Large reference displacement analysis of composite plates part I: Finite element formulation

This investigation concerns itself with the dynamic analysis of thin, laminated composite plates consisting of layers of orthotropic laminae that undergo large arbitrary rigid body displacements and small elastic deformations. A non-linear finite element formulation is developed which utilizes the assumption that the bonds between the laminae are infinitesimally thin and shear non-deformable. Using the expressions for the kinetic and strain energies, the lamina mass and stiffness matrices are identified. The non-linear mass matrix of the lamina is expressed in terms of a set of invariants that depend on the assumed displacement field. By summing the kinetic and strain energies of the laminae of an element, the element mass and stiffness matrix can be defined in terms of the set of element invariants. It is shown that the element invariants can be expressed explicitly in terms of the invariants of its laminae. By assembling the finite elements of the deformable body, the body invariants can be identified and expressed explicitly in terms of the invariants of the laminae of its elements. In the dynamic formulation presented in this paper, the shape functions of the laminae are assumed to have rigid body modes that need to describe only large rigid body translations. The computer implementation and the use of the formulation developed in this investigation in multibody dynamics are discussed in the second part of this paper.     

INFLUENCE OF GEOMETRIC NON-LINEARITIES ON THE FREE VIBRATIONS OF ORTHOTROPIC OPEN CYLINDRICAL SHELLS

This paper presents a general approach to predict the influence of geometric non-linearities on the free vibration of elastic, thin, orthotropic and non-uniform open cylindrical shells. The open shells are assumed to be freely simply supported along their curved edges and to have arbitrary straight edge boundary conditions. The method is a hybrid of finite element and classical thin shell theories. The solution is divided into two parts. In part one, the displacement functions are obtained from Sanders' linear shell theory and the mass and linear stiffness matrices are obtained by the finite element procedure. In part two, the modal coefficients derived from the Sanders–Koiter non-linear theory of thin shells are obtained for these displacement functions. Expressions for the second- and third-order non-linear stiffness matrices are then determined through the finite element method. The non-linear equation of motion is solved by the fourth-order Runge–Kutta numerical method. The linear and non-linear natural frequency variations are determined as a function of shell amplitudes for different cases. The results obtained reveal that the frequencies calculated by this method are in good agreement with those obtained by other authors. © 1997 by John Wiley & Sons, Ltd.     

Matrix analysis of local instability in plates,stiffened panels and columns

A displacement method of matrix analysis for local instability of plates, stiffened panels and thin-walled columns is presented. The analysis is applicable to stiffened panel and columns for which the cross-section is made up of thin flat plates. For these cases it may be assumed that during buckling deformation no flat component of the cross-section is translated in its own plane and the edge lines at the junctions between flats remain fixed in space. The analysis leads to the standard eigenvalue equation from which the buckling stress can be determined. The elastic and geometrical stiffness matrices derived for this analysis depend on the wavelength of the buckled pattern and this dependence is of a simple form since all coefficients in the resulting stiffness matrices contain the buckling wavelength only as a common factor allowing for considerable simplification in any numerical computations. With this new formulation of local instability analysis very few elements are required to obtain high accuracy for the buckling stress. Several examples illustrating typical applications of this new method have been included.     

Non-linear analysis of thin-walled structures using plate elements

This paper presents a simple and efficient rectangular thin plate element for elastic geometric non-linear analysis of thin-walled structures. Based on the updated Lagrangian formulation and small strain hypothesis, this element can accurately predict the non-linear pre- and post-buckling load–deflection paths of a plate structure. Using symbolic manipulation, the linear and geometric stiffness matrices for the rectangular thin plate element are derived explicitly, thereby eliminating the need for numerical integration. The element contains 30 degrees of freedom (d.o.f.): 14 d.o.f. for the in-plane (membrane) action and 16 d.o.f, for the out-of-plane (bending) action. Several examples involving the large deflection behaviour of thin-walled structures are presented to demonstrate the accuracy, efficiency and versatility of the method.     

几何非线性复合材料层合固体壳单元

通过定义广义应力,提出了一个改进的刚度矩阵,以克服固体壳元的厚度自锁问题,并能保证沿复合材料层合结构厚度方向上的连续应力分布;将应力插值函数分为低阶和高阶两部分,建议了一个新的非线性变分泛函,推导了一个用于几何非线性分析的九节点固体壳单元,该单元的计算精度和效率基本上与九节点减缩积分单元相当,与同类型其他单元相比,该单元显著提高了计算效率。     

A formulation for the 4-node quadrilateral element

A formulation for the plane 4-node quadrilateral finite element is developed based on the principle of virtual displacements for a deformable body. Incompatible modes are added to the standard displacement field. Then expressions for gradient operators are obtained from an expansion of the basis functions into a second-order Taylor series in the physical co-ordinates. The internal degrees of freedom of the incompatible modes are eliminated on the element level. A modified change of variables is used to integrate the element matrices. For a linear elastic material, the element stiffness matrix can be separated into two parts. These are equivalent to a stiffness matrix obtained from underintegration and a stabilization matrix. The formulation includes the cases of plane stress and plane strain as well as the analysis of incompressible materials. Further, the approach is suitable for non-linear analysis. There, an application is given for the calculation of inelastic problems in physically non-linear elasticity. The element is efficient to implement and it is frame invariant. Locking effects and zero-energy modes are avoided as well as singularities of the stiffness matrix due to geometric distortion. A high accuracy is obtained for numerical solutions in displacements and stresses.     

Exact solutions for extensible circular curved timoshenko beams with nonhomogeneous elastic boundary conditions

Summary A generalized Green function ofnth-order ordinary differential equation with forcing function composed of the delta function and its derivatives is obtained. The generalized Green function can be easily and effectively applied to both the boundary value problems and the initial value problems. The generalized Green function is expressed in terms ofn linearly independent normalized homogeneous solutions. It is the generalization of those given by Pan and Hohenstein, and Kanwal. Accordingly, the exact solution for static analysis of an extensible circular curved Timoshenko beam with general nonhomogeneous elastic boundary conditions, subjected to any transverse, tangential and moment loads is obtained. The three coupled governing differential equations are uncoupled into one complete sixth-order ordinary differential characteristic equation in the tangential displacement. The explicit relations between the angle of rotation due to bending, the transverse displacement and the tangential displacement are obtained. The deflection curves due to a unit generalized displacement at nodal coordinate, and the exact element stiffness matrix are derived based on the solution for the general system. A finite element method can be developed based on the results for the dynamic analysis. Meanwhile, the stiffness locking phenomena accompanied in some other curved beam element methods does not exist in the proposed method.     

Stability conditions for non-conservative dynamical systems

The present paper treats dynamic instability problems of non-conservative elastic systems. Starting from general equations of motion, the equations of the perturbed motion are derived. The boundedness of the perturbed motions is studied and sufficient conditions for instability and a necessary condition for stability are deduced. These conditions may determine the instability of non-conservative systems and they are expressed in terms of the properties of generalized tangent damping and stiffness matrices of the systems. Thus, they can easily be incorporated with finite element computations of arbitrary structures.     

Explicit thickness integration for three-dimensional shell elements applied to non-linear analysis

The problem of multilayered degenerated 3-D shell elements for which the numerical integration is performed for each ply is that of the high generation time in non-linear analysis when the number of plies is important. But these elements give accurate results for thin and moderately thick shells, so in order to reduce the generation time explicit thickness integration is investigated. We first write an expansion of the strain-displacement matrix in power series of the thickness variable in order to obtain explicit expressions of the tangent stiffness matrix and internal force vector, appearing in the non-linear formulation. Explicit expressions of non-linear stiffness matrices are presented, using the explicit integration-first approximation. Simple expressions of several matrices, sub-matrices and vectors appearing in the formulation are given here in order to obtain an important computing-time gain. Next, some numerical validation tests comparing the classical element with numerical thickness integration and this one are discussed to prove validity of this formulation.     

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.     

Axisymmetric analytical stiffness matrices with Green-Lagrange strains

Stiffness matrices based on the non-linear Green-Lagrange definition seem complicated, but for the case of a linear displacement ring-element with triangular cross-section, closed form final results are listed, directly suited for coding in a finite element program. These analytical secant and tangent element stiffness matrices are obtained by separating the dependence on the material constitutive parameters and on the stress/strain state from the dependence on the initial geometry and the displacement assumption. As an example of application, numerical results for a circular plate problem show the indirect severe errors that may result from a linear strain model. It is difficult to predict the indirect errors that follow from the erroneous displacement field, and the explanations behind such predictions are attempted. The nodal positions of an element and the displacement assumption give six basic matrices that do not depend on material and stress strain state, and thus are unchanged during the necessary iterations for obtaining a solution based on Green-Lagrange strain measure. The presented resulting stiffness matrices are especially useful in design optimization, because analytical sensitivity analysis can then be performed.     

Semi-implicit transient analysis procedures for structural dynamics analysis

A semi-implicit direct time integration procedure is presented which avoids factorization of the implicit difference solution matrix. The procedure, if properly implemented, requires only vectorial calculations and hence needs the same computer core space as explicit integration procedures. Guidelines for splitting the stiffness matrix into upper and lower matrices are established, which among other things are designed to satisfy a correct transmission of rigid-body motions from element (or grid) to its adjacent elements.     

Numerical simulation of instability behaviour of thin-walled frames with flexible connections

A one-dimensional finite element formulation for numerical simulation of instability behaviour of thin-walled frames containing flexible connections is presented. Stiffness matrices of a conventional 14-degree of freedom beam element are derived by applying the linearized virtual work principle and Vlasov's assumption. The structural material is assumed to be homogeneous, isotropic and linear-elastic. Flexible connection behaviour and different warping deformation conditions are introduced into the numerical model by modifying stiffness matrices of a conventional beam element. For that purpose a special transformation matrix is derived. The effectiveness of the numerical algorithm discussed is validated through the test problem.     

新型协同转动3节点三边形壳单元

发展了一种新型3节点三边形壳单元。计算单元在局部坐标系下的节点变量时,通过采用协同转动法,预先扣除节点整体变量中的刚体转动成分,从而简化了单元的计算公式。不同于现有的其他协同转动单元,在该单元中采用了增量可以直接累加的矢量型转动变量,单元的切线刚度矩阵可以通过直接计算能量泛函对节点变量的二阶偏微分得到,且对节点变量的偏微分次序是可以互换的,因而在局部和整体坐标系下都得到了对称的单元切线刚度矩阵。为消除单元中可能出现的闭锁现象,引入了MacNeal提出的线积分法,分别用沿单元边线方向的膜应变和剪切应变构造新的假定应变场。最后,通过对几个产生了大位移与大转角变形的板壳问题进行分析,检验了该单元的可靠性、计算精度和计算效率。     

A new approach to the dynamic analysis of structures using fixed frequency dynamic stiffness matrices

The dynamic analysis of structures by the standard finite element method introduces additional inaccuracies into the solution which are not present when the method is used for static analyses. These inaccuracies can arise from two sources: (i) the element formulation and (ii) the reduction of the size of the matrices by a static condensation (i.e. using the Guyan method^1,2). The errors in both cases are caused by neglecting frequency-dependent terms in the functions relating the displacements at any point in the structure to the displacements at certain fixed points (i.e. nodes in the element formulation and ‘masters’ in the condensation). A new method of solution is proposed in this paper in which the frequency-dependent terms are retained implicitly by using dynamic stiffness matrices defined at a number of fixed frequencies. The dynamic stiffness matrices may be condensed efficiently to a relatively small number of master degrees-of-freedom using a front solution algorithm. The final stage in the solution uses these matrices to synthesize a high-order eigenvalue problem. A method of solving such an eigenvalue problem, of arbitrary order, is described in a separate paper.³ Numerical examples are given to show the accuracy and efficiency of the proposed method compared with conventional methods of solution.     

Nonlinear buckling analysis of hygrothermoelastic composite shell panels using finite element method

Hygrothermal stresses due to the change in environmental condition may induce buckling and dynamic instability in the composite shell structures. In the present investigation, the hygrothermoelastic buckling behavior of laminated composite shells are numerically simulated using geometrically nonlinear finite element method. The orthogonal curvilinear coordinate is used for modeling a general doubly curved deep or shallow shell surface. The geometrically nonlinear finite element formulation is based on general nonlinear strain–displacement relations in the orthogonal curvilinear coordinate system. The present theory can be applicable to thin and moderately thick shells. The mechanical linear and nonlinear stiffnesses, and the nonmechanical nonlinear geometric stiffness matrices and the hygrothermal load vector are presented. It is also observed that during the present numerical solution of nonlinear equilibrium equation, in order to construct the nonlinear stiffness matrices for the first load step, the initial deformation can be assumed as zero or any computer generated small random number or the properly scaled fundamental buckling mode shape. To verify the present formulations and finite element code, the present results are compared well with those available in the open literature. Parametric studies such as thickness ratio and shallowness ratio on buckling are performed for spherical, truncated conical and cylindrical composite shell panels. The buckling behavior and deflection shapes are characterized by multiple wrinkles along unreinforced direction at higher moisture concentrations or temperature rise.     

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。     

Exact static element stiffness matrices of non‐symmetric thin‐walled curved beams

An evaluation procedure of exact static stiffness matrices for curved beams with non‐symmetric thin‐walled cross section are rigorously presented for the static analysis. Higher‐order differential equations for a uniform curved beam element are first transformed into a set of the first‐order simultaneous ordinary differential equations by introducing 14 displacement parameters where displacement modes corresponding to zero eigenvalues are suitably taken into account. This numerical technique is then accomplished via a generalized linear eigenvalue problem with non‐symmetric matrices. Next, the displacement functions of displacement parameters are exactly calculated by determining general solutions of simultaneous non‐homogeneous differential equations. Finally an exact stiffness matrix is evaluated using force–deformation relationships. In order to demonstrate the validity and effectiveness of this method, displacements and normal stresses of cantilever thin‐walled curved beams subjected to tip loads are evaluated and compared with those by thin‐walled curved beam elements as well as shell elements. Copyright © 2004 John Wiley & Sons, Ltd.     

Non-linear vibration of frames by the incremental dynamic stiffness method

The dynamic stiffness method is extended to large amplitude free and forced vibrations of frames. When the steady state vibration is concerned, the time variable is replaced by the frequency parameter in the Fourier series sense and the governing partial differential equations are replaced by a set of ordinary differential equations in the spatial variables alone. The frequency-dependent shape functons are generated approximately for the spatial discretization. These shape functions are the exact solutions of a beam element subjected to mono-frequency excitation and constant axial force to minimize the spatial discretization errors. The system of ordinary differential equations is replaced by a system of non-linear algebraic equations with the Fourier coefficients of the nodal displacements as unknowns. The Fourier nodal coefficients are solved by the Newtonian algorithm in an incremental manner. When an approximate solution is available, an improved solution is obtained by solving a system of linear equations with the Fourier nodal increments as unknowns. The method is very suitable for parametric studies. When the excitation frequency is taken as a parameter, the free vibration response of various resonances can be obtained without actually computing the linear natural modes. For regular points along the response curves, the accuracy of the gradient matrix (Jacobian or tangential stiffness matrix) is secondary (cf. the modified Newtonian method). However, at the critical positions such as the turning points at resonances and the branching points at bifurcations, the gradient matrix becomes important. The minimum number of harmonic terms required is governed by the conditions of completeness and balanceability for predicting physically realistic response curves. The evaluations of the newly introduced mixed geometric matrices and their derivatives are given explicitly for the computation of the gradient matrix.     

Adaptively shifted integration technique for finite element collapse analysis of framed structures

The present study is concerned with the improvement of the previously proposed ‘shifted integration technique’ for the plastic collapse analysis of framed structures using the linear Timoshenko beam element or the cubic beam element based on the Bernoulli-Euler hypothesis. In the newly proposed ‘adaptively shifted integration technique’, the numerical integration points for the evaluation of the stiffness matrices are automatically shifted immediately after. the occurrence of plastic hinges according to the previously established relations between the locations of numerical integration points and those of plastic hinges. By using the adaptively shifted integration technique, sufficiently accurate solutions can be obtained in the non-linear frame analysis by two-linear-element or only one-cubic-element idealization for each structural member. The present technique can easily be implemented in the existing finite element codes utilizing the linear or the cubic beam element.