Optimization of shear-deformable laminated plates under buckling and strength criteria

A shear deformable laminated theory is used to study the optimal design of rectangular plates under biaxial compressive loads. Such loads lead to plate failure by buckling or material failure which corresponds to the violation of the selected strength criterion. The minimum of the two loads (buckling load or material failure load) determines the critical failure load for a given set of problem parameters. At the optimum values of the ply angles, buckling or both failure criteria may be operational depending on the laminate thickness. The present study investigates the effect of laminate thickness on the optimal design and gives numerical results for symmetrically laminated angle-ply plates.     

Thermomechanical buckling of hybrid cross-ply laminated rectangular plates

A thermomechanical buckling analysis is presented for simply supported rectangular symmetric cross-ply laminated composite plates that are integrated with surface-mounted piezoelectric actuators and are subjected to the combined action of in-plane compressive edge loads, two types of thermal loads, and constant applied actuator voltage. The formulation of equations is based on the classical laminated plate theory and the von-Karman non-linear kinematic relations. The analysis uses an exact method to obtain closed-form solutions for the buckling load. The effects of applied actuator voltage, thermal and mechanical loads, plate geometry, and lay-up configuration of the laminated plates are investigated. The novelty of the present work is to obtain closed-form solutions for electro-thermomechanical buckling of hybrid composite plates, and to cover non-uniform temperature distribution loading. The results for various states are verified with known data in the literature.     

Optimization of laminated composites subject to uncertain buckling loads

Optimal design of composite laminates under buckling load uncertainty is presented. The laminates are subjected to biaxial compressive loads and the buckling load is maximized under worst case in-plane loading which is computed using an anti-optimization approach. The magnitudes of the in-plane loads are not known a priori resulting in load uncertainty subject to the only constraint that the loads belong to a given uncertainty domain. Results are given for continuous and discrete fibre orientations which constitute the optimization problem coupled to load anti-optimization problem leading to a nested solution method. It is observed that the stacking sequence of a laminate designed for a deterministic load case only differs considerably from that of a robust laminate designed taking load uncertainties into account. Consequently the buckling load carried by a deterministic design is considerably less than the one carried by a robust design when both are subjected to uncertain loads.     

一种基于复合材料加筋板结构效率的稳定性优化方法

提出一种以承载效率最高作为目标的新设计方法, 对复合材料加筋板的承载能力进行优化。讨论了不同压缩与弯曲刚度的匹配模式与加筋板临界失稳载荷的关系。将全局失稳载荷、局部失稳载荷与静载荷的接近程度作为结构承载效率的量化标准, 通过静载荷的控制, 使结构的稳定性向着效率最高的方向优化。以宏观的加筋板压缩与弯曲刚度参数作为设计变量, 构建了一种可用于结构效率优化的代理模型, 避免了局部最优点的出现, 更便于数值寻优。通过有限元分析验证, 优化后壁板的临界失稳载荷与所施加的静载荷基本一致, 反映出较高的效率, 从而验证了该方法的可靠性。      

Elastic-plastic buckling analysis of rectangular plates subjected to biaxial loads

In order to calculate the buckling load of a rectangular plate, the analytical approach is used in this study. The plate is assumed to be simply supported on four edges and loaded by uniform stresses along the edges. If the plate is slender, the buckling is elastic. However, if the plate is sturdy, it buckles in the plastic range. Then, the instantaneous moduli in the constitutive equations depend on the external loading. In this study, the elastic and plastic buckling equations are derived for rectangular plates under biaxial loading, and the corresponding interaction curves are presented. The influences of aspect ratios, load ratios and hardening factors on the buckling stresses are investigated for rectangular plates. From the plastic buckling analysis, the optimal combination of loads is given for the buckling strength.     

Elastic-plastic buckling analysis of rectangular plates subjected to biaxial loads

In order to calculate the buckling load of a rectangular plate, the analytical approach is used in this study. The plate is assumed to be simply supported on four edges and loaded by uniform stresses along the edges. If the plate is slender, the buckling is elastic. However, if the plate is sturdy, it buckles in the plastic range. Then, the instantaneous moduli in the constitutive equations depend on the external loading. In this study, the elastic and plastic buckling equations are derived for rectangular plates under biaxial loading, and the corresponding interaction curves are presented. The influences of aspect ratios, load ratios and hardening factors on the buckling stresses are investigated for rectangular plates. From the plastic buckling analysis, the optimal combination of loads is given for the buckling strength.     

Isogeometric shape optimization for quasi‐static processes

A framework to solve shape optimization problems for quasi‐static processes is developed and implemented numerically within the context of isogeometric analysis (IGA). Recent contributions in shape optimization within IGA have been limited to static or steady‐state loading conditions. In the present contribution, the formulation of shape optimization is extended to include time‐dependent loads and responses. A general objective functional is used to accommodate both structural shape optimization and passive control for mechanical problems. An adjoint sensitivity analysis is performed at the continuous level and subsequently discretized within the context of IGA. The methodology and its numerical implementation are tested using benchmark static problems of optimal shapes of orifices in plates under remote bi‐axial tension and pure shear. Under quasi‐static loading conditions, the method is validated using a passive control approach with an a priori known solution. Several applications of time‐dependent mechanical problems are shown to illustrate the capabilities of this approach. In particular, a problem is considered where an external load is allowed to move along the surface of a structure. The shape of the structure is modified in order to control the time‐dependent displacement of the point where the load is applied according to a pre‐specified target. Copyright © 2015 John Wiley & Sons, Ltd.     

考虑屈曲的复合材料加筋壁板铺层顺序优化

提出了一种考虑屈曲的复合材料加筋壁板铺层顺序优化方法。基于复合材料加筋壁板屈曲载荷求解的能量法,系统推导了轴压载荷作用下复合材料加筋壁板蒙皮、筋条局部屈曲载荷的显示表达式,考虑了加筋壁板各板元之间的弹性支持作用及筋条下缘条的影响,引入工程法求解了加筋壁板整体屈曲载荷。基于国产自主结构分析软件HAJIF中的复合材料铺层工程数据库,以铺层参数为中间变量,利用本文提出的复合材料加筋壁板屈曲载荷求解方法,构建了考虑屈曲的复合材料加筋壁板铺层顺序优化设计流程并完成程序实现,将最小二乘法用于最优铺层顺序与工程铺层数据库的匹配。相比于传统有限元计算方法,本文提出的复合材料加筋壁板屈曲载荷求解方法具备较好的求解精度及求解效率。复合材料加筋壁板优化算例表明,采用本文提出的加筋壁板屈曲载荷分析及其优化方法,在结构重量不变的前提下,屈曲载荷提高约17%,且铺层顺序优化结果可直接从铺层工程数据库中提取并用于工程实际。      

Numerical analysis of dynamic buckling of rectangular plates subjected to intermediate-velocity impact

This paper numerically investigates the dynamic buckling of thin imperfect rectangular plates subjected to intermediate-velocity impact loads. From numerical results obtained, a dynamic buckling and a dynamic yielding critical condition are defined, and the corresponding critical dynamic loads are estimated. Numerical model employed in the present study is validated by experimental data reported earlier. Results from parametric study indicate that initial imperfection and load duration have significant influence on the dynamic buckling of the plates. The smaller the initial imperfection and the load duration, the higher the dynamic buckling critical loads of the plates. Moreover, different hardening ratios of plate material also affect the elastic–plastic dynamic buckling properties of the plates. If the plate buckles plastically, the dynamic buckling load increases as the hardening ratio of plate material increases. Unlike thin plates under high-velocity impact that buckling always occur after load application, plates under intermediate-velocity impact analyzed in the present study all buckle during the loading phase.     

复合材料加筋板屈曲/后屈曲分析的应用

研究了复合材料加筋板翼面结构稳定性问题,分析了加筋板在压缩和剪切等载荷作用下的稳定性安全裕度。利用计算复合材料加筋板屈曲及后屈曲承载能力的方法,验证复杂受载情况下结构的稳定性。验证对象是一个优化后的满足强度、刚度和工艺制造要求的复合材料机翼。该机翼在各种载荷工况下的内力分布情况由MSC.NASTRAN分析得到,通过本文提出的方法得到每块蒙皮的稳定性承载能力。然后给出复合材料层合板在复杂载荷下的屈曲及后屈曲安全裕度的计算准则,验证优化后的机翼加筋板是否满足稳定性设计要求。该方法可作为约束集成到结构优化系统平台中。     

船体开孔薄板面内剪切屈曲特性研究

船体板材不可避免地存在不同形式的开孔,开孔破坏了结构的连续性,对结构的强度、稳定性具有重要影响,因此研究船体板开孔结构的屈曲特性对保证船舶安全十分重要。在面内载荷作用下,通过画框型剪切夹具、3D全场变形测量-分析系统等对两种不同形式的船体开孔薄板进行剪切屈曲试验,获得了圆形开孔板和方形开孔板的临界屈曲载荷、全场位移/应变信息和拉力-伸长率曲线等;考虑试验夹具的影响,基于Abaqus对不同形式的开孔板进行数值仿真,通过对开孔板进行特征值屈曲分析和非线性屈曲分析,获得了两种不同形式船用开孔薄板的屈曲、后屈曲力学响应信息。通过数值仿真与试验结果的对比研究,验证了数值仿真方法的有效性、准确性。在此基础上,重点剖析载荷-伸长率关系、典型时刻板面全场位移、临界屈曲载荷以及开孔边缘的应力分布响应特征,明确了面积等效情况下圆孔和倒圆角方孔对方形薄板剪切稳定性的影响。为船用薄板面内剪切稳定性的试验和仿真研究以及大型船体结构的设计优化、力学性能评估等提供有益参考。     

A nonlinear double-superposition global–local theory for dynamic buckling of imperfect viscoelastic composite/sandwich plates: A hierarchical constitutive model

Nonlinear dynamic thermo-mechanical buckling and postbuckling analyses of imperfect viscoelastic composite laminated/sandwich plates are performed by a proposed theory that takes into account all the interlaminar kinematic and transverse stress continuity conditions, for the first time. Even the dynamic buckling analysis of the multi-layered/sandwich plates employing the hierarchical constitutive model has not been performed before. The proposed theory is a double-superposition high-order global–local theory that is calibrated based on the nonlinear strain–displacement expressions for the thermoelastic loadings taking into account the structural damping. The buckling loads are determined based on a criterion previously published by the author. Various complex sensitivity analyses evaluating effects of the relaxation parameters, rate of the loading, sudden heating, and pre-stress on thermo-mechanical buckling of the viscoelatic multi-layered/sandwich plates are performed. Results show that the viscoelastic behavior may decrease the buckling load. Sudden dynamic buckling loads are higher due to the reflected stress waves.     

Optimal design of hybrid laminates with discrete ply angles for maximum buckling load and minimum cost

The design of hybrid symmetric laminated plates consisting of high-stiffness surface and low-stiffness core layers is presented. In the first problem the maximization of buckling load is carried out over a discrete set of ply angles. In the second problem the minimum number of high-stiffness plies is determined for a given buckling load to minimize the material cost. Boolean variables are introduced to specify stacking sequence. Solution of the linear optimization problem yields an optimal stacking sequence. The effect of hybridization is investigated for various problem parameters such as the aspect ratio of the laminate and the number of plies. The optimal designs are obtained with upper bound constraints on the effect of bending-twisting coupling stiffnesses. Results are given for hybrid graphite-epoxy/glass-epoxy laminates under both uniaxial and biaxial loadings.     

Effect of in-plane boundary restraints on optimal design of skew laminates under buckling loads

Symmetrically laminated cross-ply and angle-ply skew plates subject to uniaxial buckling loads and various combinations of in-plane boundary restraints are studied using a shear deformable theory. For this purpose a finite element code is developed and applied to a couple of verification problems. The formulation of the parabolic iso-parametric plate element is briefly given and numerical results obtained for the verification problems related to stability analysis and stress diffusion are presented. The effect of in-plane restraints on the non-uniform distribution of in-plane stresses is studied by means of contour graphs. Next the buckling loads are maximized with respect to layer thicknesses in the case of cross-ply laminates and with respect to fiber orientations in the case of angle-ply laminates. The optimization results show that the exclusion of the in-plane restraints, which arise in several engineering applications, may lead to errors in the stability analysis and consequently in the design of laminated plates against buckling.     

Surrogate-based multi-objective optimization of a composite laminate with curvilinear fibers

A variable stiffness design can increase the structural performance of a composite plate and provides flexibility for trade-offs between structural properties. In this paper, we examine the simultaneous optimization of stiffness and buckling load of a composite laminate plate with curvilinear fiber paths. The problem, which falls in the area of multi-objective optimization, is formulated and solved through a surrogate-based optimization algorithm capable of finding the set of optimum Pareto solutions. We integrate surrogate modeling into an evolutionary algorithm to reduce the high computational cost required to solve the optimization process. The results show that a curvilinear fiber path can increase both buckling load and stiffness simultaneously over the quasi-isotropic laminate. Furthermore, the optimum direction for varying the fiber angle is dependent on the loading direction and boundary conditions. The results for a plate under uniform compression with free transverse edges shows that varying the fiber orientation perpendicular to the loading direction can increase the buckling load by 116% with respect to that of a quasi-isotropic laminate.     

Influence of a central straight crack on the buckling behaviour of thin plates under tension,compression or shear loading

Thin-walled structural components, such as plates and shells, are used in several aerospace, naval, nuclear power plant, pressure vessels, mechanical and civil structures. Due to their high slenderness, the safety assessment of such structural components requires to carefully assess the buckling collapse which can strongly limit their bearing capacity. For very thin plate, buckling collapse can occur under shear, compression or even under tension. In the latter case, fracture or plastic failure can also take place instead of elastic instability. In the present paper, the effects of a central straight crack on the buckling collapse of rectangular elastic thin-plates—characterized by different boundary conditions, crack length and orientation—under compression, tension or shear loading are analysed. Accurate FE numerical parametric analyses have been performed to get the critical load multipliers in such loading cases. Moreover the effect of crack faces contact is examined and discussed. Some useful conclusions related to the sensitivity to cracks of the buckling loads for thin plates, especially in the case of shear stresses, are drawn. Cracked plates under tension are finally considered in order to determine the most probable collapse mechanism among fracture, plastic flow or buckling and some failure-type maps are determined.     

Prebuckling and buckling analysis of variable angle tow plates with general boundary conditions

The concept of variable angle tow (VAT) placement is explored for enhancing the buckling resistance of composite plates subjected to axial compression loading. The problem is relatively difficult to solve because of varying stiffness properties and requires prior prebuckling analysis to determine the non-uniform stress variation followed by the buckling analysis of VAT plates. A stress function formulation for in-plane analysis and displacement formulation for buckling analysis was employed to derive the governing differential equations of VAT plates based on classical laminated plate theory. The Differential Quadrature Method (DQM) is applied to solve the differential equations. The novel aspect of the present work is the use of Airy’s stress function to model the prebuckling analysis of VAT plates which considerably reduces the problem size, computational effort and provides generality to model pure stress and mixed boundary conditions. DQM was applied first to solve the prebuckling problem of VAT plates subjected to cosine distributed compressive load/ uniform end shortening. Then, DQM was applied to solve the buckling problem of rectangular VAT plates subjected to axial compression under different plate boundary conditions. Results were validated with detailed finite element analysis and the relative accuracy and efficiency of the DQM approach is discussed.     

Curvature effects in the buckling of symmetrically-laminated rectangular plates with transverse shear deformation

It is shown that the effect of stress discontinuities from ply-to-ply must be taken into account when curvature terms are included along with shear deformation in the buckling analysis of rectangular, symmetrically-laminated plates. Such ply-stress discontinuities lead to curvature terms in the governing equations which differ considerably from those derived for homogeneous plates. Critical buckling loads are determined for orthotropic laminates subjected to biaxial inplane loading and for cylindrical bending of anisotropic plates subjected to uniaxial compression loading. Simply-supported boundary conditions are considered in conjunction with the rectangular, orthotropic laminate, while simply-supported and clamped boundaries are considered for the case of cylindrical bending of anisotropic plates. Numerical results indicate that the curvature terms have little effect on critical buckling loads for the laminates investigated. The effect of transverse shear deformation is shown to depend on the degree of boundary constraint, laminate stacking geometry, and inplane load ratio.     

Non-linear buckling analysis of tapered curved composite plates based on a simplified methodology

Very large number of load steps is required to determine the buckling load based on the non-linear analysis in which the stability limit load is calculated from the non-linear load–deflection curve. In the present work, a simplified methodology is developed to predict the stability limit load that requires the consideration of only two load steps. The stability limit loads of the tapered curved plates are calculated using this methodology. Based on the first-ply failure analysis and the simplified non-linear buckling analysis, the critical sizes and parameters of the tapered curved plates that will not fail before global buckling are determined. The stability limit loads calculated using the present simplified methodology are shown to have good agreement with that calculated based on the non-linear load–deflection curve using the conventional non-linear buckling analysis methodology.     

Buckling analysis of composite plates using moving least squares differential quadrature method

This work concerns with buckling and vibration analysis of composite plates based on a transverse shear theory. A numerical scheme is introduced to determine the angular frequencies and critical buckling loads of such plates. Moving least square differential quadrature method is employed to reduce the problem to that of eigen value problem. The accuracy and efficiency of the proposed scheme is examined with different computational characteristics, (radius of support domain, basis completeness order, and scaling factors). The obtained results agreed, at less execution time, with the previous ones. Further, a parametric study is introduced to investigate the influence of elastic and geometric characteristics, (Young's modulus gradation ratio, shear modulus gradation ratio, Poisson's ratio, loading parameter, and aspect ratio), of the composite on the values of critical buckling load, natural frequencies, and behavior of mode shape functions.