On hybrid stress,hybrid strain and enhanced strain finite element formulations for a geometrically exact shell theory with drilling degrees of freedom

The paper is concerned with the finite element formulation of a recently proposed geometrically exact shell theory with natural inclusion of drilling degrees of freedom. Stress hybrid finite elements are contrasted by strain hybrid elements as well as enhanced strain elements. Numerical investigations and comparison is carried out for a four-node element as well as a nine-node one. As far as the four-node element is concerned it is shown that the stress hybrid element and the enhanced strain one are equivalent. The hybrid strain formulation corresponds to the hybrid stress formulation only in shear dominated problems, that is the case of the plate. © 1998 John Wiley & Sons, Ltd.     

Planar four node piezoelectric elements with drilling degrees of freedom

Several new planar four node piezoelectric elements with drilling degrees of freedom are presented. We begin by deriving two families of variational formulations accounting for piezoelectricity and in‐plane rotations. The first family retains the skew‐symmetric part of the stress tensor, while in the second, the skew part of stress is eliminated from the functional. The finite elements derived from two of the variational formulations derived in this paper are investigated. The first element is based on an ‘irreducible’ form, while the other is based on a fully mixed functional, with both stress and electric flux density assumed. Our new elements are shown to be accurate and robust in comparison with a number of existing elements, for several benchmark test problems. Copyright © 2005 John Wiley & Sons, Ltd.     

A displacement scheme with drilling degrees of freedom for plane elements

A scheme of boundary displacements with drilling degrees of freedom for plane elements is presented. The scheme is free from zero displacement modes and allows the development of hybrid finite elements with vertex and mid-side nodes, each node including a drilling degree of freedom besides the translational ones. Four quadrilateral isoparametric hybrid stress elements are implemented, and numerical results for some current test problems are given.     

A new enhanced assumed strain quadrilateral membrane element with drilling degree of freedom and modified shape functions

A new four‐node quadrilateral membrane finite element with drilling rotational degree of freedom based on the enhanced assumed strain formulation is presented. A simple formulation is achieved by five incompatible modes that are added to the Allman‐type interpolation. Furthermore, modified shape functions are used to improve the behaviour of distorted elements. Numerical results show that the proposed new element exhibits good numerical accuracy and improved performance, and in many cases, superior to existing elements. In particular, Poisson's locking in nearly incompressible elasticity fades and the element performs well when it becomes considerably distorted even when it takes almost triangular shape. Copyright © 2016 John Wiley & Sons, Ltd.     

The least‐squares finite element method in elasticity—Part I: Plane stress or strain with drilling degrees of freedom

A new least‐squares finite element method (LSFEM) for plane elasticity problems is developed based on the first‐order displacement–stress–rotation formulation which includes two new first‐order compatibility constraints among the stresses and the drilling rotation. This LSFEM can accommodate all kinds of equal‐order interpolations. Numerical experiments on various examples including incompressible materials show that the method achieves an optimal rate of convergence for all variables. Copyright © 2001 John Wiley & Sons, Ltd.     

双自由度离心振动系统的动力耦合分析

本文揭示了刚体转动对由质量和弹簧组成的双自由度振动系统的频率、模态和动力响应造成的影响,可为一般弹性体和结构的离心振动耦合动力学分析提供重要指导。离心振动系统存在两阶动频,随刚体转速的增加,第一阶动频线性减小而第二阶动频线性增加。质点的动力响应表现为质点运动轨迹的动态演化,决定于振动初始条件、刚体转速、初始偏心位置、振动刚度等四个因素。质点的运动轨迹由非偏心振动和偏心振动两部分构成,非偏心振动依赖于振动初始条件和刚体转速以及振动刚度,偏心振动取决于初始偏心位置和刚体转速以及振动刚度但与振动初始条件无关。根据对动频和质点运动的分析,提出了双自由度离心振动系统的最大刚体转速和临界转速。     

A systematic development of ‘solid-shell’ element formulations for linear and non-linear analyses employing only displacement degrees of freedom

In the present contribution we propose a so-called solid-shell concept which incorporates only displacement degrees of freedom. Thus, some major disadvantages of the usually used degenerated shell concept are overcome. These disadvantages are related to boundary conditions—the handling of soft and hard support, the need for special co-ordinate systems at boundaries, the connection with continuum elements—and, in geometrically non-linear analyses, to a complicated update of the rotation vector. First, the kinematics of the so-called solid-shell concept in analogy to the degenerated shell concept are introduced. Then several modifications of the solid-shell concept are proposed to obtain locking-free solid-shell elements, leading also to formulations which allow the use of general three-dimensional material laws and which are also able to represent the normal stresses and strains in thickness direction. Numerical analyses of geometrically linear and non-linear problems are finally performed using solely assumed natural shear strain elements with a linear approximation in in-plane direction. Although some considerations are needed to get comparable boundary conditions in the examples analysed, the solid-shell elements prove to work as good as the degenerated shell elements. The numerical examples show that neither thickness nor shear locking are present even for distorted element shapes. © 1998 John Wiley & Sons, Ltd.     

考虑界面转角自由度的频域子结构法研究

采用Jetmundsen频域子结构法,针对空间刚架结构的综合进行了数值仿真,研究了界面自由度的完整性对于综合结果的影响,讨论了一种间接获取转角自由度相关频响函数信息的方法.仿真结果显示,频域子结构综合法能够有效地预报结构的动力学性能,而通过引入间接获取的转角自由度的频响信息能在一定程度上改善预报质量.     

三自由度动态角度实时测量系统

提出一种基于二维PSD和激光自准直测量原理的三自由度动态角度实时测量系统。介绍激光自准直测量的工作原理,采用一种新颖的标定方法建立测量模型,并进行系统动态和静态测试。结果表明,测量范围为±1°,响应频率为4kHz,静态测试RMS误差为0.35′,动态测试RMS误差为0.36′。系统已成功应用于某型号仪器半实物仿真实验中,并取得了满意的结果。     

Dynamic analysis of framed structures with statistical uncertainties

The forced harmonic vibration analysis of portal frames consisting of viscously damped beams with spatial stochastic variation of mass and stiffness properties is considered. The analysis is based on the assembly of element stochastic dynamic stiffness matrices. The solution involves inversion of the global dynamic stiffness matrix, which, in this case, turns out to be a complex‐valued symmetric random matrix. Three alternative approximate procedures, namely, random eigenfunction expansion method, complex Neumann expansion method and combined analytical and simulation method are used to invert the matrix. The performance of these approximate procedures is evaluated using Monte Carlo simulation results. Copyright © 1999 John Wiley & Sons, Ltd.     

Non‐linear exact geometry 12‐node solid‐shell element with three translational degrees of freedom per node

This paper presents the finite rotation exact geometry (EG) 12‐node solid‐shell element with 36 displacement degrees of freedom. The term ‘EG’ reflects the fact that coefficients of the first and second fundamental forms of the reference surface and Christoffel symbols are taken exactly at each element node. The finite element formulation developed is based on the 9‐parameter shell model by employing a new concept of sampling surfaces (S‐surfaces) inside the shell body. We introduce three S‐surfaces, namely, bottom, middle and top, and choose nine displacements of these surfaces as fundamental shell unknowns. Such choice allows one to represent the finite rotation higher order EG solid‐shell element formulation in a very compact form and to derive the strain–displacement relationships, which are objective, that is, invariant under arbitrarily large rigid‐body shell motions in convected curvilinear coordinates. The tangent stiffness matrix is evaluated by using 3D analytical integration and the explicit presentation of this matrix is given. The latter is unusual for the non‐linear EG shell element formulation. Copyright © 2011 John Wiley & Sons, Ltd.     

三转动数控台跃度逆解分析

针对机械加工对数控转台的需求,研究3PSS RU并联构型三转动数控台的跃度逆解,以分析驱动系统对数控转台机械结构的柔性冲击和振荡．首先对提出的数控台进行描述,利用矢量方程法建立三转动数控台运动学模型,对其进行位置、速度、加速度逆解计算,在加速度分析的基础上进一步求导进行跃度分析,最后通过给定末端动平台的运动规律,计算出三转动数控台滑块的位置、速度、加速度与跃度随时间变化曲线．结果证实,在给定的运动规律中滑块2的移动范围、速度、加速度与跃度均较滑块1与滑块3的大,表明滑块2所受的冲击和振颤也较大．     

离心场中含旋转梯度影响的弹性体动力特性分析

摘要：以含偶应力的弹性理论为基础,考虑小变形状态下弹性体的平动变形与旋转梯度,推导并给出了离心环境中一般弹性体运动与变形耦合动力学方程,以转角为独立变量,利用罚方法得到方程组的约束变分形式,构造了8结点48自由度的实体等参元,建立了离心场中含旋转梯度的一般弹性体动力特性分析的有限元模型。对绕中轴线旋转的悬臂梁进行动力特性分析,频率随转速的关系表明离心力和科氏力降低了该梁的一阶频率,但科氏力导致了二阶频率的上升,旋转梯度效应提高了弹性体的静力刚度,导致各阶频率出现刚化效应。     

8-node hexahedral unsymmetric element with rotation degrees of freedom for modified couple stress elasticity

A new C⁰ 8-node 48-DOF hexahedral element is developed for analysis of size-dependent problems in the context of the modified couple stress theory by extending the methodology proposed in our recent work (Shang et al., Int J Numer Methods Eng 119(9): 807-825, 2019) to the three-dimensional (3D) cases. There are two major innovations in the present formulation. First, the independent nodal rotation degrees of freedom (DOFs) are employed to enhance the standard 3D isoparametric interpolation for obtaining the displacement and strain test functions, as well as to approximatively design the physical rotation field for deriving the curvature test function. Second, the equilibrium stress functions instead of the analytical functions are used to formulate the stress trial function whilst the couple stress trial function is directly obtained from the curvature test function by using the constitutive relationship. Besides, the penalty function is introduced into the virtual work principle for enforcing the C¹ continuity condition in weak sense. Several benchmark examples are examined and the numerical results demonstrate that the element can simulate the size-dependent mechanical behaviors well, exhibiting satisfactory accuracy and low susceptibility to mesh distortion.     

Eight‐node membrane element with drilling degrees of freedom for analysis of in‐plane stiffness of thick floor plates

The development of eight‐node arbitrary quadrilateral membrane elements with drilling degrees of freedom is presented using the compatible displacement interpolation within the element. The element is considered to develop specifically for analysing the in‐plane stiffness of thick floor plates in building systems, particularly the transfer plates in tall buildings as well as pile caps. With a new set of shape functions and following the displacement‐based element procedure, the element stiffness and force vector are derived and nodal displacements are obtained after solving the simultaneous equations; the element stresses are then determined. A wide range of patch tests is conducted to evaluate the consistency and stability of the proposed element. The test results show very good agreement with the exact solutions of the beam theory. Numerical investigations are carried out, showing that the analyses using the proposed elements provide better results than those from the existing methods. Copyright © 2008 John Wiley & Sons, Ltd.     

两自由度干摩擦振动系统的粘滑运动分析

干摩擦结构广泛存在于机械系统中,由此引起的振动行为属于典型的非光滑动力学研究范畴。通过将含干摩擦非线性振动系统看作一分段线性系统,提出了求解各粘滑状态阶段解析解的方法,并以两自由度干摩擦振动系统的力学模型为例进行了数值模拟,进而分析了由干摩擦导致的粘滑振动行为。     

Improved assumed-stress hybrid shell element with drilling degrees of freedom for linear stress,buckling and free vibration analyses

An improved 4-node quadrilateral assumed-stress hybrid shell element with drilling degrees of freedom is presented. The formulation is based on Hellinger–Reissner variational principle and the shape functions are formulated directly for the 4-node element. The element has 12 membrane degrees of freedom and 12 bending degrees of freedom. It has 9 independent stress parameters to describe the membrane stress resultant field and 13 independent stress parameters to describe the moment and transverse shear stress resultant field. The formulation encompasses linear stress, linear buckling and linear free vibration problems. The element is validated with standard test cases and is shown to be robust. Numerical results are presented for linear stress, buckling, and free vibration analyses.     

含干摩擦的二自由度制动系统颤振分析

以二自由度颤振系统为对象,对含有干摩擦的制动系统的颤振分岔、粘滑等复杂的非线性动力学行为进行研究。采用非线性摩擦系数、用二自由度的制动模型取代单自由度模型,建立相应的系统动力学模型;利用Runge-Kutta-Fehlberg(RK45)法数值分析与研究了系统动力学特性,得到随制动速度、支撑刚度和阻尼变化的质量块颤振振幅的变化曲线,分析发现:系统在低速下出现颤振现象,系统的支撑刚度和阻尼对其颤振的剧烈程度具有一定的影响,合理的系统参数能够减轻系统的颤振。     

Constraint Loss under Dynamic Loading in Rate Independent Plastic Solids

The objectives of this paper are to examine the loss of crack tip constraint in dynamically loaded fracture specimens and to assess whether it can lead to enhancement in the fracture toughness at high loading rates which has been observed in several experimental studies. To this end, 2-D plane strain finite element analyses of single edge notched (tension) specimen and three point bend specimen subjected to time varying loads are performed. The material is assumed to obey the small strain J ₂ flow theory of plasticity with rate independent behaviour. The results demonstrate that a valid J–Q field exists under dynamic loading irrespective of the crack length and specimen geometry. Further, the constraint parameter Q becomes strongly negative at high loading rates, particularly in deeply cracked specimens. The variation of dynamic fracture toughness K _dc with stress intensity rate K for cleavage cracking is predicted using a simple critical stress criterion. It is found that inertia-driven constraint loss can substantially enhance K _dc for .     

4-Node combined shell element with semi-EAS-ANS strain interpolations in 6-parameter shell theories with drilling degrees of freedom

A 4-node C ⁰ shell element with drilling degrees of freedom is presented in this paper. The element is developed within the nonlinear 6-field shell theory. Kinematics of the shell is described by two independent fields: the vector field for translations and the proper orthogonal tensor field for rotations. Within the theoretical formulation no restriction is applied on magnitudes of displacements and rotations. To avoid locking phenomena the proposed element combines two interpolation schemes: the assumed natural strain (ANS) for transverse shear strains and the enhanced assumed strain (EAS). The latter interpolation is used with asymmetric (in-plane) membrane strains. The performance of the element is evaluated by example of benchmark problems with special emphasis on shell structures containing orthogonal intersections.