Thickness optimization of circular annular plate at buckling

This work deals with thickness optimization of a circular annular plate at buckling. The plate is loaded with uniform, axially symmetric, in-plane loads on the inner and outer edge. The variable thickness of the plate is approximated by a function of radial coordinates and design variables. An optimization problem is defined to find optimal sets of design variables which maximize buckling loads at constant weight/volume of the plate. The required buckling loads are determined according to the standard linear buckling equations, and the material is modelled by the small strain J₂ flow and deformation theories of plasticity where an elastic linear hardening rheological model is considered. Optimal thickness functions are determined for different support and load cases and the numerical results show that buckling loads can be increased significantly.     

Optimum design of laminated composite plates to maximize buckling load using MFD method

This paper presents optimal design of simply supported laminated composite plates subject to given in-plane static loads for which the critical failure mode is buckling. The objective function is to maximize the buckling load capacity of laminated plates and the fiber orientation is considered as design variable. The first-order shear deformation theory is used for the finite element analysis. In this paper, the effects of bending–twisting coupling are also included for the buckling optimization. The modified feasible direction method is used as an optimization method. Also, computer programs are coded in MATLAB and Golden Section method is adapted in this program for the optimal design of laminated plates for maximum buckling load. The effect of width-to-thickness ratio, aspect ratio, number of layers, material anisotropy, load ratios (N_y/N_x), uncertainties in material properties and functionally graded materials on the results is investigated and compared.     

Shear-flexible thin-walled element for composite I-beams

A shear-flexible finite element based on an orthogonal Cartesian coordinate system is developed for the flexural and buckling analyses of thin-walled composite I-beams with both doubly and mono-symmetrical cross-sections. Using the first-order shear deformable beam theory, the derived element includes both the transverse shear and the restrained warping induced shear deformations. Governing equations are derived from the principle of minimum total potential energy. Three different types of finite elements, namely, linear, quadratic and cubic elements are developed to solve the governing equations. The geometric stiffness for the buckling analysis of axially loaded, thin-walled composite beams is developed. The resulting linearized buckling problem is solved using a shifted inverse iteration algorithm. A parametric study of the effects of the aspect ratio and the fibre orientation on the tip displacement is presented. The convergence of the elements is also investigated. The elastic buckling loads for mono- and doubly-symmetric I-beam cross-sections are compared with other results available in the literature and with solutions using shell elements in a commercially available finite element program.     

Flexural–torsional buckling of thin-walled composite box beams

Buckling of an axially loaded thin-walled laminated composite is studied. A general analytical model applicable to the flexural, torsional and flexural–torsional buckling of a thin-walled composite box beam subjected to axial load is developed. This model is based on the classical lamination theory, and accounts for the coupling of flexural and torsional modes for arbitrary laminate stacking sequence configuration, i.e. unsymmetric as well as symmetric, and various boundary conditions. A displacement-based one-dimensional finite element model is developed to predict critical loads and corresponding buckling modes for a thin-walled composite bar. Governing buckling equations are derived from the principle of the stationary value of total potential energy. Numerical results are obtained for axially loaded thin-walled composites addressing the effects of fiber angle, anisotropy and boundary conditions on the critical buckling loads and mode shapes of the composites.     

Buckling strength of the cylindrical shell and tank subjected to axially compressive loads

This paper aims to develop practical design equations and charts estimating the buckling strength of the cylindrical shell and tank subjected to axially compressive loads. Both geometrically perfect and imperfect shells and tanks are studied. Numerical analysis is used to evaluate buckling strength. The modeling method, appropriate element type and necessary number of elements to use in numerical analysis are recommended. According to the results of the parametric study of the perfect shell, the buckling strength decreases significantly as the diameter-to-thickness ratio increases, while it decreases slightly as the height-to-diameter ratio increases. These results are different from those in the case of columns. The buckling strength of the perfect tank placed on an extremely soft foundation and a stiff foundation increases by up to 1.6% and 5.6%, respectively, compared with that of the perfect shell. The buckling strength of the shell and tank decreases significantly as the amplitude of initial geometric imperfection increases. Convenient and sufficiently accurate design equations and charts used for estimating buckling strength are provided.     

薄壁复合箱型梁的弯曲-扭转屈曲分析

对一轴心受压薄壁复合构件的屈曲进行研究。提出一个广义的分析模型,可用于分析轴心受压薄壁复合箱型梁的弯曲、扭转以及弯扭屈曲作用。此模型基于经典层压理论,考虑了任意层压堆积规律,结构的弯曲和扭转模式的耦合问题,如非对称以及对称和各种边界条件。采用一个基于位移的一维有限元模型来预测薄壁复合钢筋的临界荷载和随后的屈曲模式。从总势能的平稳值原则中推导出屈曲控制方程。轴心受压薄壁复合件的数值计算结果可用于估测纤维角、各向异性和边界条件对临界屈曲荷载和复合件模态的影响。     

Re-interpreting simultaneous buckling modes of axially compressed isotropic conical shells

Elastic stability of shell structures under certain loading conditions is characterized by a dramatically unstable postbuckling behavior. The presence of simultaneous ‘competing’ buckling modes (corresponding to the same critical buckling load) is understood to be largely responsible for such behavior. In this paper, within the framework of linear bifurcation eigenvalue analysis and Donnell shallow shell theory, the presence of simultaneous buckling modes in axially compressed isotropic cones is determined using the semi-analytical method of Galerkin. The results are presented in the plane of the dimensionless reciprocal meridional and circumferential buckling half wavelengths, and are compared with the locus of simultaneous buckling modes of axially compressed cylinders, described by the so-called Koiter circle. By using an optimizing procedure, it is shown that the cluster of simultaneous buckling modes in cones is well described by the Koiter circle of an equivalent cylinder with appropriate length and radius. Such optimizing values of length and radius allow us to gain some insight into the simplifying treatment of the buckling of cones through the concept of equivalent cylinder.     

Buckling optimization of variable thickness prismatic folded plates

This paper deals with the structural shape optimization of prismatic folded plates under buckling load consideration. Buckling loads are determined using linear, quadratic and cubic, variable thickness, C(0) continuity, Mindlin-Reissner finite strips. The whole structural optimization process is carried out by integrating finite strip analysis, cubic spline shape and thickness definition, semi analytical sensitivity analysis and mathematical programming algorithm. The objective is either the maximization of the critical buckling load or minimization of the cross-section of the prismatic folded plate with constraints on the volume and buckling loads. Several examples are included to illustrate various features of the optimization algorithm, including plates and stiffened panels.     

Elastic buckling of an axially compressed cylindrical panel with three edges simply supported and one edge free

The paper is devoted to a circular cylindrical panel with three edges simply supported and one edge free. This panel is axially compressed by a load uniformly distributed at both ends. The Donnell equations for linear buckling of shells are assumed and the adequate boundary conditions are defined. The deflection function and the force function–the Airy function are formulated with respect to the boundary conditions. The Donnell equations are reduced to a generalized eigenvalue problem with the use of Galerkin method. The results of numerical investigation are presented in figures.     

Inelastic buckling analyses of beams, columns and plates using the spline finite strip method

A method of inelastic buckling analysis of thin-walled structural members and plates is described. The method is based on the spline finite strip method of structural analysis. The analysis takes into account the non-linear material stress-strain properties, strain hardening and residual stresses. The plastic theories used in the study are the flow theory of plasticity and the deformation theory of plasticity. The method of inelastic buckling analysis is applied to a variety of instability problems including plates, cold-formed columns, hot-rolled columns and welded tee section beams. The buckling modes and loads computed are compared with theoretical values and test results.     

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。     

Exact and approximate solutions for the flexural buckling of columns with oblique rotational end restraints

This paper is concerned with the elastic flexural buckling of doubly symmetric columns with oblique restraints under concentric loading. Oblique restraints cause coupling between the principal axis deflections and rotations, and the flexural buckling mode involves simultaneous bending about both principal axes.The paper discusses the nature of oblique restraints, and presents exact and approximate solutions for the buckling loads of columns with rigid or elastic restraints against rotations at the ends. The exact solutions are obtained by solving the governing differential equations and boundary conditions, while the approximate solutions are based on energy solutions with assumed buckling displacements. The approximate solutions are sufficiently simple that they can be used in design, and are shown to be within 1% of the exact solutions.     

Stability of elastic–plastic conical shells under shear loading

A method of determination of critical loads for thin-walled conical shells loaded by shear forces developed by moment of twist is presented. The three governing equations of neutral equilibrium with respect to basic displacement vector components u, v, and w are used. It is assumed that effective stress in the prebuckling state of stress in the shell can exceed the yield limit of the shell material. The use is made both of physical relations of Nadai–Hencky small elastic–plastic deformation theory of plasticity, and Prandtl–Reuss J₂ incremental plastic flow theory. Also, a bilinear stress–strain material model, material compressibility and Shanley approach will be accepted in the analysis. Galerkin method is applied to solve the problem equations and iterative techniques are accepted in numerical algorithm to determine critical loads for elastic–plastic shells.     

均布压力作用下板和柱面塑性屈曲和初始后屈曲的分析结果

主要分析均布压力作用下板和柱面弹塑性屈曲和初始后屈曲的理论特性。分析基于3D塑性分歧理论提出,这个分歧理论中假定了von Mises屈服标准和线性各向同性硬化的J2塑性流动理论。提议的方法具有系统性和统一性,可用于计算多种边界条件下的单轴或双轴(拉)压力作用的矩形板,和轴压作用下的柱面的临界荷载、屈曲模式和分歧初始斜率。     

Elastic stability of bi-axially loaded rectangular plates with a single circular hole

The finite-element method has been employed to determine the elastic buckling stresses of a bi-axially loaded perforated rectangular plate with dimensions a and b in the x- and y-directions, respectively. The considered perforation is a single circular hole whose center is located along the longitudinal axis of the plate. The considered plate has simply supported edges in the out-of-plane direction and is subjected to bi-axial uniformly distributed end loads (compressive load σ_x in the x-direction, and compressive or tensile load σ_y in the y-direction). Parameters considered in the study are the plate's aspect ratio a/b, the stress ratio ξ between the applied stresses in the y- and x-directions (ξ=σ_y/σ_x), the circular hole size d and location e_x.The study shows that, in most of the considered cases, the bigger the hole size d, the lesser the plate stability and the lesser the buckling stresses. It also shows that the plate aspect ratios, a/b, between 0.6 and 1.2 should be avoided for plates with large holes and negative ξ, due to the large reduction in the buckling stresses. The hole location should also be selected to be away from the loaded edge of the plate as much as possible (better to have e_x/b>1.0) to increase the buckling stresses and improve stability. The study demonstrates also that the increase in tension in the y-direction in bi-axially loaded plate with large hole (d/b>0.4) reduces its stability. This is in contrary to the expected increase in the stability due to the increase in tension which can be seen clearly in the cases of solid plates and plates with small holes.     

3D buckling analysis of a cylindrical metal bin composed of corrugated sheets strengthened by vertical stiffeners

The paper presents 3D results of a quasi-static buckling analysis of a funnel-flow cylindrical metal bin composed of horizontally corrugated sheets strengthened by vertical columns. A linear buckling and a non-linear analysis with geometric and material non-linearity were carried out with a perfect and an imperfect real silo shell by taking into account axisymmetric and non-axisymmetric loads imposed by a bulk solid following Eurocode 1 and different initial geometric imperfections and load non-uniformities around the circumference. The calculated buckling forces were compared with the permissible one given by Eurocode 3.     

Modification of -equation in the SSRC Guide for buckling of monosymmetric I-beams under transverse loads

Finite element buckling analyses of monosymmetric I-beams subjected to transverse loading applied at different heights with respect to the mid-height of the cross-section were conducted. Transverse loads consisting of a mid-span point load and a uniformly distributed load were considered in the investigation. Four types of end restraint also were considered. The method suggested in SSRC Guide was compared with the finite element method (FEM) results. In order for C-equation in the SSRC Guide to be applicable for monosymmetric I-beams, it must be modified to contain three variables; A_m, B_m, and D_m. The first two variables A_m and B_m are modified variables A and B in the SSRC Guide whereas the third variable, D_m, developed herein, is the new modification factor for monosymmetric I-beams. The applicability of this new design rule is limited to monosymmetric I-beams in which the degree of monosymmetry, ρ, is within the range from 0.1 to 0.9.     

Numerical and experimental investigations of the response of aluminum cylinders with a cutout subject to axial compression

Understanding how a cutout influences the load bearing capacity and buckling behavior of a cylindrical shell is critical in the design of structural components used in automobiles, aircrafts, and marine applications. Numerical simulation and analysis of moderately thick and thin unstiffened aluminum cylindrical shells (D/t=45, 450 and L/D=2, 5, 10), having a square cutout, subjected to axial compression were systematically carried out in this paper. The investigation examined the influence of the cutout size, cutout location, and the shell aspect ratio (L/D) on the prebuckling, buckling, and postbuckling responses of the cylindrical shells.An experimental investigation on the moderately thick-walled shells was also carried out. A good correlation was observed between the results obtained from the finite element simulation and the experiments. Furthermore, empirical equations, in the form of a ‘buckling load reduction factor’ were developed using the least square regression method. These simple equations could be used to predict the buckling capacities of several specific types of cylindrical shells with a cutout.     

基于广义梁理论的非线性材料薄壁受压构件屈曲荷载计算方法

提出适用于非线性材料的广义梁理论屈曲荷载计算方法,并对不锈钢薄壁受压构件屈曲荷载进行计算验证。通过定义材料非线性应力应变关系和瞬时弹性模量,对传统线弹性广义梁理论进行修正,建立非线性材料薄壁构件受压屈曲荷载计算方法,推导不锈钢薄板受压局部屈曲、冷弯薄壁不锈钢卷边槽形柱畸变屈曲及箱形不锈钢长柱弯曲屈曲荷载计算公式,并与既有试验数据对比。经验证,线弹性分析方法不适用于不锈钢材料;提出的修正GBT法具有较高精度,且本构关系采用变形法则结果偏于安全,可用于不锈钢等非线性金属材料薄壁构件受压屈曲荷载的确定,为研究和设计提供理论指导。     

Recent advances on postbuckling analyses of thin-walled structures: Beams, frames and cylindrical shells

The lateral postbuckling response of thin-walled structures such as bars and frames with members having steel rolled shapes as well as circular cylindrical shells under axial compression is thoroughly reconsidered. More specifically via a simple and very efficient technique it is found that beams with rolled shapes (symmetric or non symmetric) under uniform bending and axial compression exhibit a stable lateral-torsional secondary path with limited margins of postbuckling strength. New findings for the static and dynamic stability of frames with crooked steel members-due to the presence of residual stresses-are also reported. It is comprehensively established that the coupling effect due to initial crookedness and loading eccentricity may have a beneficial effect on the load-carrying capacity of the frames. Moreover, simple mechanical models are proposed for simulating the buckling mechanism of axially compressed circular cylindrical shells. Very recently Bodner and Rubin proposed an 1-DOF mechanical model whose buckling parameters correlated to those of the shells by using an empirical formula based on experimentally observed shell buckling loads. In the present analysis a new 2-DOF model for the static and dynamic buckling of axially compressed circular cylindrical shells, which can include mode coupling, is presented.