Buckling load optimization of beams

In this paper, shape optimization is used to optimize the buckling load of a Euler–Bernoulli beam having constant volume. This is achieved by varying appropriately the beam cross section so that the beam buckles with the maximum or a prescribed buckling load. The problem is reduced to a nonlinear optimization problem under equality and inequality constraints as well as specified lower and upper bounds. The evaluation of the objective function requires the solution of the buckling problem of a beam with variable stiffness subjected to an axial force. This problem is solved using the analog equation method for the fourth-order ordinary differential equation with variable coefficients. Besides its accuracy, this method overcomes the shortcomings of a possible FEM solution, which would require resizing of the elements and recomputation of their stiffness properties during the optimization process. Several example problems are presented that illustrate the method and demonstrate its efficiency.     

考虑弯扭形变的薄壁箱形梁的非线性后屈曲

在大位移和扭转的前提下,通过一中等弯曲扭转的位移场描述了薄壁箱形梁在偏心载荷作用下的静稳定性问题.该非线性公式可用于分析简支薄壁箱形梁在不同载荷作用下的屈曲和后屈曲行为.采用伽辽金方法将非线性微分系统离散,并通过牛顿-拉普森增量迭代法求解得代数方程组.数值计算结果表明,当前屈曲位移不可忽略时,经典的横向屈曲预测是保守的...     

Imperfection sensitivity of ultimate buckling strength of elastic-plastic square plates under compression

The mechanism of imperfection sensitivity of elastic-plastic plates under compression is complex as they undergo elastic and/or plastic buckling, dependent on their width-thickness ratio. For elastic buckling, the Koiter power law is an established means to describe the imperfection sensitivity. Yet, for plastic buckling, there is no such an established way to describe it. In this paper, the quadratic power law is advanced to describe imperfection-insensitive plastic buckling behavior. The Koiter power law is extended by implementing the quadratic law so as to describe the elastic and plastic buckling in a synthetic manner. The finite-displacement, elastic-plastic analysis was conducted on simply-supported square plates under compression by varying the plate thickness and the initial deflection of a sinusoidal form. In association with an increase of the plate slenderness parameter (decrease of plate thickness), the predominant buckling is shown to change from (1) plastic buckling to (2) unstable elastic-plastic buckling and to (3) elastic stable bifurcation followed by a maximum point of load. In accordance with the change of the mechanism of buckling, the power law is changed pertinently to describe the complex imperfection sensitivity of the compression plates in a synthetic manner. The extended imperfection sensitivity law is thus advanced as a simple and strong tool to describe the ultimate buckling strength of elastic-plastic plates.     

Buckling of a circular plate made of a shape memory alloy due to a reverse thermoelastic martensite transformation

We solve the axisymmetric buckling problem for a circular plate made of a shape memory alloy undergoing reverse martensite transformation under the action of a compressing load, which occurs after the direct martensite transformation under the action of a generally different (extending or compressing) load. The problem was solved without any simplifying assumptions concerning the transverse dimension of the supplementary phase transition region related to buckling. The mathematical problem was reduced to a nonlinear eigenvalue problem. An algorithm for solving this problem was proposed. It was shown that the critical buckling load under the reverse transition, which is obtained by taking into account the evolution of the phase strains, can be many times lower than the same quantity obtained under the assumption that the material behavior is elastic even for the least (martensite) values of the elastic moduli. The critical buckling force decreases with increasing modulus of the load applied at the preliminary stage of direct transition and weakly depends on whether this load was extending or compressing. In shape memory alloys (SMA), mutually related processes of strain and direct (from the austenitic into the martensite phase) or reverse thermoelastic phase transitions may occur. The direct transition occurs under cooling and (or) an increase in stresses and is accompanied by a significant decrease (nearly by a factor of three in titan nickelide) of the Young modulus. If the direct transition occurs under the action of stresses with nonzero deviator, then it is accompanied by accumulation of macroscopic phase strains, whose intensity may reach 8%. Under the reverse transition, which occurs under heating and (or) unloading, the moduli increase and the accumulated strain is removed. For plates compressed in their plane, in the case of uniform temperature distribution over the thickness, one can separate trivial processes under which the strained plate remains plane and the phase ratio has a uniform distribution over the thickness. For sufficiently high compressing loads, the trivial process of uniform compression may become unstable in the sense that, for small perturbations of the plate deflection, temperature, the phase ratio, or the load, the difference between the corresponding perturbed process and the unperturbed process may be significant. The results of several experiments concerning the buckling of SMA elements are given in [1, 2], and the statement and solution of the corresponding boundary value problems can be found in [3–11]. The experimental studies [2] and several analytic solutions obtained for the Shanley column [3, 4], rods [5–7], rectangular plates under direct [8] and reverse [9] transitions showed that the processes of thermoelastic phase transitions can significantly (by several times) decrease the critical buckling loads compared with their elastic values calculated for the less rigid martensite state of the material. Moreover, buckling does not occur in the one-phase martensite state in which the elastic moduli are minimal but in the two-phase state in which the values of the volume fractions of the austenitic and martensite phase are approximately equal to each other. This fact is most astonishing for buckling, studied in the present paper, under the reverse transition in which the Young modulus increases approximately half as much from the beginning of the phase transition to the moment of buckling. In [3–9] and in the present paper, the static buckling criterion is used. Following this criterion, the critical load is defined to be the load such that a nontrivial solution of the corresponding quasistatic problem is possible under the action of this load. If, in the problems of stability of rods and SMA plates, small perturbations of the external load are added to small perturbations of the deflection (the critical force is independent of the amplitude of the latter), then the critical forces vary depending on the value of perturbations of the external load [5, 8, 9]. Thus, in the case of small perturbations of the load, the problem of stability of SMA elements becomes indeterminate. The solution of the stability problem for SMA elements also depends on whether the small perturbations of the phase ratio and the phase strain tensor are taken into account. According to this, the problem of stability of SMA elements can be solved in the framework of several statements (concepts, hypotheses) which differ in the set of quantities whose perturbations are admissible (taken into account) in the process of solving the problem. The variety of these statements applied to the problem of buckling of SMA elements under direct martensite transformation is briefly described in [4, 5]. But, in the problem of buckling under the reverse transformation, some of these statements must be changed. The main question which we should answer when solving the problem of stability of SMA elements is whether small perturbations of the phase ratio (the volume fraction of the martensite phase q) are taken into account, because an appropriate choice significantly varies the results of solving the stability problem. If, under the transition to the adjacent form of equilibrium, the phase ratio of all points of the body is assumed to remain the same, then we deal with the “fixed phase atio” concept. The opposite approach can be classified as the “supplementary phase transition” concept (which occurs under the transition to the adjacent form of equilibrium). It should be noted that, since SMA have temperature hysteresis, the phase ratio in SMA can endure only one-sided small variations. But if we deal with buckling under the inverse transformation, then the variation in the volume fraction of the martensite phase cannot be positive. The phase ratio is not an independent variable, like loads or temperature, but, due to the constitutive relations, its variations occur together with the temperature variations and, in the framework of connected models for a majority of SMA, together with variations in the actual stresses. Therefore, the presence or absence of variations in q is determined by the presence or absence of variations in the temperature, deflection, and load, as well as by the system of constitutive relations used in this particular problem. In the framework of unconnected models which do not take the influence of actual stresses on the phase ratio into account, the “fixed phase ratio” concept corresponds to the case of absence of temperature variations. The variations in the phase ratio may also be absent in connected models in the case of specially chosen values of variations in the temperature and (or) in the external load, as well as in the case of SMA of CuMn type, for which the influence of the actual stresses on the phase compound is absent or negligible. In the framework of the “fixed phase ratio” hypothesis, the stability problem for SMA elements has a solution coinciding in form with the solution of the corresponding elastic problem, with the elastic moduli replaced by the corresponding functions of the phase ratio. In the framework of the supplementary phase transition” concept, the result of solving the stability problem essentially depends on whether the small perturbations of the external loads are taken into account in the process of solving the problem. The point is that, when solving the problem in the connected setting, the supplementary phase transition region occupies, in general, not the entire cross-section of the plate but only part of it, and the location of the boundary of this region depends on the existence and the value of these small perturbations. More precisely, the existence of arbitrarily small perturbations of the actual load can result in finite changes of the configuration of the supplementary phase transition region and hence in finite change of the critical values of the load. Here we must distinguish the “fixed load” hypothesis where no perturbations of the external loads are admitted and the “variable load” hypothesis in the opposite case. The conditions that there no variations in the external loads imply additional equations for determining the boundary of the supplementary phase transition region. If the “supplementary phase transition” concept and the “fixed load” concept are used together, then the solution of the stability problem of SMA is uniquely determined in the same sense as the solution of the elastic stability problem under the static approach. In the framework of the “variable load” concept, the result of solving the stability problem for SMA ceases to be unique. But one can find the upper and lower bounds for the critical forces which correspond to the cases of total absence of the supplementary phase transition: the upper bound corresponds to the critical load coinciding with that determined in the framework of the “fixed phase ratio” concept, and the lower bound corresponds to the case where the entire cross-section of the plate experiences the supplementary phase transition. The first version does not need any additional name, and the second version can be called as the "all-round supplementary phase transition" hypothesis. In the present paper, the above concepts are illustrated by examples of solving problems about axisymmetric buckling of a circular freely supported or rigidly fixed plate experiencing reverse martensite transformation under the action of an external force uniformly distributed over the contour. We find analytic solutions in the framework of all the above-listed statements except for the case of free support in the “fixed load” concept, for which we obtain a numerical solution.     

Buckling of a flush-mounted plate in simple shear flow

The buckling of an elastic plate with arbitrary shape flush-mounted on a rigid wall and deforming under the action of a uniform tangential load due to an overpassing simple shear flow is considered. Working under the auspices of the theory of elastic instability of plates governed by the linear von Kármán equation, an eigenvalue problem is formulated for the buckled state resulting in a fourth-order partial differential equation with position-dependent coefficients parameterized by the Poisson ratio. The governing equation also describes the deformation of a plate clamped around the edges on a vertical wall and buckling under the action of its own weight. Solutions are computed analytically for a circular plate by applying a Fourier series expansion to derive an infinite system of coupled ordinary differential equations and then implementing orthogonal collocation, and numerically for elliptical and rectangular plates by using a finite-element method. The eigenvalues of the resulting generalized algebraic eigenvalue problem are bifurcation points in the solution space, physically representing critical thresholds of the uniform tangential load above which the plate buckles and wrinkles due to the partially compressive developing stresses. The associated eigenfunctions representing possible modes of deformation are illustrated, and the effect of the Poisson ratio and plate shape is discussed.     

Nonlinear buckling optimization of composite structures considering “worst” shape imperfections

Nonlinear buckling optimization is introduced as a method for doing laminate optimization on generalized composite shell structures exhibiting nonlinear behaviour where the objective is to maximize the buckling load. The method is based on geometrically nonlinear analyses and uses gradient information of the nonlinear buckling load in combination with mathematical programming to solve the problem. Thin-walled optimal laminated structures may have risk of a relatively high sensitivity to geometric imperfections. This is investigated by the concepts of “worst” imperfections and an optimization method to determine the “worst” shape imperfections is presented where the objective is to minimize the buckling load subject to imperfection amplitude constraints. The ability of the nonlinear buckling optimization formulation to solve the laminate problem and determine the “worst” shape imperfections is illustrated by several numerical examples of composite laminated structures and the application of both formulations gives useful insight into the interaction between laminate design and geometric imperfections.     

Modified Monte Carlo method for buckling analysis of nonlinear imperfect structures

In this paper, we propose a modified Monte Carlo method for analysis of buckling of an imperfect beams on softening nonlinear elastic foundation. Such structures exhibit considerable imperfection sensitivity, i.e. reduction in the maximum load that the structure is able to support in contrast to classical buckling load of the perfect structure. The initial imperfections are treated as random functions of axial coordinate. In order to reduce the needed number of simulations, the Monte Carlo method is coupled with maximum likelihood methodology and the Kolmogorov–Smirnov test.     

Long-term non-linear behaviour and buckling of shallow concrete-filled steel tubular arches

This paper presents a theoretical analysis for the long-term non-linear elastic in-plane behaviour and buckling of shallow concrete-filled steel tubular (CFST) arches. It is known that an elastic shallow arch does not buckle under a load that is lower than the critical loads for its bifurcation or limit point buckling because its buckling equilibrium configuration cannot be achieved, and the arch is in a stable equilibrium state although its structural response may be quite non-linear under the load. However, for a CFST arch under a sustained load, the visco-elastic effects of creep and shrinkage of the concrete core produce significant long-term increases in the deformations and bending moments and subsequently lead to a time-dependent change of its equilibrium configuration. Accordingly, the bifurcation point and limit point of the time-dependent equilibrium path and the corresponding buckling loads of CFST arches also change with time. When the changing time-dependent bifurcation or limit point buckling load of a CFST arch becomes equal to the sustained load, the arch may buckle in a bifurcation mode or in a limit point mode in the time domain. A virtual work method is used in the paper to investigate bifurcation and limit point buckling of shallow circular CFST arches that are subjected to a sustained uniform radial load. The algebraically tractable age-adjusted effective modulus method is used to model the time-dependent behaviour of the concrete core, based on which solutions for the prebuckling structural life time corresponding to non-linear bifurcation and limit point buckling are derived.     

含内埋矩形脱层复合材料圆柱壳屈曲分析的混合变量条形传递函数方法

提出了一种分析含内埋矩形脱层正交各向异性圆柱壳稳定性问题的混合变量条形传递函数方法。首先基于Mindlin一阶剪切壳理论,通过定义圆柱壳的广义力变量和混合变量,建立了壳的改进混合变量能量泛函;然后,为了便于脱层壳的分区求解,通过引入条形单元,创建了基于混合变量条形传递函数解的含脱层和不合脱层两种超级壳单元;在此基础上,将含内埋矩形脱层的复合材料层合壳划分成两种超级壳单元的组合体,通过各超级壳单元相互之间连接结点处的位移连续和力平衡条件得到脱层壳的屈曲方程;最后由屈曲方程计算含内埋矩形脱层壳的屈曲载荷和屈曲模态。算例分析的结果验证了本方法的正确性,并给出了几种因素对屈曲载荷和屈曲模态的影响。     

Buckling of a stiff film bound to a compliant substrate—Part II:: A global scenario for the formation of herringbone pattern

We study the buckling of a thin compressed elastic film bonded to a compliant substrate. We focus on a family of buckling patterns, such that the film profile is generated by two functions of a single variable. This family includes the unbuckled configuration, the classical primary mode made of straight stripes, as well the pattern with undulating stripes obtained by a secondary instability investigated in the first companion paper, and the herringbone pattern studied in last companion paper. A simplified buckling model relevant for the analysis of these patterns is introduced. It is solved analytically for moderate or for large residual compressive stress in the film. Numerical simulations are presented, based on an efficient implementation. Overall, the analysis provides a global picture for the formation of herringbone patterns under increasing residual stress. The film shape is shown to converge at large load to a developable shape with ridges. The wavelength of the pattern, selected in a first place by the primary buckling bifurcation, is frozen during the subsequent increase of loading.     

Buckling analysis of functionally graded arbitrary straight-sided quadrilateral plates on elastic foundations

As a first endeavor, the buckling analysis of functionally graded (FG) arbitrary straight-sided quadrilateral plates rested on two-parameter elastic foundation under in-plane loads is presented. The formulation is based on the first order shear deformation theory (FSDT). The material properties are assumed to be graded in the thickness direction. The solution procedure is composed of transforming the governing equations from physical domain to computational domain and then discretization of the spatial derivatives by employing the differential quadrature method (DQM) as an efficient and accurate numerical tool. After studying the convergence of the method, its accuracy is demonstrated by comparing the obtained solutions with the existing results in literature for isotropic skew and FG rectangular plates. Then, the effects of thickness-to-length ratio, elastic foundation parameters, volume fraction index, geometrical shape and the boundary conditions on the critical buckling load parameter of the FG plates are studied.     

Nonlinear dynamic buckling of pinned–fixed shallow arches under a sudden central concentrated load

The structural behavior of a shallow arch is highly nonlinear, and so when the amplitude of the oscillation of the arch produced by a suddenly-applied load is sufficiently large, the oscillation of the arch may reach a position on its unstable equilibrium paths that leads the arch to buckle dynamically. This paper uses an energy method to investigate the nonlinear elastic dynamic in-plane buckling of a pinned–fixed shallow circular arch under a central concentrated load that is applied suddenly and with an infinite duration. The principle of conservation of energy is used to establish the criterion for dynamic buckling of the arch, and the analytical solution for the dynamic buckling load is derived. Two methods are proposed to determine the dynamic buckling load. It is shown that under a suddenly-applied central load, a shallow pinned–fixed arch with a high modified slenderness (which is defined in the paper) has a lower dynamic buckling load and an upper dynamic buckling load, while an arch with a low modified slenderness has a unique dynamic buckling load.     

含轴对称初始缺陷的具有正交各向异性表层的夹层圆柱壳的非线性屈曲

夹层圆柱壳具有很高的结构效能。在许多工程结构中被广泛采用。本文研究分析了含有轴对称初始缺陷的夹层圆柱壳在轴压下的非线性屈曲问题。该夹层壳具有正交各向异性表层和各向同性可承剪的夹心．利用Stein的前屈曲一致理论得出了前屈曲挠度随轴向载荷及缺陷参数的变化情况，运用Galerkin法导出了屈曲控制方程，并进行了数值计算，得到了屈曲载荷、缺陷幅值、缺陷波数、夹心模量等参量之间的关系．结果表明与壳体实际屈曲模态相同的初始缺陷具有很大的危害性，可以通过增加壳体表层的轴向弹性模量或优化夹心的有关参数等途径来提高屈曲载荷，改善壳体屈曲性能。     

In-plane buckling of prismatic funicular arches with shear deformations

The in-plane buckling behavior of funicular arches is investigated numerically in this paper. A finite strain Timoshenko beam-type formulation that incorporates shear deformations is developed for generic funicular arches. The elastic constitutive relationships for the internal beam actions are based on a hyperelastic constitutive model, and the funicular arch equilibrium equations are derived. The problems of a submerged arch under hydrostatic pressure, a parabolic arch under gravity load and a catenary arch loaded by overburden are investigated. Buckling solutions are derived for the parabolic and catenary arch. Subsequent investigation addresses the effects of axial deformation prior to buckling and shear deformation during buckling. An approximate buckling solution is then obtained based on the maximum axial force in the arch. The obtained buckling solutions are compared with the numerical solutions of Dinnik (Stability of arches, 1946) [1] and the finite element package ANSYS. The effects of shear deformation are also evaluated.     

Sampling-Based RBDO of Ship Hull Structures Considering Thermo-Elasto-Plastic Residual Deformation

We present a shape optimization method using a sampling-based RBDO method linked with a commercial finite element analysis (FEA) code ANSYS, which is applicable to residual deformation problems of the ship hull structure in welding process. The programming language ANSYS Parametric Design Language (APDL) and shell elements are used for the thermo-elasto-plastic analysis. The shape of the ship hull structure is modeled using the bicubic Ferguson patch and coordinate components of vertices, tangential vectors of boundary curves are selected as design variables. The sensitivity of probabilistic constraint is calculated from the probabilistic sensitivity analysis using the score function and Monte Carlo Simulation (MCS) on the surrogate model constructed by using the Dynamic Kriging (DKG) method. The sequential quadratic programming (SQP) algorithm is used for the optimization. In two numerical examples, the suggested optimization method is applied to practical residual deformation problems in welding ship hull structures, which proves the sampling-based RBDO can be successfully utilized for obtaining a reliable optimum design in highly nonlinear multi-physics problem of thermo-elasto-plasticity.     

受轴向冲击有限长弹性直杆中应力波引起的分叉问题

研究了有限长理想直杆受阶跃载荷作用的弹性动力屈曲问题。将直杆的屈问题归结为由轴向应力波传播而导致的杆的分叉问题，并考虑了应力波反射的影响。给出了分叉发生的临界条件并对具体实例进行了计算。最后对阶跃载荷及脉冲载荷对杆的动力屈曲的影响进行了讨论。     

The asymptotic solution of a dynamic buckling problem in elastic columns

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Semi-analytical buckling analysis of heterogeneous variable thickness viscoelastic circular plates on elastic foundations

Buckling analysis of the functionally graded viscoelastic circular plates has not been carried out so far. In the present paper, a series solution is developed for buckling analysis of radially graded FG viscoelastic circular plates with variable thickness resting on two-parameter elastic foundations, based on Mindlin's plate theory. The complex modulus approach in combination with the elastic–viscoelastic correspondence principle is employed to obtain the solution for various edge conditions. A comprehensive sensitivity analysis is carried out to evaluate effects of various parameters on the buckling load. Results reveal that the viscoelastic behavior of the materials may postpone the buckling occurrence and the stiffness reduction due to the section variations may be compensated by the graded material properties.     

受均布载荷压杆后屈曲形态的数值计算

对受均布载荷压杆的屈曲及后屈曲行为进行了分析.基于杆的大变形理论,考虑杆的轴向伸长,建立了受均布载荷作用下细长压杆的几何非线性平衡方程.采用打靶法和解析延拓法数值求解非线性两点边值问题,得到了杆的后屈曲平衡路径和平衡构形.     

Nonlinear analysis on dynamic buckling of eccentrically stiffened functionally graded material toroidal shell segment surrounded by elastic foundations in thermal environment and under time-dependent torsional loads

The nonlinear analysis with an analytical approach on dynamic torsional buckling of stiffened functionally graded thin toroidal shell segments is investigated. The shell is reinforced by inside stiffeners and surrounded by elastic foundations in a thermal environment and under a time-dependent torsional load. The governing equations are derived based on the Donnell shell theory with the von K′arm′an geometrical nonlinearity,the Stein and McE lman assumption, the smeared stiffeners technique, and the Galerkin method. A deflection function with three terms is chosen. The thermal parameters of the uniform temperature rise and nonlinear temperature conduction law are found in an explicit form. A closed-form expression for determining the static critical torsional load is obtained. A critical dynamic torsional load is found by the fourth-order Runge-Kutta method and the Budiansky-Roth criterion. The effects of stiffeners, foundations, material,and dimensional parameters on dynamic responses of shells are considered.