Reduced basis technique for evaluating the sensitivity of the nonlinear vibrational response of composite plates

A reduced basis technique and a computational procedure are presented for generating the nonlinear vibrational response, and evaluating the first-order sensitivity coefficients of composite plates (derivatives of the nonlinear frequency with respect to material and geometric parameters of the plate). The analytical formulation is based on a form of the geometrically nonlinear shallow shell theory with the effects of transverse shear deformation, rotatory inertia and anisotropic material behavior included. The plate is discretized by using mixed finite element models with the fundamental unknowns consisting of both the nodal displacements and the stress-resultant parameters of the plate. The computational procedure can be conveniently divided into three distinct steps. The first step involves the generation of various-order perturbation vectors, and their derivatives with respect to the material and lamination parameters of the plate, using Linstedt-Poincaré perturbation technique. The second step consists of using the perturbation vectors as basis vectors, computing the amplitudes of these vectors and the nonlinear frequency of vibration, via a direct variational procedure. The third step consists of using the perturbation vectors, and their derivatives, as basis vectors and computing the sensitivity coefficients of the nonlinear frequency via a second application of the direct variational procedure. The effectiveness of the proposed technique is demonstrated by means of numerical examples of composite plates.     

Tracing post-limit-point paths with reduced basis technique

A reduced basis technique and a problem-adaptive computational algorithm are presented for predicting the post-limit-point paths of structures. In the proposed approach the structure is discretized by using displacement finite element models. The nodal displacement vector is expressed as a linear combination of a small number of vectors and a Rayleigh-Ritz technique is used to approximate the finite element equations by a small system of nonlinear algebraic equations.To circumvent the difficulties associated with the singularity of the stiffness matrix at limit points, a constraint equation, defining a generalized arc-length in the solution space, is added to the system of nonlinear algebraic equations and the Rayleigh-Ritz approximation functions (or basis vectors) are chosen to consist of a nonlinear solution of the discretized structure and its various order derivatives with respect to the generalized arc-length. The potential of the proposed approach and its advantages over the reduced basis-load control technique are outlined. The effectiveness of the proposed approach is demonstrated by means of numerical examples of structural problems with snap-through and snap-back phenomena.     

Comparison of boundary element and finite element methods for dynamic analysis of elastoplastic plates

Boundary and finite element methodologies for the determination of the response of inelastic plates are compared and critically discussed. Flexural dynamic plate bending problems are considered and a hardening elastoplastic constitutive model is used to describe material behaviour. The domain/boundary element methodology using linear boundary and quadratic interior elements and the finite element method with quadratic Mindlin plate elements are used in this work. The discretized equations of motion in both methodologies are solved by an efficient step-by-step time integration algorithm. Numerical results obtained are presented and compared in order to access the accuracy and computational efficiency of the two methods. In order to make the comparison as meaningful as possible, boundary and finite element computer codes developed by the author are used in this paper. In general, boundary elements appear to be a better choice than finite elements with respect to computational efficiency for the same level of accuracy.     

Collapse of long, noncircular, cylindrical shells under pure bending

This paper describes an extension of a method developed in a previous paper to determine the moment carrying capacity of elastoplastic noncircular cylindrical shells with infinite length by the finite element method. As a result of the shape change in the cross section of a shell during deformation, the bending moment reaches a global maximum value and then decreases as the bending curvature further increases. The shell would consequently collapse at the maximum moment. However, a bifurcation buckling may occur before the maximum moment can be developed. This bifurcation buckling could induce collapse of the shell under a moment less than the maximum. Determination of the likelihood that the bifurcation buckling would generate shell collapse may be made from the initial post-buckling behavior. An initial post-buckling analysis based on the J₂ deformation theory of plasticity has been developed in this paper. The finite element method with one spatial variable is used to locate the bifurcation point as well as to analyze the initial post-buckling behavior. Numerical examples of cylindrical shells with various cross-sectional shapes are shown. In particular, for a shell of square cross section, the moment at the bifurcation is much lower than the maximum value; however, the initial post-buckling analysis reveals that the state of equilibrium is still stable. Deep post-buckling analysis is required to determine the moment carrying capacity of a shell with such cross section.     

An Inverse Procedure for Crack Detection in Anisotropic Laminated Plates Using Elastic Waves

. This paper presents a computational inverse technique to detect the location and length of cracks in anisotropic laminated plates. The scattered elastic harmonic wave fields in the laminated plates with horizontal or vertical crack are calculated using the strip element method, whereby the anisotropic laminated plate is discretized into strip elements in the thickness direction. By applying the principle of virtual work, the governing differential equations of the wave propagation are derived for the field variables. These differential equations are solved analytically together with the vertical boundary conditions. The crack length and its location are then identified by minimizing an error function, which is defined as the difference between the scattered wave fields in plates with actual and searched parameters. A uniform micro-genetic algorithm is employed to search for the correct parameters that minimize the error function. Numerical examples are given to demonstrate the efficiency of the procedure in the detection of the location and the length of both horizontal and vertical cracks in composite laminates.     

A finite element method for construction of dynamical theories of layered plates

A procedure is presented for reduction of equations of three-dimensional elasticity to a twodimensional theory for elastodynamic behavior of a transversely heterogeneous plate. The method is based upon the elimination of the thickness coordinate by using, in conjunction with a variational principle, a set of finite element basis functions along the axis normal to the plate; an asymptotic analysis of the resulting semi-discrete equations yields a hierarchy of reduced order models for the plate. Some numerical results illustrate the applicability of the proposed methodology to layered plates.     

A finite element large displacement analysis of stiffened plates

A finite element analysis of the large deflection behaviour of stiffened plates using the isoparametric quadratic stiffened plate bending element is presented. The evaluation of fundamental equations of the stiffened plates is based on Mindlin's hypothesis. The large deflection equations are based on von Kármán's theory. The solution algmrithm for the assembled nonlinear equilibrium equations is based on the Newton-Raphson iteration technique. Numerical solutions are presented for rectangular plates and skew stiffened plates.     

A generalised technique for fracture analysis of cracked plates under combined tensile,bending and shear loads

The objective of this paper is to propose a generalized technique called numerically integrated modified virtual crack closure integral (NI-MVCCI) technique for fracture analysis of cracked plates under combined tensile, bending and shear loads. NI-MVCCI technique is used for post-processing the results of finite element analysis (FEA) for computation of strain energy release rate (SERR) components and the corresponding stress intensity factor (SIF) for cracked plates. NI-MVCCI technique has been demonstrated for 4-noded, 8-noded (regular and quarter-point) and 9-noded (regular and quarter-point) isoparametric plate finite elements. These elements are based on Mindlin’s plate theory that considers shear deformation. For all the elements, reduced integration/selective reduced integration techniques have been employed in the studies. In addition, for 9-noded element assumed shear interpolation functions have been used to overcome the shear locking problem. Numerical studies on fracture analysis of plates subjected to tension–moment and tension–shear loads have been conducted employing these elements. It is observed that among these elements, the 9-noded Lagrangian plate element with assumed shear interpolation functions exhibits better performance for fracture analysis of cracked plates.     

A reduction method for nonlinear structural dynamic analysis

A computational algorithm for predicting the nonlinear dynamic response of a structure is presented. The nonlinear system of ordinary differential equations resulting from the finite element discretization is highly reduced by means of a Rayleigh-Ritz analysis. The basis vectors are chosen to be the current tangent eigenmodes together with some modal derivatives that indicate the way in which the spectrum is changing. Only a few basis updatings are required during the whole time integration.The truncation error introduced at every change of basis is pointed out as the cause for a divergence-type behaviour, and some means for eliminating it are discussed.Results for examples involving large displacements are shown and compared to the results obtained by integrating the complete system of equations.     

Geometric nonlinear analysis of stiffened plates by the spline finite strip method

Geometric nonlinear analysis of stiffened plates is investigated by the spline finite strip method. von Karman’s nonlinear plate theory is adopted and the formulation is made in total Lagrangian coordinate system. The resulting nonlinear equations are solved by the Newton–Raphson iteration technique. To analyse plates having any arbitrary shapes, the whole plate is mapped into a square domain. The mapped domain is discretised into a number of strips. In this method, the displacement interpolation functions used are: the spline functions in the longitudinal direction of the strip and the finite element shape functions in the other direction. The stiffener is elegantly modelled so that it can be placed anywhere within the plate strip. The arbitrary orientation of the stiffener and its eccentricity are incorporated in the formulation. All these aspects have ultimately made the proposed approach a most versatile tool of analysis. Plates and stiffened plates are analysed and the results are presented along with those of other investigators for necessary comparison and discussion.     

Exploiting symmetries in the modeling and analysis of tires

Three basic types of symmetry (and their combinations) exhibited by tire response are identified. A simple and efficient computational strategy is presented for reducing both the size of the model and the cost of the analysis of tires in the presence of symmetry-breaking conditions (e.g., unsymmetry of the tire material, geometry, and/or loading). The strategy is based on approximation of the unsymmetric response of the tire with a linear combination of symmetric and antisymmetric global approximation vectors (or modes).The three main elements of the computational strategy are as follows: (1) use of three-field mixed finite element models having independent shape functions for stress resultants, strain components, and generalized displacements, with the stress resultants and the strain components allowed to be discontinuous at interelement boundaries; (2) use of operator splitting (additive decomposition of some of the matrices and vectors in the finite element model) to delineate the symmetric and antisymmetric contributions to the response; and (3) successive use of the finite element method and the classic Rayleigh-Ritz technique to substantially reduce the number of degrees of freedom. The finite element method is first used to generate a few global approximation vectors (or modes). Then the amplitudes of these modes are computed with the Rayleigh-Ritz technique.The proposed computational strategy is applied to three quasi-symmetric problems of tires, namely, (1) linear analysis of anisotropic tires through the use of semianalytic finite elements, (2) nonlinear analysis of anisotropic tires through the use of two-dimensional shell finite elements, and (3) nonlinear analysis of orthotropic tires subjected to unsymmetric loading. In the first two applications, the anisotropy (nonorthotropy) of the tire is the source of the symmetry breaking; in the third application, the quasi-symmetry is due to the unsymmetry of the loading. The effectiveness of the proposed computational strategy is also demonstrated with numerical examples, and its potential for handling practical tire problems is outlined.     

Dynamics of a large scale rigid–flexible multibody system composed of composite laminated plates

A novel computational approach for the dynamic analysis of a large scale rigid–flexible multibody system composed of composite laminated plates is proposed. The rigid parts in the system are described through the Natural Coordinate Formulation (NCF) and the flexible bodies in the system are modeled via the finite elements of Absolute Nodal Coordinate Formulation (ANCF), which can lead to a constant mass matrix for the derived system equation of motion. For modeling composite laminated plates accurately, a new composite laminated plate element of ANCF is proposed and the corresponding efficient formulations for evaluating both the elastic force and its Jacobian of the element are derived from the first Piola–Kirchhoff stress tensor. To improve computational efficiency, the sparse matrix technology and graph theory are used to solve the huge set of linear algebraic equations in the process of integrating the equations of motion by using the generalized-a method, and an OpenMP based parallel scheme is also introduced. Finally, the effectiveness of the proposed approach is validated through two numerical examples. One is the static simulation of a single composite laminated plate under gravity and the other is the dynamic simulations of unfolding process of a satellite system with a pair of complicated antennas.     

Application of boundary-domain element method to the free vibration problem of plate structures

The boundary-domain element method is applied to the free vibration problem of thin-walled plate structures. The static fundamental solutions are used for the derivation of the integral equations for both in-plane and out-of-plane motions. All the integral equations to be implemented are regularized up to an integrable order and then discretized by means of the boundary-domain element method. The entire system of equations for the plate structures composed of thin elastic plates is obtained by assembling the equations for each plate component satisfying the equilibrium and compatibility conditions on the connected edge as well as the boundary conditions. The algebraic eigenvalue equation is derived from this system of equations and is able to be solved by using the standard solver to obtain eigenfrequencies and eigenmodes. Numerical analysis is carried out for a few example problems and the computational aspects are discussed.     

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.     

Vibration analysis of laminated anisotropic shells of revolution

An efficient computational procedure is presented for the free vibration analysis of laminated anisotropic shells of revolution, and for assessing the sensitivity of their response to anisotropic (nonorthotropic) material coefficients. The analytical formulation is based on a form of the Sanders-Budiansky shell theory including the effects of both the transverse shear deformation and the laminated anisotropic material response. The fundamental unknowns consist of the eight stress resultants, the eight strain components, and the five generalized displacements of the shell. Each of the shell variables is expressed in terms of trigonometric functions in the circumferential coordinate and a three-field mixed finite element model is used for the discretization in the meridional direction.The three key elements of the procedure are: (a) use of three-field mixed finite element models in the meridional direction with discontinuous stress resultants and strain components at the element interfaces, thereby allowing the elimination of the stress resultants and strain components on the element level; (b) operator splitting, or decomposition of the material stiffness matrix of the shell into the sum of an orthotropic and nonorthotropic (anisotropic) parts, thereby uncoupling the governing finite element equations corresponding to the symmetric and antisymmetric vibrations for each Fourier harmonic; and (c) application of a reduction method through the successive use of the finite element method and the classical Bubnov-Galerkin technique.The potential of the proposed procedure is discussed and numerical results are presented to demonstrate its effectiveness.     

Elasto-plastic buckling of stiffener plates in beam-to-column flange connections

The problem of plastic buckling of steel plates is reviewed in relation to the load carrying capacity of stiffener plates in beam-to-column flange connections. Due to the non-uniformity of the stress distribution in these plates, the finite element method is used to compute the stresses in the elastic and plastic ranges. A bifurcation analysis is performed using both flow and deformation theory to evaluate the elasto-plastic buckling of the stiffener. A scaled inverse iterative version of the power method is employed to evaluate the bifurcation load. A parametric study is conducted on stiffeners and design curves are obtained showing the relationship between the critical stress and the slenderness ratio for different plate aspect ratios.     

Effective stiffness and microscopic deformation of an orthotropic plate containing arbitrary holes

Many engineering materials and structures, such as cellular structures, sandwich core structures and laminated plates with holes, can be modeled by an inclusion problem with anisotropic matrix. The paper studies the effective properties and the microscopic deformation of anisotropic plates with periodic holes by using direct and mathematical homogenization. The effective stiffnesses are calculated by different homogenization methods and the microscopic deformation of a RVE is modeled by the finite element method for the plate with arbitrarily shaped holes. All of the effective stiffness coefficients, especially stretching-shear coupling coefficients are evaluated.     

Thermal post-buckling of circular plates

The thermal post-buckling behaviour of isotropic circular plates has been studied in this paper through a simple finite element formulation. The accuracy of the solution by finite element method is established through a solution by Rayleigh-Ritz method for simply-supported circular plates. Effect of non-linearity on the load parameter is found to be much higher in the case of thermal loading than in the case of mechanical loading.     

A reinvestigation of post-buckling behaviour of elastic circular plates using a simple finite element formulation

A modified finite element formulation to study the post-buckling behaviour of elastic circular plates is presented in this paper. A discussion on the derivation of nonlinear stiffness matrix for post-buckling analysis is included and the present results are compared with continuum solutions.