bluuebunny/crossref_metadata_2025 huggingface dataset:
Application of the simultaneous iteration method to undamped vibration problems
Authors: Jennings Alan , Orr D. R. L.
Published: August 2005

The simultaneous iteration method of obtaining eigenvalues and eigenvectors is employed for the solution of undamped vibration problems. This method is of significance when a few of the dominant eigenvalues and eigenvectors are required from a large matrix, and hence is particularly suitable for vibration problems involving a large number of degrees of freedom. It is shown that advantage may be taken of both the symmetry and the band form of the mass and stiffness matrices, thus making it feasible to process on a computer larger order vibration problems than can be processed using transformation methods. A method of allowing for body freedom is given and some numerical tests are discussed.
An economical method for determining the smallest eigenvalues of large linear systems
Authors: Wright G. C. , Miles G. A.
Published: August 2005

The general eigenvalue problem Ax = λBx as arising from vibration problems tackled by the finite element method is often solved by the economization method in which the two matrices are reduced to a more manageable size and approximate answers are obtained.

This paper analyses the method in a more mathematical way than previous accounts and leads to a definition of the optimum set of variables to be retained during the reduction. An extension to the method is suggested which allows the approximate eigenvalue to be improved and bounds are obtained on the eigenvalues of the full problem before reduction.

An estimate is made of the calculation involved in the method and it is concluded, by reference to examples, that the extended method leads to a more efficient algorithm.
Solution of the compressible flow equations
Authors: Gelder D.
Published: August 2005

A technique is described for solving the compressible flow equations in subsonic flow. The general quasi‐linear equation ∇.g∇v = 0 is considered with g a function of ∇v. ∇v, and iterations of the form ∇.gn∇vn+1 = 0 are analysed, where g0 is suitably chosen and gn defined from vn for n≥1. This approach is applied to the compressible flow equations in terms of a velocity potential ø: monotonic convergence is predicted and at each iteration the error is multiplied by a factor less than the square of the greatest Mach number in the solution.

Reliable convergence has been obtained in practice solving the linear equation for øn+1 by a finite difference method. The alternative of working in terms of the stream function ψ is discussed, and also discretization by the finite element method.
The piecewise‐linear analysis of two connecting structures including the effect of clearance at the connections
Authors: Scholes A. , Strover E. M.
Published: August 2005

A method of analysis is presented for two structures which have connections with clearance. It is assumed that the behaviour of the structures is described, by the movement of discrete nodes and that as the load on the structure changes, clearances open or close between corresponding nodes on the structures; each time this occurs the stiffness of the combined structure changes because its configuration has changed. Between these changes, the structure will behave linearly and the overall behaviour can be described as piecewise linear. Applications include diesel engine crankshafts with bearing clearances and the wrap‐around problem in pin‐and‐eye connections.
Finite element method applied to the problem of stability of a non‐conservative system
Authors: Barsoum Roshdy S.
Published: August 2005

The finite element method is applied to the stability analysis of structural systems subject to non‐conservative forces. The development of the method is general, but the specific application considered here is the stability of thin‐walled members subject to follower forces. The method predicts the type of instability, whether it be buckling or flutter. Example problems, for which exact solutions are known, illustrate the accuracy and convergence characteristic of the finite element formulation.
An annular segment finite element for plate bending
Authors: Sawko F. , Merriman P. A.
Published: August 2005

A stiffness matrix for a finite element having the planform of an annular segment is derived using the displacement approach. Numerical problems involved in the derivation are discussed and rapid convergence to exact solutions is demonstrated on three sample problems. It is concluded that the new element will be of great value to engineers concerned with the analysis of slabs of bridge decks curved in plan.
A static and dynamic finite element shell analysis with experimental verification
Authors: Klein Stanley
Published: August 2005

A system of finite element shell analysis codes, called SABOR/DRASTIC, is used to analyse a complex two‐layered shell of revolution under static and dynamic asymmetric loads. The dynamic analysis is compared with experimentally measured response. In this linear elastic analysis, emphasis is placed on the inherent flexibility of the finite element method in modelling the complex structural geometry of a given test specimen. Static studies, which involve variations in important shell parameters, and dynamic studies, which provide a successful correlation with experiment, are used to illustrate both the detail and the generality with which shell analyses may now be performed with confidence.
On the calculation of consistent stress distributions in finite element approximations
Authors: Oden J. T. , Brauchli H. J.
Published: August 2005

The theory of conjugate approximations1 is used to obtain consistent approximations of stress fields in finite element approximations based on displacement assumptions. These consistent stresses are continuous across interelement boundaries and involve less mean error than those computed by the conventional approach.
The solution of structural equilibrium equations by the conjugate gradient method with particular reference to plane stress analysis
Authors: Yettram A. L. , Hirst M. J. S.
Published: August 2005

A computer program is described for the analysis of plates in extension using the equivalent plane‐framework idealization. The conjugate gradient method is adopted to solve the resulting large number of simultaneous equations. Performance figures are quoted for the program when used on an ICL KDF9 machine with a restriction of 18K on the available core store. The method is generally shown to provide a satisfactory solution to a heavy computational task and thus permits the solution of very large structural systems on a small computer.
An equation solver of very large capacity
Authors: Cantin Gilles
Published: August 2005

A very simple equation solver of very large capacity is developed. The method described here does not require a very large in‐core memory; however, a fast random access device like a magnetic disk drive is required.
A powerful hybrid method in finite element analysis
Authors: Rao A. K. , Raju I. S. , Murty A. V. Krishna
Published: August 2005

An important limitation of finite element analysis, namely, the need for a large number of small elements in regions of finite or infinite stress concentrations and the difficulties of convergence in such cases, is well known. Rao1 suggested a possibility of overcoming this by developing hybrid techniques combining continuum and finite element concepts. In such techniques, each region of stress concentration is covered by one large ‘primary element’ whose description includes term(s) identifying the type and order of concentration, while the remaining structure is split into a few ‘secondary elements’ which are conventional finite elements. In this paper a procedure incorporating this concept is developed and its effectiveness is clearly demonstrated by successful application to two important examples, one of them with stress singularities. The concept, in fact, can be applied equally well to other two‐ and three‐dimensional problems of continua with discontinuities and concentrations.
Linear programming methods of displacement analysis in elastic‐plastic frames
Authors: Baronas R. P. , Chyras A. A.
Published: August 2005

The analysis is aimed at determination of displacements in elastic‐perfectly plastic frames, in the case of monotonically increasing loads. Spread of partial plasticity from hinges and change of geometry on the conditions of equilibrium are not included. Extremum energy principles and linear programming methods are employed to solve the problem. Both the kinematic and the static formulations of the problem are presented. The mathematical models of the formulations constitute a dual pair of linear programming problems.
Network‐topological formulation of analysis and design of rigid‐plastic framed structures
Authors: Fenves S. J. , Gonzalez‐Caro A.
Published: August 2005

A unified network‐topological approach to the formulation and solution of the analysis and minimum weight design of rigid‐plastic structural systems is presented. It is shown that both analysis and design can be formulated using network concepts, but that, in contrast to elastic networks, the formulation results in a mathematical programming problem. Formulations based both on the static and kinematic theorems of plasticity are presented, and it is shown that duality guarantees that the two types of formulation yield identical results. The problem is first formulated on the basis of a simple definition of yielding, and is then generalized to the case where yielding is defined in terms of a yield surface.
A note on convergence of finite element solutions
Authors: Yamamoto Yoshiyukl , Tokuda Naoaki
Published: August 2005

The convergence of nodal displacements obtained by the finite element method is investigated for special element patterns in plane elasticity with the aid of the differential equation derived from the equilibrium condition at each node. The convergence of strains is also discussed. It is shown that averaging is an effective tool for obtaining approximations to strain for some kinds of mesh arrangement. It is also shown that the approach becomes inaccurate along force boundaries.
Numerical methods for evaluation of equilibrium composition of ideal gases
Authors: Williams T. J.
Published: August 2005

This paper considers methods of solution of the relevant equations based on the Newton‐Raphson process. Expressions are derived for calculation with both good and very poor initial guesses and for calculation of pressure, volume and temperature during an equilibrium process. A method of programming the calculations is outlined. Variations and modifications to suit specific situations are discussed.
An automatic mesh generation scheme for plane and curved surfaces by ‘isoparametric’ co‐ordinates
Authors: Zienkiewicz O. C. , Phillips D. V.
Published: August 2005

A computer orientated method is presented which generates meshes of triangular elements in plane and curved surfaces. Depending on geometrical and material variations, the region to be discretized is divided into a number of four sided zones. By using curvi‐linear co‐ordinate systems, nodes within and on the boundary of each zone are automatically positioned and referenced to a global Cartesian co‐ordinate system. Elements are automatically assembled from these nodes. Input data is required to specify the positions and material properties of each zone and how they are connected. Examples are given which illustrate the range of meshes that can be generated. Extension of the method to generate three‐dimensional tetrahedral elements is indicated.
Evaluation of orthogonal damping matrices
Authors: Wilson E. L. , Penzien J.
Published: August 2005

Two methods for the numerical evaluation of orthogonal damping matrices are developed. The first relates the modal damping ratios to the coefficients of the Caughey series. The second is a direct approach which expresses the damping matrix as a sum of a series of matrices each of which produces damping in a particular mode. It is found that the direct approach is more convenient to apply and is less numerically sensitive than the series approach.
Non‐dimensional surge tank analysis
Authors: Lin Cheng‐Nan , Gladwell John S.
Published: August 2005

A computer program for surge tank analysis, which gives a complete history of the water level in the tank and the fluid velocity in the pressure conduit from the time of initiation of value closure, is described and presented. The governing equations are solved by third‐order Runge–Kutta procedure and losses in head in the pressure conduit and surge tank constriction. A major aspect in the paper is the non‐dimensional from of the governing parameters, facilitating the use of the program in preliminary design situations.
A numerical method for the thermodynamic properties of monatomic ionized gases
Authors: Schmidt W.
Published: August 2005

A numerical method for the direct computation of the particle densities, mass density, enthalpy and specific heat at constant pressure of a monatomic ionized gas in thermodynamic equilibrium is presented. This method permits calculation of the thermodynamic properties without iteration or numerical differentiation of any of the formulae.
A mixed finite element formulation for plate problems
Authors: Chatterjee A. , Setlur A. V.
Published: August 2005

A mixed triangular finite element model has been developed for plate bending problems in which effects of shear deformation are included. Linear distribution for all variables is assumed and the matrix equation is obtained through Reissner's variational principle. In this model, interelement compatibility is completely satisfied whereas the governing equations within the element are satisfied ‘in the mean’. A detailed error analysis is made and convergence of the scheme is proved. Numerical examples of thin and moderately thick plates are presented.
The application of dual minimum theorems to the finite element solution of potential problems with special reference to seepage
Authors: Ponter A. R. S.
Published: August 2005

By formulating the potential boundary value problems in terms of physical (in contrast to purely mathematical) variables, the existence of dual energy principles may be clearly distinguished. In direct analogy with elasticity, two possible energy methods present themselves giving rise to two possible finite element procedures. General upper and lower bounds on an energy quantity may be obtained, which in the case of seepage become the total seepage rate for a certain class of seepage problem. For this case of seepage the derivation of the finite element procedure and the computing of the bounds is described.
Finite element analysis of perturbed compressible flow
Authors: Leonard John W.
Published: August 2005

The finite element representation of the linearized equations governing the steady compressible flow of an isentropic perfect gas is considered. The fully non‐linear system of equations is linearized on the basis of small perturbation theory. The finite element matrix equations for an arbitrary polygonal element are generated by the method of weighted residuals: Galerkin's criterion is used. As an example, a triangular element in two‐dimensional flow is treated in detail and numerical result for a sample problem are given.
Mathematical programming methods for the inelastic analysis of reinforced concrete frames allowing for limited rotation capacity
Authors: De Donato Osvaldo , Maier Giulio
Published: August 2005

The inelastic flexural behaviour of reinforced concrete beams is described, as usual, by a trilinear (or, in general, multilinear) moment–rotation relationship. The analysis of beams and frames subject to given external actions is formulated as a ‘linear complementary problem’ and, in two dual ways, as a ‘quadratic programming problem’. These problems are solved by means of classical and recent algorithms in use in operations research. By the procedures pointed out it is also possible to calculate the distribution of plastic deformations along the beams. Two methods are proposed for the evaluation of the safety factor against local failure due to limited rotation capacity: the solution technique of the direct one essentially consists of an application of the simplex method for linear programming. Finally, a procedure is established for determining, allowing for the irreversibility of plastic rotations, the structural response to non‐proportional loading histories which consist of individually proportional stages. Illustrative examples are given.
An iterative approach to strucural optimization
Authors: Dupuis G.
Published: August 2005

An iterative procedure of structural optimization is discussed that combines finite element analysis with the use of explicit optimality conditions. The procedure is illustrated by examples concerning sandwich beams with constraints on the deflection at a specified section, the maximum deflection or the sum of the maximum deflection in all spans of a continuous beam. The examples suggest that, starting from a uniform beam, by far the greatest saving in weight is obtained in the first two or three iterations.
Stochastic plastic analysis of a shell
Authors: Muspratt M. A.
Published: August 2005

A method is proposed for the inclusion of stochastic parameters, using a Monte Carlo simulation technique, in the lower bound plastic analysis of a structural continuum by non‐linear mathematical programming. The particular example chosen is a circular, cylindrical concrete silo subject to loading by its contents.
A conforming finite element with internal equilibrium for two‐dimensional problems
Authors: Yam Lloyd C. P.
Published: August 2005

A method is presented for the derivation of displacement fields which satisfy compatibility of normal slopes at inter‐element boundaries and the condition of internal equilibrium. The displacement field is obtained by integrating the governing partial differential equation over the element, using the initial value approach. The method is applied to calculate the stiffness matrix of a triangular bending element with nine degrees of freedom. Comparison with published solutions based on various element models shows a high degree of accuracy and convergence of the method. The advantage of internal equilibrium is illustrated by an example involving slab‐column interaction. The solution correlates satisfactorily with existing experimental results.
Finite displacement plate flexure by the use of matrix incremental approach
Authors: Yang T. Y.
Published: August 2005

A finite element incremental stiffness formulation for determining finite displacement flexure behaviour of thin elastic plates is presented. The solution of the finite displacement problems is based on a piecewise linear incremental approach. The procedural aspects of the incremental approach, the control of incremental sizes and the accuracy of the incremental approximation are investigated. The details concerning how to obtain in‐plane stresses and how to treat edge conditions are discussed. Results include rectangular plates under lateral pressure with various edge conditions and aspect ratios. The laterally loaded plate with edges movable in the plane of the plate is also treated.
Finite elements for field problems in cylindrical co‐ordinates
Authors: Daly P.
Published: August 2005

This paper deals with the extension of the finite element method as applied to the solution of the Laplace or wave equation to cylindrical co‐ordinate systems. The base matrices required for solving problems governed by these equations are derived for circular polar, elliptic cylinder and parabolic cylinder co‐ordinates. The matrices allow problems whose boundaries are described as co‐ordinate surfaces in cylinder co‐ordinates to be attacked directly by the finite element method. The subject is discussed from the point of view of one interested in electromagnetic wave propagation in uniform waveguide structures.
Finite element method applied to analysis of flow over a spillway crest
Authors: Ikegawa M. , Washizu K.
Published: August 2005

Numerical analysis of fluid flow over a spillway is treated in the present paper. A variational principle is introduced for a flow with a free surface boundary under gravity, where the stream function and the profile of the free boundary are independent quantities subjected to variation. A new iteration method is formulated by the combined use of the variational principle and the finite element method. A numerical example based on the iterative method is illustrated. Results thus obtained show good agreement with those obtained from an empirical formula.
A convergent method for cyclic plasticity analysis with application to nuclear components
Authors: Barsoum Roshdy S.
Published: August 2005

A new method for elastic‐plastic analysis of pressure vessels subject to cyclic mechanical and thermal loading is proposed. The method is shown to lead to a highly convergent solution when applied in conjunction with Prager's complete kinematic hardening rule. The plastic strain increments are written in terms of the total strains, thus leading to a consistent formulation when the analysis is done by an incremental iterative process.

The proposed method is applied to the analysis of a long cylinder in the state of generalized plane strain and subject to cyclic loading. Cyclic plasticity and creep are considered in one of the problems analysed. The finite difference method is used in the solution of the compatibility and equilibrium equations. It is shown that the computation time using the proposed method is very small compared to other techniques.
An integrated force method for discrete analysis
Authors: Patnaik Suryanarayan
Published: August 2005

An integrated force method is developed for the analysis of discrete structures in which all the element forces are taken as the independent variables instead of the conventional way of treating the redundants as the prime unknowns. Computationally the integrated force method is superior to the standard force method. The analogy between Beltrami–Michell formulation for continuous problems and the present formulation for discrete structures is presented. Illustrative examples for the determination of forces and displacements are presented for pin‐connected and rigid‐connected frame structures for various load conditions.
A note on some variational statements for the slow flow of a Navier‐Poisson fluid
Authors: Bettess P.
Published: August 2005

Three variational statements for the slow time‐independent flow of a Navier‐Poisson fluid are derived. All of these statements are suitable as the bases of finite elements. The first statement is applied successfully to several flow situations.
Body oriented (natural) co‐ordinates for generating three‐dimensional meshes
Authors: Cook William A.
Published: August 2005

Body oriented co‐ordinates are defined for three‐dimensional regions bounded by six surfaces. From this definition of body oriented co‐ordinates, two transformations are derived:

Body oriented surface co‐ordinates to the Cartesian co‐ordinates, and

Body oriented co‐ordinates for a three‐dimensional region to Cartesian co‐ordinates.

With these transformations a mesh generator is developed based on the principle that the mesh required to describe a region will be calculated according to the placement of the discrete points that describe the boundary edges. This is accomplished with two versions of the mesh generator: the surface generator and the volume generator. The surface generator maps the four boundaries of the surface onto the four boundaries of a unit square. Next, the interior points for the surface mesh are established in the mapped space. Then, using the first transformation, the co‐ordinates for the surface nodal points are calculated. Similarly, the volume generator maps the six surfaces of the body onto the surfaces of a unit cube, establishes the interior nodal points in the mapped space, and using the second transformation, calculates the co‐ordinates for the interior nodal points.
Boussinesq problem of plane micropolar elasticity
Authors: Korman T. , Goldberg J. E. , Morghem F. , Baluch M. H.
Published: August 2005

This paper deals with a finite element formulation and solution of the boundary value problem of a concentrated force acting on a semi‐infinite micropolar solid. A stiffness matrix is derived for a flat micropolar rectangular element in a plane state and use is made of this element to solve the case of the semi‐infinite micropolar solid both in a state of plane stress and plane strain. An immediate application of the plane strain case is the important practical problem of a long line load acting on a soil mass which may be simulated as a micropolar body.
A finite element analysis for time‐dependent problems
Authors: Holzlöhner Ulrich
Published: August 2005

The finite element method is applied to dynamic linear viscoelastic analysis. The arising system of differential equations is integrated by means of the Laplace transformation. The direct numerical evaluation of the inversion integral seems to be appropriate in case of vibrational problems. The method is applied to examples of transient vibrations of elastic bodies. Initial conditions and arbitrary damping matrices can be considered.
Finite elements formulated by the weighted discrete least squares method
Authors: Lynn Paul P. , Arya Santosh K.
Published: August 2005

By use of the least squares error criterion, an alternate finite element formulation is presented. The method is based on the discrete or element‐wise minimization of square and weighted differential equation residuals which are expressed in terms of element nodal quantities. In order to overcome the stringent inter‐element continuity requirement, a major stumbling block, on the element trial functions two practical schemes are proposed. One is the reduction of the original governing differential equation to a system of equivalent first order differential equations; the other is a method of smoothing discontinous trial functions. The latter essentially relaxes the continuity requirement and yields efficient non‐conforming finite elements. This paper also demonstrates the use of constant weights which significantly improves the rates of convergence. Several numerical examples illustrate the proposed method. From these examples, it may be concluded that the use of constant weights and the relaxation of the inter‐element continuity requirement are two indispensable features of the weighted discrete least square method.
On some numerical schemes for the solution of wave propagation problems
Authors: Smith Paul D.
Published: August 2005

This paper deals with the numerical solution of wave propagation problems in general although the primary area of interest is elastodynamics. The work is applicable to a variety of wave propagation phenomena of classical physics. Explicit schemes of Strang type are used, which allow many simplifications over existing methods. The Strang schemes allow in a very real but perhaps unexpected way the splitting of the elasticity operator.
On the accuracy of finite element solutions to the transient heat‐conduction equation
Authors: Donea J.
Published: August 2005

This paper discusses three numerical integration schemes connected with a finite element solution to the transient heat‐conduction equation. The main emphasis is on the achievable accuracy. A simple recurrence relation derived from the Galerkin process is suggested for the treatment of fast‐varying boundary conditions as being significantly better than the usual Crank‐Nicholson scheme in matters of short‐time accuracy.
A direct matrix method for the elastic stability analysis of deep circular arches
Authors: Awadalla E. S.
Published: August 2005

A direct method is described for the evaluation of the critical loads on deep circular arches having various forms of end support. The method is applicable to arches of non‐uniform flexural rigidity and may cater for arbitrary forms of applied load distribution. The results calculated for two examples are given, and compared with those obtained analytically and experimentally by other investigators. A complete listing of a computer program for one of the examples is provided in the Appendices.
Matrix structural analysis with mixed variables
Authors: Wiberg Nils‐Erik
Published: August 2005

Use of mixed force and displacement variables may be the key to a reliable solution of many physical problems. For the analysis of elastic structures with large rigid motion the displacement method may fail due to illconditioning, but use of mixed variables (diacoptics) may work due to the possibility of using relative displacements within substructures. Analysis with mixed variables complicates the topology (the number of nodes and the number of variables at the nodes may vary in the same problem) so a careful study of the system analysis is required. Some general computer routines are given for handling topology, merging and solution of the sparse system equation. The solution routine handles constraints in a rational way and can also be used for decomposition of the system equation or for releasing inner nodes in superelements.
The static condensation algorithm
Authors: Wilson Edward L.
Published: August 2005

The application of the static condensation method is discussed in general. The technique is presented as an extension of the Gauss elimination algorithm. An efficient FORTRAN subroutine is given which simultaneously reduces both the stiffness matrix and the stress‐displacement transformation matrix.
Finite element buckling analysis of homogeneous and sandwich plates
Authors: Cook Robert D.
Published: August 2005

Finite element buckling analyses of plates are performed by means of quardrilateral elements having 12 degrees‐of‐freedom. It is found that if the plate is capable of appreciable transverse shear deformation, initial‐stress stiffness matrices that represent the lateral dispalcement by cubic polynomials do not yield convergence to correct results. An initial‐stress stiffness that works satisfactorily for both homogeneous and sandwich plates is formulated. A Fortran listing of the associated subroutine is provided.
Finite difference solution for circular plates on elastic foundations
Authors: Chakravorty A. K. , Ghosh A.
Published: August 2005

In the present work the authors have developed a finite difference method of analysis for any circular plate with any kind of loading on semi‐infinite elastic foundations. No assumption regarding the contact pressure distribution has been made. The equations have been developed in non‐dimensional form and also the results have been obtained in non‐dimensional form. These results have been compared with the available experimental results and the agreement between them is found to be much better than that of the previous works. The same method with slight modification can be applied for Winkler type foundations and problems of circular plates with varying thickness.
Numerical method for time‐delay problems
Authors: Hanson Jerry L.
Published: August 2005

Analytical solutions for linear systems with time‐delays are difficult, if not impossible, to obtain. This is particularly true in the synthesis of an optimal control for a system with time‐delays or of optimal strategies for a differential or difference game with delays. Some of these problems are discrete in nature while many others can be converted into a discrete format. Probably the most direct method of handling linear, discrete problems with time delays is by using an enlarged state vector to eliminate the delays. The main disadvantage of this technique is the increased dimensions of the resulting matrices. This paper shows, through an illustrative example, that these dimensions of the resulting matrices. This paper shows, through an illustrative example, that these dimensions (and thus the storage requirements and computational time) can be effectively reduced. Thus the method is computationally feasible for a larger class of problems than if standard matrix algebra routines were used.
Finite elements on the basis of continuum mechanics
Authors: Kärcher Hans Jürgen
Published: August 2005

The formulation of finite element models on the basis of different variational principles is reviewed. The degrees‐of‐freedom of the elements are defined in an abstract way without the help of nodal points. In this manner it is possible to describe elements of arbitrary shape and accuracy. The formulation is confined to linear elasto‐statics. For two‐dimensional structures two hybrid element models are developed using Legendre polynomials on the element boundaries. Examples of plane stress problems are used to test the generation of convergence by increasing the accuracy of the elements vs. by increasing the number of elements.
Thermomechanical response of inelastic fibre composites
Authors: Chung T. J.
Published: August 2005

A theoretical formulation and the analysis of heat conducting fibre‐reinforced structures are shown herein. Viscoelastoplastic behaviour at infinitesimal strain is considered in the context of non‐equilibrium irreversible thermodynamics. To demonstrate the theory, a fibre‐reinforced cylindrical rod subjected to a dynamic axial load is solved via the finite elements. Strain hardening of both fibres and matrix media is incorporated in the analysis. The results clarify the relationships of energy dissipation resulting from viscoelasticity, plasticity, or the combined effects with respect to orientation of the fibres.
On the method of virtual crack extensions
Authors: Hellen T. K.
Published: August 2005

The measurement of energy release rates using virtual crack extensions has been made using finite element techniques. An economic and accurate technique for calculating energy changes due to any number of virtual tip changes is presented. Mixed‐mode situations can be dealt with by observing the direction of maximum energy release rate. Examples are given including various cracks in a plate in tension, a curving crack in a general two‐dimensional shape, and a three‐dimensional crack in a plate.
The direct numerical integration of linear matrix differential equations using pade approximations
Authors: Trujillo D. M.
Published: August 2005

A class of approximations to the matrix linear differential equation is presented. The approximations range, in accuracy, from the simplest forward difference scheme to the exact solution. The infinite series defining the exponential matrix is used to ascertain the accuracy of the various approximations. A clear distinction is made between approximations to the system equations and the forcing function, with the forcing term being represented by a piecewise linear function. Special application is given to the equations arising in structural dynamics of the form
equation image
For these structural dynamic equations, the measure of the energy of the system is used to analyse the stability of the numerical approximations.
Uniform rational approximation of functions of several variables
Authors: Kaufman E. H. , Taylor G. D.
Published: August 2005

We present a program which has given excellent results for uniform approximation of functions by polynomials, rational functions, generalized polynomials, and generalized rational functions. The algorithm is described in detail and several examples are discussed. The approximation is done over a finite point set, which is commonly a set of real numbers or points in the plane (in the latter case we are doing what is often known as surface fitting). Input to and output from the program is in tabular form. The method used is a linear programming approach known as the differential correction algorithm, which has been shown by several authors to always converge in theory (quadratically in some situations). In practice, we have obtained convergence in nearly every case, and quadratic convergence in most cases. The program can also be used for simultaneous approximation of several functions.
Finite element analysis of the motion of a gas‐liquid interface in a forous medium
Authors: Lewis R. W. , Norris V. A. , France P. W.
Published: August 2005

A numerical method is outlined for determining the movement of a gas‐liquid interface in a porous medium following the withdrawal of gas from an underground reservoir. The governing equations for the flow of both gas and the underlying liquid are formulated in terms of the finite element process using the Galerkin approach. The method consists of an iterative technique in which solution of the basic equations is dependent on the location of the interface. Cubic isoparametric elements are employed to allow accurate modelling of the curved boundary and to provide a relatively large number of interface points for a small number of elements. Extensions of the method are indicated in the text.
Power series expansion of the general stiffness matrix for beam elements
Authors: Paz Mario , Dung Lam
Published: August 2005

The general stiffness matrix for a beam element is derived from the Bernoulli–Euler differential equation with the inclusion of axial forces. The terms of this matrix are expanded into a power series as a function of the two variables: the axial force, and; the vibrating frequency. It is shown that the first three terms of the resulting series, which are derived in the technical literature from assumed static displacement functions, correspond respectively to the elastic stiffness matrix, the consistent mass matrix, and the geometric matrix. Higher order terms up to the second order terms of the series expansion are obtained explicitly. Also a discussion is presented for establishing the region of convergence of the series expansion for the dynamic stiffness matrix, the stability matrix, and the general stiffness matrix.
Minimization of load reduction in pipe networks subject to interrupted supply
Authors: Brameller A. , Li P. M.
Published: August 2005

The paper considers the question of reduction of loads on a pipe network, which may be necessary as a result of failure in supply or interruption of a pipe. The non‐linear relationship between pressure loss and flow along a pipe makes the problem of minimizing the reduction of loads also non‐linear. An iterative linear approximation technique is therefore adopted. When the question is formulated as a problem in linear programming, the constraints include nodal pressures as well as loads. It is shown how the number of these constraints may be reduced.
Optimal frequency‐weight computations for a disc
Authors: De Silva B. M. E. , Grant G. N. C.
Published: August 2005

This paper describes the computational aspects of an earlier analytical study1,2 of the problem of maximizing a linear combination of the natural frequencies of vibration of a turbine disc idealization subject to weight and fabrication constraints. The problem was formulated as an optimal control problem and solutions to the adjoint equations of the Pontryagin formalism obtained using perturbation techniques. These techniques transformed the problem to a non‐linear program. This paper gives detailed computational solutions to the non‐linear program which is characterized by pockets of relative optima. In addition, the dual problem of minimizing the weight subject to a frequency constraint is considered within the optimal control framework. The duality results are confirmed computationally for the two non‐linear programs.

A description of some practical procedures for improving the convergence characteristics of these highly non‐linear programs is included.
Linear equality constraints in finite element approximation
Authors: Szabo Barna A. , Kassos Tony
Published: August 2005

The typical numerical problem associated with finite element approximations is a quadratic programming problem with linear equality constraints. When nodal variables are employed, the coefficient matrix of the constraint equations, [A], acquires a block‐diagonal structure. The transformation from polynomial coefficients to nodal variables involves finding a basis for [A] and computing its inverse.

Simultaneous satisfaction of completeness and C1 (or higher) continuity requirements establishes linear relationships among the nodal variables and precludes inversion of the basis by exclusively element‐level operations.

Linear dependencies among the constraint equations and among the nodal variables can be evaluated by the simplex method. The computational procedure is outlined.
Alternate hybrid stress finite element models
Authors: Wolf John P.
Published: August 2005

Alternate hybrid stress finite element models in which the internal equilibrium equations are satisfied on the average only, while the equilibrium equations along the interelement boundaries and the static boundary conditions are adhered to exactly a priori, are developed. The variational principle and the corresponding finite element formulation, which allows the standard direct stiffness method of structural analysis to be used, are discussed. Triangular elements for a moderately thick plate and a doubly‐curved shallow thin shell are developed. Kinematic displacement modes, convergence criteria and bounds for the direct flexibility‐influence coefficient are examined.
An iterative improvement for non‐conforming bending elements
Authors: Crisfield M. A.
Published: August 2005

A method is described whereby the formulation for non‐conforming bending elements includes the work dissipated across the inter‐element boundaries. For the examples considered, this correction is shown to lead to a significant improvement in accuracy at little extra cost. Since the solution procedure is based on an iterative technique, the method would be particularly suitable if applied to non‐linear problems.
A computational method for optimal structural design II: Continuous problems
Authors: Haug E. J. , Pan K. C. , Streeter T. D.
Published: August 2005

A method for solving structural design problems that allows a continuous distribution of material along structural elements is presented. The method is an extension of the generalized steepest descent method presented in Reference 1. Inequality constraints on design variables, displacement, natural frequency, and buckling are explicitly treated and a minimum weight cost function is employed. A steepest descent method for boundary‐value state equations is developed and a computational algorithm is given. Several example problems in minimum weight structural design are solved and compared with results obtained by discretization techniques.
Solution of stiff boundary value problems by orthogonal collocation
Authors: Serth R. W.
Published: August 2005

The utility of the method of orthogonal collocation for solving stiff boundary value problems is investigated. To this end, the method is applied to a number of very stiff problems from the engineering and applied mathematics literature. The examples include both linear and non‐linear problems defined on both finite and infinite intervals, a singular perturbation problem, and an inherently unstable problem. The examples demonstrate the usefulness of the method for a wide variety of stiff problems, as well as some of its limitations.
Difference operator for variable stiffness plates
Authors: Chugh A. K. , Gesund H.
Published: August 2005

A symmetric central finite difference operator is derived for the small deflection analysis of transversely loaded, ‘thin’, elastic plates, whose stiffnesses vary as functions of the in‐plane co‐ordinates. The operator is written here in Cartesian co‐ordinates for a square mesh, but the same technique can also be used to construct an operator for a rectangular mesh.
On implementation of the extended interior penalty function
Authors: Cassis Juan H. , Schmit Lucien A.
Published: August 2005

The extended interior penalty function formulation is implemented. A rational method for determining the transition between the interior and extended parts is set forth. The formulation includes a straightforward method for avoiding design points with some negative components, which are physically meaningless in structural analysis. The technique, when extended to problems involving parametric constraints, can facilitate closed form integration of the penalty terms over the most important parts of the parameter interval. The method lends itself well to the use of approximation concepts, such as design variable linking, constraint deletion and Taylor series expansions of response quantities in terms of design variables.

Examples demonstrating the algorithm, in the context of planar orthogonal frames subjected to ground motion, are included.
On the use of isoparametric finite elements in linear fracture mechanics
Authors: Barsoum Roshdy S.
Published: August 2005

Quadratic isoparametric elements which embody the inverse square root singularity are used in the calculation of stress intensity factors of elastic fracture mechanics. Examples of the plane eight noded isoparametric element show that it has the same singularity as other special crack tip elements, and still includes the constant strain and rigid body motion modes. Application to three‐dimensional analysis is also explored. Stress intensity factors are calculated for mechanical and thermal loads for a number of plane strain and three‐dimensional problems.
Panel flutter optimization—a refined finite element approach
Authors: Weisshaar Terrence A.
Published: August 2005

This paper presents the results of an aeroelastic optimization study. In this study the weight of a large aspect ratio panel immersed in high Mach number supersonic flow is minimized subject to the requirement that a critical aeroelastic parameter for flutter be held within specified limits. A refined finite element technique is used to model the panel flutter equilibrium equations, equations which act as constraints on the design search. The optimization mechanism itself is studied and discussed to provide qualitative results which may be useful to the solution of other aeroelastic optimization problems. The numerical results of this investigation are compared to converged numerical results to illustrate finite element accuracy. The results of the study show that the weight savings and material distribution found with an effective finite element model are comparable to those found by more complicated numerical methods.
A new finite element scheme for instability analysis of thin shells
Authors: Matsui Tetsuya , Matsuoka Osamu
Published: August 2005

This paper describes a new finite element scheme for the analysis of instability phenomena of arbitrary thin shells. A computationally efficient procedure is proposed for calculating the non‐linear stiffness and tangential stiffness matrices for a doubly‐curved quadrilateral element defined by co‐ordinate lines. The essential feature is the explicit addition of the non‐linear terms into the rigid‐body motion of the element. Thus the non‐linear and tangential element stiffness matrices can easily be generated by transforming the generalized element stiffness matrix for linear analysis, and the non‐linear terms of these matrices are separated into a number of component terms multiplied by the rigid‐body rotations. These component terms can be stored permanently and used to calculate efficiently the non‐linear and tangential stiffness matrices at each iteration. Illustrative examples are presented which confirm the validity of the present approach in the analysis of instability phenomena of thin plates and shells.
A computer solution for two‐dimensional fluid‐particle flows
Authors: Crowe C. T. , Stock D. E.
Published: August 2005

A computer solution for two‐dimensional fluid‐particle flows using the cellular approach is presented. The euqations describing the flow of the continuous (fluid) phase are formulated using stream function, vorticity and enthalpy as the dependent variables. The effect of the disperse (particle) phase on the continous phase is represented by vorticity and energy sources in each cell. To illustrate the capability of the solution scheme, the flow of gas laden with hot particles issuing into a sudden expansion is analysed.