Design Sensitivity Analysis of Truss Structures with Elastoplastic Material*

ABSTRACT

A continuum-based design sensitivity analysis (DSA) method is presented for configuration (or layout) design of nonlinear structural systems with rate-independent elastoplastic material. Configuration design variables are characterized by shape and orientation changes of the structural component. A continuum-based shape DSA method that utilizes the material derivative of continuum mechanics is extended to account for effects of shape and orientation variations. The incremental analysis method, with updated Lagrangian formulation, is used to derive the design sensitivity for the nonlinear structural system.

To derive the design sensitivity, incremental energy and load forms are utilized. The first variations of energy and load forms and the static response with respect to configuration design variables are described using the material derivative. Direct differentiation is utilized to obtain the first variation of the performance measure explicitly in terms of variations of configuration design variables. With the consistent tangent stiffness matrix employed at the end of each load step to compute the design sensitivity, it is found that no iterations are necessary to compute design sensitivity. In addition, the linear design velocity is used to account for configuration design changes, with the velocity field being updated at each load step of the incremental analysis.     

A Bending-Gradient model for thick plates,Part II: Closed-form solutions for cylindrical bending of laminates

In the first part (Lebée and Sab, 2010a) of this two-part paper we have presented a new plate theory for out-of-plane loaded thick plates where the static unknowns are those of the Kirchhoff–Love theory (3 in-plane stresses and 3 bending moments), to which six components are added representing the gradient of the bending moment. The new theory, called Bending-Gradient plate theory is an extension to arbitrarily layered plates of the Reissner–Mindlin plate theory which appears as a special case when the plate is homogeneous. Moreover, we demonstrated that, in the general case, the Bending-Gradient model cannot be reduced to a Reissner–Mindlin model. In this paper, the Bending-Gradient theory is applied to laminated plates and its predictions are compared to those of Reissner–Mindlin theory and to full 3D (Pagano, 1969) exact solutions. The main conclusion is that the Bending-Gradient gives good predictions of deflection, shear stress distributions and in-plane displacement distributions in any material configuration. Moreover, under some symmetry conditions, the Bending-Gradient model coincides with the second-order approximation of the exact solution as the slenderness ratio L/h goes to infinity.     

A Bending-Gradient model for thick plates. Part I: Theory

This is the first part of a two-part paper dedicated to a new plate theory for out-of-plane loaded thick plates where the static unknowns are those of the Kirchhoff–Love theory (3 in-plane stresses and 3 bending moments), to which six components are added representing the gradient of the bending moment. The new theory, called the Bending-Gradient plate theory is described in the present paper. It is an extension to arbitrarily layered plates of the Reissner–Mindlin plate theory which appears as a special case of the Bending-Gradient plate theory when the plate is homogeneous. However, we demonstrate also that, in the general case, the Bending-Gradient model cannot be reduced to a Reissner–Mindlin model. In part two (Lebée and Sab, 2011), the Bending-Gradient theory is applied to multilayered plates and its predictions are compared to those of the Reissner–Mindlin theory and to full 3D Pagano’s exact solutions. The main conclusion of the second part is that the Bending-Gradient gives good predictions of both deflection and shear stress distributions in any material configuration. Moreover, under some symmetry conditions, the Bending-Gradient model coincides with the second-order approximation of the exact solution as the slenderness ratio L/h goes to infinity.     

复合材料层合厚板锯齿理论及有限元分析

对于较厚复合材料弯曲问题,已有锯齿型厚板理论最大误差超过35%。为了合理地分析较厚复合材料弯曲问题,发展了准确高效的锯齿型厚板理论。此理论位移变量个数独立于层合板层数,其面内位移不含有横向位移一阶导数,构造有限元时仅需C0插值函数,故称此理论为C0型锯齿厚板理论。基于发展的锯齿理论,构造了六节点三角形单元并推导了复合材料层合/夹层板弯曲问题有限元列式。为验证C0型锯齿厚板理论性能,分析了复合材料层合/夹层厚板弯曲问题,并与已有C1型锯齿理论对比。结果表明,本文的C0型锯齿厚板理论最大误差15%,比已有锯齿型厚板理论准确高效。     

An advanced higher-order theory for laminated composite plates with general lamination angles

This paper proposes a higher-order shear deformation theory to predict the bending response of the laminated composite and sandwich plates with general lamination configurations.The proposed theory a priori satisfies the continuity conditions of transverse shear stresses at interfaces.Moreover,the number of unknown variables is independent of the number of layers.The first derivatives of transverse displacements have been taken out from the inplane displacement fields,so that the C 0 shape functions are only required during its finite element implementation.Due to C 0 continuity requirements,the proposed model can be conveniently extended for implementation in commercial finite element codes.To verify the proposed theory,the fournode C 0 quadrilateral element is employed for the interpolation of all the displacement parameters defined at each nodal point on the composite plate.Numerical results show that following the proposed theory,simple C 0 finite elements could accurately predict the interlaminar stresses of laminated composite and sandwich plates directly from a constitutive equation,which has caused difficulty for the other global higher order theories.     

Mindlin 矩形微板的热弹性阻尼解析解

首次给出了四边简支的 Mindlin 矩形微板热弹性阻尼的解析解. 基于考虑一阶剪切变形的 Mindlin 板理论和单向耦合热传导理论建立了微板热弹性耦合自由振动控制微分方程. 忽略温度梯度在面内的变化,在上下表面绝热边界条件下求得了用变形几何量表示的温度场的解析解. 进一步将包含热弯曲内力的结构振动方程转化为只包含挠度振幅的四阶偏微分方程. 利用特征值问题之间在数学上的相似性,在四边简支条件下给出了用无阻尼 Kirchhoff 微板的固有频率表示的 Mindlin 矩形微板的复频率解析解,从而利用复频率法求得了反映热弹性阻尼水平的逆品质因子. 最后,通过数值结果定量地分析了剪切变形、材料以及几何参数对热弹性阻尼的影响 规律. 结果表明,Mindlin 板理论预测的热弹性阻尼小于 Kirchhoff 板理论预测的热弹性阻尼. 两种理论预测的热弹性阻尼之间的差值在临界厚度附近十分显著. 另外,随着微板的边/厚比增大,Mindlin 微板的热弹性阻尼最大值单调增大,而 Kirchhoff 微板的热弹性阻尼最大值却保持不变.      

A quadrilateral element based on refined global-local higher-order theory for coupling bending and extension thermo-elastic multilayered plates

In this paper a refined higher-order global-local theory is presented to analyze the laminated plates coupled bending and extension under thermo-mechanical loading. The in-plane displacement fields are composed of a third-order polynomial of global coordinate z in the thickness direction and 1,2–3 order power series of local coordinate ζ_k in the thickness direction of each layer, which is identical to the 1,2–3 global-local higher-order theory by Li and Liu [Li, X.Y., Liu, D., 1997. Generalized laminate theories based on double superposition hypothesis. Int. J. Numer. Methods Eng. 40, 1197–1212] Moreover, a second-order polynomial of global coordinate z in the thickness direction is chosen as transverse displacement field. The transverse shear stresses can satisfy continuity at interfaces, and the number of unknowns does not depend on the layer numbers of the laminate.Based on this theory, a quadrilateral laminated plate element satisfying the requirement of C¹ continuity is presented. By solving both bending and thermal expansion problems of laminates, it can be found that the present refined theory is very accurate and obviously superior to the existing 1,2–3 global-local higher-order theory. The most attractive feature of this theory is that the transverse shear stresses can be accurately predicted from direct use of constitutive equations without any post-processing method. It is also shown that the present quadrilateral element possesses higher accuracy.     

ANLYTICAL SOLUTION OF THERMOELASTIC DAMPING IN RECTANGULAR MINDLIN MICRO PLATES 1)

首次给出了四边简支的 Mindlin 矩形微板热弹性阻尼的解析解. 基于考虑一阶剪切变形的 Mindlin 板理论和单向耦合热传导理论建立了微板热弹性耦合自由振动控制微分方程. 忽略温度梯度在面内的变化,在上下表面绝热边界条件下求得了用变形几何量表示的温度场的解析解. 进一步将包含热弯曲内力的结构振动方程转化为只包含挠度振幅的四阶偏微分方程. 利用特征值问题之间在数学上的相似性,在四边简支条件下给出了用无阻尼 Kirchhoff 微板的固有频率表示的 Mindlin 矩形微板的复频率解析解,从而利用复频率法求得了反映热弹性阻尼水平的逆品质因子. 最后,通过数值结果定量地分析了剪切变形、材料以及几何参数对热弹性阻尼的影响 规律. 结果表明,Mindlin 板理论预测的热弹性阻尼小于 Kirchhoff 板理论预测的热弹性阻尼. 两种理论预测的热弹性阻尼之间的差值在临界厚度附近十分显著. 另外,随着微板的边/厚比增大,Mindlin 微板的热弹性阻尼最大值单调增大,而 Kirchhoff 微板的热弹性阻尼最大值却保持不变.     

Shear deformable bending solutions for nonuniform beams and plates with elastic end restraints from classical solutions

Summary The relationships of bending solutions between Timoshenko beams and Euler-Bernoulli beams are derived for uniform and non-uniform beams with elastic rotationally restrained ends. Extensions of these relationships for the cylindrical bending of Mindlin and Kirchhoff plates and for the bending of symmetrically laminated beams are also discussed. The new set of general relationships is useful because the more complex Timoshenko beam and Mindlin plate solutions may be readily obtained from their simpler Euler-Bernoulli beam and Kirchhoff plate solutions respectively, without much tedious mathematics. Received 16 March 1997; accepted for publication 26 November 1997     

考虑横法向热应变的C~0型整体-局部高阶层合梁理论

通过考虑横法向热变形,本文建立了预先满足层间应力连续的C0型整体-局部高阶层合梁理论,并用于分析复合材料层合梁热膨胀和热弯曲问题。虽然考虑了横法向应变,不增加额外的位移变量。此理论位移场不含有横向位移一阶导数,便于构造多节点高阶单元。基于虚功原理推导了复合材料层合梁平衡方程,并分析了简支多层复合材料梁热膨胀和热弯曲问题。数值结果表明,建立的模型能准确分析复合材料层合梁热膨胀和热弯曲问题,忽略横法向应变的理论分析热膨胀问题误差较大。     

Configuration Design Sensitivity Analysis of Kinematic Responses of Mechanical Systems∗

Abstract

A continuum-based configuration design sensitivity analysis method is presented for kinematically driven mechanical systems. Configuration design variable for mechanical systems are defined. The 3-1-3 Euler angle is used as the orientation design variable. The reassembly process for a mechanical system after a configuration design change is eliminated by imposing kinematic admissibility conditions of the velocity field. The direct differentiation method is used to derive design sensitivity equations. Numerical examples are presented to demonstrate the validity and effectiveness of the proposed method.     

考虑电场梯度的挠曲电纳米板弯曲性能分析

具有尺度依赖的挠曲电效应在器件的设计中扮演着越来越关键的角色, 研究人员在微纳米尺度多物理场分析中进行了大量工作. 基于考虑挠曲电和电场梯度效应的弹性介电材料非经典理论, 以二维纳米板为例, 通过理论建模, 分析纳米板在弯曲问题中的力?电耦合行为. 根据Mindlin假设给出板的位移场和电势场的一阶截断, 选取板的材料为立方晶体(m3m点群), 将广义三维本构方程代入到高阶应力、高阶偶应力、高阶电位移和高阶电四极矩的表达式中得到相应的二维本构方程, 利用弹性电介质变分原理得到板的控制方程和边界上的线积分等式, 分别将二维本构方程和边界上外法线的方向余弦代入, 得到板的高阶弯曲方程、高阶电势方程以及对应的四边简支边界条件. 利用四边简支矩形板的高阶弯曲方程、高阶电势方程和相应的边界条件, 根据Navier解理论, 求解纳米板的电势场, 重点分析电场梯度对板内一阶电势的影响. 数值计算结果表明: 电场梯度对纳米板中由挠曲电效应产生的一阶电势有削弱作用, 且材料参数g₁₁越大, 一阶电势受到的削弱越大; 同时电场梯度的存在消除了纳米板在受横向集中载荷作用时一阶电势的奇异性. 本文是对具有挠曲电效应和电场梯度效应的纳米板结构分析理论的一个扩展, 为微纳米尺度器件的结构设计提供参考.      

How to use reactive stresses to improve plate-theory approximations of the stress field in a linearly elastic plate-like body

All plate theories allow for an approximate evaluation of the stress field in the associated three-dimensional plate-like bodies. We here show how such an approximation can be bettered by an appropriate use of the reactive stress fields that maintain in the three-dimensional body the kinematical constraints implicit in the formulation of a given plate theory. In particular, we discuss transversely extensible, linearly elastic plates and Reissner–Mindlin plates, two theories where the built-in second-order internal constraints turn out to be of importance to improve the evaluation of the stress fields in the corresponding three-dimensional bodies. In addition to arguing our point of view in general, we work out a few explicit examples, both analytically and numerically.     

Thick Lévy plates re-visited

This paper is concerned with the bending problem of Lévy plates which are simply supported on two opposite edges with any combination of simply supported, clamped or free conditions at the remaining two edges. This study attempts to solve thick Lévy plate problems in a novel way by establishing bending relationships that allow the prediction of Mindlin plate results using the corresponding Kirchhoff solutions. Based on the concept of load equivalence, these relationships obviate the need for complicated thick plate analyses that involve significant computation time and effort. Numerical plate solutions are then determined from these relationships and the validity of these results is verified using other known results and those generated using the abaqus software. It is through this study that the only analytical Mindlin plate solutions by Cooke and Levinson (Int. J. Mech. Sci. 25 (1983) 207) are found to contain errors. In this study, it is found that there are important distinctions between the Mindlin and Reissner plate theories. These differences will also be substantiated by numerical comparison.     

应变梯度Mindlin板边值问题的讨论

实验和分子动力学计算结果表明,当材料/结构的特征尺寸降为微纳米量级时,他们将表现出明显的尺度效应,因此能否建立精确表征其力学行为的连续介质力学模型具有重要的理论和现实意义.尽管现有文献对非经典Mindlin板的力学行为进行了大量研究,但该模型的变分自洽的边值问题是近年来未攻克的科学问题之一.基于简化的应变梯度理论给出了各向同性Mindlin板应变能的表达式,通过变分原理和张量分析,得到了Mindlin板变分自洽的边值问题及其对应角点条件的位移微分表达式.本文Mindlin板模型的边值问题可退化为相应的Timoshenko梁和Kirchhoff板模型的边值问题,验证了本文结果的有效性.研究结果发现,该Mindlin板模型的控制方程是一个解耦后横向振动具有12阶的偏微分方程,因此需要每个板边提供6个边界条件.角点条件由双应力(double stress)产生,并与经典的剪力、弯矩和扭矩沿截面的法向梯度有关.本文首次澄清了应变梯度Mindlin板存在角点条件这一事实,所得的变分结果有望为其有限元法和伽辽金法等数值方法提供理论依据.     

Laser polarimeter for measuring angular displacement of bend bars

A two-channel differential polarimeter suitable for the continuous measurement of the angular displacement of a rigid body about a single axis is described. A prototype of this design was assembled to make angular displacement measurements on a bend-bar specimen during a fracture-mechanics experiment. A half-wave plate attached to the bend bar rotates the linear polarization of a He–Ne laser beam as the specimen is deformed during the test. The sensitivity of the prototype instrument was limited by inhomogeneity in the wave plate, and thus did not approach the theoretically predicted values for resolution on the order of 10^–7 deg. Based on preliminary results, however, this technique shows promise for quasistatic as well as dynamic bend-bar testing.     

钢筋混凝土梁冲击动力响应和破坏模式转化试验研究

随着结构配置和冲击能量等主要影响因素的变化,钢筋混凝土梁的冲击动力响应和破坏模式会发生转化。开展不同配置的钢筋混凝土梁的落锤冲击试验,综合测量获得冲击力、支座反力、钢筋与混凝土应变、冲击局部与结构整体变形等参数,重点分析不同混凝土强度、不同纵筋/箍筋配置以及不同冲击速度对钢筋混凝土梁的动力响应以及破坏模式的影响规律。试验表明：低速撞击下钢筋混凝土梁的位移峰值、残余位移随冲击速度的提高而增大,均与冲击动能与极限静承载力之比存在近似线性关系;混凝土强度越高、纵筋配筋率越高,相同冲击条件下梁所受的撞击力峰值越大,但整体位移响应越小;配箍率的变化对结构的局部响应和整体响应的影响均较小;结构受到撞击时剪切效应在前,弯曲效应在后,斜裂缝先于垂直裂缝出现;依据结构的破坏极限状态,判断梁在冲击作用下存在的弯曲破坏、弯剪破坏、剪切破坏和冲切破坏等4种破坏模式,结果表明：相同结构配置条件下,随冲击速度的不断提高,钢筋混凝土梁由弯曲破坏向弯剪破坏、剪切破坏和冲切破坏转化;冲击速度相同时,提高混凝土强度、配箍率或降低纵向钢筋配筋率,梁的破坏模式逐步由冲切、剪切破坏向弯曲破坏模式转化。结构的冲击破坏模式及其转化规律能够为结构的抗撞设计与防护提供参考。

     

On canonical bending relationships for plates

In recent years, a series of papers have appeared on algebraic relationships between the solutions (e.g., deflections, buckling loads and frequencies) of a given higher-order plate theory and the classical plate theory. The bending relationships, for example, can be used to generate the transverse deflection of a plate according to the particular higher-order theory from the known deflection of the same problem according to the classical plate theory. In the present study relationships between the bending solutions of several higher-order plate theories and the classical plate theory are obtained in a canonical form (i.e., one set of relationships contain several theories and they can be specialized to a specific theory by assigning values to the constants appearing in the relationships). Numerical examples of bending solutions for rectangular plates with various boundary conditions are presented to show how the relations can be used to determine the deflections and bending moments for various theories. The relationships are validated by comparing the numerical results obtained using the relationships for the Mindlin plate theory against those computed using the ABAQUS finite element program.     

Higher-order shear deformable theories for flexure of sandwich plates—Finite element evaluations

A simple isoparametric finite element formulation based on a higher-order displacement model for flexure analysis of multilayer symmetric sandwich plates is presented. The assumed displacement model accounts for non-linear variation of inplane displacements and constant variation of transverse displacement through the plate thickness. Further, the present formulation does not require the fictitious shear correction coefficient(s) generally associated with the first-order shear deformable theories. Two sandwich plate theories are developed: one in which the free shear stress conditions on the top and bottom bounding planes are imposed and another, in which such conditions are not imposed. The validity of the present development(s) is established through, numerical evaluations for deflections/stresses/stress-resultants and their comparisons with the available three-dimensional analyses/closed-form/other finite element solutions. Comparison of results from thin plate. Mindlin and present analyses with the exact three-dimensional analyses yields some important conclusions regarding the effects of the assumptions made in the CPT and Mindlin type theories. The comparative study further establishes the necessity of a higher-order shear deformable theory incorporating warping of the cross-section particularly for sandwich plates.     

Polygon-wise constant curvature natural element approximation of Kirchhoff plate model

A polygon-wise constant curvature natural element approximation is presented in this paper for the numerical implementation of the abstract Kirchhoff plate model. The strict C¹-continuity requirement in the displacement field is relaxed by converting the area integral of the curvatures into the boundary integral along the Voronoi boundary. Curvatures and bending moments are assumed to be constant within each Voronoi polygon, and the Voronoi-polygon-wise constant curvatures are derived in a selective manner for the sake of the imposition of essential boundary conditions. The numerical results illustrating the proposed method are also given.