A fast multipole boundary element method for solving the thin plate bending problem

A fast multipole boundary element method (BEM) for solving large-scale thin plate bending problems is presented in this paper. The method is based on the Kirchhoff thin plate bending theory and the biharmonic equation governing the deflection of the plate. First, the direct boundary integral equations and the conventional BEM for thin plate bending problems are reviewed. Second, the complex notation of the kernel functions, expansions and translations in the fast multipole BEM are presented. Finally, a few numerical examples are presented to show the accuracy and efficiency of the fast multipole BEM in solving thin plate bending problems. The bending rigidity of a perforated plate is evaluated using the developed code. It is shown that the fast multipole BEM can be applied to solve plate bending problems with good accuracy. Possible improvements in the efficiency of the method are discussed.     

Plate bending for frame analysis by coupling method of singular integral operators method with finite elements

The coupling of the Singular Integral Operators Method (SIOM) and the Finite Element Method (FEM) is proposed and investigated for the solution of plate bending problems. Such combination of the above two numerical methods gives more accurate results for the solution of plate bending problems in structural analysis. The plates are basic units of building frame structures, as well as to underground structures and thus the determination of bending moments around the plates is very important for the analysis of all kinds of structures. An application of plate bending problems is finally given to the determination of the bending moments in a rectangular plate with a rectangular hole in the center of the plate.     

基于平面弹性-板弯曲比拟理论的Wilson型板弯曲单元

应用平面弹性板弯曲比拟关系,可以避开1c连续性的困难,为板单元的构造提供了一种新的途径。这一新方法已成功地将一些平面弹性协调单元转化为新型板弯曲单元。根据比拟理论将著名的平面弹性Wilson元QM6转化为板弯曲单元,从而将新方法应用到平面弹性非协调元。单元构造简单,数值结果表明具有很好的收敛性和精度。     

应用Pian-Sumihara杂交列式的板弯曲模拟元

把典型的平面弹性杂交应力元-Pian-Sumihara 元-转化为板弯曲单元,从而初步探讨了将平面弹性杂交元转化为板弯曲单元的方法。应用板弯曲多类变量变分原理和弯矩函数空间中的Pian-Sumihara 列式,再通过基于平面弹性-板弯曲相似性的单元转化过程,得到一个四节点八自由度板弯曲位移元。该单元为显式单元,计算量少。数值结果表明该单元能通过常曲率分片试验,收敛稳定并具有较好的精度。     

基于平面弹性-板弯曲比拟理论的Wilson型平板壳单元

平面弹性—板弯曲比拟关系为充分利用现有的平面弹性单元构造新型板弯曲单元提供了有效的途径。按膜、板两部分平行列式的观点,新方法还可以利用平面弹性单元构造性能良好的新型平板壳单元。按此观点根据著名的平面弹性Wilson元(QM6)的列式方式构造出一个新的十六自由度平板壳单元。单元构造简单,无虚假自由度,数值结果表明具有很好的收敛性和精度。     

Analytical Solution for an Orthotropic Elastic Plate Containing Cracks

The problem of estimating the bending stress distribution in the neighborhood of a crack located on a single line in an orthotropic elastic plate of constant thickness subjected to bending moment or twisting moment is examined. Using classical plate theory and integral transform techniques, the general formulae for the bending moment and twisting moment in an elastic plate containing cracks located on a single line are derived. The solution is obtained in a closed form for the case in which there is a single crack in an infinite plate and the results are compared with those obtained from the literature.     

一个基于膜板比拟理论的四边形双线性薄板单元

平面弹性与板弯曲的相似性理论为构造薄板单元提供了一条有效的新途径。根据这一理论,现有的平面弹性单元原则上可以转化为板弯曲单元。从平面弹性四节点双线性等参元Q4出发,根据相似性理论构造出一个新的四边形八自由度双线性薄板单元。该单元构造简单,节点自由度少,可以视为最简单的四边形薄板单元。数值结果表明,该单元能通过分片试验,满足坐标不变性,具有良好的收敛性和精度。是一个良好的低阶薄板单元。     

钢筋混凝土板弯曲问题的各向同性化域外奇点解法

本文提出了求解钢筋混凝土板弯曲问题的各向同性化域外奇点法。引入简单的坐标变换，将该问题转化成相应的各向同性板的弯曲问题，利用后者的简单格林函数，按域外奇点法求解。算例表明，这种方法简单，计算时间短，精度高。     

Stochastic behavior of Mindlin plate with uncertain geometric and material parameters

The Mindlin assumption in the plate bending allows the shear deformation together with the bending one. The constitutive relation for shear behavior, however, has different order of contribution from the plate thickness t than that in bending. Therefore, when we have concern on the stochastic behavior of Mindlin plate taking into account the uncertain plate thickness, the bending and shear parts have to be handled in different ways. Furthermore, if the concern is put on the multiple uncertain parameters such as between elastic modulus and plate thickness, the influence of correlation between these parameters also has to be resolved. In this study, a formulation for stochastic analysis is given in the context of the weighted integral method for Mindlin plate with uncertainties in elastic modulus and plate thickness. The effects of correlation between these parameters on the structural behavior are examined. In addition, the stability of response variability in terms of the aspect ratio of a plate is investigated. Several square plates with various boundary conditions subjected to uniformly distributed or concentrated loads are taken as examples, and the results are compared with those of precedent works as well as those of classical Monte Carlo simulation.     

Stress intensity factors for curvilinear cracks loaded by bending and torsional moments

The geometry of a thin flat plate containing a curvilinear crack of finite size is investigated. The plate is loaded by uniform bending and torsional moments at infinite distance from the crack contour, while distributed normal bending moments and torques exist at the upper and lower crack surfaces. The bending stress intensity factors for the curvilinear crack are calculated on the basis of classical plate theory. The displacements and internal moments are represented by two complex analytic functions. Extra conditions are imposed to ensure the univalence of the displacements, which is not evident because the plate comprises a multiply connected domain due to the presence of the crack. A linearization with respect to the crack-curvature function has been performed and the bending stress intensity factors are calculated as the first-order solutions for slightly curved cracks. The results are illustrated with a few examples, such as uniform loading configurations and the geometry of a crack along a circular arc. The loading of thin flat plates by a combination of tensile forces and bending moments is also investigated. In analogy with the variation of the stress components over the cross section of the plate, two combined stress intensity factors are introduced having the same dependence on the perpendicular coordinate and being related to the symmetric and anti-symmetric stress intensity factors of the separate plane-stress and bending problems. The resulting energy release rate is shown to be in full agreement with known results in the literature.     

Bending mode effect on sensitivity of plate surface acoustic wave pressure sensors

The bending mode effect on sensitivity of pressure sensors operating with surface acoustic waves (SAW) propagating over the surface of a plate is analyzed using Tiersten's perturbation integral for frequency shifts in a piezoelectric resonator due to the presence of initial fields. For a plate subjected to nonpure bending, the SAW speed, whose shift in response to pressure determines the sensitivity, can be defined locally through the local variations of the effective material constants. Three common bending modes of pure bending, bending under a concentrate load, and bending under a uniformly load are analyzed and compared.     

A finite element alternating approach for the bending analysis of thin cracked plates

Based on the classical plate theory, the analytical solution for an infinte thin plate containing a crack subjected to arbitrary symmetric bending moments on the crack surfaces is first derived. Using this solution, an efficient and accurate finite element alternating procedure is then devised to deal with symmetric plate bending problems with single or multiple cracks. The interaction effect among cracks and the influence of the geometric boundaries on the calculation of bending stress intensity factors are also presented in detail. Several numerical examples are solved to demonstrate the validity of the approach.     

Bending of simply-supported,antisymmetrically laminated rectangular plate under transverse loading

The deformation of antisymmetric angle-ply laminated plate under transverse loading is analysed using the bending theory of laminated plate presented by Ren.^1,2 By expanding the load in a double Fourier series, a closed-form solution is obtained for special types of simply-supported boundary condition. The results are compared with similar results from the classical laminated plate theory and Mindlin theory. This shows that the theory presented by Ren is suitable not only for cylindrical bending but also for general bending problems.     

动物骨折治疗用金属骨板局部增量成形新工艺

目的目前动物骨折常用的锁定骨板内固定技术(Point-contact Reconstruction Compress Locking, PRCL)需要采用多个工具配合手动完成骨板成形,针对该过程中精度不可控、效率低等问题,提出一种弯扭复合成形模具,发展一种局部增量成形金属骨板的方法。方法 PRCL骨板固定治疗中,为了贴合受伤骨骼,治疗前骨板需经过面内弯曲、弯曲以及扭转3类变形。通过调整弯扭复合成形模具的空间位置及模具不同组成部分的相对位置,实现不同区域内不同变形量的面内弯曲、弯曲或扭转。应用数值方法分析验证弯扭复合成形模具及成形方法的适用性,基于DEFORM软件建立工业纯钛TA2骨板局部增量成形过程有限元模型,分析具有两个成形区的TA2骨板局部增量成形特征。结果塑性变形仅发生在复合模具附近,对已变形区无影响,会引起未成形区的刚性位移;骨板长度方向受力小于其宽度和厚度方向受力,面内弯曲需要较大的成形载荷。结论所发展的模具和方法可实现预期的骨板成形,也适用于其他PRCL金属骨板的成形。     

Kirchhoff型裂纹板的边界元法

本文用的复变函数理论,导出了含裂板弯曲问题的基本解。该基本解满足自由裂纹的边界条件。将其引入直接或间接积分方程中,只要对板的外边界进行离散,就可计算有限尺寸裂纹板的弯曲问题。算例表明,本文所得到的基本解用以求解裂纹板弯曲问题划分的单元较少,精度较高。本文的方法还可用以求解含有形状比较复杂的裂纹或孔洞板弯曲问题的基本解。     

弯曲矩形板的广义位移解及其边界值

本文应用功的互等定理,给出了弯曲矩形板的广义位移解及其边界值.从广义位移解可导出在各种载荷作用下具有各种边界条件矩形板的弯曲位移公式.本文导出的广义位移解的边界值是求解各种复杂边界条件矩形板弯曲的理论基础.     

高阶剪切变形板理论下FG-GRC板的屈曲和弯曲分析

基于三阶剪切变形板理论(TSDPT)和正弦剪切变形板理论(SSDPT),研究了功能梯度石墨烯增强复合材料(FG-GRC)板的屈曲和弯曲行为,并通过与一阶剪切变形板理论(FSDPT)计算结果的比较,分析了TSDPT、SSDPT与FSDPT在FG-GRC板屈曲和弯曲力学行为研究过程中的差异。材料的有效杨氏模量通过修正的Halpin-Tsai微观力学模型估算,有效泊松比通过混合律确定。利用最小势能原理推导出了包含五个未知量的控制方程,并获得了简支FG-GRC矩形板弯曲挠度和临界屈曲载荷Navier形式的解析解。数值结果表明：与TSDPT和SSDPT相比,FSDPT明显高估了FG-X型FG-GRC板的临界屈曲载荷而明显低估了其弯曲挠度,且略微低估了FG-O型FG-GRC板的临界屈曲载荷而略微高估了其弯曲挠度,而UD型和FG-A型FG-GRC板在三种理论下的计算结果几乎完全一致;TSDPT和SSDPT在计算FG-GRC板的弯曲挠度和临界屈曲载荷时结果十分相近;当板的总层数NL小于10层～15层时,弯曲载荷比率和临界屈曲载荷比率的变化非常显著,当总层数NL超过10层～15层时,弯曲载荷比率和临界屈...     

功能梯度Levinson圆板弯曲解的均匀化和经典化表示

基于Levinson三阶剪切变形理论,研究了材料性质沿厚度任意连续变化的功能梯度材料圆板的轴对称弯曲问题。首先,建立了功能梯度材料圆板在Levinson板理论下轴对称弯曲问题位移形式的控制微分方程,其中考虑了拉-弯耦合和三阶剪切变形效应。然后,利用载荷等效关系以及均匀板的经典理论控制微分方程,导出功能梯度圆板在Levinson剪切变形理论下弯曲解与经典理论下均匀圆板的挠度之间的解析转换关系,给出了转换系数的计算公式。由此,可将功能梯度材料圆板在Levinson三阶剪切理论下的弯曲问题转化为相应均匀薄圆板在经典理论下的弯曲问题求解,以及转换系数的计算问题。     

蜂窝夹层修理结构的弯曲性能试验分析

采用四点弯加载方式研究分析了含损伤蜂窝夹层修理结构的弯曲性能,该夹层结构由碳纤维增强的聚合物面板和蜂窝芯子组成。进一步分析了挖补斜度、挖补方式、损伤程度、修理设备和修理材料对修理板弯曲性能的影响。研究表明,修理板的破坏模式可分为补片边缘折断、补片中面折断和胶层破坏三种,相同破坏模式修理板的名义弯曲强度相近,其中前两种破坏模式修理板的名义弯曲强度与完好板相近,而第三种破坏模式修理板的名义弯曲强度相对较低。所有修理板的名义弯曲强度恢复率基本处于95%以上,同时修理后抗弯刚度也满足修理准则。     

钢‒CFRP异质复合B柱的弯曲实验分析

目的 为提高B柱的抗弯性能,通常会在B柱上额外焊接补丁板,但同时也增加了B柱的重量。在原始B柱上铺设碳纤维增强复合材料(CFRP),获得钢?CFRP异质复合B柱,取消B柱的钢制加强板,实现B柱的轻量化。方法 通过热冲压制备原始B柱及带补丁板B柱,并以原始B柱为凹模,采用真空袋压工艺制备钢?CFRP异质复合B柱。基于2018版C?NCAP侧面碰撞实验要求,设计B柱三点弯曲夹具,进行原始B柱、带补丁板B柱及钢?CFRP异质复合B柱的三点弯曲实验,并对其重量及弯曲性能进行分析。结果 原始B柱重量4.1 kg,三点弯曲实验测得其刚度为0.763 kN/mm,最大载荷为21.59 kN,平均力为14.52 kN;带补丁板B柱质量为5.6 kg,三点弯曲实验测得其刚度为1.095 kN/mm,最大载荷为31.08 kN,平均力为18.38 kN;钢?CFRP异质复合B柱总质量4.7 kg,三点弯曲试验测得其刚度为1.071 kN/mm,最大载荷为31.76 kN,平均力为19.58 kN。结论 在保持刚度、最大载荷及平均力等弯曲力学性能不变的前提下,相对于带补丁板B柱,钢?CFRP异质复合B柱可以减轻质量0.9 kg,并且吸能更优,实现了B柱的轻量化。