柔性框架结构动力非线性分析的刚体准则法

大跨、高层等柔性结构,其动力响应往往表现出大位移、大转动等非线性特征。动力非线性问题的分析关键在于运动方程的高效稳定求解,以及单元大转动产生的结点力增量的有效计算。动力时程分析通常采用直接积分法,但对于强非线性动力问题,直接积分法难以兼顾计算精度与稳定性。该文基于几何非线性分析的刚体准则,针对杆件结构大转动小应变的非线性问题,提出了一种新型空间杆系结构动力非线性分析的刚体准则法。该方法采用满足刚体准则的空间非线性梁单元,结合HHT-α法求解结构运动方程,并将刚体准则植入动力增量方程的迭代求解过程以计算结点力增量。通过典型柔性框架算例结果表明,该文方法可以有效分析柔性框架结构的强动力非线性行为。与高精度单元相比,该文采用的单元刚度矩阵构造简明,计算过程简洁;与商业软件所用方法相比,单元数和迭代步少,精度高,适于工程应用。     

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。     

索穹顶结构非线性分析的杆索有限元法

索穹顶结构是大变形柔性组合结构，其受力分析属于几何非线性问题，求解较复杂。本文应用有限元法，结合索穹顶结构特征，提出三节点等参数索单元与两节点杆单元相结合的混合有限元模式。基于Lagrangian坐标描述法和虚功原理推导了空间拉索和杆单元的大位移刚度矩阵，并建立了索穹顶结构非线性分析的增量平衡方程，利用NewtonRaphson法进行了实例计算。结果表明，本文方法精度很高，适合于索穹顶结构的空间非线性分析。     

杆系结构非线性后屈曲分析的增量割线刚度法

基于结构构件刚体运动与其变形抗力无关原理,假定构件经历与经典Updated-Lagrangian列式隐含的“微小自然变形-刚体运动”顺序相反的运动过程,建立空间杆系结构几何非线性分析的势能增量列式,推导了适用于典型增量步有限位移、有限应变分析的割线刚度矩阵。克服了Updated-Lagrangian列式下高阶非线性刚度矩阵推导过程繁琐及表达式不唯一等问题。该文提出的增量割线刚度既能预测位移（与协同转动法使用的割线刚度相比）,又能较精确校正变形恢复力,列式简便而易于实际应用（与拉格朗日列式使用的割线刚度相比）。为了提升数值追踪算法追踪各类型平衡路径的通过能力及计算效率,提出非线性方程求解的增量割线刚度法:应用增量割线刚度矩阵作为非线性分析“预测”和“校正”算子,建立基于柱面弧长约束的直接迭代策略,提出适应多回路路径的荷载因子自动调整算法实现自动加载。经典算例验证了增量割线刚度法能有效防止路径追踪“回溯”,快速收敛到正确解,可靠地反映杆系结构受力全过程行为。     

空间薄壁截面梁非线性有限单元模型

基于Timoshenko梁理论和Vlasov薄壁杆件理论,通过设置单元内部节点并对弯曲转角和翘曲角采取独立插值的方法,建立了可考虑横向剪切变形和扭转剪切变形及其耦合作用、弯扭耦合、以及二次剪应力影响的空间薄壁梁非线性有限元模型。以更新的拉格朗日格式描述的几何非线性应变推得几何刚度矩阵。同时考虑了材料非线性,假定材料为理想塑性体,服从Von Mises屈服准则和Prandtle-Reuss增量关系,采用有限分割法,由数值积分得到空间薄壁梁的弹塑性刚度矩阵。算例表明该文所建梁单元模型具有良好的精度,适用于空间薄壁结构的有限元分析。     

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。     

柔性梁几何非线性/后屈曲分析的改进势能列式方法研究

针对基于Updated-Lagrangian列式的能量方法存在:1) 由于位移模型的近似性而带来虚假节点力;2) 在分析节点空间转动效应上存在争议;3) 势能高阶项由于物理概念不明确给简化列式带来困难等问题,提出描述柔性梁构件有限位移过程受力状态变化的势能列式方法。根据连续介质力学极分解定理,将典型增量步内单元内力势能分解为刚体变位下初始节点力势能和自然变形中积累的初始节点力势能和应变能,推导了满足刚体运动检验和变形后节点受力平衡的空间梁单元几何刚度矩阵。建立全面反映构件非线性大位移行为的增量割线刚度矩阵显式列式。数值分析结果表明,势能列式能准确预测任意荷载作用下结构非线性平衡路径,物理概念清晰,适应工程实践对一般杆系结构非线性分析需求。     

铰接板机构运动分析的简便协调方程

一些新型空间结构的施工分析模型可以简化为铰接板机构,如Pantadome。利用三角形形状稳定性特点,将三条边长变化为零来表征三角形板单元的刚体位移。进而基于杆单元的协调方程,建立了一个简便的顶点铰接三角形板单元的机构位移协调方程,且给出了1阶和2阶协调矩阵。将四边形板单元划分为2个三角形板单元,通过引入单元四顶点共面条件,推导出平面四边形板单元的协调矩阵。理论上,利用该思路可构建任意平面多边形板单元的协调矩阵。针对该简便协调方程,进一步给出了求解铰接板机构运动路径的计算策略。对1个顶升施工的Pantadome和1个顶推施工的双坡网架的成形过程进行了数值模拟,结果表明该方法对于此类铰接板机构的运动路径分析有很高的精度。     

带转动弹簧的杆单元平面钢结构二阶弹性分析

用两端带转动弹簧的杆单元代替一般的杆单元,提出了一种新的平面杆系钢结构二阶弹性分析计算模型。通过对杆单元的转角位移方程进行修正,推导了这种计算模型的结构弹性刚度矩阵和几何刚度矩阵,并编制了该种模型的一阶和二阶弹性分析计算程序,采用该程序对两个半刚性连接钢框架进行了计算分析,并将部分结果与有限元计算结果进行了比较。算例结果表明:该模型计算结果精度高,可用于结构初步设计与分析。     

充气薄膜褶皱分析的高效互补有限元列式

褶皱变形是柔性薄膜结构的一种常见的失稳模式,其数值模拟具有挑战性。基于连续体和张力场理论,提出了一种适用于充气薄膜结构褶皱分析的互补共旋有限元方法。采用共旋坐标法,将物体的大变形分解为结构整体坐标系下的刚体运动和单元局部坐标系下小应变变形,推导了一个空间三节点三角形膜单元的切线刚度矩阵。该刚度矩阵包含材料刚度、旋转刚度和平衡投影刚度矩阵三个部分,涵盖了随动载荷对单元刚度的影响。在单元局部坐标系下,依据双模量材料本构关系构造了一个褶皱模型,能够判断单元处于"张紧""褶皱"或"松弛"状态。进一步通过建立等价的线性互补问题,消除了迭代求解过程中的内力振荡,改善了算法的稳定性。数值算例表明:该文方法能够准确地预测充气薄膜结构的位移、应力以及褶皱区域。较之已有的"拟动态"和"惩罚"方法,该方法不需要引入额外的求解技术来保证收敛,具有良好的稳定性。     

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.     

An efficient co‐rotational formulation for curved triangular shell element

A 6‐node curved triangular shell element formulation based on a co‐rotational framework is proposed to solve large‐displacement and large‐rotation problems, in which part of the rigid‐body translations and all rigid‐body rotations in the global co‐ordinate system are excluded in calculating the element strain energy. Thus, an element‐independent formulation is achieved. Besides three translational displacement variables, two components of the mid‐surface normal vector at each node are defined as vectorial rotational variables; these two additional variables render all nodal variables additive in an incremental solution procedure. To alleviate the membrane and shear locking phenomena, the membrane strains and the out‐of‐plane shear strains are replaced with assumed strains in calculating the element strain energy. The strategy used in the mixed interpolation of tensorial components approach is employed in defining the assumed strains. The internal force vector and the element tangent stiffness matrix are obtained from calculating directly the first derivative and second derivative of the element strain energy with respect to the nodal variables, respectively. Different from most other existing co‐rotational element formulations, all nodal variables in the present curved triangular shell formulation are commutative in calculating the second derivative of the strain energy; as a result, the element tangent stiffness matrix is symmetric and is updated by using the total values of the nodal variables in an incremental solution procedure. Such update procedure is advantageous in solving dynamic problems. Finally, several elastic plate and shell problems are solved to demonstrate the reliability, efficiency, and convergence of the present formulation. Copyright © 2007 John Wiley & Sons, Ltd.     

Rigid body considerations for non-linear finite element analysis

The purpose of the paper is to demonstrate how the concept of rigid body motions can be employed to derive the external stiffness matrix for an initially stressed finite element. Such a matrix is as important as the elastic and geometric stiffness matrices. It can be used not only in an eigenvalue analysis for testing the zero energy modes of a finite element under initial loadings, but for calculating the element forces in a step-by-step non-linear analysis. The two-dimensional beam element presented in this paper serves as a vehicle to demonstrate the concept involved. The principle of virtual work in its updated Lagrangian form has been adopted as the method of formulation. Several examples are provided to illustrate the adequacy of the present approach.     

A reliable three‐node triangular plate element satisfying rigid body rule and incremental force equilibrium condition

Abstract

This paper proposes a simple method for deriving the geometric stiffness matrix (GSM) of a three‐node triangular plate element (TPE). It is found that when the GSM of the element is combined into the global one of the structure, this structural stiffness matrix becomes symmetric and satisfies both the rigid body rule and incremental force and moment equilibrium (IFE) conditions, which are basically two fundamental conditions for analysis of mechanics. The former condition has been widely used in the community of mechanics; while the latter one, to our best knowledge, has never been considered. Advantages with the GSM derived are that derivations only need simple matrix operations without cumbersome non‐linear virtual strain energy derivations and tedious numerical integrations and more appealingly, this derived GSM can be explicitly given for applications. In addition, based on IFE and the rigid body rule conditions, a reasonable GSM for the three‐node TPE must be asymmetric; however, an asymmetric matrix usually gives rise to tedious numerical calculation especially in geometrically nonlinear problems and further, greatly influences computation efficiency. Fortunately, the skew‐symmetric parts of the derived GSM can be canceled out once they are merged into the global stiffness matrix of the structure. In this regard, this structural stiffness matrix becomes a symmetric one and thus enhances its effectiveness. Finally, several examples are provided for validating the robustness of the derived GSM.     

Three-dimensional absolute nodal co-ordinate formulation: plate problem

In this investigation, an absolute nodal co-ordinate dynamic formulation is developed for the large deformations and rotations of three-dimensional plate elements. In this formulation, no infinitesimal or finite rotations are used as nodal co-ordinates, instead global displacements and slopes are used as the plate coordinates. Using this interpretation of the plate coordinates the new method does not require the use of co-ordinate transformation to define the global inertia properties of the plates. The resulting mass matrix is the same constant matrix that appears in linear structural dynamics. The stiffness matrix, on the other hand, is a non-linear function of the nodal co-ordinates of the plate even in the case of a linear elastic problem. It is demonstrated in this paper that, unlike the incremental finite element formulations, the proposed method leads to an exact modelling of the rigid body inertia when the plate element moves as a rigid body. It is also demonstrated that by using the proposed method the conventional plate element shape function has a complete set of rigid body modes that can describe an exact arbitrary rigid body displacement. Using this fact, plate elements in the proposed new formulation can be considered as isoparametric elements. As a consequence, an arbitrary rigid body motion of the element results in zero strain. © 1997 John Wiley & Sons, Ltd.     

基于矩阵位移法的桩锚结构分析方法

采用将桩锚结构视为荷载作用-侧向弹簧支承的连续梁模型,支护桩为直立的弹性地基梁,主动土体转换为土压力,锚杆简化成为可逐步施加的弹簧支承,而被动土体则用侧向支承的土体弹簧替代.土体弹簧的刚度K是通过半无限大弹性空间内部水平荷载作用下的Mindlin 解来确定,这种方法降低了因深度加深而带来弹簧刚度的增长率,更加符合实际情况.针对这种弹性地基梁模型,提出了以矩阵分析为手段的计算方法,该法优点在于将桩锚视为桩-锚-土三者协同作用的支护结构,同时在计算过程中充分考虑了基坑分步开挖的施工过程对支护桩、锚杆以及被动土体的变形影响.通过实例分析表明:矩阵位移法为桩锚结构设计提供了一种合理、有效的计算方法.     

改进共旋坐标法的Timoshenko梁单元非线性分析

为提高空间Timoshenko梁单元非线性问题的计算精度,在共旋坐标法的基础上,提出了一种改进的Timoshenko梁单元几何非线性分析方法。利用虚功原理得到改进空间梁单元的刚度矩阵;使用有限质点法中的逆向运动思路计算单元局部坐标系下的刚体旋转矩阵;根据整体坐标系与局部坐标系之间旋转角度的转化以及微分关系,求得空间梁单元的切线刚度矩阵;编制了相应的有限元程序,对多个经典的大变形结构进行几何非线性分析。计算结果印证了该文所提出改进方法的正确性,同时与传统共旋坐标法相比,具有更高的精度。