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.     

Simple solution for buckling of orthotropic circular cylindrical shells

Most papers dealing with the analysis of the buckling behaviour of orthotropic circular cylindrical shells provide solutions which are complex in nature and difficult to use. In this paper the theory has been developed to the point that relatively simple solutions of a general nature have been formulated. Based on Flugge's linear theory for isotropic cylindrical shells, a general buckling solution under combined axial compression and external pressure was derived. For moderate-length orthotropic cylindrical shells loaded by either external pressure or axial compression, buckling loads are formulated in a simple form.     

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。     

Stability of multi-storey unbraced steel frames subjected to variable loading

Proposed in this paper is an approach of evaluating the elastic buckling loads for multi-storey unbraced steel frames subjected to variable loading or non-proportional loading. In the case of variable loading, the conventional assumption of proportional loading is abandoned, and different load patterns may cause the frame to buckle at different levels of critical loads. In light of the use of the storey-based buckling concept to characterize the lateral sway buckling of unbraced framed structures, the problems of determining the lower and upper bounds among all of the frame buckling loads associated with different load patterns are presented as a pair of minimization and maximization problems subjected to elastic stability constraints. The problems take into account the semi-rigid behaviour of beam-to-column connections and the lateral stiffness reduction of columns due to the presence of an axial compressive load. The minimization and maximization problems are then solved by a linear programming method; thus, the lower and upper bounds of the frame buckling loads subjected to variable loading are obtained. Parametrical studies on the influence of the connection rigidity to the lower and upper bounds of critical loads and the comparisons to the conventional proportional loading are also presented in this paper.     

Thermal and mechanical instability of functionally graded truncated conical shells

Thermal and mechanical instability of truncated conical shells made of functionally graded material (FGM) is studied in this paper. It is assumed that the shell is a mixture of metal and ceramic that its properties changes as a function of the shell thickness. The governing equations are based on the first-order shell theory and the Sanders nonlinear kinematics equations. The results are obtained for a number of thermal and mechanical loads and are validated with the known data in the literature.     

Buckling and postbuckling behaviors of imperfect cylindrical shells subjected to torsion

Buckling and postbuckling behaviors of imperfect cylindrical shell subjected to torsion are investigated. The governing equations are based on the Karman–Donnell-type nonlinear differential equations. A boundary layer theory of shell buckling is applied to obtain the analytic solutions that meet the boundary conditions strictly. A singular perturbation technique is employed to determine the buckling loads and postbuckling equilibrium paths. Numerical results reveal that the current theory gives quite good estimates of the postbuckling paths of cylindrical shells. The effects of the geometric parameters on the buckling and postbuckling behaviors of the cylindrical shells are analyzed. It is confirmed that the postbuckling equilibrium paths of cylindrical shells subjected to torsion are unstable and the relatively shorter shells have higher postbuckling equilibrium paths. Finally, the effects of the initial imperfections on the buckling and postbuckling behaviors of the cylindrical shells are clarified. The illustrated results of the imperfect shells with different initial transverse deflections show that extremely small imperfections do indeed reduce the buckling loads and make the postbuckling equilibrium paths be lower. The buckling and postbuckling of cylindrical shells under torsion exhibit obvious imperfect sensitivity. Furthermore, the effects become greater following with the larger imperfections.     

轴向冲击下柱形壳局部和整体动力屈曲

对轴向受压冲击荷载下,柱形壳局部和整体屈曲进行研究。引进哈密尔顿系统,通过哈密尔顿二元方程,采用4对对偶变量,对系统中的基础问题进行数学描述,并归纳为临界荷载的特征值和特征解问题。通过各自的阶数对屈曲模式进行描述,将其分为两类:短波或局部屈曲以及长波或整体屈曲,得到解析与数值的解决方案并给出了一些规则。     

Buckling of thin-walled conical shells under uniform external pressure

Shells are for the most part the deep-seated structures in manufacturing submarines, missiles, tanks and their roofs, and fluid reservoirs; therefore it is a matter of concern to bring about some basic regulations associated with the existing codes. Above all, truncated conical shells (frusta) and shallow conical caps (SCC) subjected to external uniform pressure when discharging liquids or wind loads are discussed closely in this paper concerning and thrashing out their empirical nonlinear responses along with envisaging numerical methods in contrast. The buckling aptitude of shells is contingent upon two leading geometric ratios of “slant-length to radius” (L/R) and “radius to thickness” (R/t). In this paper, developing six frusta and four shallow cap specimens and their relevant FE models, use is made of laboratory modus operandi to enumerate buckling elastic and plastic responses and asymmetric imperfection sensitivity, whose adequacy has been reckoned through comparisons with arithmetical and numerical data correspondingly. These obtained upshots were aimed at validating and generalizing the data for unstiffened truncated cones and SCC in full scale.     

Dynamic local and global buckling of cylindrical shells under axial impact

In this paper, the local and global buckling of cylindrical shells under axial, compressive impact loads is studied. A Hamiltonian system is introduced in the problem. The fundamental problem in the system can be described mathematically by the Hamiltonian dual equations, which are expressed in four pairs of dual variables. The problem is reduced to a problem of eigenvalues and eigensolutions for critical loads and buckling modes, respectively. The buckling modes can be described by their respective orders and they are grouped into two classes, the short-wave or local buckling and the long-wave or global buckling. The solutions are obtained analytically and numerically, and some rules observed are indicated.     

Buckling of cracked cylindrical thin shells under combined internal pressure and axial compression

Linear eigenvalue analysis of cracked cylindrical shells under combined internal pressure and axial compression is carried out to study the effect of crack type, size and orientation on the buckling behavior of cylindrical thin shells. Two types of crack are considered; through crack and thumbnail crack. Our calculations indicate that depending on the crack type, length, orientation and the internal pressure, local buckling may precede the global buckling of the cylindrical shell. The internal pressure, in general, increases the buckling load associated with the global buckling mode of the cylindrical shells. In contrast, the effect of internal pressure on buckling loads associated with the local buckling modes of the cylindrical shell depends mainly on the crack orientation. For cylindrical shells with relatively long axial crack, buckling loads associated with local buckling modes of the cylindrical shell reduce drastically on increasing the shell internal pressure. In contrast, the internal pressure has the stabilizing effect against the local buckling for circumferentially cracked cylindrical shells. A critical crack length for each crack orientation and loading condition is defined as the shortest crack causing the local buckling to precede the global buckling of the cylindrical shell. Some insight into the effect of internal pressure on this critical crack length is provided.     

四块组合型双曲抛物面扁扭壳的稳定分析

本文用脉冲函数处理四边简支四块组合型双曲抛物面扁扭壳脊线处的曲率不连续性,根据线性稳定理论导出了与单块四边简支双曲抛物面扁扭壳具有相同形式的临界荷载计算公式,并给出了有关参数的图表。研究表明,这种壳体可能有两种失稳模态,即关于两个对称轴对称或反对称失稳。反对称失稳时,其临界荷载与单壳相同。     

Large deflection orthotropic plate approach to develop ultimate strength formulations for stiffened panels under combined biaxial compression/tension and lateral pressure

This paper uses the large deflection orthotropic plate approach to develop the ultimate strength formulations for steel stiffened panels under combined biaxial compression/tension and lateral pressure loads, considering the overall (grillage) buckling collapse mode. The object panel has a number of one-sided small stiffeners in either one or both orthogonal directions. The stiffened panel is then modeled as an equivalent orthotropic plate, for which the various elastic constants characterizing structural orthotropy are determined in a consistent systematic manner using classical theory of elasticity. The panel edges are considered to be simply supported. The influence of initial deflections is taken into account. The membrane stress distribution inside the panel under combined uniaxial loading (in either longitudinal or transverse direction) and lateral pressure is analyzed by solving the nonlinear governing differential equations of large deflection orthotropic plate theory. It is presumed that the panel collapses when the most highly stressed boundary location yields, resulting in closed-form expressions for the ultimate strength of the stiffened panel. Based on the insights previously developed through numerical studies, the panel ultimate strength interaction formulation between biaxial loads, with lateral pressure regarded as a secondary load component is then proposed as a relevant combination of the two sets of panel ultimate strength formulations, i.e. one for combined longitudinal axial load and lateral pressure and the other for combined transverse axial load and lateral pressure. The validity of the proposed ultimate strength formulations is verified by a comparison with nonlinear finite element and other numerical solutions.     

复合圆柱形壳在径向冲击荷载和轴压荷载作用下的瞬时动力响应

基于首次逼近理论,利用第一阶剪切变形理论对圆柱形壳的自由和强迫振动进行分析。边界条件（BCs）考虑为悬臂状态。分析复合壳体在横向冲击和轴向压力作用下的动力响应（轴压荷载小于临界屈曲荷载）,同时对复合柱形壳进行了建模分析。利用卷积积分对给定荷载状况下的壳体进行分析。揭示纤维方向、轴向荷载及一些几何参数对壳体时间响应的影响。结果表明：动力响应主要由结构的自振周期所控制。     

Dynamic analysis of functionally graded truncated conical shells subjected to asymmetric moving loads

The dynamic behavior of functionally graded (FG) truncated conical shells subjected to asymmetric internal ring-shaped moving loads is studied. The material properties are assumed to have continuous variations in the shell thickness direction. The equations of motion are derived based on the first-order shear deformation theory (FSDT) using Hamilton׳s principle. The finite element method (FEM) together with Newmark׳s time integration scheme is employed to discretize the equations of motion in the spatial and temporal domain, respectively. The formulation and method of solution are validated by studying their convergence behavior and carrying out the comparison studies in the limit cases with existing solutions in the literature. Then, the influences of material graded index, radius-to-length ratio, semi-vertex angle, thickness, boundary conditions and moving load velocity on the dynamic behavior of the FG truncated conical shells are studied. In addition, the difference between the responses of the FG shells under symmetric and asymmetric loadings is compared.     

Elephant's foot buckling in pressurised cylindrical shells

Metal cylindrical bins, silos and tanks are thin shell structures subject to internal pressure from stored materials together with axial compression from the frictional drag of stored materials on the walls and horizontal loads. The governing failure mode is frequently buckling under axial compression. The internal pressure exerted by the stored fluids or solids can significantly enhance the buckling strength, but high internal pressures lead to severe local bending near the base. Local yielding then precipitates an early elastic‐plastic buckling failure. This failure mode, commonly known as “elephant's foot buckling”, has received relatively little attention to date and until recently was often ignored in tank and silo design. This problem is an unusual buckling condition, because it involves very high tensile stresses in one direction, coupled with rather small compressive stresses in the orthogonal direction. Thus, although it is a buckling failure involving considerable plasticity, it occurs at low buckling stresses and under conditions that appear to be classically “slender”. The normal concatenation of “slender” with “elastic” in buckling formulations does not apply at all here. This paper describes alternative approaches to the formulation of design rules for the elastic‐plastic instability and collapse of axially‐loaded internally‐pressurised thin cylindrical shells adjacent to the base support. The differences between the different approaches arise from different conceptual models for the manner in which an elastic‐plastic slender structure instability should be treated.     

厂房中吊车柱的稳定设计

讨论厂房中变截面柱设计中的屈曲问题。这些柱子的顶端和变截面处通常承受轴向偏心荷载,因此,柱子会产生弯矩。横向荷载产生的附加屈曲也必须考虑。基于这些因素,对几何非线性模型进行屈曲性能分析,并给出平衡微分方程。成功应用了两个柱段的等效(或有效)长度概念。考虑以下参数范围给出了有效长度系数:端部固定,端部与中间轴向荷载比,上、下段长度与惯性矩的比值以及连接处的力学性能。对临界荷载的缺陷敏感性进行研究。讨论不同偏心距和一次性施加横向荷载时柱的承载力变化。结果表明:上、下段可分别按照梁、柱单独处理。以交互方程的形式给出一些设计建议。     

Buckling of elastic cylindrical shells considering the effect of localized axisymmetric imperfections

The effect of localized axisymmetric initial imperfections on the critical load of elastic cylindrical shells subjected to axial compression is studied through analytical modeling. Some classical results regarding sensitivity of shell buckling strength with respect to distributed defects having axisymmetric or asymmetric forms are recalled. Special emphasis is placed after that on the more severe case of localized defects satisfying axial symmetry by displaying an analytical solution to the Von Kármán–Donnell shell equations under specific boundary conditions. The obtained results show that the critical load varies very much with the geometrical parameters of the localized defect. These variations are not monotonic in general. They indicate, however, a clear reduction of the shell critical load for some defects recognized as the most hazardous isolated ones. Reduction of the critical load is found to reach a level which is up to two times lower than that predicted by general distributed defects.     

Effects of imperfections of the buckling response of composite shells

The results of an experimental and analytical study of the effects of initial imperfections on the buckling response and failure of unstiffened thin-walled compression-loaded graphite-epoxy cylindrical shells are presented. The shells considered in the study have six different shell-wall laminates two different shell-radius-to-thickness ratios. The shell-wall laminates include four different orthotropic laminates and two different quasi-isotropic laminates. The shell-radius-to-thickness ratios includes shell-radius-to-thickness ratios equal to 100 and 200. The numerical results include the effects of traditional and nontraditional initial imperfections and selected shell parameter uncertainties. The traditional imperfections include the geometric shell-wall mid-surface imperfections that are commonly discussed in the literature on thin shell buckling. The nontraditional imperfections include shell-wall thickness variations, local shell-wall ply-gaps associated with the fabrication process, shell-end geometric imperfections, nonuniform applied end loads, and variations in the boundary conditions including the effects of elastic boundary conditions. The cylinder parameter uncertainties considered include uncertainties in geometric imperfection measurements, lamina fiber volume fraction, fiber and matrix properties, boundary conditions, and applied end load distribution. Results that include the effects of these traditional and nontraditional imperfections and uncertainties on the nonlinear response characteristics, buckling loads and failure of the shells are presented. The analysis procedure includes a nonlinear static analysis that predicts the stable response characteristics of the shells, and a nonlinear transient analysis that predicts the unstable response characteristics. In addition, a common failure analysis is used to predict material failures in the shells.     

Nonlinear mechanical,thermal and thermo-mechanical postbuckling of imperfect eccentrically stiffened thin FGM cylindrical panels on elastic foundations

This paper presents an analytical approach to investigate the nonlinear stability analysis of eccentrically stiffened thin FGM cylindrical panels on elastic foundations subjected to mechanical loads, thermal loads and the combination of these loads. The material properties are assumed to be temperature-dependent and graded in the thickness direction according to a simple power law distribution. Governing equations are derived basing on the classical shell theory incorporating von Karman–Donnell type nonlinearity, initial geometrical imperfection, the Lekhnitsky smeared stiffeners technique and Pasternak type elastic foundations. Explicit relations of load–deflection curves for FGM cylindrical panels are determined by applying stress function and Galerkin method. The effects of material and geometrical properties, imperfection, elastic foundations and stiffeners on the buckling and postbuckling of the FGM panels are discussed in detail. The obtained results are validated by comparing with those in the literature..     

反对称层合板屈曲性态分析

应用屈曲控制方程，导出了反对称层合板在面内双向压缩下的屈曲方程，计算了相应的临界荷载并讨论了层合角对临界应力的影响，分析表明层合角对材料的临界应力有较大的影响。