Buckling loads of variable thickness thin isotropic plates

This work presents accurate solutions for bifurcation buckling loads of rectangular thin plates with thickness that varies in the directions parallel to the two sides. The plates are subjected to biaxial compression and various combinations of boundary conditions are considered. The calculation of the critical loads was carried out by using the extended Kantorovich method. For the resulting ordinary differential equation, an exact method for the stability analysis of compressed members with variable flexural rigidity is used. The buckling load is found as the inplane load that makes the determinant of the stiffness matrix equal zero. New, exact results are given for many cases of uni-directional and bi-directional variation in thickness. The results are very accurate and exact for cases where an exact solution is available.     

Buckling of plates with strengthenings

This paper studies the buckling behaviour of simply supported square plates, which have weakening or strengthening bands. The weakening/strengthening bands are equally spaced and their thickness is either decreased or increased. The analysis assumes that the stress state in the plate before and during the buckling process remains in the elastic range. Two cases of plate loading are studied, one with compressive forces and one with tangential forces. The buckling coefficients are calculated for different numbers and thicknesses of strengthening/weakening bands. The thickness of strengthening/weakening bands and the thickness of the remaining plate is varied so that the volume/weight of the plate remains constant. In one case it is found that the buckling load at constant weight of the plate can be increased by 118% if an optimal ratio of the thickness of strengthening bands and the thickness of the remaining plate is used.     

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.     

Effect of transverse shear strains on plates resting on elastic foundation using modified Vlasov model

A four-noded quadrilateral (PBQ4) and an eight-noded quadrilateral (PBQ8) plate bending element based on Mindlin plate theory are adopted for the analysis of thin and thick plates resting on elastic foundation using modified Vlasov model. The terms of vertical deflection stiffness matrix and shear deformation stiffness matrix of the subsoil are evaluated using finite element method, and presented in explicit forms. Selective reduced integration technique is used in addition to full integration technique for both the types of the elements to avoid shear-locking problem that occurs under the thin plate limit. It has been demonstrated that the performance of the eight-noded quadrilateral element is excellent for thin and thick plates on elastic foundation when selective-reduced integration technique is used. General conclusion can be drawn from the results that the effect of the shear strain on the behavior of the plate resting on subsoil is always quite small for free plates compare to the supported ones.     

Strengthening of perforated plates under uniaxial compression: Buckling analysis

This paper focuses on the cutout-strengthening of perforated steel plates subjected to uniaxial compressive loads. The square plates considered each has a centrally placed circular hole and four simply supported edges in the out-of-plane direction. Four types of stiffeners named ringed stiffener (RS), flat stiffener (FS), longitudinal stiffener (LS) and transverse stiffener (TS) are mainly discussed. The finite element method (FEM) has been employed to analyse the elastic and elasto-plastic buckling behaviors of strengthened and unstrengthened perforated plates. The results show that the strengthened perforated plates have higher buckling strengths than those of the unstrengthened ones, while the elevations in elastic buckling stress and elasto-plastic ultimate strength are closely related to stiffener types (i.e., RS, FS, LS and TS) as well as plate geometric parameters (i.e., a plate slenderness ratio and a hole diameter ratio). Furthermore, comparisons of strengthening efficiency considering the variations of buckling stress with stiffener weight are carried out, and recommendations on the most efficient cutout-strengthening methods for the uniaxially compressed perforated square plates with centric circular holes are proposed.     

双参数弹性地基上的FGM壳体的热屈曲性能

分析双参数弹性地基上的FGM壳体的热屈曲性能。根据幂律分布,从厚度方向对各个构件的材料性能进行分级。周边的弹性介质被模拟为Pasternak弹性地基。在假定基本关系之后,对温度荷载下及位于双参数弹性地基上的FGM截顶圆锥壳体的稳定性和相容方程式进行分析。通过求解特征值,得到沿壳体厚度方向非线性分布的温度荷载下的基于或不基于弹性地基上的FGM截顶圆锥壳体临界温差。作为一个特殊案例,提出基于或不基于弹性地基上的FGM截顶圆锥壳体的方程。为了保证目前研究的正确性,详细地评价了临界温度的收敛性。     

Nonlinear vibration of laminated plates on an elastic foundation

This paper studies the effects of initial stresses on the nonlinear vibrations of laminated plates on elastic foundations. The nonlinear partial differential equations based on Mindlin plate theory are derived for the nonlinear vibration of laminated plates in a general state of nonuniform initial stress. Both rotary inertia and transverse stress are considered. Using the derived governing equations, the nonlinear vibration behavior of an initially stressed cross-ply plate on a Pasternak or Winkler elastic foundation is studied. The Galerkin's approximate method is applied to the governing partial differential equations to yield ordinary differential equations. The ordinary differential equations are solved by Runge–Kutta method to obtain the ratio of nonlinear frequency to linear frequency. The initial stress is taken to be a combination of a pure bending stress and an extensional stress in the plane of the plate. It is found that the foundation stiffness and initial stresses may result in a drastic change of the nonlinear vibration behavior. The frequency responses of nonlinear vibration are sensitive to the initial stress, vibration amplitude, modulus ratio and foundation stiffness. The effects of various parameters on the nonlinear vibration are discussed.     

Thermal buckling of FGM shells resting on a two-parameter elastic foundation

This paper focuses on the thermal buckling analysis of FGM shells resting on the two-parameter elastic foundation. Material properties of the constituents are graded in the thickness direction according to the power-law distribution. The surrounding elastic medium is modeled as an elastic foundation of the Pasternak-type. After giving the fundamental relations, the stability and compatibility equations of an FGM truncated conical shell subjected to thermal load and resting on a two-parameter elastic foundation have been derived. Critical temperature differences of FGM truncated conical shells with or without elastic foundations subjected to non-linearly distributed temperature across the thickness of the shells are obtained by solving eigenvalue problems. The appropriate formulas for FGM cylindrical shells with or without elastic foundations are found as a special case. In order to assure the accuracy of the present study, convergence properties of the critical temperature are examined in detail.     

Two-dimensional analysis of stacked geosynthetic tubes on deformable foundations

Stacked geosynthetic tubes resting on a deformable foundation such as soil are analyzed. The tubes contain a slurry which applies hydrostatic pressure. The material of the tubes is assumed to act like an inextensible membrane and to have negligible weight, and the foundation is assumed to exert a normal upward pressure which is proportional to the downward deflection. Friction is neglected between tubes and at the foundation interface. Two configurations are considered: (a) one tube on top of another and (b) one tube straddling two tubes underneath it. For the latter formation, the case of external fluid acting on one side is analyzed, to simulate an application as a dike, and rigid blocks are utilized to prevent sliding of the tubes. Equilibrium shapes of the tubes are obtained numerically from a closed-form integral formulation, and the tension in each tube is computed.     

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。     

Free vibrations of fluid-filled cylindrical shells on elastic foundations

The free vibration characteristics of fluid-filled cylindrical shells on elastic foundations are presented by a semi-analytical finite element method. A shell is discretized into cylindrical finite elements where shell governing equations based shape functions in the longitudinal direction are used instead of the usual simple polynomials. Non-uniformities of the foundations in the circumferential and longitudinal directions are handled by the Fourier series and an element mesh strategy, respectively. The fluid domain is described by the potential flow theory. The hydrodynamic pressure acting on shells is derived from the condition for dynamic coupling of the fluid-structure. The effect of fluid in a shell, shell geometries, and foundation parameters on the dynamic behavior of fluid-containing shells is investigated. Numerical results based on the present method converge more rapidly than those obtained by the simple polynomial formulation. The method is suitable for the problem considered due to its generality, simplicity, and potential for further development.     

Eccentric low-velocity impact analysis of transversely graded plates with Winkler-type elastic foundations and fully or partially supported edges

Impact analysis of plates with either elastic foundations or partially supported edges has not been accomplished, even for the homogeneous isotropic plates. In the present paper, eccentric low-velocity impact analysis of a functionally graded plate with fully/partially supported edges and a Winkler-type elastic foundation is performed. In contrast to the available researches, stiffness of the underneath layers is also taken into account in determination of the apparent stiffness of the contact region, employing Mori–Tanaka micromechanics-based model instead of the traditional rule of mixtures. The governing equations are derived based on the principle of minimum total potential energy and the first-order shear-deformation plate theory. Some of the included novelties are: modification of the contact law, incorporation of effects of the elastic foundation, considering the material heterogeneity and partially supporting the edges, and proposing various semi-analytical and (Galerkin-type) numerical solutions. Numerical time integration schemes are used to trace time variations of the responses. A sensitivity analysis is performed to investigate influences of the specifications of the plate and the indenter, the eccentricity, and the supporting length ratio on time histories of the indentation and contact force.     

Nonlinear analysis of imperfect laminated thin plates under transverse and in-plane loads and resting on an elastic foundation by a semi-analytical approach

The large deflection and postbuckling behavior of imperfect composite laminated plates exposed to a combined action of transverse loads and in-plane edge compressions and resting on an elastic foundation are investigated in this paper by a semi-analytical approach. The formulations are based on the classical laminated plate theory (CLPT), and include the plate–foundation interaction effects via a two-parameter model (Pasternak-type) from which Winkler elastic foundation can be recovered as a limiting case. The present approach employs a perturbation technique, one-dimensional differential quadrature approximation and the Galerkin procedure to model the nonlinear performance of the plate with arbitrary combination of simply supported, clamped or elastic rotational edge constraints. Studies concerning its accuracy and convergence characteristics are carried out through some numerical illustrations. Effects of foundation stiffness, plate aspect ratio, total number of plies, fiber orientation, initial geometrical imperfection, the character of boundary conditions, and load patterns on the nonlinear behavior of the plate are studied. Typical numerical results are given in dimensionless graphical forms.     

The effect of elastic foundations on the nonlinear buckling behavior of axially compressed heterogeneous orthotropic truncated conical shells

The buckling problem of a heterogeneous orthotropic truncated conical shell subjected to an axial load and surrounded by elastic media is analyzed based on the finite deformation theory. Using von-Karman nonlinearity, the governing equations of elastic buckling of heterogeneous orthotropic truncated conical shells surrounded by elastic media are derived. The governing equations are solved using superposition and Galerkin methods and obtained expressions for upper and lower critical axial loads. The influences of elastic foundations, heterogeneity, orthotropy and geometric characteristics on the upper and lower critical loads of conical shells with and without elastic foundations are studied in detail.     

基于Winkler弹性地基模型格构梁节点荷载分配研究

根据格构梁的受力特性,利用Winkler弹性地基模型推导出有限长梁受集中力的内力解答,根据此解答再计算出节点荷载分配系数kT和kL,进而得到格构梁的内力。结合实际工程,对理论解与ABAQUS有限元软件模拟所得数值解进行比较,验证了理论解答的正确性,并分析了其不足之处。     

岩溶地区刚性桩复合地基宏观本构模型研究

针对岩溶地区刚性桩复合地基承载力与沉降关系的复杂性,对4类典型地质条件下的44个复合地基现场压板试验数据样本进行数理统计分析,在此基础上提出基于Winkler地基模型的修正二折线和四折线弹簧本构模型。为了得到偏于安全的用于工程设计的复合地基弹簧本构模型,采用共同作用分析方法对一栋典型的高层建筑结构进行多折线参数敏感性分析。分析结果表明:二折线弹簧本构模型比四折线弹簧本构模型得到的筏板内力大,在工程设计中,可采用二折线本构模型;对于二折线弹簧本构模型,承载力特征值对应的沉降越小,筏板的内力越大,综合考虑筏板设计的安全性和经济性,在实际工程中,建议承载力特征值点对应的沉降取10 mm,承载力极限点对应的沉降取40 mm。     

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.     

Applicability of simple intuitive formulation for predicting thermal post-buckling behaviour of columns resting on hardening nonlinear elastic foundation

The paper demonstrates the applicability of a simple intuitive formulation, the development of which started around a decade ago in obtaining the thermal post-buckling behaviour of uniformly heated slender columns that rest on hardening nonlinear Winkler foundation. This foundation, which is firmly bonded to the column, is idealised by a number of closely spaced hardening nonlinear elastic springs. The nonlinear foundation stiffness of the nonlinear elastic foundation is dependent on the square of the deflection of the columns. Analytical expressions for the thermal post-buckling loads, for the buckled mode shapes with wave number one, by applying the concept of smart structures, of the commonly used columns on hardening nonlinear foundation are derived, by using a simple, intuitive formulation. The numerical results presented are validated, by comparing from those obtained for the limiting cases and non-violation of the physical and logical behaviour, in the absence of any similar results available in literature. These results reveal, the relative contributions of the various parameters existing in the hardening nonlinear Winkler foundation, on the thermal post-buckling behaviour of hinged–hinged and clamped–clamped columns.     

BEM analysis of elastic foundation beams on multilayered isotropic soils

This paper presents a boundary element method to analyze the elastic foundation finite beams on 2D plane-strain and 3D multilayered isotropic soils. Starting with the basic equations of elasticity and based on the Fourier transform, the transfer matrix of a single soil layer is derived. According to the boundary conditions and continuity conditions between two adjacent soil layers, the solution of multilayered elastic soils is obtained to be a kernel function of BEM analysis. The elastic foundation beam is modeled as a Bernoulli–Euler beam using the finite difference method. With the displacement and stress condition of coordination between beam and soil, the solution is acquired for beams resting on multilayered soils. Comparing the solution with the published data shows that the solution is in good agreement, and some numerical examples show that the beam behavior is affected vitally by soil–beam stiffness ratio and the stratification of soils.     

分层地基模型|桩筏基础|沉降|非线性|离心模型试验

基于改进的分层地基模型,考虑桩–土–筏的共同作用,提出了刚性桩筏基础非线性分析方法。地基附加应力计算考虑土层参数的变化,采用层状弹性体系的Burmister应力解;地基变形采用分层总和法,所采用的计算参数压缩模量和一维回弹模量试验确定方便。通过简单的室内一维加卸载试验确定其随加卸载次数的变化,进一步得到桩筏基础在重复加卸载下的沉降规律。对单桩和桩筏基础进行离心模型试验,并将计算结果和试验结果进行对比分析,验证了本文方法的正确性,可应用于实际工程问题的分析。