The geometric stiffness of thick shell triangular finite elements for large rotations

This paper is concerned with the development of the geometric stiffness matrix of thick shell finite elements for geometrically nonlinear analysis of the Newton type. A linear shell element that is comprised of the constant stress triangular membrane element and the triangular discrete Kirchhoff Mindlin theory (DKMT) plate element is ‘upgraded’ to become a geometrically nonlinear thick shell finite element. Perturbation methods are used to derive the geometric stiffness matrix from the gradient, in global coordinates, of the nodal force vector when stresses are kept fixed. The present approach follows earlier works associated with trusses, space frames and thin shells. It has the advantage of explicitness and clear physical insight. A special procedure, tailored to triangular elements is used to isolate pure rotations to enable stress recovery via linear elastic constitutive relations. Several examples are solved. The results compare well with those available in the literature. Copyright © 2005 John Wiley & Sons, Ltd.     

A new finite element scheme for instability analysis of thin shells

This paper describes a new finite element scheme for the analysis of instability phenomena of arbitrary thin shells. A computationally efficient procedure is proposed for calculating the non-linear stiffness and tangential stiffness matrices for a doubly-curved quadrilateral element defined by co-ordinate lines. The essential feature is the explicit addition of the non-linear terms into the rigid-body motion of the element. Thus the non-linear and tangential element stiffness matrices can easily be generated by transforming the generalized element stiffness matrix for linear analysis, and the non-linear terms of these matrices are separated into a number of component terms multiplied by the rigid-body rotations. These component terms can be stored permanently and used to calculate efficiently the non-linear and tangential stiffness matrices at each iteration. Illustrative examples are presented which confirm the validity of the present approach in the analysis of instability phenomena of thin plates and shells.     

A resultant 8-node solid-shell element for geometrically nonlinear analysis

A new resultant force formulation of 8-node solid element is presented for the linear and nonlinear analysis of thin-walled structures. The global, local and natural coordinate systems were used to accurately model the shell geometry. The assumed natural strain methods with plane stress concept were implemented to remove the various locking problems appearing in thin plates and shells. The correct warping behavior in the very thin twisted beam test was obtained by using an improved Jacobian transformation matrix. The 2 × 2 Gauss integration scheme was used for the calculation of the element stiffness matrix. From the computational viewpoint, the present solid element is very efficient for a large scale of nonlinear modeling. A lot of numerical tests were carried out for the validation of the present 8-node solid-shell element and the results are in good agreement with references.An erratum to this article can be found at     

钢管混凝土结构材料非线性的一种有限元分析方法

为了更简单地考虑梁单元的材料非线性受力性能,把断面广义力和广义应变的概念运用于单元分析中,将单元的弹塑性刚度矩阵分离为弹性刚度矩阵和塑性刚度矩阵。这样,梁单元的变形可以由弹性变形和塑性变形简单地迭加,结构内力可通过弹性应变能的斜率(弹性刚度矩阵)与位移的乘积求得,从而在增量-迭代计算时可较准确且较快地计算出结构变形后的不平衡力。应用这一计算方法,推导了基于纤维模型的三维梁单元的钢管混凝土结构的有限元基本公式,并将其植入能考虑几何非线性的三维梁单元非线性计算程序NL_Beam3D中以计算结构的双重非线性问题。算例分析表明该方法和程序能较准确地反映钢管混凝土结构的双重非线性特性。     

基于共旋三角形厚薄通用壳元的几何非线性分析

基于共旋列式方法发展了一种用于复合材料层合板结构几何非线性分析的简单高效的三结点三角形平板壳元。该壳元由具有面内转动自由度的广义协调膜元GT9与假设剪切应变场和假设单元转角场的广义协调厚薄通用板元TMT组合而成。为避免薄膜闭锁而采用单点积分计算与薄膜应变有关的项, 同时增加一个稳定化矩阵以消除单点积分导致的零能模式。基于层合板一阶剪切变形理论, 给出了考虑层合板具体铺层顺序的修正的横向剪切刚度, 使该壳元可用于中厚层合板结构的分析。由于共旋列式大转动小应变的假设, 共旋列式内核的几何线性的单元刚阵可仅计算一次而保存下来用于整个几何非线性求解的过程以提高计算效率。数值算例表明提出的壳元进行包括复合材料层合板结构的厚薄壳结构的几何非线性分析的精度高且效率高。     

弹性膜结构几何非线性分析的刚体准则法

提出了一种新型弹性空间膜结构几何非线性分析方法。根据刚体准则的思想,初始受力平衡的单元在经历刚体位移后,其单元结点力方向随单元发生转动而大小不变,单元仍保持平衡。建立了新型三角形空间膜单元,该膜单元由三根空间杆件组成铰接三角形,并在中间张拉薄膜而成,杆件的材料与薄膜相同。所建立的空间膜单元的整体位形由杆单元空间铰接三角形确定,而膜单元的有限弹性变形由内部张拉的薄膜变形确定。由满足刚体准则的杆单元几何刚度矩阵推导了空间膜单元的几何刚度矩阵。根据刚体准则思想,认为膜单元在变形过程中,其刚体位移对其整体变形的贡献较大,而单元的弹性变形贡献较小。采用更新的拉格朗日格式的增量迭代法,在分析的每个阶段充分考虑刚体转动效应,利用小变形线性化理论处理自然变形的剩余效应。该方法几何刚度矩阵推导简单,无需引入对单元大变形的人为假定,可容易地退化为平面膜单元,增量迭代计算过程充分考虑刚体准则,对若干典型空间膜结构算例的分析及与已有方法的比较,验证了所建单元与方法的准确性以及计算效率。     

A simple approach to bifurcation and limit point calculations

A simple approach is proposed to calculate the bifurcation and limit points of structures, talcing into account the pre-unstable behaviour, by the finite element method. The approach is as follows: at each load step, the triangular factorization of the tangent stiffness matrix is checked to determine if the matrix is positive definite or not. When the tangent stiffness matrix is positive definite at a certain load step and non-positive definite at the next load step, the structure is considered to become unstable between the two load steps and an eigenproblem is constructed based on the difference of the tangent stiffness matrices at the two load steps. The critical load and corresponding mode of the structure are then derived from solving the eigenproblem. The proposed procedure is simple and economical, and it can be easily incorporated into a conventional geometric nonlinear analysis computer program. It is implemented in the ADINA program and some sample calculations are shown.     

多边形比例边界有限元非线性高效分析方法

比例边界有限元法作为一种高精度的半解析数值求解方法,特别适合于求解无限域与应力奇异性等问题,多边形比例边界单元在模拟裂纹扩展过程、处理局部网格重剖分等方面相较于有限单元法具有明显优势。目前,比例边界有限元法更多关注的是线弹性问题的求解,而非线性比例边界单元的研究则处于起步阶段。该文将高效的隔离非线性有限元法用于比例边界单元的非线性分析,提出了一种高效的隔离非线性比例边界有限元法。该方法认为每个边界线单元覆盖的区域为相互独立的扇形子单元,其形函数以及应变-位移矩阵可通过半解析的弹性解获得;每个扇形区的非线性应变场通过设置非线性应变插值点来表达,引入非线性本构关系即可实现多边形比例边界单元高效非线性分析。多边形比例边界单元的刚度通过集成每个扇形子单元的刚度获取,扇形子单元的刚度可采用高斯积分方案进行求解,其精度保持不变。由于引入了较多的非线性应变插值点,舒尔补矩阵维数较大,该文采用Woodbury近似法对隔离非线性比例边界单元的控制方程进行求解。该方法对大规模非线性问题的计算具有较高的计算效率,数值算例验证了算法的正确性以及高效性,将该方法进行推广,对实际工程分析具有重要意义。     

基于共旋法与稳定函数的几何非线性平面梁单元

建立一个准确、高效的几何非线性梁单元对于描述杆系结构的非线性行为至关重要。该文基于共旋坐标法和稳定函数提出了一种几何非线性平面梁单元。该单元在形成中把变形和刚体位移分开,局部坐标系内采用稳定函数以考虑单元P-δ效应的影响,从局部坐标系到结构坐标系的转换则采用共旋坐标法以及微分以考虑几何非线性,给出了几何非线性平面梁单元在结构坐标系下的全量平衡方程和切线刚度矩阵;在此基础上根据带铰梁端弯矩为零的受力特征,导出了能考虑梁端带铰的单元切线刚度矩阵表达式。通过多个典型算例验证了算法与程序的正确性、计算精度和效率。     

空间钢框架结构的改进双重非线性分析

为了探讨三维结构的高等分析方法,给出基于UL法的严格三维单元虚功增量方程,详细推导了考虑剪切变形影响的三维梁柱单元的几何非线性切线刚度矩阵和基于三维单元简化塑性区模型的双重非线性刚度方程,并编制空间钢框架结构的双重非线性分析程序。使用包括几何与材料双重非线性的数值算例来验证方法和计算机程序的可行性、精确度与有效性。利用该程序,只需一个单元/构件即可准确预测空间钢框架结构的极限承载力与失稳模态,提高了结构的非线性分析效率。     

基于界面单元的子结构协调技术

该文提出一种用于协调子结构的界面单元方法。基于广义变分原理,将两子域的刚度与界面单元刚度组装成耦合结构的整体刚度矩阵,求解新的平衡方程即可得到各个耦合子域的位移。界面单元的意义是对子域引入边界力,并建立边界上平衡关系和位移协调关系。该文利用悬臂梁单轴受拉案例验证了界面单元方法的精确性。为了使得界面单元能够应用到子结构混合试验中,引入静力凝聚与BFGS方法,这样只需通过提取边界上的力与位移即可实现多子域不共节点的边界协调问题。该文最终以悬臂梁案例验证了界面单元在解决非线性静、动力加载工况下的正确性。     

基于场一致性的可带铰空间梁元几何非线性分析

虽然关于几何非线性分析的空间梁单元研究成果较多,但这些单元均是基于几何一致性得到的单元刚度矩阵,而基于场一致性的单元研究则较少,该文基于局部坐标系(随转坐标系)下扣除结构位移中的刚体位移得到的结构变形与结构坐标系下的总位移的关系,直接利用微分方法导出两者增量位移之间的关系,再基于场一致性原则,最终获得空间梁单元在大转动、小应变条件下的几何非线性单元切线刚度矩阵,在此基础上根据带铰梁端受力特征,导出了能考虑梁端带铰的单元切线刚度矩阵表达式,利用该文的研究成果编制了程序,对多个梁端带铰和不带铰的算例进行了空间几何非线性分析,计算结果表明这种非线性单元列式的正确性,实用价值较强。     

Geometrically nonlinear static and transiently dynamic behavior of laminated composite plates based on a higher order displacement field

In this work a finite element analysis of geometrically nonlinear static and transiently dynamic behavior of laminated composite plates is presented. A higher order displacement field allowing both transverse shear and transverse normal strains is adopted through the analysis. The mass matrix, linear stiffness matrix, and nonlinear stiffness matrix for a typical discretized element are derived and presented. Several numerical examples are included, where comparisons among linear and nonlinear theories, the current displacement field and other displacement fields are made. Effects of laminate thickness, stacking sequences and stacking angles are also studied. One interesting result is that even for symmetric cross ply, linear theory can not provide accurate enough results.     

任意截面预应力混凝土细长柱的非线性分析

提出了轴力和双向弯曲作用下任意截面混凝土和预应力混凝土细长柱的非线性有限元计算模型。分析时既考虑了由单元变形和轴力二次矩引起的几何非线性效应,也考虑了由材料非线性应力应变关系和截面刚度矩阵引起的材料非线性效应。推导了非线性全过程分析的标准有限元公式,得到的单元刚度矩阵可分割成三个子矩阵,分别反映了材料非线性、材料非线性和单元大位移的耦合、轴力二次矩等三种不同的非线性作用效应。计算分析结果和试验结果吻合较好。     

基于静力凝聚的高精度变截面梁单元及其几何非线性分析研究

基于Euler-Bernoulli梁单元基本假定,通过静力凝聚获得截面特性沿单元轴向连续变化的变截面梁单元高精度刚度矩阵,并提出一种基于随动坐标法求解变截面梁杆结构大位移、大转动、小应变问题的新思路。首先依据插值理论和非线性有限元理论推导出三节点变截面梁单元的切线刚度矩阵,然后使用静力凝聚方法消除中间节点自由度,从而得到一种新型非线性两节点变截面梁单元。结合随动坐标法,在变形后位形上建立随动坐标系,得到变截面梁单元的大位移全量平衡方程。实例计算表明,该新型变截面梁单元具有较高的计算精度,可应用于变截面梁杆系统大位移几何非线性分析。     

Refined triangular discrete Mindlin flat shell elements

In this paper flat shell elements are formed by the assemblage of discrete Mindlin plate elements RDKTM and either the constant strain membrane element CST or the Allmans membrane element with drilling degrees of freedom LST. The element RDKTM is a robust Mindlin plate element, which can perform uniformly thick and thin plate bending analysis. It also passes the patch test for thin plate bending, and its convergence for very thin plates can be ensured theoretically. The singularity of the stiffness matrix and membrane locking are studied for the present elements. Numerical examples are presented to show that the present models indeed possess properties of simple formulations, high accuracy for thin and thick shells, and it is free from shear locking for thin plate/shell analysis.     

Finite rotation exact geometry solid-shell element for laminated composite structures through extended SaS formulation and 3D analytical integration

A nonlinear exact geometry hybrid-mixed four-node solid-shell element using the sampling surfaces (SaS) formulation is developed for the analysis of the second Piola-Kirchhoff stress that extends the authors' finite element (Int J Numer Methods Eng. 2019;117:498-522) to laminated composite shells. The SaS formulation is based on choosing inside the layers the arbitrary number of SaS parallel to the middle surface and located at Chebyshev polynomial nodes in order to introduce the displacements of these surfaces as basic shell unknowns. The external surfaces and interfaces are also included into a set of SaS. The proposed hybrid-mixed solid-shell element is based on the Hu-Washizu variational principle and is completely free of shear and membrane locking. The tangent stiffness matrix is evaluated by efficient three-dimensional (3D) analytical integration. As a result, the developed exact geometry solid-shell element exhibits a superior performance in the case of coarse meshes and allows the use of load increments, which are much larger than possible with existing displacement-based solid-shell elements. It could be useful for the 3D stress analysis of thick and thin doubly curved laminated composite shells because the SaS formulation gives the possibility to obtain the 3D solutions with a prescribed accuracy.     

Refined triangular discrete Kirchhoff plate element for thin plate bending,vibration and buckling analysis

A refined triangular discrete Kirchhoff thin plate bending element RDKT which can be used to improve the original triangular discrete Kirchhoff thin plate bending element DKT is presented. In order to improve the accuracy of the analysis a simple explicit expression of a refined constant strain matrix with an adjustable constant can be introduced into its formulation. The new element displacement function can be used to formulate a mass matrix called combined mass matrix for calculation of the natural frequency and in the same way a combined geometric stiffness matrix can be obtained for buckling analysis. Numerical examples are presented to show that the present methods indeed, can improve the accuracy of thin plate bending, vibration and buckling analysis. © 1998 John Wiley & Sons, Ltd.     

Flexural response to impulsive loads of a fluid-saturated thin poroelastic circular plate

Biot's poroelastic theory is used with classic plate theory and plane stress theory to determine the constitutive relationships for a thin poroelastic plate. The dynamic equations for the thin poroelastic plate are derived from the extended Hamilton's principle. The dynamic equations are then transformed to frequency domain and Galerkin's finite element method is used to derive the stiffness matrix of a triangular plate element. When impulsive loads and elastic boundary conditions are applied, the finite element frequency domain analysis for the thin poroelastic plates is achieved. Vibration behavior of thin elastic and poroelastic circular plates is accurately predicted.     

Application of the boundary-domain element method to the pre/post-buckling problem of von Kármán plates

Based on the BEM formulations for the finite deflection problem of von-Kármán-type plates, this paper presents an incremental boundary-domain element method for the pre/post-buckling problem of thin elastic plates. As the governing equations involve the coupled in-plane and out-of-plane deformations as the nonlinear terms, the boundary integral equations are formulated in terms of the increment by using the fundamental solutions for the linear parts of the differential operators. Some of the innovations are made in order to improve the accuracy and accelerate the convergence of the solution procedure. The load-incrementation method and also the arc-length-incrementation method are employed for each incremental step. Numerical analysis is carried out and the results are compared with the available analytical solutions to demonstrate the effectiveness of the proposed method.