Finite element dynamic analysis of geometrically exact planar beams

We present a new strain-based finite element formulation for the dynamic analysis of highly flexible elastic planar beams. The formulation employs the geometrically exact Reissner planar beam theory which accounts for finite displacements and rotations, and finite membrane, shear and bending strains. The system of semi-discrete dynamic equations of motion is derived from the modified Hamilton principle in which only the strain variables are interpolated. Such a choice of the interpolated variables is an advantage over approaches, in which the displacements and rotations are interpolated, since the field consistency problem and related locking phenomena do not arise. The performance and accuracy of the formulation are illustrated by several numerical examples.     

Nonlinear dynamic analysis of curved beams

A computational procedure is presented for predicting the dynamic response of curved beams with geometric nonlinearities. A mixed formulation is used with the fundamental unknowns consisting of stress resultants, generalized displacements and velocity components. The governing semidiscrete finite element equations consist of a mixed system of algebraic and differential equations. The temporal integration of the differential equations is performed by using an explicit half-station central difference method. A procedure is outlined for lumping both the flexibilities and masses of the mixed model, thereby uncoupling all the equations of the system. The advantages of the proposed computational procedure over explicit methods used with the displacement formulation are discussed. The effectiveness and versatility of the proposed approach are demonstrated by means of numerical examples.     

Finite element stress formulation for dynamic elastic-plastic analysis

The solution to wave propagation problems in solids with elastic-plastic material properties is obtained by using the finite element method directly in terms of the stresses. A variational principle due to Gurtin is modified by including a plastic strain tensor in the constitutive relationship. The resulting finite element equations, which represent the strain-displacement equations written in terms of the stresses, are simultaneous integral equations in time. With a transformation of variables, a set of simultaneous differential equations is obtained of the form$\text{H}\text{s}\text{?}\text{+ Qs+ Ve}{}_{\mathrm{p}}\text{= q(t)}$, where $\text{H}$ is a symmetric positive-semidefinite matrix, and $\text{Q}$ is a symmetric positive-definite matrix. The stresses and the plastic strains are represented by $\text{s}\text{?}$ and $\text{e}{}_{\mathrm{p}}$, respectively.Finite element equations are developed for an axisymmetric ring element with an arbitrary quadrilateral cross section in which the stresses and the plastic strains vary linearly along the sides of the elements. The equations are numerically integrated with respect to time by Newmark's generalized acceleration method.An iterative procedure is presented, which uses the finite element strain-displacement equations and the plasticity relationships, to determine the state of stress at the end of the time step. Several examples are used to demonstrate the solution technique for elastic and elastic-plastic problems.     

Finite difference analysis of composite beams with partial interaction

A general formulation for the analysis of composite beams with partial interaction, in which the basic equilibrium and compatibility equations are expressed in terms of displacements, is presented. Numerical solutions of the basic equations are obtained by expressing the displacement derivatives in finite difference form, and the solutions so obtained show close agreement with existing analytical solutions for linear material and shear connector behaviour.     

Finite element analysis of laminated anisotropic beams of bimodulus materials

This paper presents finite element analysis of laminated anisotropic beams of bimodulus materials. The finite element has 16 d.o.f. and uses the displacement field in terms of first order Hermite interpolation polynomials. As the neutral axis position may change from point to point along the length of the beam, an iterative procedure is employed to determine the location of zero strain points along the length. Using this element some problems of laminated beams of bimodulus materials are solved for concentrated loads/moments perpendicular and parallel to the layering planes as well as combined loads.     

Finite-element dynamic inelastic analysis of beams and plates under combined loading

The response of beams and plates under multiple loading is considered taking into account the inelastic interactions between moments and axial (in-plane) forces. The plastic analysis is based on distribution of stresses and strains in three dimensions so that yielding remains a function of the uniaxial state of stress in beams and of the biaxial state of stress in plates. The interaction effects between dynamic transverse loads and static in-plane loads is studied by using initial stress stiffness matrices that modify the original stiffness matrices. Detailed response behavior of beams and plates in combined loading is presented. The response characteristics are found to exhibit interaction instability properties.     

Static, dynamic and stability analysis of structures composed of tapered beams

A new numerical method is proposed for the static, dynamic and stability analysis of linear elastic plane structures consisting of beams with constant width and variable depth. It is a finite element method based on an exact flexural and axial stiffness matrix and approximate consistent mass and geometric stiffness matrices for a linearly tapered beam element with constant width. Use of this method provides the exact solution of the static problem with just one element per member of a structure with linearly tapered beams and excellent approximate solutions of the dynamic and stability problems with very few elements per member of the structure in a computationally very efficient way. Very detailed comparison studies of the proposed method against a number of other known finite element methods with respect to accuracy and computational efficiency for cantilever tapered beams of rectangular and I cross section clearly favor the proposed method. A continuous beam, a gable frame and a portal frame consisting of tapered members are analyzed by the proposed method as well as by other known methods to illustrate the use of the method to structures composed of tapered beams.     

Theoretical and numerical convergence rates for some conforming finite elements

One of the major requirements of an approximate solution method is to rapidly converge to the exact solution. This paper is concerned with the rapidity of solution convergence for many conforming finite element models. Convergence rates are determined by theoretical analyses and checked by numerical investigation. Solution error plots are made for each element and in most cases these plots indicate a numerical convergence rate which is in agreement with that predicted by theory. The investigation considers only discretization errors; and the solution quantities referred to are for the system energy quantities which correspond to the eigenvalues for eigenvalue problems. The convergence rates determined are applicable to eigenvalue and static problems devoid of stress singularities.     

Uncertainty-based loads analysis for spacecraft: Finite element model validation and dynamic responses

The stochastic correlation and validation of spacecraft structural dynamic models and the stochastic launcher-satellite coupled loads analysis present several points of interest and involve novel aspects. This paper describes the main objectives of two studies in progress under the technical management of the European Space Agency and performed by consortia of European industries and university. The paper presents an overview of the major aspects related to the implementation of the approach, the identified methods and tools and the future developments.     

Finite element analysis of dynamic quasi-bifurcations in elastic-plastic structures

Some numerical results related to the critical behavior of inelastic structures subjected to dynamic step loading are given. The underlying theory of dynamic quasi-bifurcations is adapted to the analysis of elastic-plastic solids. Two examples, a 2D truss and a plane stress column-like structure, are discussed at length.     

Finite elements for post-buckling analysis. I—The W-formulation

This paper presents a convenient formulation for the stability analysis of structures using the finite element method. The main assumptions are linear elasticity, a linear fundamental path, and the existence of distinct critical loads (i.e. no coupling between buckling modes occurs). The formulation developed is known as W-formulation, in which the energy is written in terms of a sliding set of incremental coordinates measured with respect to the fundamental path. In the presentation developed here, the only ingredients required to carry out the analysis are the strain-displacement and the constitutive matrices at the element level. The present formulation is compared with the so called V-formulation, in which the displacements refer to the unloaded state. It is shown that under the present assumptions of linear fundamental path, the advantages of the V-formulation are lost and both approaches are similar. An example of a circular plate under in-plane loading illustrates the procedures. Part II of this paper deals with the application to the post buckling analysis of plate assemblies made of composite materials.     

Discrete element analysis of dynamic response of Timoshenko beams under moving mass

In this paper, dynamic response of Timoshenko beams under moving mass is analyzed using a numerical method called discrete element technique (DET). In DET, continuous flexible beam elements are replaced by a system of rigid bars and flexible joints. We present a DET model of Timoshenko beams under moving mass. The results of our DET model are compared with the solutions obtained by PAFEC (programs for automatic finite element calculations) for Euler–Bernoulli beams and finite difference method for Timoshenko beams. The effects of beam thickness and moving mass velocity on dynamic response of beams under moving mass are numerically studied.     

An engineering approach to impact analysis of notched beams by conventional beam elements

A new method for dynamic analysis of pre-cracked three point bend specimens using conventional beam finite elements is presented. The model is evaluated for both static and dynamic situations with analytical solutions. The applicability of the method in fracture toughness determination is demonstrated by comparing the model predictions with impact experiments conducted on aluminum and polymeric materials.     

新型光学动态靶标结构刚度有限元分析

为了更好的对光电经纬仪进行室内检测,提出了一种新型光学动态靶标,并对其优点和结构进行分析。基于材料力学原理建立靶标三维实体造型,以有限元分析软件NXNastran为辅助工具,对新型动态靶标的结构刚度进行分析,得出各个部件在靶标工作过程中的变形量,并计算靶标整体精度。结果表明,新型靶标结构形式合理、刚度良好,其俯仰角误差3.18″、方位角误差2.98″,能够初步满足光电经纬仪动态测角性能检测的需要。     

动态海缆机械性能有限元分析

为探究动态海缆在剧烈交变环境载荷下的疲劳寿命,基于环境载荷响应,开展动态海缆系统的整体分析。基于有限元方法,研究海水中悬浮的动态海缆在服役过程中承受的极端载荷,分析破断力和侧压力工况,提取关键部位的应力、应变分布并进行校核。研究结果可为动态海缆材料选型、结构优化以及疲劳分析和测试提供理论依据。     

Finite element dynamic analysis of anisotropic elastic solids with voids

The paper is concerned with dynamic problems of the linear theory of elasticity for anisotropic porous materials according to the Cowin–Nunziato model. In order to take into account the structural damping in elastic media, a new attenuation model which extends the Rayleigh models is used. A set of new finite element schemes is proposed for finding numerical solutions of transient, harmonic, modal and static problems in the context of elastic media with voids. Three-dimensional, plane (plane stress and plane strain) and axisymmetric problems are considered.     

A dynamic stiffness element for free vibration analysis of composite beams and its application to aircraft wings

The dynamic stiffness matrix of a composite beam that exhibits both geometric and material coupling between bending and torsional motions is developed and subsequently used to investigate its free vibration characteristics. The formulation is based on Hamilton’s principle leading to the governing differential equations of motion in free vibration, which are solved in closed analytical form for harmonic oscillation. By applying the boundary conditions the frequency dependent dynamic stiffness matrix that relates the amplitudes of loads to those of responses is then derived. Finally the Wittrick-Williams algorithm is applied to the resulting dynamic stiffness matrix to compute the natural frequencies and mode shapes of an illustrative example. The results are discussed and some conclusions are drawn. The theory can be applied for modal analysis of high aspect ratio composite wings and can be further extended to aeroelastic studies.     

Finite elements for post-buckling analysis. II—Application to composite plate assemblies

In a companion paper the authors presented a convenient formulation for the stability analysis of structures using the finite element method. The main assumptions are linear elasticity, a linear fundamental path and the existence of distinct critical loads. The formulation developed is known as the W-formulation, where the energy is written in terms of a sliding set of incremental coordinates measured with respect to the fundamental path. In the present paper a number of applications of finite elements for post-buckling analysis on composite plate assemblies are presented. Thin-walled composite plates, I-beams, angle sections, and a specially designed box-beam with flanges (unicolumn) are studied in post-buckling when axially loaded. The results are in good agreement with previous studies. Moreover, a parametric study involving critical buckling load and geometry is presented for the case of the unicolumn.     

Finite elements for elliptic problems with stochastic coefficients

We describe a deterministic finite element (FE) solution algorithm for a stochastic elliptic boundary value problem (sbvp), whose coefficients are assumed to be random fields with finite second moments and known, piecewise smooth two-point spatial correlation function. Separation of random and deterministic variables (parametrization of the uncertainty) is achieved via a Karhunen–Loève (KL) expansion. An O(N log N) algorithm for the computation of the KL eigenvalues is presented, based on a kernel independent fast multipole method (FMM). Truncation of the KL expansion gives an (M, 1) Wiener polynomial chaos (PC) expansion of the stochastic coefficient and is shown to lead to a high dimensional, deterministic boundary value problem (dbvp). Analyticity of its solution in the stochastic variables with sharp bounds for the domain of analyticity are used to prescribe variable stochastic polynomial degree r = (r₁, …, r_M) in an (M, r) Wiener PC expansion for the approximate solution. Pointwise error bounds for the FEM approximations of KL eigenpairs, the truncation of the KL expansion and the FE solution to the dbvp are given. Numerical examples show that M depends on the spatial correlation length of the random diffusion coefficient. The variable polynomial degree r in PC-stochastic Galerkin FEM allows to handle KL expansions with M up to 30 and r₁ up to 10 in moderate time.     

Finite element solution for beams using an element averaging technique

The article investigates the error in the computation of natural frequency through finite element models of the structures when elements of unequal length are chosen and then goes on to suggest an averaging technique for unequal length elements, with different structural properties. Using the element averaging technique, a typical structure has been analysed for its natural frequencies and the results are described.