesses. It is observed that the amount of reduction in the buckling load depends on the stacking sequence as well as on the relative thicknesses of layers. This reduction is found to be substantially less than the reduction in the affected elastic moduli of the material.     

Modeling of Coriolis mass flow meter of a general plane-shape pipe

The vibration of a general plane tube with a flowing fluid, which is the measuring element in a Coriolis mass flow meter (CMF), is studied. The dynamic stiffness matrix method is used to model such a tube. The tube is divided into straight and circular elements. The elements dynamic stiffness matrices are derived from the equations of motion. By assembling the elements matrices into a global matrix the natural frequencies are obtained. The mode shapes are obtained by applying the boundary conditions at the supports and the compatibility conditions at the nodes. The effects of the flow velocity on the natural frequency and the relative phase difference are modeled. The method is applied to different tube shapes. The results are compared to the published data which reveals good agreement.     

Effects of solid distribution on the elastic buckling of honeycombs

The elastic buckling strengths of honeycombs depend on their relative density, cell geometry and the elastic modulus of solid cell edges. In this study, we consider the effect of the distribution of solid between three cell edges and a vertex on elastic buckling using a semi-analytical integral-equation approach. At first, the geometry of three cell edges connected at a vertex with Plateau borders is analyzed and then employed to represent a repeating element for regular hexagonal honeycombs. The bending moments, rotational angle and the stiffness of a rotational spring corresponding to the constraint from inclined adjacent cell edges are derived for the vertical cell edge within the repeating element. Consequently, the elastic buckling strength of regular hexagonal honeycombs can be numerically obtained. Moreover, the effects of the distribution of the solid on the elastic buckling strengths of regular hexagonal honeycombs are presented and evaluated.     

Large deflection analysis of elasto-plastic beams and frames

A general method for analysis of elasto-plastic beams and frames with large displacements is described in this paper. A hybrid-type beam element with 3df at each end is used in the analysis. The element stiffness matrix is obtained by inversion of a flexibility matrix, which is computed from an assumed distribution of internal forces along the element axis. This approach with approximations of the stress fields better imitates varying stiffness along the beam than the traditional approach with assumed displacement fields.Large displacement effects are taken into account by updating the geometry (both node co-ordinates and cross-section shapes) and accurately computing the elongations of each element. Partial yielding is considered across the height of the beam as well as along the element axis.     

A composite view on Windenburg's problem: Buckling and minimum stiffness requirements of compressively loaded orthotropic plates with edge reinforcements

This paper discusses the buckling behaviour of orthotropic composite plates under uniform uniaxial compression with one free reinforced unloaded edge. A typical application example for use of such a mechanical model is the web of stiffeners and frames attached to the fuselage skin of an aircraft. The considered plates are rectangular and simply supported at the loaded transverse edges. One of the longitudinal unloaded edges is also simply supported, while the second unloaded edge is not supported at all but is reinforced by a flange of arbitrary cross-section. At first, an exact solution for the elastic buckling problem is derived from the governing differential equation by imposing the underlying boundary conditions. Thereafter, two approximate closed-form solutions for the buckling load are derived, which can be conveniently used for practical application purposes. Generic buckling curves using characteristic non-dimensional quantities are also presented. Finally, the question of the required bending stiffness EI_min of the flange is treated, to ensure that the flange withstands buckling and provides simply supported boundary conditions to the free reinforced plate edge.     

新型PRRR+PURU+S球面并联人形机器人踝关节机构静刚度性能分析

提出了一种新型PRRR+PURU+S球面并联人形机器人踝关节机构,对该8次超静定机构进行了静刚度分析。首先基于各构件的弹性变形,利用小变形叠加原理,推导出机构动平台角位移、球心点线位移与各作用力的解析关系式;然后利用该机构静力学分析的结果,即各构件上的力与机构外载荷的关系方程,建立机构外载荷与机构动平台角位移、球心点线位移的解析关系,进而得到机构的静刚度矩阵;最后采用正交变换方法,得到机构的6个主刚度及其所在方向。研究结果表明,与UP+R踝关节机构相比,新型踝关节机构的角位移刚度、线位移刚度均有增大,其中线位移刚度大幅增大,基本实现刚度均衡。     

Vibration characteristics and power transmission of coupled rectangular plates with elastic coupling edge and boundary restraints

Coupled-plate structures are widely used in the practical engineering such as aeronautical,civil and naval engineering etc.Limited works can be found on the vibration of the coupled-plate structure due...     

纤维增强FGM梁的自由振动和临界屈曲载荷分析

基于经典梁理论（CBT）研究轴向力作用下纤维增强功能梯度材料（FGM）梁的横向自由振动和临界屈曲载荷问题。首先考虑由混合律模型来表征纤维增强FGM梁的材料属性,其次利用Hamilton原理推导轴向力作用下纤维增强FGM梁横向自由振动和临界屈曲载荷的控制微分方程,并应用微分变换法（DTM）对控制微分方程及边界条件进行变换,计算了纤维增强FGM梁在固定-固定（C-C）、固定-简支（C-S）和简支-简支（S-S）3种边界条件下横向自由振动的无量纲固有频率和无量纲临界屈曲载荷。退化为各向同性梁和FGM梁,并与已有文献结果进行对比,验证了本文方法的有效性。最后讨论在不同边界条件下纤维增强FGM梁的刚度比、纤维体积分数和无量纲压载荷对无量纲固有频率的影响以及各参数对无量纲临界屈曲载荷的影响。     

Vibration of rectangular Mindlin plates resting on non-homogenous elastic foundations

This paper is concerned with the vibration behaviour of rectangular Mindlin plates resting on non-homogenous elastic foundations. A rectangular plate is assumed to rest on a non-homogenous elastic foundation that consists of multi-segment Winkler-type elastic foundations. Two parallel edges of the plate are assumed to be simply supported and the two remaining edges may have any combinations of free, simply supported or clamped conditions. The plate is first divided into subdomains along the interfaces of the multi-segment foundations. The Levy solution approach associated with the state space technique is employed to derive the analytical solutions for each subdomain. The domain decomposition method is used to cater for the continuity and equilibrium conditions at the interfaces of the subdomains. First-known exact solutions for vibration of rectangular Mindlin plates on a non-homogenous elastic foundation are obtained. The vibration of square Mindlin plates partially resting on an elastic foundation is studied in detail. The influence of the foundation stiffness parameter, the foundation length ratio and the plate thickness ratio on the frequency parameters of square Mindlin plates is discussed. The exact vibration solutions presented in this paper may be used as benchmarks for researchers to check their numerical methods for such a plate vibration problem. The results are also important for engineers to design plates supported by multi-segment elastic foundations.     

剪式机构阵列可展结构的静力学分析方法与应用研究

针对大型可展结构提出了一种新的结构静力学分析方法,克服了常规有限元方法在进行大型可展结构分析时计算量大、计算耗时长等缺点。首先建立了剪式单元机构的刚度矩阵,组装得到整体结构的刚度矩阵和有限元方程;然后通过等价变形,将总刚度矩阵表示成循环矩阵,有限元方程表示成离散卷积的形式;最后采用一种新的方法求解了卷积形式的平衡方程,得到了各铰链点的位移和约束力,提高了计算效率。     

轴—壳体系统耦合振动控制原理分析及试验

螺旋桨脉动力是引起轴系—壳体振动的主要因素。基于频域模型,研究轴—壳系统耦合振动特性及参数影响,从原理上分析在推进轴系上施加控制力来抑制耦合系统振动的控制方法,并讨论基于自适应前馈控制策略的主动控制方案。采用自适应对消及归一化最小均方算法,在轴上施加纵向控制力,实时抵消脉动推力,降低轴系纵向振动,由此抑制由于轴系振动所引起的壳体振动。仿真结果表明：控制力施加位置对控制效果影响很小,不同轴承刚度对控制力大小的需求不同,刚度较小时误差收敛速度较慢,刚度达到一定程度以后收敛速度不变。通过轴—壳体试验系统对原理分析进行验证,结果也表明推力轴承刚度对轴系纵振频率的影响,所提出的控制方法能够有效地抑制壳体振动。     

Improved flexural–torsional stability analysis of thin-walled composite beam and exact stiffness matrix

A simple but efficient method to evaluate the exact element stiffness matrix is newly presented in order to perform the spatially coupled stability analysis of thin-walled composite beams with symmetric and arbitrary laminations subjected to a compressive force. For this, the general bifurcation-type buckling theory of thin-walled composite beam is developed based on the energy functional, which is consistently obtained corresponding to semitangential rotations and semitangential moments. A numerical procedure is proposed by deriving a generalized eigenvalue problem associated with 14 displacement parameters, which produces both complex eigenvalues and multiple zero eigenvalues. Then the exact displacement functions are constructed by combining eigenvectors and polynomial solutions corresponding to non-zero and zero eigenvalues, respectively. Consequently exact element stiffness matrices are evaluated by applying member force–displacement relationships to these displacement functions. As a special case, the analytical solutions for buckling loads of unidirectional and cross-ply laminated composite beams with various boundary conditions are derived. Finally, the finite element procedure based on Hermitian interpolation polynomial is developed. In order to verify the accuracy and validity of this study, the numerical, analytical, and the finite element solutions using the Hermitian beam elements are presented and compared with those from ABAQUS's shell elements. The effects of fiber orientation and the Wagner effect on the coupled buckling loads are also investigated intensively.