Horizontally curved isoparametric beam element with or without elastic foundation including effect of shear deformation

A new horizontally curved three-noded isoparametric beam element with or without an elastic base throughout its length, is formulated, in which the same set of parabolic shape functions in natural coordinates is used to interpolate beam geometry as well as displacements at any point of the beam. The formulation includes shear deformation effects and is also capable of accounting for torsional loading. The elastic base is assumed to offer distributed vertical reaction and torque at any point of the beam proportional to the vertical deflection and angle of twist respectively at that point. Development of the stiffness matrix is presented. A two point Gaussian quadrature rule has been followed for the necessary numerical integration. Numerical results for a few sample problems have been presented and comparison with the analytical solutions indicates the suitability of the element.     

Finite element analysis of laminated shells of revolution

A finite element formulation for the analysis of axisymmetric fibre reinforced laminated shells subjected to axisymmetric load is presented. The formulation includes arbitrary number of bonded layers each of which may have different thicknesses, orientation of elastic axes, and elastic properties. Superparamatric curved elements[17] having four degrees of freedom per node including the normal rotation, are used. Stress-strain relation for an arbitrary layer is obtained from the consideration of three dimensional aspect of the problem. The element stiffness matrix has been obtained by using Gauss quadrature numerical integration, even though the elasticity matrix is different for different layers. The formulation is checked for a cylindrical tube subjected to internal pressure and axial tension, and the results are found to compare very well with the elastic solution [9].     

A high precision triangular laminated anisotropic cylindrical shell finite element

The stiffness matrix for a high precision triangular laminated anisotropic cylindrical shell finite element has been formulated and coded into a composite structural analysis program. The versatility of the element's formulation enables its use in the analysis of multilayered composite plate and cylindrical shell type structures taking into account actual lamination parameters. The example applications presented demonstrated that accurate predictions of stresses as well as displacements are obtained with modest number of elements.     

Finite element analysis of buckling and post-buckling behaviors of arches with geometric imperfections

The buckling and post-buckling behavior of arches is very sensitive to their geometric imperfections. The purpose of this paper is to develop a refined curved finite element that might accurately represent the actual geometry of arches so that the imperfection effects on their buckling behavior could be properly investigated. For an arch with known geometric imperfections, the element stiffness matrix is precisely formulated in terms of Lagrangian variables for a perfect arch from a general incremental variational principle. In general, the element stiffness matrix contains Lagrangian strain, first and second order incremental strain and imperfection terms. For any general planar imperfect arch with a variable curvature, the element stiffness matrix is evaluated by numerical integration; however, for a nominally circular arch, it can be represented in closed form. Numerical results in terms of load-deformation curves are presented for a number of circular arches with and without imperfections and compared with existing solutions.     

Deflection analysis of expanded open-web steel beams

The finite element method is very appropriate for calculating stresses at isolated areas of expanded open-web steel beams. For deflection analysis, however, the entire beam, or one-half the beam if the system is symmetrical, must be included in the idealization and therefore it is not practical to obtain deflections by a direct application of the finite element method. In this paper, the authors present a method of deflection analysis which treats the castellated beam as an assemblage of typical segments and utilizes the finite element method to form the stiffness matrix for the typical segment. The beam deflections at the panel points are then computed by the conventional stiffness method. Two different idealizations were tried for the finite element analysis of a typical segment. These idealizations resulted in a 13 × 13 and a 7 × 7 stiffness matrix. Deflection values calculated by using the 7 × 7 stiffness matrix showed close agreement with those obtained experimentally.     

Numerically integrated finite elements as assemblies of bars

An eight node plane stress element is formulated using a system of bars. These bars are located at the Gauss points and their widths are equal to the weights of the Gauss rule considered. It is shown that an (n × n) integration rule yields, for the stiffness matrix, exactly the same values as a grid of (n × n) bars located at the corresponding Gauss points. This analogy is used to justify the existence of mechanisms with underintegration.     

Geometric nonlinear analysis of flexible spatial beam structures

An updated Lagrangian formulation of the spatial beam element is presented for a purely geometric nonlinear analysis in which the geometric stiffness matrix is expressed either by a one-dimensional integration of the stress resultants or by a closed form of element-end forces. A computer code, NACS, is developed based on this formulation which has a number of facilities to meet the special requirements for the analysis of suspension and cable-stayed bridges. Several example problems are reported.     

Dynamic stiffness matrix of an axisymmetric shell and response to harmonic distributed loads

This paper describes a procedure for taking into account distributed loads in the dynamic stiffness matrix formulation, also known as the continuous element method. In that formulation, concentrated or linearly distributed loads are taken into account by considering the necessary number of elements in such a way that these loads are applied on element boundaries. As for distributed loads such as pressure and surface forces, they were not provided for in the formulation. The validation is achieved by comparison of accuracy and computational time of harmonic response calculations performed by the continuous element model and a finite element model.     

Analysis of Large Flexible Body Deformation in Multibody Systems Using Absolute Coordinates

To consider large deformation problems in multibody system simulations afinite element approach, called absolute nodal coordinate.formulation,has been proposed. In this formulation absolute nodal coordinates andtheir material derivatives are applied to represent both deformation andrigid body motion. The choice of nodal variables allows a fullynonlinear representation of rigid body motion and can provide the exactrigid body inertia in the case of large rotations. The methodology isespecially suited for but not limited to modeling of beams, cables andshells in multibody dynamics.This paper summarizes the absolute nodal coordinate formulation for a 3D Euler–Bernoulli beam model, in particular the definition of nodal variables, corresponding generalized elastic and inertia forces and equations of motion. The element stiffness matrix is a nonlinear function of the nodal variables even in the case of linearized strain/displacement relations. Nonlinear strain/displacement relations can be calculated from the global displacements using quadrature formulae.Computational examples are given which demonstrate the capabilities of the applied methodology. Consequences of the choice of shape.functions on the representation of internal forces are discussed. Linearized strain/displacement modeling is compared to the nonlinear approach and significant advantages of the latter, when using the absolute nodal coordinate formulation, are outlined.     

Nonlinear finite element analysis of elastic systems under nonconservative loading-natural formulation. part I. Quasistatic problems

The paper presents a nonlinear finite element analysis of elastic structures subject to nonconservative forces. Attention is focused on the stability behaviour of such systems. This leads mathematically to non-self-adjoint boundary-value problems which are of great theoretical and practical interest, in particular in connection with alternative modes of instability like divergence of flutter. Only quasistatic effects are however considered in the present part.The methodology of our theory is general, but the specific thrust of the present research is directed towards the analysis of structures acted upon by displacement-dependent nonconservative (follower) forces. In a finite element formulation the analysis of geometrically nonlinear elastic systems subject to such forces gives, in general, rise to a contributory nonsymmetric stiffness matrix known as the load correction matrix. As a result, the total tangent stiffness matrix becomes unsymmetric - an indication of the non-self-adjoint character of the problem. Our theory is based on the natural mode technique [1, 2, 3]and permits i.a. a simple but elegant derivation of the load correction matrix. The application of the general theory as evolved in this paper is demonstrated on the beam element in space. A number of numerical examples are considered including divergence and flutter types of instability, for which exact analytic solutions are known. The problems demonstrate the efficiency of the present finite element formulation.The paper furnishes also a novel and concise formulation of finite rotations in space which may be considered as a conceptual generalization of the theory presented in [2, 3].     

Stiffness of beam-columns on elastic foundation with exact shape functions

Exact shape functions from the solution of the governing differential equation are used to determine the stiffness and equivalent joint load matrices for a beam-column finite element resting on a Winkler-type elastic foundation. The degrees of freedom at the nodes are assumed to be lateral displacement and flexural rotation. The formulation is verified by analyzing a continuous beam-column, and the results are compared with an existing solution. A FORTRAN subroutine that generates the stiffness matrix and equivalent joint forces is appended. This subroutine can be easily incorporated into existing finite element or frame analysis programs.     

Optimal design of steel frames subject to gravity and seismic codes' prescribed lateral forces

Allowable stress design of two-dimensional braced and unbraced steel frames based on AISC specifications subject to gravity and seismic lateral forces is formulated as a structural optimization problem. The nonlinear constrained minimization algorithm employed is the feasible directions method. The objective function is the weight of the structure, and behaviour constraints include combined bending and axial stress, shear stress, buckling, slenderness, and drift. Cross-sectional areas are used as design variables. The anylsis is performed using stiffness formulation of the finite element analysis method. Equivalent static force and response spectrum analysis methods of seismic codes are considered. Based on the suggested methodology, the computer program OPTEQ has been developed. Examples are presented to illustrate the capability of the optimal design approach in comparative study of various types of frames subjected to gravity loads and seismic forces according to a typical code.     

A linear quadrilateral shell element with fast stiffness computation

A new quadrilateral shell element with 5/6 nodal degrees of freedom is presented. Assuming linear isotropic elasticity a Hellinger–Reissner functional with independent displacements, rotations and stress resultants is used. Within the mixed formulation the stress resultants are interpolated using five parameters for the membrane forces as well as for the bending moments and four parameters for the shear forces. The hybrid element stiffness matrix resulting from the stationary condition is integrated analytically. This leads to a part obtained by one point integration and a stabilization matrix. The element possesses the correct rank, is free of locking and is applicable within the whole range of thin and thick shells. The in-plane and bending patch tests are fulfilled and the computed numerical examples show that the convergence behaviour of the stress resultants is very good in comparison to comparable existing elements. The essential advantage is the fast stiffness computation due to the analytically integrated matrices.     

Nonlinear finite element analysis of elastic frames

A very simple and effective formulation and numerical procedure to remove the restriction of small rotations between two successive increments for the geometrically nonlinear finite element analysis of in-plane frames is presented. A co-rotational formulation combined with small deflection beam theory with the inclusion of the effect of axial force is adopted. A body attached coordinate is used to distinguish between rigid body and deformational rotations. The deformational nodal rotational angles are assumed to be small, and the membrane strain along the deformed beam axis obtained from the elongation of the arc length of the deformed beam element is assumed to be constant. The element internal nodal forces are calculated using the total deformational nodal rotations in the body attached coordinate. The element stiffness matrix is obtained by superimposing the bending and the geometric stiffness matrices of the elementary beam element and the stiffness matrix of the linear bar element. An incremental iterative method based on the Newton-Raphson method combined with a constant arc length control method is employed for the solution of the nonlinear equilibrium equations. In order to improve convergence properties of the equilibrium iteration, a two-cycle iteration scheme is introduced. Numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method.     

Interval Finite Elements as a Basis for Generalized Models of Uncertainty in Engineering Mechanics

Latest scientific and engineering advances have started to recognize the need for defining multiple types of uncertainty. Probabilistic modeling cannot handle situations with incomplete or little information on which to evaluate a probability, or when that information is nonspecific, ambiguous, or conflicting [12], [47], [50]. Many interval-based uncertainty models have been developed to treat such situations. This paper presents an interval approach for the treatment of parameter uncertainty for linear static structural mechanics problems. Uncertain parameters are introduced in the form of unknown but bounded quantities (intervals). Interval analysis is applied to the Finite Element Method (FEM) to analyze the system response due to uncertain stiffness and loading. To avoid overestimation, the formulation is based on an element-by-element (EBE) technique. Element matrices are formulated, based on the physics of materials, and the Lagrange multiplier method is applied to impose the necessary constraints for compatibility and equilibrium. Earlier EBE formulation provided sharp bounds only on displacements [32]. Based on the developed formulation, the bounds on the system’s displacements and element forces are obtained simultaneously and have the same level of accuracy. Very sharp enclosures for the exact system responses are obtained. A number of numerical examples are introduced, and scalability is illustrated.     

A co-rotational formulation for nonlinear dynamic analysis of curved euler beam

A co-rotational finite element formulation for the dynamic analysis of a planar curved Euler beam is presented. The Euler-Bernoulli hypothesis and the initial curvature are properly considered for the kinematics of a curved beam. Both the deformational nodal forces and the inertial nodal forces of the beam element are systematically derived by consistent linearization of the fully geometrically nonlinear beam theory in element coordinates which are constructed at the current configuration of the corresponding beam element. An incremental-iterative method based on the Newmark direct integration method and the Newton-Raphson method is employed here for the solution of the nonlinear dynamic equilibrium equations. Numerical examples are presented to demonstrate the effectiveness of the proposed element and to investigate the effect of the initial curvature on the dynamic response of the curved beam structures.     

An efficient facet shell element for corotational nonlinear analysis of thin and moderately thick laminated composite structures

In the present work, an efficient facet shell element for the geometrically nonlinear analysis of laminated composite structures using the corotational approach is developed. The facet element is developed by combining the discrete Kirchhoff-Mindlin triangular bending element (DKMT), and the optimal membrane triangular element (OPT). The membrane-bending coupling effect of composite laminates is incorporated in the formulation, and inconsistent stress stiffness matrix is formulated. Using corotational formulation and the proposed facet element, some example laminated composite structures with geometric nonlinearity are analyzed, and the results are compared with those found using other facet elements.     

The Galerkin element method applied to the vibration of rectangular damped sandwich plates

The Galerkin element method (GEM), which combines Galerkin orthogonal functions with the traditional finite element formulation, has previously been applied successfully to the vibration analysis of damped sandwich beams, and an improved iteration method was developed for its eigen solution. In the current paper, this promising method is extended to the vibration of damped sandwich plates. A quite different model is formulated which has both nodal coordinates and edge coordinates, while in the case of beams, there are only nodal coordinates. Displacement compatibility over the interfaces between the damping layer and the elastic layers is taken account of in order to ensure a conforming element and thereby guarantee good accuracy. The seed matrix method is proposed for simplifying the building of the element mass, stiffness and damping matrices. Numerical examples show that the application of the GEM to sandwich plate structures is computationally very efficient, while providing accurate estimates of natural frequencies and modal damping over a wide frequency range.     

Finite element buckling analysis of stiffened plates

An isoparametric stiffened plate bending element for the buckling analysis of stiffened plates has been presented. In the present approach, the stiffener can be positioned anywhere within the plate element and need not necessarily be placed on the nodal lines. The element, being isoparametric quadratic, can readily accommodate curved boundaries, laminated materials and transverse shear deformation. The formulation is applicable to thin as well as thick plates. The buckling loads for various rectangular and skew stiffened plates with varying skew angles and stiffness parameters have been indicated. The results show good agreement with those published.     

A coupled formulation of finite and boundary element methods in two-dimensional elasticity

A symmetric stiffness formulation based on a boundary element method is studied for the structural analysis of a shear wall, with or without cutouts. To satisfy compatibility requirements with finite beam elements and to avoid problems due to the eventual discontinuities of the traction vector, different interpolation schemes are adopted to approximate the boundary displacements and tractions. A set of boundary integral equations is obtained with the collocation points on the boundary, which are selected by the error minimization technique proposed in this paper, and the stiffness matrix is formulated with those equations and symmetric coupling techniques of finite and boundary element methods. The newly developed plane stress element can have the openings in its interior domain and can be easily linked with the finite beam/column elements.