Abstract: A simple finite element approach to problems of dynamic structural instability under step loading is discussed. The method proposed is believed to yield important information about the structural behaviour in the non‐linear range. Incorporation of the method into existing finite element codes is straightforward.

Abstract: The paper describes the derivation and application of a range of numerical algorithms for implementing the Mohr—Coulomb yield criterion in a non‐linear finite element computer program. Emphasis is placed on the difficulties associated with the corners of the yield surface. In contrast to the more conventional forward‐Euler procedures, a backward‐Euler integration technique is adopted. A range of methods, including a ‘consistent approach’ are used to derive the tangent modular matrix. Numerical experiments are presented which involve solution algorithms including the modified and full Newton—Raphson procedures, ‘line‐searches’ and the arc‐length method. It is shown that the introduction of efficient integration and tangency algorithms can lead to very substantial improvements in the convergence characteristics.

Abstract: A review is made of methods for calculating parameters characterizing crack tip behaviour in non‐linear materials. Convenient methods of calculating J‐integral type quantities are reviewed, classified broadly into two groups, as domain integrals and virtual crack extension techniques. In addition to considerations of how such quantities may be calculated by finite elements, assessment methods of conducting the actual incremental analyses are described.

Abstract: The evaluation of linear dynamic response analysis of large structures by vector superposition requires, in its traditional formulation, the solution of a large and expensive eigenvalue problem. A method of solution based on a Ritz transformation to a reduced system of generalized coordinates using load dependent vectors generated from the spatial distribution of the dynamic loads is shown to maintain the high expected accuracy of modern computer analysis and significantly reduces the execution time over eigensolution procedures. New computational variants to generate load dependent vectors are presented and error norms are developed to control the convergence characteristics of load dependent Ritz solutions. Numerical applications on simple structural systems are used to show the relative efficiency of the proposed solution procedures.

Abstract: The comparative efficiency of three flat triangular shell elements is being assessed for analysing non‐linear behaviour of general shell structures. The bending formulation of the three elements is based on a discrete Kirchhoff model (namely the well‐known 3‐node DKT element and a new 6‐node DKTP element). The in‐plane behaviour is defined by constant (CST), linear (LST)and quadratic (QST) strain approximations. The super‐position of bending and membrane elements leads to the 3‐node DCT element (DKT plus CST), the 3‐node DQT element (DKT plus QST) and the 6‐node DLT element (DKTP plus LST). The geometrically non‐linear formulation is based on an approximate updated Lagrangian formulation (AULF) and the solution is obtained by using the Newton‐Raphson method with an automatic arc‐length control method. Illustrative examples include pre‐ and post‐buckling of different shell structures showing, in particular, some bifurcation points, large rotations and displacements and very important membrane‐bending coupling.

Abstract: A three‐dimensional computer simulation of a combustion chamber used in the glass production industry is presented. A numerical solution technique is used to solve the governing time‐averaged partial differential equation and the physical modelling for turbulence, combustion and thermal radiation. A two‐equation turbulence model is employed along with a combustion model based on a fast kinetics statistical approach. A radiation model is used along with the Hottel mixed grey gas model. To solve the governing differential equations an implicit technique of finite‐difference kind is applied. The economy of the computations is very considerably enhanced by the separate calculation of the burner and bulk glass combustion chamber regions, in a manner which takes account of the differing physical nature of their flows. The burner outlet region is calculated with an axisymmetric model. Such two‐dimensional calculations allowed a good resolution of the burner outlet, and provide the inlet conditions for the three‐dimensional calculations of the glass furnace. The prediction procedure is applied to an industrial glass furnace, which operates with oxy‐fuel conditions. Measurements of mean gas temperature and concentrations were performed at different locations in the furnace. The calculated flame length, temperature field and concentrations are with satisfactory agreement with the measured ones.

Abstract: An alternative stabilization approach has been developed for the 9‐node Lagrange plane and plate elements. In this approach, a stabilization stiffness is formulated using functions associated with the spurious zero‐energy modes. Efficiency has been increased by employing the same uniformly‐reduced integration scheme on the stabilization and underintegrated stiffness matrices. The results obtained using this rank‐sufficient element, termed the γ‐ψ element, appear to surpass those obtained with other rank‐sufficient 9‐node elements in accuracy.

Abstract: This paper presents a new 9 DOF triangular element for plate bending. It is an analytically integrated improved version of the simplest member in the hierarchy of numerically integrated elements. These elements have been based on the so‐called Hybrid—Trefftz model (HT), a recently developed hybrid model associated with enforcing interelement continuity on locally based displacement fields chosen such that they a priori verify the Lagrange plate equation over the element. In the process of development of the element stiffness matrix in a standard HT model, one has to invert the so‐called natural stiffness matrix, a 7 × 7 matrix associated with the expression of the strain energy in terms of the Trefftz's functions of the element. The inversion of this fully populated matrix represents the most expensive part of the calculation of the element. The basic improvement of the standard Hybrid—Trefftz 9 DOF triangle consists in replacing the original Trefftz's functions by new ones which are energy orthogonal and consequently, result in a diagonal natural stiffness matrix. This not only alleviates considerably the computer cost, but also significantly simplifies the algebra making analytical integrations possible. The practical efficiency of the new element which passes the patch test is demonstrated through numerical examples including the difficult simply supported skew plate problem with a strong singularity at its 150° obtuse corner.

Abstract: This paper deals initially with a new algorithm for generating automatically, from solid models of mechanical parts, finite element meshes that are organized as spatially addressable quaternary trees (for 2D work) or octal trees (for 3D work). Because such meshes are inherently hierarchical as well as spatially addressable, they permit efficient substructuring techniques to be used for both global analysis and incremental re‐meshing and re‐analysis. The paper summarizes the global and incremental techniques, and presents some results from an experimental closed loop 2D system in which meshing, analysis, error evaluation, and re‐meshing and re‐analysis are done automatically and adaptively. The paper concludes with a progress report on a 3D implementation.

Abstract: A computational procedure is presented for the efficient non‐linear dynamic analysis of quasi‐symmetric structures. The procedure is based on approximating the unsymmetric response vectors, at each time step, by a linear combination of symmetric and antisymmetric vectors, each obtained using approximately half the degrees of freedom of the original model. A mixed formulation is used with the fundamental unknowns consisting of the internal forces (stress resultants), generalized displacements and velocity components. The spatial discretization is done by using the finite element method, and the governing semi‐discrete finite element equations are cast in the form of first‐order non‐linear ordinary differential equations. The temporal integration is performed by using implicit multistep integration operators. The resulting non‐linear algebraic equations, at each time step, are solved by using iterative techniques. The three key elements of the proposed procedure are: (a) use of mixed finite element models with independent shape functions for the stress resultants, generalized displacements, and velocity components and with the stress resultants allowed to be discontinuous at interelement boundaries; (b) operator splitting, or restructuring of the governing discrete equations of the structure to delineate the contributions to the symmetric and antisymmetric vectors constituting the response; and (c) use of a two‐level iterative process (with nested iteration loops) to generate the symmetric and antisymmetric components of the response vectors at each time step. The top‐ and bottom‐level iterations (outer and inner iterative loops) are performed by using the Newton—Raphson and the preconditioned conjugate gradient (PCG) techniques, respectively. The effectiveness of the proposed strategy is demonstrated by means of a numerical example and the potential of the strategy for solving more complex non‐linear problems is discussed.

Abstract: An analysis of the plate and folded plate structures is carried out, taking into account the geometrical non‐linearities and the effects of creep, using the finite strip method. An assumption is made that only small deformations and large displacements and rotations exist. Creep of concrete has an important influence on some structures and cannot be neglected in such analysis, especially when geometrical non‐linearities are taken into account. The stiffness matrices (classical and geometrical) and the vector of equivalent nodal loading for the finite strips are obtained using the variation approach. The interpolation functions used are multiples of polynomial and trigonometric functions. Numerical examples showing the theoretical considerations are presented.

Abstract: A hexahedral 8‐node element based on the Hellinger—Reissner principle is formulated with the γ projection operator so that it can achieve engineering accuracy for plate and beam problems with a single layer of elements. It passes the patch test and is less sensitive to mesh shape since the local coordinates are used to describe the stress fields. The resulting element stiffness is simple and only 3×3 submatrix inversions are needed. Numerical results show that the new element is both accurate and efficient.

Abstract: The application of numerical techniques to the solution of practical problems which exist in rubber technology is described. Structures and components in the form of reinforced rubber shells are widely used in industry and prediction of their performance is complicated by both the anisotropic nature of composite construction and the incompressible behaviour of the basic material. A layered shell element is developed for the solution of such problems with general anisotropic behaviour independently permitted in each layer. The approach adopted permits the easy location of reinforcement patterns. Numerical solution is based on a single field formulation by eliminating at integrating point level the Lagrange multiplier imposing the incompressible constraint. Large deformation, including large rotation, behaviour is accommodated and a total Lagrangian solution process is adopted. The code developed permits the simulation of non‐conservative loading and its versatility is demonstrated by the solution of some practical examples.

Abstract: This paper suggests some improvements to the computer‐aided estimation of the torsional properties of open cross‐sections as presented in a previous paper. The complete listing is given herein.

Abstract: The accuracy of finite element discretizations modelling one‐dimensional wave propagation problems is presented. The spurious reflections arising from finite/infinite element discretizations for unbounded domain problems are quantified. The error curves, numerically obtained, yield a criterion for rational mesh design. Formulae for minimum discretization ratios are given.

Abstract: A new triangular shell finite element ‘TNTE.1’ (Ten Node Triangular Element, Model 1) is presented. The formulation is based on Sanders' theory which involves the inclusion of the normal rotation Φn in the bending‐strain relations only. The element displacement functions are complete cubic polynomials for inplane displacements u and v. For out‐of‐plane displacement w, three new singular rational shape functions were added at the element corners. Thus a conforming triangular element with twenty seven degrees‐of‐freedom is obtained after eliminating the internal displacements by static condensation. The formulation of this element is new in that an integration technique is developed and applied to the element stiffness matrix and load vector. This technique is based on performing all the necessary integrations externally (i.e. outside the main computer program) and then modifying the formulation of the element matrices to account for this change. Hence, such a method allows the inclusion of higher‐order integration rules without any loss of economy, due to computer time, in the main program. Results using this element showed good agreement with other finite element and closed form solutions.

Abstract: In conjunction with the usual rate forms of the conservation equations, the time derivative form of the equation of state is investigated from a numerical consideration point of view. First, the derivation of the rate form of the equation of state is presented. Systematic comparison between the new method and the traditional iterative method is made by applying the method to a simple flow problem. The comparison is then extended to a practical engineering problem requiring accurate prediction of pressure. The rate method is found to be more advantageous in many aspects. It is more intuitive for system analysis, more appropriate for eigenvalue extraction, as well as easier to program and to implement. Numerically, the rate method is found to be more efficient and accurate than the traditional method.

Abstract: The general structure of geometrically‐based automatic finite element modelling systems is discussed. The development of a specific system employing the modified‐quadtree and modified‐octree mesh generators is presented. The application of this approach to metal forming analysis is then given.

Abstract: The applicability of a non‐linear finite element method for the determination of the static strength of tubular joints is examined. In order to establish static strength, non‐linear elasto‐plastic models are implemented. Techniques for automatically generating finite element meshes in stress analysis of tubular intersections are used. The analysis is carried out on a typical X‐joint under axial brace loads and the model represents only one‐eighth of the joint. The results are obtained by two different element procedures: three node flat shell element (Ilyushin yield criterion); eight node isoparametric shell element (von Mises yield criterion). The objective of this work is to discuss the modelling and computational aspects which are required for dealing with this elasto‐plastic analysis and to determine the necessary degree of refinement in order to obtain reliably the loads at which ultimate failure occurs.

Abstract: In the last twenty years, the methods for structural system reliability evaluation have evolved considerably. Since these methods are based on different assumptions, it is necessary to evaluate their capabilities. For this reason, a research study was initiated at the University of Colorado at Boulder in order to investigate the accuracy and reliability of various methods for structural system reliability evaluation. This paper emphasizes certain important parametric studies on system reliability methods under uncertainty. The sensitivity of the accuracy of the methods to changes in both strength and load correlations is demonstrated by means of numerical examples.

Abstract: Implicit solutions in viscoplastic analysis of structures are very often avoided due to the fact that they lead to non‐symmetric systems of equations. On the other hand, structures such as soils have a marked non‐associate behaviour that has to be taken into consideration. Here the implicit solutions are formulated in a way that, even for non‐associate models, the system of equations is kept as symmetric. Apart from an ‘exact’ formulation of the algorithm other possibilities are considered and tested in the numerical examples given.

Abstract: The numerical treatment of coupled field interaction problems frequently uses mixed time integration methods. These methods permit different time integration methods (implicit, explicit) and/or different timesteps to be used simultaneously in different parts of the mesh. This paper summarizes the various mixed time integration methods and provides a unified presentation. Computer implementation of the generalized scheme is provided through a 1D linear structural dynamics program (GEMIX). Two common examples illustrate the use of GEMIX program.

Abstract: Two efficient computational procedures are presented for generating the global approximation vectors used in conjunction with the reduction methods for the large‐deflection non‐linear analysis of symmetric structures with unsymmetric boundary conditions. Both procedures are based on restructuring the governing equations for each of the unsymmetric global approximation vectors to delineate the different contributions to the symmetric and antisymmetric components of this vector. In the first procedure the unsymmetric global approximation vectors are approximated by linear combinations of symmetric and antisymmetric modes, which are generated by using the finite element method. The amplitudes of these modes are computed by using the classical Rayleigh‐Ritz technique. The second procedure is based on using a preconditioned conjugate gradient (PCG) technique for generating the global approximation vectors, and selecting the preconditioning matrix to be the matrix associated with the symmetric response. In both procedures the size of the analysis model used in generating the global approximation vectors is identical to that of the corresponding structure with symmetric boundary conditions. The similarities between the two procedures are identified, and their effectiveness is demonstrated by means of two numerical examples of large‐deflection, non‐linear static problems of shells.

Abstract: A solution algorithm for the transient analysis of bodies undergoing creep under constant or time varying loads is presented. The constitutive equation adopted is of the form: έc=γσm. The finite element formulation is carried out in terms of displacements and creep strains as internal variables. The time discretization is achieved with a trapezoidal time integration scheme. The creep strains are condensed out to give an equation for displacement increments involving a modified stiffness matrix and force vector. A Newton—Raphson iterative scheme is used for the non‐linear creep strain rate‐stress relation, and creep strains are updated at the end of the time step. The algorithm has been implemented in NOSTRUM for two‐dimensional structural and plane continuum problems, with a von Mises type potential function governing the multiaxial creep constitutive relationship. Numerical results are presented.

Abstract: The application of computational plasticity to the very practical engineering problem of crash‐worthiness of vehicle safety cab frames during impact and rollover incidents is described. The resulting behaviour of these structures cannot be determined solely by strict elastic analyses as plastic behaviour is intrinsic to the problem. The importance of predicting the deformations of the cab structure under extreme loadings lies in a consideration for the safety of its occupants. Physical testing is extremely valuable here but is costly and time‐consuming. The role of the computer is thus evident in producing rapid estimates of cab collapse modes, loadings and deformations. Three main causes of non‐linearity are identified in the behaviour of ductile framed structures under static loading, i.e. the effects of plasticity in the structural material, the effects of axial forces and the effects of large displacements of the structure under load. The paper describes and compares two computer programs which have been used to model the behaviour of vehicle cab frames when subjected to roof crush loading in a static test. Both programs employ non‐linear beam finite elements to model the behaviour of a framed structure. One program runs in an iterative fashion while the other runs in an incremental fashion.

Abstract: A class of exact solutions to the elastohydrodynamic problem to be used as test cases is presented. A numerical solution to the elastohydrodynamic problem according to the Petrov—Galerkin method is developed. The appearance of spurious numerical undulations in the film profile is examined. A comparison between analytical and numerical results is employed to determine which numerical schemes limit the outcome of numerical oscillations without compromising the solution accuracy.

Abstract: We compare potential‐based (o‐U‐P0) and displacement‐based finite element methods for static analysis of contained fluids A general transient formulation may be specialized to static analysis in both cases In the potential‐based method velocity potentials (o) and a single pressure (P0) variable are the unknowns in the fluid region Displacements are the unknowns in the fluid for displacement‐based methods Higher‐order displace‐ment‐based elements may produce singular matrices for some static analyses, restricting us to four‐node elements for reliability While both methods can yield excellent results when compared with experimental data, potential‐based methods appear to have computational advantages over displacement‐based methods

Abstract: The modelling of soil‐foundation systems and the evaluation of the foundation compliance functions through a finite/infinite element technique is presented. By means of in situ measurements performed upon heavy machinery foundations, the actual dynamic foundation loading condition is computed. The results obtained are used for redesigning the foundations according to maximum allowable vibration patterns. The procedure clearly shows the advantages of the technique, which leads to efficient designs for all types of foundations.

Abstract: The computational efficiency of subspace iteration is addressed relative to the data structures adopted for the very large and generally sparse coefficient matrices. The frequent triangulations and matrix multiplications demand that access to the terms in the coefficient matrices be unbiased. Reliance on virtual memory (paging) operating systems with no special considerations for localized data access is not adequate. Specific data structures must be designed that accommodate the needs of the numerical algorithm yet eliminate unnecessary paging. An implementation of the subspace iteration method using hypermatrix data structures is presented. Use of hypermatrices is shown to provide unbiased and localized data access. The various modifications to the conventional formulation are described and an example problem illustrates the potential benefits of the hypermatrix formulation. Possibilities for adapting hypermatrix data structures to new supercomputer architectures are discussed.