A unified approach to three finite element theories for geometric nonlinearity

The paper considers theoretical and numerical questions associated with finite element analysis of geometrically nonlinear structural problems. A variety of finite element theories have recently arisen leading to geometric stiffness matrices, initial displacement matrices, and other similar forms. The present analysis determines an integrated unified approach to these geometrically nonlinear problems. The development leads very readily to the resolution of questions pertaining to highly nonlinear problems, and the interrelations of recent emergent theories become apparent. The investigation includes evaluation of alternative approaches for numerical computations of incremental and single-step analyses and advocates specific basic approximations for particular applications.     

Hybrid strain based three node flat triangular shell elements—I. Nonlinear theory and incremental formulation

Aspects and theories of nonlinear analysis of structures, with special emphasis on structures that are discretized by the finite element method, are discussed. The updated Lagrangian formulation and the incremental Hellinger-Reissner variational principle are adopted. The independently assumed fields employed are the incremental displacements and incremental strains. Accordingly, the incremental second Piola-Kirchhoff stress and the incremental Washizu strain are selected as the incremental stress and strain measures. Various schemes for the transformation of the second Piola-Kirchhoff stress to Cauchy stress are included. Two versions of linear and nonlinear element stiffness and mass matrices are considered. These are the director and simplified versions. Variable thickness of the shell is considered so as to account for the ‘thinning effect’ due to large strain. Material nonlinearity studied in this paper is of elasto-plastic type with isotropic strain hardening. Cases in which small elastic but large plastic strain condition applies are considered and the J₂ flow theory of plasticity, in conjunction with Ilyushin's yield criterion, is employed. To simplify the derivation of (small displacement) stiffness matrix and to facilitate the derivation of explicit expressions for the element matrices, the non-layered approach has been applied.     

A simple and effective element for analysis of general shell structures

A simple flat three-node triangular shell element for linear and nonlinear analysis is presented. The element stiffness matrix with 6 degrees-of-freedom per node is obtained by superimposing its bending and membrane stiffness matrices. An updated Lagrangian formulation is used for large displacement analysis. The application of the element to the analysis of various linear and nonlinear problems is demonstrated.     

Finite element analysis of the in-plane behaviour of annular disks

In-plane analysis of annular disks using the finite element method is presented. A semi-analytical, one-dimensional finite element model is developed using a Fourier series approach to account for the circumferential behaviour. Using displacement functions which are exact solutions of the two dimensional elasticity plane stress problem, the shape functions, stiffness matrices and mass matrices corresponding to the 0th, 1st and nth harmonics are derived. To show the utility of this new element, example probelms have been solved and compared with the exact solution. The present element can be readily coded into any general purpose finite element program.     

FINGER: A Symbolic System for Automatic Generation of Numerical Programs in Finite Element Analysis

FINGER is a LISP-based system to derive formulas needed in finite element analysis, and to generate FORTRAN code from these formulas. The generated programs can be used with existing, FORTRAN-based finite element analysis packages. This approach aims to replace tedious hand computations that are time consuming and error prone. The design and implementation of FINGER are presented. Techniques for generating efficient code are discussed. These include automatic intermediate expression labelling, interleaving formula derivation with code generation, exploiting symmetry through generated functions and subroutines. Current capabilities include generation of material matrices, strain-displacement matrices and stiffness matrices. FINGER contains a package, called GENTRAN, that translates symbolic formulas into FORTRAN. GENTRAN can generate functions, subroutines and entire programs. Thus, it is also of interest as a general-purpose FORTRAN code generator, Aside from the finite element application, the techniques developed and employed are useful for automatic code generation in general.     

Large displacement extension of consistent shell element for static and dynamic analysis

The superior performance of the consistent shell element in the small deflection range has encouraged the authors to extend the formulation to large displacement static and dynamic analyses. The nonlinear extension is based on a total Lagrangian approach. A detailed derivation of the non-linear extension is based on a total Lagrangian approach. A detailed derivation of the non-linear stiffness matrix and the unbalanced load vector for the consistent shell element is presented in this study. Meanwhile, a simplified method for coding the nonlinear formulation is provided by relating the components for the nonlinear B-matrices to those of the linear B-matrix. The consistent mass matrix for the shell element is also derived and then incorporated with the stiffness matrix to perform large displacement dynamic and free vibration analyses of shell structures. Newmark's method is used for time integration and the Newton-Raphson method is employed for iterating within each increment until equilibrium is achieved. Numerical testing of the nonlinear model through static and dynamic analyses of different plate and shell problems indicates excellent performance of the consistent shell element in the nonlinear range.     

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.     

The finite element solution of a class of elastica problems

A finite element procedure for the analysis of an inextensional elastica bent through frictionless supports is presented. Element displacements are expressed in terms of cubic Hermite polynomials with nodal displacements and derivatives being determined to minimise the strain energy. It is shown that the axial force at any point is proportional to the square of the bending moment, which enables member incremental stiffness matrices to be expressed in a similar form to those used in stability analysis. The iterative procedure proposed for solving the nonlinear stiffness equations may readily be incorporated into any continuous beam program by the modification and addition of very few statements. An example is considered which indicates that accurate solutions may be obtained to a wide range of practical problems using the proposed technique.     

An improved formulation of space stiffeners

This paper presents a displacement based finite element model for predicting the constraint torsion effect of stiffeners. In structural modelling, the plate/shell and the stiffeners are treated as separate elements where the displacement compatibility transformation between these two types of elements takes into account the constraint torsional warping effect in the stiffeners. The development is based on a general beam theory which includes flexural-torsion coupling, constrained torsion warping, and shear-centre location. The virtual work principle includes the second order terms of finite beam rotations. For finite element analysis, cubic Hermitian polynomials are used as shape functions of the straight space frame element with two nodes. Elastic stiffness and geometric stiffness matrices for an arbitrary cross-section are evaluated in a closed form, and load correction stiffness for eccentric stiffener loads are considered. To demonstrate the importance of torsion warping constraints and to illustrate the accuracy of this formulation, finite element solutions are presented and compared with available solutions.     

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.     

An energy balancing strategy for solution of combined geometrical and material nonlinearity problems

This paper describes an energy based line-search technique for large displacement and nonlinear stability analysis of planar truss and frame structures which, in addition to geometric non-linearities, simultaneously exhibit nonlinear material behavior. To improve the convergence characteristics of incremental-iterative procedures, a line search seeks a scalar factor which, at fixed load level, scales the displacement vector in such a way that the fundamental energy identity is satisfied. Furthermore, this factor can also be used for guiding the load incrementation. Aiming at computational simplicity and economy, efficient “section-type” beam elements are introduced. Using the total Lagrangian formulation, consideration of geometric nonlinearities yields, however, stiffness matrices which are linear and quadratic functions of the displacements. The material nonlinearity is treated by bilinear laws. The inelastic behavior is defined for several cross sections: thus, by forming the element stiffness matrices for bending the resulting variable stiffness must be accounted for. The effectiveness of the simple solution procedure has been tested using planar truss, beam, and cable-beam-type structures with known analytical and/or alternative solutions. Good agreement with these existing solutions demonstrates the applicability of the proposed method for routine engineering analysis.     

Buckling of pressure loaded rings and shells by the finite element method

The correct formulations for solving nonlinear structural problems by the finite element method have now been established. Numerous investigators have given the derivation for the solution of problems by the incremental tangent stiffness method and total formulation methods. These derivations have been applied to many problems and the results have been shown to be quite accurate for the problems that have been selected. However there is one area of application that has received practically no attention. This is in the investigation of the buckling strength of pressure loaded rings and shells. The effect of pressure loading where the loading changes direction as the structure deforms has been included in several previous derivations, by what is known as the load stiffness matrix, but to the author's knowledge no one has investigated problems where this effect has been included in the solution procedure. For rings and some buckling modes of shells, the results can be in error by as much as 50%.This paper will describe an iterative process for solving the nonlinear equilibrium equations and correcting the loads to include the effect of changing geometry at each load level. This approach is different from the classical eigenvalue or bifurcation method. Several case studies will be described which were performed on ring and shell problems. The geometry of these example problems were axisymmetric and in order to apply a nonlinear collapse analysis, the structure had to be perturbed out of its axisymmetric pattern into a buckling pattern. Imperfect geometry and very small concentrated loads were used to cause this perturbation and this will be described in the paper. The sensitivity of the computed collapse pressure to the finite element mesh gradation will be discussed. A comparison will be made between results obtained by including the effect of following pressure load and those obtained by not including this effect.     

Vibration and buckling analysis of axisymmetric polar orthotropic circular plates

The vibration and stability analysis of polar orthotropic circular plates using the finite element method is discussed. In order to formulate the eigenvalue problems associated with the vibration and stability analyses, the clement stiffness, mass, and stability coefficient matrices are presented. By assuming the static displacement function, which is an exact solution of the polar orthotropic circular plate equation, approximates the vibration and buckling modes, the mass and stability coefficient matrices are readily derived from the given displacement function. Results showing the effects of orthotropy on natural frequencies and buckling loads are compared with their isotropic counterpart.     

Analytic formulation of stiffness matrix for axisymmetric solids

Stiffness matrices for axisymmetric solids with arbitrary loading were derived through the application of variational principles in analytic form. The singularities in the stiffness matrices were removed through displacement constraints along the axis of symmetry for each circumferential mode. The shear stresses and maximum deflections of a set of Saint-Venant flexural problems were obtained both analytically and numerically. The results indicate that the finite element analysis with the analytic stiffness matrix provides a very good solution. The same problems were solved with a commercial code, ANSYS. and showed that the analytic stiffness matrix contributed to a faster convergence rate as the number of elements increases in an analysis.     

Static,vibration and buckling analysis of axisymmetric circular plates using finite elements

The static, vibration, and buckling analysis of axisymmetric circular plates using the finite element method is discussed. For the static analysis, the stiffness matrix of a typical annular plate element is derived from the given displacement function and the appropriate constitutive relations. By assuming that the static displacement function, which is an exact solution of the circular plate equation ?²?²W = 0, closely represents the vibration and buckling modes, the mass and stability coefficient matrices for an annular element are also constructed. In addition to the annular element, the stiffness, mass, and stability coefficient matrices for a closure element are also included for the analysis of complete circular plates (no center hole). As an extension of the analysis, the exact displacement function for the symmetrical bending of circular plates having polar orthotropy is also given.     

On large displacement-small strain analysis of structures with rotational degrees of freedom

The matrix displacement analysis of geometrically nonlinear structures becomes an intricate task as soon as finite elements in space with rotational degrees of freedom are considered. The fundamental reason for these difficulties lies in the non-commutativity of successive finite rotations about fixed axes with different directions. In order to circumvent this difficulty, a new definition of rotations — the so-called semitangential rotations — is introduced in this paper. Our new definition leads to a reformulation of the theory of [1,2]which in itself is clearly consistent and correct.In contrast to rotations about fixed axes these semitangential rotations which correspond to the semitangential torques of Ziegler [3]possess the most important property of being commutative. In this manner, all complexities involved in the standard definition of rotations are avoided ab initio.A specific aspect of this paper is a careful exposition of semitangential torques and rotations, as well as the consequences of the semitangential definitions for the geometrical stiffness of finite elements. In fact, these new definitions permit a very simple and consistent derivation of the geometrical stiffness matrices. Moreover, the semitangential definition automatically leads to a symmetric geometrical stiffness which clearly expresses that the nonlinear strain-displacement relations must satisfy the condition of conservativity of the structure itself — independently of any loading.The general theory of geometrical stiffness matrices as evolved in this paper is applied to beams in space. The consistency of the theory is demonstrated by a large number of numerical examples not only of straight beams but also of the lateral and torsional buckling and post-buckling behaviour of stiff-joined frames. Most of the former developments appear to be inadequate.     

Consistent tangent stiffness for nonlocal damage models

This paper deals with the computational analysis of strain localization problems using nonlocal continuum damage models of the integral type. The general framework for a consistent derivation of the “nonlocal” tangent stiffness is presented. The properties of the tangent stiffness matrix are discussed and the corresponding assembly procedure is described. The quadratic rate of convergence of the Newton–Raphson iteration procedure is demonstrated and the efficiency of the proposed technique is compared to the standard approach based on the secant or elastic stiffness matrices. In this context, performance of direct and iterative solvers for the linearized equilibrium equations is also examined.     

A finite difference variational method for bending of plates

The method of analysis for bending of plates presented in this paper combines a finite difference scheme for the plate strain components and a variational derivation of the equations of motion or equilibrium. The plate strain components are expressed in terms of discrete nodal displacements with the aid of the two dimensional Taylor expansion. Consequently, the virtual work, or the first variation of the strain energy, in an area element is found as a function of the nodal displacements. The derivation of the element forces or the element stiffness matrices and the assembly of the equations of motion or equilibrium follows closely the steps of the finite element method.     

Large deflection analysis of plates and shells by discrete energy method

The discrete energy method—a special form of finite difference energy approach—is presented as a suitable alternative to the finite element method for the large deflection elastic analysis of plates and shallow shells of constant thickness. Strain displacement relations are derived for the calculation of various linear and nonlinear element stiffness matrices for two types of elements into which the structure is discretized for considering separately energy due to extension and bending and energy due to shear and twisting. Large deflection analyses of plates with various edge and loading conditions and of a shallow cylindrical shell are carried out using the proposed method and the results compared with finite element solutions. The computational efforts required are also indicated.