An efficient recursive rotational-coordinate-based formulation of a planar Euler–Bernoulli beam

Multibody System Dynamics - A recursive rotational-coordinate-based formulation of a planar Euler–Bernoulli beam is developed, where large displacements, deformations, and rotations are...     

An asymptotic variational formulation for dynamic analysis of multilayered anisotropic plates

A multilevel variational formulation for dynamic analysis of multilayered anisotropic plates is developed within the framework of three-dimensional elasticity. By means of asymptotic expansions the Hellinger-Reissner functional for the elastodynamic problem is decomposed into a series of functionals with which a computational model can be constructed. In the formulation multiple time scales are introduced so that the secular terms can be eliminated systematically in obtaining a uniform expansion leading to valid asymptotic solution. Modifications to the approximation of various orders are determined by considering the solvability conditions of the higher-order equations. The model is adaptive, when combined with the finite element method, it has many appealing features, including that the displacements and transverse stresses may be interpolated independently, that the nodal degree-of-freedom (DOF) at each level is less than that of Kirchhoff plates, and that the mass and stiffness matrices generated at the leading-order level are always used at subsequent levels. Above all, the solution is three-dimensional in effect yet requires only two-dimensional interpolation. The through-thickness variations of the field variables are determined analytically with no need of interpolating in the thickness direction.     

Optimal design of dynamically loaded reinforced concrete frames under displacement and rotation constraints

The paper deals with reinforced concrete beams and frames subjected to short-time, high intensity dynamic pressure. The shape and geometry of the structure and the layout of the longitudinal reinforcement are given and the areas of reinforcement are design variables.The determination of the plastic displacements and deformations caused by pressure is based on the plastic hinge theory and on the assumption that during the dynamic response the structure undergoes stationary displacements. The problem is to minimize the total amount of reinforcement such that the plastic displacements do not exceed the allowable displacements prescribed at certain points of the structure, or alternatively, that the plastic rotations in the plastic hinges do not reach the limits at which brittle failure occurs.A variational formulation of the problem is presented and the solution is based on the optimality criteria approach which requires an iterative procedure. A few examples illustrate the application of the method.     

A simple method to impose rotations and concentrated moments on ANC beams

Recently introduced ANC beam elements furnish a simple formulation that allows to solve nonlinear problems of beams, including those with large displacements and strains, as well as complex nonlinear (inelastic) materials. The success and simplicity of these finite elements is mainly due to the fact that the only nodal degrees of freedom that they employ are displacements, and rotations are thus completely avoided. This in turn makes it very difficult to apply concentrated moments or to impose rotations at specific nodes of a finite element mesh. In this article, we present a simple enhancement to this beam formulation that allows to apply these two types of boundary conditions in a simple manner, making ANC beam elements more versatile for both multibody and structural applications.     

A large displacement analysis of a beam using a CAD geometric definition

In industrial applications, the design of beam and shell structures is generally based on global representations for curves and surfaces, such as Bezier curves, splines and NURBS. On the other hand, the mechanical analysis of these structures by the finite element method generally requires a refined discretization of the geometry. It is shown here that this analysis can be performed directly from the CAD representation which involves a small number of variables. Assuming a Bezier representation, the theory is developed in the framework of large displacements and then is applied to the static analysis of a beam undergoing large rotations. A compatible parametrization of the beam displacements is discussed. A formulation for the large displacement beam mechanics is summarized and applied to a straight beam submitted to various loading cases. Finally, three illustrative examples are presented.     

The soft way method and the velocity formulation

In this paper the new approach to dynamic contact problems is described. The velocity formulation was assumed and a new time integration scheme was elaborated. The space-time finite element method used in derivation enables control of the accuracy (order of the error) and stability. Methods for the solution of contact problems were discussed. A discretized approach, prepared for large displacements and large rotations, enabled real engineering problems to be solved in a relatively short time.     

A mixed computational algorithm for plane elastic contact problems—I. formulation

A mixed variational statement and corresponding finite element model are developed for an arbitrary plane body undergoing large deformations (i.e. large displacements, large rotations and small strains) under external loads using the updated Lagrangian formulation. The mixed finite element formulation allows the nodal displacements and stresses to be approximated independently. Two different contact algorithms are presented for the separate cases of a thin plate in contact with a rigid pin and a flexible pin, and the algorithms account for the computational difficulties that arise from the unknown contact area and the presence of friction between the pin and the plate.     

A C finite element method for the analysis of inextensible pipe lines

A C⁰ finite element method in which rotations and displacements are coupled by a penalty function formulation is developed for pipe line analysis. Inextensibility is also handled by a penalty method. Simplicity and effectiveness of this approach is illustrated by three dimensional computations including contact problems and an appendix giving all necessary stiffness matrices.     

On the role of geometrically exact and second-order theories in buckling and post-buckling analysis of three-dimensional beam structures

Several geometrically nonlinear beam models are evaluated with respect to their utility in the analysis of buckling and post-buckling behavior of three-dimensional beam structures. The first two models are based on the so-called geometrically exact beam theory capable of representing finite rotations and finite displacements. The principal difference between these models concerns only the chosen parameterization of finite rotations, with the orthogonal matrix used in the first and the rotation vector used in the second one. The third beam model based on the second-order approximation of finite rotations is also discussed along with its application to constructing a consistent formulation of the linear eigenvalue problem for computing an estimate of the critical load. Exact linearized forms, which are crucial for facilitating the buckling load computation and assuring a robust performance of a Newton-method-based continuation strategy, are presented for all three beam models. An elaborate set of numerical simulations of buckling and post-buckling analysis of beam structures is given in order to illustrate the performance of each of the presented models. Finally, some conclusions are drawn.     

A finite element formulation for shells of negative Gaussian curvature

An expression for the strain energy of a shell of negative Gaussian curvature, including thickness shear deformations and without neglecting z/R in comparison with unity, is derived. Then a curved trapezoidal finite element formulation based on the principle of minimum potential energy is obtained. The shell element has eight nodes with 40 degrees of freedom and at each node there are three displacements and two rotations. The formulation is applicable for both thin and moderately thick shell analysis. The performance of this finite element is verified by applying it to some problems existing in the literature.     

Slope discontinuities in the finite element absolute nodal coordinate formulation: gradient deficient elements

In this paper, the treatment of the slope discontinuities in the finite element absolute nodal coordinate formulation (ANCF) is discussed. The paper explains the fundamental problems associated with developing a constant transformation that accounts for the slope discontinuities in the case of gradient deficient ANCF finite elements. A procedure that allows for the treatment of slope discontinuities in the case of gradient deficient finite elements which do not employ full parameterization is proposed for the special case of commutative rotations. The use of the proposed procedure leads to a constant orthogonal element transformation that describes the element initial configuration. As a consequence, one obtains in the case of large deformation and commutative rotations, a constant mass matrix for the structures. In order to achieve this goal, the concept of the intermediate finite element coordinate system is invoked. The intermediate finite element coordinate system used in this investigation serves to define the element reference configuration, follows the rotation of the structure, and maintains a fixed orientation relative to the structure coordinate system. Since planar rotations are always commutative, the procedure proposed in this investigation is applicable to all planar gradient deficient ANCF finite elements.     

A hybrid strain technique for finite element analysis of plates and shells

A specialization of the Hu-Washizu [1] functional wherein strains and displacements are taken as independent variables is employed in the formulation of ‘hybrid’ elements. Both the strains and displacements are independently interpolated with the strains being eliminated at the element level, leaving displacement variables only to be assembled into the global system of equations. This distinguishes such elements as ‘hybrid’, in contrast to ‘mixed’ wherein the global system of equations contains all the discretized variables. Applications including ‘thick’ plate and shell elements are considered. In many applications the hybrid strain technique appears more natural than the hybrid stress technique since stress discontinuities are accommodated quite conveniently.     

Three-dimensional finite elements for large deformation micropolar elasticity

In this paper, we propose a three-dimensional finite element formulation for micropolar elasticity dealing with large displacements and small strains (or equivalently small strains and finite rotations). A comprehensive outline of the theory’s characteristical features is given and we try to elucidate the set-up of a possible non-linear finite element implementation. One focus of the present study is on a sound verification process, featuring the construction of an enhanced Patch Test and the assessment of quadratic asymptotic rates of convergence. Aspects of performance and validity are discussed at a set of numerical examples. We show that, the proposed model is able to reproduce the transition between micropolar and classical continua highly accurate. Finally, we present results underlining the implementation’s applicability in the realm of finite deformation with arbitrarily large rotations.     

Nonlinear dynamic analysis of frame structures

A geometrically nonlinear dynamic analysis method is presented for frames which may be subjected to finite rotations in three-dimensional space. The proposed method is based on the static geometrically nonlinear analysis method reported by Yoshida et al., in which the governing incremental equilibrium equation is represented by the coordinates after the deformation themselves rather than conventional displacements. The governing dynamic equilibrium equation for each element is obtained from the static equation by adding the inertia term. In the solution procedure, a modified Steffensen's iteration process is introduced and combined with the two-step approximation and iterative correction solution procedure developed for static analysis. A numerical example of a curved cantilever beam under lateral loads indicates the effectiveness of the proposed method in cases with three-dimensional finite rotations. Forced vibration analyses of a two-hinged shallow arch are conducted under centrally concentrated loading with several loading amplitudes. The resulting dynamic buckling load is compared with that given by Gregory and Plaut in 1982, who used Galerkin method, and shows good agreement.     

Triangular thick plate elements based on a hybrid-Trefftz approach

The paper presents a family of triangular, thick plate elements derived using the hybrid-Trefftz approach. Exact solutions of the governing thick plate equations are used as interpolations for the internal element displacements. An immediate benefit of this approach is that the locking problem is avoided a priori. Independent interpolations are used to describe the displacement and rotations on the element boundaries. The element formulation is based on a modified hybrid-stress principle, leading to a standard stiffness formulation. This enables the elements to be readily implemented into existing finite element schemes. A number of examples are considered to demonstrate the accuracy achieved by the elements.     

A discrete formulation for elastic solids with damaging interfaces

An elastic solid with embedded interfaces, loci of possible displacement discontinuities, is considered here. Decohesion and quasi-brittle fracture processes are simulated by making use of softening interface laws which relate tractions to displacements jumps. A discrete formulation in terms of interface variables only, is obtained. The space discretization is carried out by means of a mixed finite element approach in which all interface variables are modelled. Study of uniqueness of the rate problem formulated in terms of interface variables and of stability of the equilibrium states is presented. Some examples are shown in order to clarify the formulation.     

On the use of shape functions in the cell centered finite volume formulation for plate bending analysis based on Mindlin–Reissner plate theory

This paper deals with a new formulation of the cell centered finite volume application for plate bending analysis based on Mindlin–Reissner plate theory. In this formulation shape functions are used to represent the variation of the unknown variables across the control volumes’ faces, which facilitates the calculation of stress resultants on the faces. The performance of the formulation for the computation of displacements and stress resultants for thin and thick plates is evaluated in a number of test problems. This testing reveals that the proposed approach enhances the predictive capability of the finite volume method in the analysis of thin to thick plates.     

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.