Nonlinear static and dynamic buckling analysis of functionally graded shallow spherical shells including temperature effects

This paper presents an analytical approach to investigate the nonlinear static and dynamic unsymmetrical responses of functionally graded shallow spherical shells under external pressure incorporating the effects of temperature. Governing equations for thin FGM spherical shells are derived by using the classical shell theory taking into account von Karman–Donnell geometrical nonlinearity. Approximate solutions are assumed and Galerkin procedure is applied to determine explicit expressions of static critical buckling loads of the shells. For the dynamical response, motion equations are numerically solved by using Runge–Kutta method and the criterion suggested by Budiansky–Roth. A detailed analysis is carried out to show the effects of material and geometrical parameters, boundary conditions and temperature on the stability and dynamical characteristics of FGM shallow spherical shells.     

Nonlinear dynamic and static analysis of laminated anisotropic thick circular plates

Summary A nonlinear shear-deformable theory is presented for dynamic behavior of generally laminated circular plate composed of rectilinearly orthotropic layers. The basic equations derived by use of Hamilton's principle and variational calculus are expressed in terms of the transverse displacement and two in-plane displacements. On the basis of a single-mode analysis a solution is formulated for clamped laminated circular plates with movable and immovable inplane boundary conditions. Two inplane equilibrium equations and all boundary conditions are satisfied exactly. The Galerkin procedure yields a nonlinear ordinary differential equation for the time function which is then solved by using the method of harmonic balance. Numerical results for the static large-deflection behavior and the amplitude-frequency response of laminated angle-ply and cross-ply graphite-epoxy circular plates are presented for various values of the number of layers and the radius-to-thickness ratio.Notation A _ij,B _ij,D _ij Plate stiffnesses defined in Eq. (4) - a radius of circular plate - E _i Young's moduli along thei principal direction - E _ij plate shear rigidities defined in Eq. (4) - f(t) function of time - f _i (x, y) functions of space coordinates defined in Eqs. (24) - G _ij shear moduli - H _mn constants defined in Eqs. (24) - h thickness of plate - I constant defined in Eq. (10) - K _j differential operators defined in Appendix - k _n Fourier coefficients defined in Eq. (27) - L _i differential operators defined in Appendix - M _x,M _y,M _xy bending and twisting moments per unit length - N _x,N _y,N _xy membrane forces per unit length - N _i differential operators defined in Appendix - P _mn,p _i constants defined in Eqs. (24) and (25) - Q _x,Q _y transverse shear forces per unit length - q, q ₀ nonuniformly and uniformly distributed loads per unit area - R _i tracing constant for rotatory inertia - R _mn,s _n,T _mn constants defined in Eqs. (24) and (21) - S _i differential operators defined in Appendix - S _ij ^(k) elastic stiffness of thek-th layer - T _s tracing constant for transverse shear - u, v inplane displacements - u ⁰,v ⁰ midsurface in-plane displacements - w ₀ maximum transverse displacement at centre of plate - w transverse displacement - , slope functions - coefficient defined in Eq. (10) - _ij ⁰ middle surface strain components - _ij total strain components - _ij Poisson's ratio - 0^(k) mass density ofk-th layer - fundamental frequency - ₀ fundamental linear frequency     

A wavelet collocation method for the static analysis of sandwich plates using a layerwise theory

A study of bending deformations of sandwich plates using a layerwise theory of laminated or sandwich plates is presented. The analysis is based on a wavelet collocation technique to produce highly accurate results. Numerical results for symmetric laminated composite and sandwich plates are presented and discussed.     

正交异性扁薄球壳的非线性轴对称振动

给出了一种研究圆柱正交异性扁薄球壳非线性轴对称自由振动的新的时间模态方法。首先,使用变分法导出了决定振动频率的一对代数-微分方程。然后,利用作者提出的改进的修正迭代法求得了渐进解。这使此种壳体的非线性振动的进展获得扩充研究。     

低速冲击下含损伤层合中厚浅球壳的非线性动力响应分析

基于中厚壳几何非线性理论和损伤理论 , 采用应变描述的失效准则 , 导出了层合浅球壳含基体损伤和纤维2基体剪切损伤的损伤本构关系 , 建立了低速冲击下轴对称正交对称铺设层合中厚浅球壳的非线性运动控制方程。对未知函数在空间域采用正交配置法离散 , 时间域采用 Newmark2 β方法离散 , 对整个问题进行迭代求解。数值结果表明 , 损伤、 冲击物的初速度以及结构的几何参数都在不同程度上影响着低速冲击下结构所受的冲击载荷和结构的非线性动力响应。      

波纹扁球壳的非线性动态屈曲

研究了用于传感器弹性元件波纹扁球壳的非线性动态屈曲问题。建立了波纹扁球壳的非线性振动微分方程,根据突变理论建立了该壳体动态屈曲的突变模型,得到了动态屈曲的临界方程。     

An indirect BIM for static analysis of spherical shells using auxiliary boundaries

Boundary integral approaches, which are known for their mathematical sophistication and elegance as well as their ability to reduce the problem dimensions by one, suffer from drawbacks associated with their performance in the vicinity of the boundaries. Such behaviour is the result of the unavoidable, in most cases, discretization of the boundary on one hand, which consequently results in the reduction of the integral problem to an algebraic one, and of the tedious evaluation of singularities that are present in most kernels on the other. The sensitivity of solutions of shell problems using a special form of boundary integral method is studied. Such an approach hopes to achieve a better representation of the solution near the boundaries by utilizing fictitious surface lines to perform the kernel integrations. Lastly, the performance of the integral formulation is examined through some representative examples.     

Nonlinear viscoelastic analysis of composite plates and shells

A finite element analysis is developed for treating nonlinear viscoelastic response of laminated composites. The analysis uses an eight-node layered shell element. The transient creep compliance in the viscoelastic model is represented as an exponential series plus a steady-flow term. This allows for a simplification of the numerical procedure for handling hereditary effects. Calculations are performed to study the time-dependent redistribution of stress in a flat plate under uniform pressure, a spherical cap under a point load, and a cylindrical shell pinched between two concentrated forces.     

考虑损伤双层扁球面网壳的非线性动力学特性

基于Lematire等效应变损伤理论,计及扁球面网壳各个杆件的损伤影响,应用拟壳法导出了具有损伤的扁球面网壳的动力学非线性控制方程。提出了以中心最大振幅为摄动参数的摄动-变分法的求解方法,对动力非线性控制方程进行了求解,得出了相应的物理量的解析式。据此进行数值分析,得出了相应的特征关系。并用Galerkin方法导出了一个含二次和三次非线性振动微分方程并求解了具有损伤扁球面网壳的的非线性动力学的自由振动方程,给出了准确解。而后利用Melnikov函数法,从理论上给出了考虑损伤的系统发生混沌运动的临界条件,并通过计算机数字仿真证实了考虑损伤的扁球面网壳在非线性强迫振动时存在混沌运动,同时发现损伤使得系统更易发生混沌运动。     

Nonlinear static/dynamic analysis for elasto-plastic laminated plates with interfacial damage evolution

A new analysis model, which includes the effects of interfacial damage, geometrical nonlinearity and material nonlinearity, is presented for elasto-plastic laminated plates. Based on the model, the nonlinear equilibrium differential equations for elasto-plastic laminated plates with interfacial damage are established. The finite difference method and iteration method are adopted to solve these equations. The nonlinear static and dynamic behaviors for the elasto-plastic laminated plates under the action of transverse loads are analyzed. Effects of interfacial damage on the stress and displacement distribution and nonlinear dynamic response are discussed in the numerical examples together with the comparison of nonlinear mechanical behaviors between the elastic and elasto-plastic laminated plates. Numerical results show that both the interfacial damage and plastic deformation put obvious influence on the mechanical properties of structures.     

Thermomechanical postbuckling analysis of functionally graded plates and shallow cylindrical shells

Summary. In this paper, an analytic solution is provided for the postbuckling behavior of plates and shallow cylindrical shells made of functionally graded materials under edge compressive loads and a temperature field. The material properties of the functionally graded shells are assumed to vary continuously through the thickness of the shell according to a power law distribution of the volume fraction of the constituents. The fundamental equations for thin rectangular shallow shells of FGM are obtained using the von Karman theory for large transverse deflection, and the solution is obtained in terms of mixed Fourier series. The effect of material properties, boundary conditions and thermomechanical loading on the buckling behavior and stress field are determined and discussed. The results reveal that thermomechanical coupling effects and the boundary conditions play a major role in dictating the response of the functionally graded plates and shells under the action of edge compressive loads.     

An alternative collocation boundary element method for static and dynamic problems

A collocation boundary element formulation is presented which is based on a mixed approximation formulation similar to the Galerkin boundary element method presented by Steinbach (SIAM J Numer Anal 38:401–413, 2000) for the solution of Laplace’s equation. The method is also applicable to vector problems such as elasticity. Moreover, dynamic problems of acoustics and elastodynamics are included. The resulting system matrices have an ordered structure and small condition numbers in comparison to the standard collocation approach. Moreover, the employment of Robin boundary conditions is easily included in this formulation. Details on the numerical integration of the occurring regular and singular integrals and on the solution of the arising systems of equations are given. Numerical experiments have been carried out for different reference problems. In these experiments, the presented approach is compared to the common nodal collocation method with respect to accuracy, condition numbers, and stability in the dynamic case.     

Axisymmetric nonlinear analysis of shallow spherical shells by a combined boundary element and finite difference method

This paper is concerned with the development of the mixed boundary element method and finite element method for the analysis of spherical annular shells under axisymmetric loads. The boundary element techniques are used to solve the equilibrium equation of shells and the central difference operator is adopted to deal with the compatibility equations. Iterative techniques are used throughout the analysis procedure. A number of numerical examples are given in the paper to illustrate the validity of the present approach.     

A refined dynamic stiffness element for free vibration analysis of cross-ply laminated composite cylindrical and spherical shallow shells

An exact free vibration analysis of doubly-curved laminated composite shallow shells has been carried out by combining the dynamic stiffness method (DSM) and a higher order shear deformation theory (HSDT). In essence, the HSDT has been exploited to develop first the dynamic stiffness (DS) element matrix and then the global DS matrix of composite cylindrical and spherical shallow shell structures by assembling the individual DS elements. As an essential prerequisite, Hamilton’s principle is used to derive the governing differential equations and the related natural boundary conditions. The equations are solved symbolically in an exact sense and the DS matrix is formulated by imposing the natural boundary conditions in algebraic form. The Wittrick–Williams algorithm is used as a solution technique to compute the eigenvalues of the overall DS matrix. The effect of several parameters such as boundary conditions, orthotropic ratio, length-to-thickness ratio, radius-to-length ratio and stacking sequence on the natural frequencies and mode shapes is investigated in details. Results are compared with those available in the literature. Finally some concluding remarks are drawn.     

Post-buckling analysis of shallow shells by the field-boundary-element method

The non-linear field-boundary-element technique is applied to the analysis of snap-through phenomena in thin shallow shells. The equilibrium path is traced by using the arc-length method and the solution strategy is discussed in detail. The results show that, as compared to the approaches based on the popular symmetric-variational Galerkin finite element formulation, the current approach based on an unsymmetric variational Petrov–Galerkin field-boundary-element formulation gives a faster convergence while using fewer degrees of freedom. The illustrative numerical examples deal with post-buckling responses of several shallow shells with different geometries.     

Nonlinear theory of hard-skin plates and shells

In paper [3] it was developed a linear theory of hard-skin plates, i.e. sandwich plates the outer layers of which are much harder than the core layer. It was shown that the leading approximation involves the transverse shear, and, a closed system of governing equations was derived. In this paper the results of [3] are extended to nonlinear theory of hard-skin plates and shells.     

Bending instability and vibration of plates and shallow shells

In the equations of the general theory of shells, the nonlinear terms accounting for membrane strains are accompanied by large coefficients (describing the extensional stiffness of the shell). Although formally infinitesimals of the second order, in the case of small but finite displacements these terms can be commensurate with, or even exceed in magnitude, the first-order terms. Disregarding such nonlinear terms can be debilitating to the linearized theory. For example, it will not reflect the fact that a plate or shell in bending prefers a deflection mode not accompanied by midsurface stretching (so-called isometric bending) and can switch to such a mode more or less abruptly, exhibiting the phenomenon of bending instability

In this study, bending instability and vibration of plates and shallow shells are investigated by simple analytical and numerical means, having in mind certain types of layered, sandwich, and truss-like structures.