A deep rod finite element for structural dynamics and wave propagation problems

In this paper, a new element for higher order rod (normally referred to as Minlin–Herrman rod) is formulated by introducing lateral contraction effects. The cross‐section is assumed to be rectangular. The stiffness and mass matrices are obtained by using interpolating functions that are exact solution to the governing static equation. The studies using this element for free vibration analysis show that lateral contractional inertia has a pronounced effect on the natural frequencies of the rod systems. The formulated element is not only able to capture the two propagating spectrums but also the dispersive effects in a deep rod. The results obtained from this element is compared with the previously formulated exact higher order spectral rod element. Copyright © 2000 John Wiley & Sons, Ltd.     

Symmetry Transformations and Exact Solutions of a Generalized Hyperelastic Rod Equation

In this paper, a nonlinear wave equation with variable coefficients is studied, interestingly, this equation can be used to describe the travelling waves propagating along the circular rod composed of a general compressible hyperelastic material with variable cross-sections and variable material densities. With the aid of Lou’s direct method1, the nonlinear wave equation with variable coefficients is reduced and two sets of symmetry transformations and exact solutions of the nonlinear wave equation are obtained. The corresponding numerical examples of exact solutions are presented by using different coefficients. Particularly, while the variable coefficients are taken as some special constants, the nonlinear wave equation with variable coefficients reduces to the one with constant coefficients, which can be used to describe the propagation of the travelling waves in general cylindrical rods composed of generally hyperelastic materials. Using the same method to solve the nonlinear wave equation, the validity and rationality of this method are verified.     

The method of ray expansions for solving boundary-value dynamic problems for spatially curved rods of arbitrary cross-section

The method of ray expansions is developed for solving boundary-value problems connected with the propagation of planes of strong and weak discontinuity in spatially curved linearly elastic rods of arbitrary cross section. The equations of the three-dimensional theory of elasticity are utilized, which are written on the wave surface using the theory of discontinuities and then are integrated over the cross-sectional area. It is assumed that the plane of discontinuity remains perpendicular to the centroidal axis of the rod all the time during its propagation; the discontinuities in the normal stresses in the sections with the normals perpendicular to the centroidal axis can be ignored as compared to the discontinuities in stresses in the sections with the normals parallel to the centroidal axis. The cross-sections of the rod remain plane during the process of the rod deformation. These assumptions lead to the generation of two wave surfaces propagating in the spatially curved rod with the velocities of longitudinal-flexural and transverse-torsional waves of elastic rods. As this takes place, on the longitudinal-flexural wave, the bulk deformations experience a discontinuity not only at the sacrifice of shortening-elongation of the medium’s element locating along the centroidal axis, but also at the expense of thickening–thinning of this element in the directions of the principle axes of the rod cross-section. For the transverse-torsional wave, there exist discontinuities in the components of the velocities directed along the principle axes of the cross-section, in the angular velocity of the cross-section rotation as a rigid whole with respect to the centroidal axis, as well as in transverse deformations occurring due to the inhomogeneity of the transverse displacements. During the solution of boundary-value problems, the values to be found are represented in terms of the power series, the coefficients of which are the discontinuities in arbitrary order partial time-derivatives of the desired functions, while the time of arrival of the wave front is the independent variable; in so doing the order of the partial time-derivative coincides with the power exponent of the independent variable. The ray series coefficients are determined from the recurrent equations of the ray method within the accuracy of arbitrary constants, while the arbitrary functions themselves are found from the boundary conditions. Examples illustrating the efficiency of the ray method for solving the problems of dynamic contact interaction, resulting in the propagation of transient waves of strong discontinuity in spatially curved rods, are presented.     

低张力缆索有限元模型及其应用

为准确计算缆索在低张力时的运动状态,该文考虑缆索低张力状态时的拉伸刚度、弯曲刚度和扭转刚度,建立了一种适用于低张力缆索的三维有限元模型。首先基于细长杆理论推导了缆索的动力学微分方程,接着以三次样条曲线为试函数,运用Galerkin加权残值法导出了单元刚度矩阵,最后对其进行组合建立了缆索的整体矩阵方程,并采用Matlab编写了求解程序。将其应用于实例中,所得结果与实验结果相一致。研究成果为拖曳缆、系泊缆、潜行器脐带缆等海洋缆索的运动分析与设计提供了理论依据。     

Exact stiffness matrix of mono-symmetric composite I-beams with arbitrary lamination

The exact stiffness matrix, based on the simultaneous solution of the ordinary differential equations, for the static analysis of mono-symmetric arbitrarily laminated composite I-beams is presented herein. For this, a general thin-walled composite beam theory with arbitrary lamination including torsional warping is developed by introducing Vlasov’s assumption. The equilibrium equations and force–deformation relations are derived from energy principles. The explicit expressions for displacement parameters are then derived using the displacement state vector consisting of 14 displacement parameters, and the exact stiffness matrix is determined using the force–deformation relations. In addition, the analytical solutions for symmetrically laminated composite beams with various boundary conditions are derived as a special case. Finally, a finite element procedure based on Hermitian interpolation polynomial is developed. To demonstrate the validity and the accuracy of this study, the numerical solutions are presented and compared with the analytical solutions and the finite element results using the Hermitian beam elements and ABAQUS’s shell element.     

曲线纤维增强复合材料圆环板的非轴对称弯曲问题

由自动铺丝机制造的结构具有面内变刚度特征。这种由空间变化引起的刚度变化使结构的控制方程成为了变系数偏微分方程,给求解非轴对称弯曲问题带来了很大挑战,难以求解其精确解。该文基于经典板壳理论,推导了柱坐标下正交各向异性变刚度圆环板非轴对称弯曲问题的控制方程。假定刚度分别随弹性模量指数函数和曲线纤维方向角连续变化,采用加权残值法计算了周边弹性约束时复合材料圆环板的挠度。通过与精确解结果的比对,验证该方法是有效的,并有较高精度。计算结果表明曲线纤维方向角的变化将使曲线纤维增强复合材料结构的相关力学性能明显优于同等比例的直线纤维增强复合材料结构。同时,结果还表明变刚度复合材料圆环板的非轴对称挠度与其周边的约束条件、材料参数、内外半径比值、纤维方向角等密切相关。     

A higher-order triangular finite element for the solution of field problems in orthotropic media

This paper presents a triangular finite element for the solution of two-dimensional field problems in orthotropic media. The element has nine degrees of freedom, these being the potential and its two derivatives at each node. The ‘stiffness’ matrix is derived analytically so that no further integration is required when computations are performed using the element. The results obtained using the element are compared with the exact mathematical solution of both a temperature distribution and a torsion problem.     

A general finite element for beams or beam-columns with or without an elastic foundation

A general finite element is derived for beams or beam-columns with or without a continuous Winkler type elastic foundation. The need to discretize members into shorter elements for convergence towards an ‘exact’ solution is eliminated by employing in the derivation of the element exact shape functions obtained from the equation of the elastic line. Inter-nodal values of deflections, bending moments and shear forces are obtained using the exact shape functions and trigonometric series. The effect of heavy compressive or tensile axial forces on bending stiffness is treated as a linear problem by considering the axial force as a constant parameter affecting the stiffness. FORTRAN subroutines to compute the stiffness matrix, equivalent nodal forces, deflected shape, bending moments and shear forces are provided and verified by an example.     

Explicit finite element artificial boundary scheme for transient scalar waves in two‐dimensional unbounded waveguide

To simulate the transient scalar wave propagation in a two‐dimensional unbounded waveguide, an explicit finite element artificial boundary scheme is proposed, which couples the standard dynamic finite element method for complex near field and a high‐order accurate artificial boundary condition (ABC) for simple far field. An exact dynamic‐stiffness ABC that is global in space and time is constructed. A temporal localization method is developed, which consists of the rational function approximation in the frequency domain and the auxiliary variable realization into time domain. This method is applied to the dynamic‐stiffness ABC to result in a high‐order accurate ABC that is local in time but global in space. By discretizing the high‐order accurate ABC along artificial boundary and coupling the result with the standard lumped‐mass finite element equation of near field, a coupled dynamic equation is obtained, which is a symmetric system of purely second‐order ordinary differential equations in time with the diagonal mass and non‐diagonal damping matrices. A new explicit time integration algorithm in structural dynamics is used to solve this equation. Numerical examples are given to demonstrate the effectiveness of the proposed scheme. Copyright © 2011 John Wiley & Sons, Ltd.     

Microscale heat conduction in homogeneous anisotropic media: a dual-reciprocity boundary element method and polynomial time interpolation approach

In this paper, a dual-reciprocity boundary element method based on some polynomial interpolations to the time-dependent variables is presented for the numerical solution of a two-dimensional heat conduction problem governed by a third order partial differential equation (PDE) over a homogeneous anisotropic medium. The PDE is derived using a non-Fourier heat flux model which may account for thermal waves and/or microscopic effects. In the analysis, discontinuous linear elements are used to model the boundary and the variables along the boundary. The systems of algebraic equations are set up to solve all the unknowns. For the purpose of evaluating the proposed method, some numerical examples with known exact solutions are solved. The numerical results obtained agree well with the exact solutions.     

The scaled boundary finite element method—alias consistent infinitesimal finite element cell method—for diffusion

The scaled boundary finite element method, alias the consistent infinitesimal finite element cell method, is developed starting from the diffusion equation. Only the boundary of the medium is discretized with surface finite elements yielding a reduction of the spatial dimension by one. No fundamental solution is necessary, and thus no singular integrals need to be evaluated. Essential and natural boundary conditions on surfaces and conditions on interfaces between different materials are enforced exactly without any discretization. The solution of the function in the radial direction is analytical. This method is thus exact in the radial direction and converges to the exact solution in the finite element sense in the circumferential directions. The semi‐analytical solution inside the domain leads to an efficient procedure to calculate singularities accurately without discretization in the vicinity of the singular point. For a bounded medium symmetric steady‐state stiffness and mass matrices with respect to the degrees of freedom on the boundary result without any additional assumption. Copyright © 1999 John Wiley & Sons, Ltd.     

Elastic buckling analysis of 3-D trusses and frames with thin-walled members

 A finite element method is presented for the determination of the elastic buckling load of three-dimensional trusses and frames with rigid joints. The beam element stiffness matrix is constructed on the basis of the exact solution of the governing equations describing the coupled flexural-torsional buckling behaviour of a three-dimensional beam with an open thin-walled section in the framework of a small deformation theory. Large deformation effects are taken into account approximately through consideration of P−Δ effects. The structural stiffness matrix is obtained by an appropriate superposition of the various element stiffness matrices. The axial force distribution in the members is obtained iteratively for every value of the externally applied loading and the vanishing of the determinant of the structural stiffness matrix is the criterion used to numerically determine the elastic buckling load of the structure. The effect of initial member imperfections is also included in the formulation. Comparisons of accuracy and efficiency of the present exact finite element method against the conventional approximate finite element method are made. Cases where the axial force distribution determination can be done without iterations are also identified. The effect of neglecting the warping stiffness of some mono-symmetric sections is also investigated. Numerical examples involving simple and complex three-dimensional trusses and frames are presented to illustrate the method and demonstrate its merits. Received: 2 May 2000 / Accepted: 15 July 2002     

A generalized layerwise finite element for multi-layer damping treatments

This paper presents a 4-node facet type quadrangular shell finite element, based on a layerwise theory, developed for dynamic modelling of laminated structures with viscoelastic damping layers. The bending stiffness of the facet shell element is based on the Reissner–Mindlin assumptions and the plate theory is enriched with a shear locking protection adopting the MITC approach. The membrane component is corrected by using incompatible quadratic modes and the drilling degrees of freedom are introduced through a fictitious stiffness stabilization matrix. Linear static tests, using several pathological tests, showed good and convergent results. Dynamic analysis evaluation is provided by using two eigenproblems with exact analytical solution, as well as a conical sandwich shell with a closed-form analytical solution and a semi-analytical ring finite element solution. The applicability of the proposed finite element to viscoelastic core sandwich plates is assessed through experimental validation.     

Modeling crack discontinuities without element‐partitioning in the extended finite element method

In this paper, we model crack discontinuities in two‐dimensional linear elastic continua using the extended finite element method without the need to partition an enriched element into a collection of triangles or quadrilaterals. For crack modeling in the extended finite element, the standard finite element approximation is enriched with a discontinuous function and the near‐tip crack functions. Each element that is fully cut by the crack is decomposed into two simple (convex or nonconvex) polygons, whereas the element that contains the crack tip is treated as a nonconvex polygon. On using Euler's homogeneous function theorem and Stokes's theorem to numerically integrate homogeneous functions on convex and nonconvex polygons, the exact contributions to the stiffness matrix from discontinuous enriched basis functions are computed. For contributions to the stiffness matrix from weakly singular integrals (because of enrichment with asymptotic crack‐tip functions), we only require a one‐dimensional quadrature rule along the edges of a polygon. Hence, neither element‐partitioning on either side of the crack discontinuity nor use of any cubature rule within an enriched element are needed. Structured finite element meshes consisting of rectangular elements, as well as unstructured triangular meshes, are used. We demonstrate the flexibility of the approach and its excellent accuracy in stress intensity factor computations for two‐dimensional crack problems. Copyright © 2016 John Wiley & Sons, Ltd.     

Explicit solutions in the stochastic dynamics of structural systems

The paper presents a procedure to derive in explicit form the stationary response of a linear structure subjected to Gaussian white noise stochastic excitation. Namely, the analytical relationship between the second order statistical moments of the response and the structural parameters (element stiffness and modal damping ratio) is furnished. The method is based on the solution of complex eigenvalue problems, one for each variable structural parameter, possessing a number of eigenvalues different from zero much smaller than the problem dimension. If a single structural quantity is treated as a parameter then the exact explicit solution is found. When more parameters are present, the explicit solution is approximate and the introduction of cross terms is suggested to obtain more accurate predictions.The aforementioned explicit solution is exploited herein in the field of uncertain structures. The structural parameters are modeled as random variables and a Monte Carlo procedure is adopted to get the conditional, given the structural parameters, probability density function of the second order moments of the response. The efficiency and the accuracy of the proposed procedure are evidenced by numerical applications.     

Bounds for element size in a variable stiffness cohesive finite element model

The cohesive finite element method (CFEM) allows explicit modelling of fracture processes. One form of CFEM models integrates cohesive surfaces along all finite element boundaries, facilitating the explicit resolution of arbitrary fracture paths and fracture patterns. This framework also permits explicit account of arbitrary microstructures with multiple length scales, allowing the effects of material heterogeneity, phase morphology, phase size and phase distribution to be quantified. However, use of this form of CFEM with cohesive traction–separation laws with finite initial stiffness imposes two competing requirements on the finite element size. On one hand, an upper bound is needed to ensure that fields within crack‐tip cohesive zones are accurately described. On the other hand, a lower bound is also required to ensure that the discrete model closely approximates the physical problem at hand. Both issues are analysed in this paper within the context of fracture in multi‐phase composite microstructures and a variable stiffness bilinear cohesive model. The resulting criterion for solution convergence is given for meshes with uniform, cross‐triangle elements. A series of calculations is carried out to illustrate the issues discussed and to verify the criterion given. These simulations concern dynamic crack growth in an Al₂O₃ ceramic and in an Al₂O₃/TiB₂ ceramic composite whose phases are modelled as being hyperelastic in constitutive behaviour. 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.     

Analytical Solution of Functionally Graded Beam Having Longitudinal Stiffness Variation

Abstract

The aim of the present paper is to developed analytical elasticity solution for a beam having longitudinal stiffness variation using recently developed multiterm extended Kantorovich (EKM) method. By applying the EKM method, the system of 4n first order ordinary differential equations (ODEs) and 1n algebraic equation are obtained along the in-plane (x) and thickness (z) directions. The system of the equations along the thickness direction (z) having constant coefficient but the set of equations along the x-direction have variable coefficients. In the thickness direction (z), exact closed-form solutions are obtained. And along the x-direction, the system of ODEs with variable coefficients are solved by employing the modified power series. In this paper, specific predefined variations are assumed in material property and their influence on the bending response of beam, subjected to mechanical loading, is investigated. Benchmark numerical results are presented for a different combination of boundary conditions. These numerical results can be used to validate approximate one-dimensional solutions and 2D numerical results.     

Theoretical aspects of the internal element connectivity parameterization approach for topology optimization

The internal element connectivity parameterization (I‐ECP) method is an alternative approach to overcome numerical instabilities associated with low‐stiffness element states in non‐linear problems. In I‐ECP, elements are connected by zero‐length links while their link stiffness values are varied. Therefore, it is important to interpolate link stiffness properly to obtain stably converging results. The main objective of this work is two‐fold (1) the investigation of the relationship between the link stiffness and the stiffness of a domain‐discretizing patch by using a discrete model and a homogenized model and (2) the suggestion of link stiffness interpolation functions. The effects of link stiffness penalization on solution convergence are then tested with several numerical examples. The developed homogenized I‐ECP model can also be used to physically interpret an intermediate design variable state. Copyright © 2008 John Wiley & Sons, Ltd.     

An analysis of alternatives for computing axisymmetric element stiffness matrices

This note compares the use of numerical and closed form integration in computation of element stiffness matrices for axisymmetric finite element analysis. Only constant strain elements are considered. Results obtained with Gaussian quadrature and closed form integrators are compared mutually and with an exact solution obtained from classical methods. The FEM global equations estimate the force-displacement behaviour of an elastic continuum with an accuracy that depends on the integration method used. Selection of an integration order minimizing error is particularly critical in the presence of high stress gradients. Best results in the vicinity of the axis of revolution may be obtained with single-point integration rather than higher order approximations or exact integration of the element stiffness matrix. This phenomenon and its consequences are subsequently discussed.