bluuebunny/crossref_metadata_2025 huggingface dataset:
Finite element modelling of transport of organic chemicals through soils
Authors: Kuppusamy T.
Published: August 2005

Aquifer contamination by organic chemicals in subsurface flow through soils due to leaking underground storage tanks filled with organic fluids is an important groundwater pollution problem. The problem involves transport of a chemical pollutant through soils via flow of three immiscible fluid phases: namely air, water and an organic fluid. In this paper, assuming the air phase is under constant atmospheric pressure, the flow field is described by two coupled equations for the water and the organic fluid flow taking interphase mass transfer into account. The transport equations for the contaminant in all the three phases are derived and assuming partition equilibrium coefficients, a single convective – dispersive mass transport equation is obtained. A finite element formulation corresponding to the coupled differential equations governing flow and mass transport in the three fluid phase porous medium system with constant air phase pressure is presented. Relevant constitutive relationships for fluid conductivities and saturations as function of fluid pressures lead to non‐linear material coefficients in the formulation. A general time‐integration scheme and iteration by a modified Picard method to handle the non‐linear properties are used to solve the resulting finite element equations. Laboratory tests were conducted on a soil column initially saturated with water and displaced by p‐cymene (a benzene‐derivative hydrocarbon) under constant pressure. The same experimental procedure is simulated by the finite element programme to observe the numerical model behaviour and compare the results with those obtained in the tests. The numerical data agreed well with the observed outflow data, and thus validating the formulation. A hypothetical field case involving leakage of organic fluid in a buried underground storage tank and the subsequent transport of an organic compound (benzene) is analysed and the nature of the plume spread is discussed.
Modelling of fluid expulsion and deformation behaviour of dewatering sediments
Authors: Chang F. Tom , Lennon Gerard P. , Pamukcu S. , Carson B.
Published: August 2005

A finite element model is developed for modelling coupled fluid expulsion/deformation behaviour of dewatering sediments subjected to external loadings under isothermal conditions. The non‐linear deformation behaviour of the sediment (soil) skeleton is based on the force equilibrium equation in which the constitutive relationship of stress and strain is implemented by the modified Cam‐Clay model in soil plasticity. The fluid flow behaviour in the model is described by the generalized porous media flow equation. The model allows temporal and spatial variations of porosity and permeability. The fluid viscosity and density are assumed to be temperature‐dependent. The model also allows the development of single and multiple faults, depending upon the material (sediment and fluid) properties, loading and boundary conditions.

Procedures are implemented for (1) updating the material properties such as porosity, permeability, fluid density and viscosity and (2) the development of faults which allow the formation of high‐permeability conduits for fluid flow. The solution algorithm for displacements of the sediments and the excess pore (fluid) pressure is based on a residual load technique to handle the non‐linear (elastic‐plastic) deformation behaviour of the sediment skeleton. The model can be applied to one‐ and two‐dimensional problems. Examples of a plane strain saturated sediment layer subjected to stepwise horizontal tractions versus time are given.
Short‐crested wave‐induced soil response in a porous seabed of infinite thickness
Authors: Hsu J. R. C. , Jeng D. S. , Tsai C. P.
Published: August 2005

Waves reflecting obliquely from structures establish short‐crested waves which are conducive to scour of the bed adjacent to them. The pore pressures and effective stresses induced by this wave system are derived analytically for an unsaturated, anisotropic soil matrix of infinite depth in a three‐dimensional domain. Verification is available by the solutions being readily reduced to the two‐dimensional cases of the progressive and standing wave, for which some closed‐form solutions are available. It is shown that wave obliquity is coupled with soil permeability. The effects of wave obliquity, relative water depth, soil stiffness and soil permeability on wave‐induced pore pressure are discussed in detail.
Fuzzy finite element analysis of a foundation on an elastic soil medium
Authors: Valliappan S. , Pham T. D.
Published: August 2005

One of the important factors that lead to errors in settlement predicitions is the degree of precision in obtaining the soil parameters. Most mathematical methods for reliability modelling offered to date in the area of geomechanics are based on the classical probabilistic approach, in which soil properties are treated as random variables. In this paper, a model based on the theory of fuzzy sets is presented to take account of the uncertainty in the soil behaviour. This proposed method considers the elastic modulus and Poisson's ratio as two fuzzy numbers in the elastic matrix. An example is given to show the possibility distributions of the displacements and the stresses at some locations in the soil medium. By means of a fuzzy inference scheme, the total possibility distributions or total membership functions of the finite element results may be obtained by considering the estimated error resulting from the mesh discretization.
On the description of localized deformation
Authors: Pietruszczak S. , Niu X.
Published: August 2005

A mathematical framework for the description of localized deformation, as outlined in Reference 14, is reviewed. The formulation is derived by estimating the average mechanical properties of a medium intercepted by a shear band. The mechanical response of the homogenized system depends on the properties of the constituents and the respective volume fractions, it remains invariant however with respect to the thickness of the shear band. The implementation of the framework in the context of finite element analysis is discussed. Numerical examples, pertaining to some geotechnical problems, are provided.
Influence of relative density and stress level on the bearing capacity of sands
Authors: Simonini Paolo
Published: August 2005

A finite element approach is presented to determine the bearing capacity of shallow footings on silica sand deposits. The approach can take into account the effects of relative density and stress level on the shear strength of granular soil and is applicable to a fairly large effective stress range. The strength of sand is characterized by a non‐linear Mohr–Coulomb criterion which depends on maximum and critical friction angles, widely used parameters in engineering practice. The results of analysis indicate that this approach yields reliable predictions of bearing capacity and, in particular, it can model the volumetric behaviour of the soil at failure.
A generalised D matrix for anisotropic elastic granular media
Authors: Du Jiang , Dusseault Maurice B.
Published: August 2005

A matrix relating stress and elastic strain tensors for anisotropic particulate materials has been derived. The magnitude of the matrix depends on the state of the material anisotropy. Anisotropy in granular materials depends on strain because normal and tangential particle contact forces, as well as the spatial distribution of the contacts, vary with stress and strain. However, the rotation tensor and the strain tensor cannot be independent; they must satisfy certain constraints to meet the requirement for macroscopic stress tensor symmetry. These conditions and constraints lead to the derivation of the matrix presented in this article. The principal directions of the stress tensor and strain tensor are generally not coincident, and the values of deformation parameters, Young's modulus and Poisson's ratio, are direction dependent; these two aspects are also discussed in this paper. Whereas this matrix can be used in static numerical analyses for elastic problems, we note that this relationship can also be used as a basis upon which to derive a fully incremental stress–strain relationship for anisotropic granular materials in the plastic state, where the anisotropy is evolving with strain.
A fully coupled non‐linear dynamic analysis procedure and its verification using centrifuge test results
Authors: Muraleetharan K. K. , Mish K. D. , Arulanandan K.
Published: August 2005

The equations governing the dynamic behavior of saturated porous media as well as a finite element spatial discretization of these equations are summarized. A three‐parameter time integration scheme called the Hilber–Hughes–Taylor α‐method is used together with a predictor/multi‐corrector algorithm, instead of the widely used Newmark's method, to integrate the spatially discrete finite element equations. The new time integration scheme possess quadratic accuracy and desirable numerical damping characteristics. The proposed numerical solution and bounding surface plasticity theory to describe the constitutive behaviour of soil have been implemented as the computer code DYSAC2. Predictions made by DYSAC2 code are verified using dynamic centrifuge test results for a clay embankment. Importance of initial state of a soil on its dynamic behaviour is demonstrated.
Conformal analysis of seepage below a hydraulic structure with an inclined cutoff
Authors: Ijam Abbas Z.
Published: August 2005

Conformal mapping techniques are used to obtain an exact solution for seepage flow below a hydraulic structure founded on permeable soil of infinite depth for a flat floor with an inclined cut‐off at the downstream end. Results are presented for different values of b/s and θ. The exit gradient decreases considerably along a distance beyond the floor end with an increase in cut‐off inclination. The maximum exit gradient decreases for θ = 10, 20 and 30° and starts increasing for θ = 45°. The uplift pressure decreases as θ increases. It is concluded that using an inclined cut‐off increases the factor of safety in design against uplift and piping.
Elastoplastic analysis of jointed rocks using a coupled finite element and boundary element method
Authors: Xiao B. , Carter J. P. , Alehossein H.
Published: August 2005

This paper presents a coupled, elastoplastic, finite element and boundary element method for the two‐dimensional, non‐linear analysis of anisotropic jointed rock. The non‐linear and anisotropic behaviour of a jointed rock mass is simulated by representing the mass as an equivalent anisotropic, elastoplastic continuum, so that the influence of the jointing system is ‘smeared’ across the continuum, i.e. the individual joints are not modelled as discrete entities. Numerical examples have been solved to verify the capability, accuracy and efficiency of the present technique. The proposed technique has also been applied to the analysis of tunnel excavation problems in plane strain. The effects of anisotropy and non‐linearity of the jointed rock mass during excavation have been investigated in some detail.
A macroscopic elastoplastic constitutive law for multilayered media: Application to reinforced earth material
Authors: Pruchnicki Erick , Shahrour Isam
Published: August 2005

This paper refers to the formulation of a macroconstitutive law for elastoplastic multilayered media, using both thermodynamical approach and homogenization procedure. The latter shows that both the microstress and strain tensors are constant in each constituent. This result permits the formulation of a macroconstitutive law involving the microplastic strains as internal parameters. Both the macrofree energy and dual dissipation potential are computed and used to derive explicit expressions for the macrocompliance tensor, yield surface and hardening rule. Finally, this law is used to analyse the behavior of a reinforced earth material under various loading paths.
A numerical analysis of the foot‐floor interaction in long wall workings
Authors: Cristescu Nicolae , Ionescu Ioan R. , Rosca I.
Published: August 2005

A mathematical and numerical approach for the description of creep and failure of coal around a longwall is given. Two coal strata are considered. They are assumed to be elasto‐viscoplastic. The variation in time of the stress and displacement around the excavation, as well as the limit thickness of the upper stratum which would avoid failure was found.
Prediction of pile capacity from stress‐wave measurements: A neural network approach
Authors: Chow Y. K. , Chan W. T. , Liu L. F. , Lee S. L.
Published: August 2005

A neural network approach for the prediction of pile bearing capacity by the stress‐wave matching technique is presented. The main advantage of this approach over the traditional manual or automated matching approach is that it avoids the time‐consuming process of iterative adjustment. This makes it feasible to determine the static pile capacity in real time in the field. Another benefit of this approach is that as more case histories become available, the neural network can be improved by learning from these new examples.

A three‐layer back‐propagation network is set up to illustrate the capability of the proposed approach for 70 dynamically tested concrete bored piles. A wave equation model developed at the National University of Singapore and coded in the NUSWAP computer program is used to formulate the problem. Up to 14 of the 70 piles (20 percent) are used in training the network. The NUSWAP program is used to generate simulation training examples based on the manually fitted training examples for further training of the network. Different network configurations are examined. The trained network produces results exhibiting good stress‐wave matching qualities compared to those obtained by manual fitting. The pile bearing capacities predicted by the two approaches agree very closely. The load‐settlement curve and axial load distribution in the pile computed using the network‐predicted soil parameters are in good agreement with the field measurements obtained from a maintained load test.
Dynamic response of an elastic and viscoelastic full‐space to a spherical source
Authors: Jiang J. , Blair D. P. , Baird G. R.
Published: August 2005

The dynamic response due to a spherical source of radius a embedded in an elastic and viscoelastic full‐space is investigated at a distance R from the source. Previous solutions to the elastic case are extended to incorporate realistic source pressure functions. The elastic solution is then cast in a scale independent form in order to generalize the application. The results show that the near‐field of the spherical source may be defined by R/a < 5. For this region the particle velocity and displacement decrease as R−2, and the risetime decreases as R−1. However. in the far‐field region (R/a > 5) the particle velocity and displacement decrease as R−1, and the risetime is independent of R. A non‐constant Q model is developed to model viscoelastic attenuation and a complete analytical solution for wave propagation is obtained by cascading the separate mechanisms of geometric attenuation and viscoelastic attenuation. A comparison of our analytical model with the results of dynamic finite element modelling shows excellent agreement. This suggests that the method of cascading the separate transfer functions is a valid approach for wave propagation in viscoelastic media.
Numerical simulation of the convergence of a bolt‐supported tunnel through a homogenization method
Authors: Bernaud D. , De Buhan P. , Maghous S.
Published: August 2005

The present paper describes the numerical implementation of constitutive relationships previously developed for modelling the elastoplastic behaviour of bolted rockmass regarded as a homogenized anisotropic medium on the macroscopic scale. Attention is more particulary focused on the iterative algorithm involved in such a numerical method, which makes use of projection formulas onto the yield surface. Those formulas are made explicit in the case of a purely cohesive rock material obeying a von Mises yield condition with associated flow rule. Combined with a finite element code, the proposed numerical procedure is then carried out for simulating the advancement of a bolt‐supported tunnel and calculating its convergence as the excavation proceeds. The results of this numerical simulation prove to agree perfectly well with those derived from an analytical model, thus validating the proposed numerical scheme. A quantitative study, varying some relevant parameters of the problem (bolt density, length of bolts, delay of placement behind the tunnel facing), is finally undertaken. It points to the versatility of the numerical approach, whose range of applicability can be further extended to various kinds of geotechnical structures reinforced by regularly distributed inclusions.
Magnetoelastic shear waves in an infinite self‐reinforced plate
Authors: Chattopadhyay A. , Choudhury S.
Published: August 2005

The propagation of magnetoelastic shear waves in an infinite self‐reinforced plate is studied in this paper. The frequency spectrum for SH‐modes in the plate is drawn and it is shown that if the dimensionless wave number is real, the branches are hyperbolas while for imaginary wave numbers, they are ellipses. It is also observed that for higher modes, the dimensionless group velocity approaches 9·9754 as the wave number increases. The group velocity is normalized with respect to the phase velocity of shear waves.

For a homogeneous and isotropic medium, on the other hand, the dimensionless group velocity approaches unity. So it is approximately 10 times larger in case of magnetoelastic shear waves in an infinite self‐reinforced plate.

It is also noted that in a reinforced medium, if both the values of the coupling parameter and the angle at which the wave crosses the primary magnetic field are increased, the values of the dimensionless frequency and group velocity are decreased.
Modelling transient heat and moisture transfer in unsaturated soil using a parallel computing approach
Authors: Thomas Hywel Rhys , Li Chi Leung Welkin
Published: August 2005

A parallel numerical model, employing a finite difference explicit scheme for the analysis of coupled heat and moisture transfer in unsaturated soil, is employed to simulate a laboratory experiment of heating of medium sand. The model, written in a two‐dimensional polar co‐ordinate formulation, is programmed in the concurrent language Occam and executed on a parallel computing network of transputers. Parallelization is adopted as a means of overcoming computing difficulties, which limited numerical solutions to those at steady state, to enable transient behaviour to be simulated.

The parallel algorithm was found to be very efficient, enabling a full solution of transient behaviour to be obtained. An investigation of the ability of the model to accurately simulate the complex, interrelated coupled nature of both two‐dimensional transient and steady‐state behaviour yielded very good correlation between experimental and numerical results. It can therefore be concluded that overall the results obtained provide confidence in the validity of the approach proposed.
An analytical solution for the transient response of saturated linear elastic porous media
Authors: Gajo A. , Mongiovi L.
Published: August 2005

The motions of fluid and solid phases in saturated porous media are coupled by inertial, viscous and mechanical interactions as described by Biot's equations. A one‐dimensional exact analytical solution of the Biot's equations for the completely general solution of the transient problem in saturated, linear, elastic, porous media is presented. The problem is solved by using the Fourier series. The transient response of porous media is shown for typical material properties of a natural granular deposit and for different degrees of viscous coupling. The analytical results show the mechanics of dispersive wave propagation in saturated porous media and they should provide a useful comparison term for the existing numerical solution methods.
Hypoplastic constitutive equation with internal variables
Authors: Kolymbas D. , Herle I. , Von Wolffersdorff P. A.
Published: August 2005

A new version of a hypoplastic constitutive equation is presented which is characterized by the introduction of a stress‐like internal parameter called back stress. The back stress is a function of the void ratio and of the hydrostatic stress. Using a unique set of material constants, the new constitutive equation describes many aspects of the behaviour of cohesionless soils including the influence of density and stress level. This is demonstrated by a series of verification tests. The determination of the material constants from laboratory tests is described analytically.
Improved computation of non‐linear advection in porous media using slightly modified basic finite element algorithms
Authors: Perrochet P.
Published: August 2005

The numerical stability of standard finite element schemes applied to the advection–diffusion equation is evaluated using a space‐time eigenvalue analysis. Unlike the usual approaches which only consider temporal aspects of stability, this analysis also describes the spatial stability of the solutions. To this end, the one‐dimensional advection–diffusion equation is put into an alternative semi‐discrete form which allows the derivation of a very practical stability condition. In multidimensional flow situations the latter is applied along the streamlines by means of a tensorial corrective function that prevents excessive numerical smearing of fronts or phase interfaces. The efficiency of the procedure is illustrated by an example which successfully simulates the coupling of two low miscible fluid phases in a variably saturated porous medium.
An anisotropic hardening constitutive model based on Generalized Isotropic Hardening rule for modelling clay behaviour
Authors: Lee Seung R. , Oh Seboong
Published: August 2005

An anisotropic hardening description is proposed on the basis of the Generalized Isotropic Hardening (GIH) rule. The key concept of the GIH rule is that any proper stress state inside or on an yield surface can be the homologic center of isotropic hardening. A concrete constitutive relationship was formulated by utilizing a simple hardening function. For verification, three sets of triaxial test results obtained from drained and undrained tests on overconsolidated clays and K0 consolidated clays were evaluated. Consequently, it was determined that the proposed model can represent any plastic deformation in reverse loading.
Kinematic hardening of soft clay in simple shear
Authors: Puzrin Alexander , Frydman Sam , Talesnick Mark
Published: August 2005

In a separate paper, the authors have proposed a normalized, non‐degrading form of the shear stress–shear strain relationship for undrained, cyclic simple shear of soft clay. This relationship is described in the present paper, and it is seen to include a single fatigue parameter—the mean effective stress. Application of the relationship therefore requires knowledge of the history of the mean effective stress during any loading history. The present paper proposes an effective stress path model which may be used for prediction of this history. The model is developed within the framework of bounding surface kinematic and isotropic hardening plasticity. It incorporates an isotropic hardening bounding surface, and a kinematic hardening yield surface, in which the elastic region vanishes, and so the yield surface reduces to the stress point. The normalized shear stress–shear strain relationship, developed on the basis of Iwan's model, is used to establish the shape of the cap of the bounding surface. A new translation rule is also incorporated in the model, allowing improved prediction of stress path development within the bounding surface during regular or irregular cyclic loading. Use of the proposed model to simulate the behaviour of soft clay in laboratory undrained cyclic simple shear tests shows excellent qualitative agreement, with most of the major features of the actual behaviour being predicted.
Settlement of a soil stratum subjected to an earthquake
Authors: Przewłócki JarosłAw , Knabe Włladysłlaw
Published: August 2005

A settlement analysis of a cohesionless soil layer subjected to an earthquake is presented. A theoretical solution utilizing Bessel functions and based on the so‐called densification model for amplitudes has been obtained. This permits to calculate the settlement under cyclic loading. The simplified formula is also presented which allows for the fast and straightforward estimation of settlement.

A numerical example is included giving the comparison of results obtained by different methods.
Studies on the behaviour of axially loaded tapered piles by the finite element method
Authors: Kurian Nainan P. , Srinivas Moola S.
Published: August 2005

Tapered piles represent a more equitable distribution of the pile material in several respects. In order to study their efficiency over piles of uniform section with the same material input, a three‐dimensional finite element analysis is developed. The numerical procedure accounts for the non‐linear elastic behaviour of both the soil and the pile‐soil interface. In order to include the latter, which involves relative slip and debonding, zero/non‐zero thickness interface elements are used. Three shapes of cross‐section, viz. circular, square and triangular, have been attempted for the piles. The load‐settlement behaviour under axial load predicted by the analysis is compared with laboratory test results obtained on instrumented model piles, installed as ‘replacement’ piles, and the fit obtained is found to be reasonably good. Also examined are interface shear and axial force in the pile, displacement and stress fields in the medium and the progression of failure in the latter.
A matrix for the solution of mixed rotating and non‐rotating vibration systems
Authors: Ramsden J. N.
Published: August 2005

A method is presented based on matrix algebra for the dynamic analysis of a mixed rotating and non‐rotating vibration system. Such systems occur in practice when turbo‐alternator sets are mounted on flexible steel foundations.

The mathematical model considered consists of a rotating non‐uniform, bi‐symmetric shaft connected via three or more flexible‐film bearings to a flexible structure throughout which mass is distributed. The shaft, which is assumed to rotate at constant speed, is heavy, flexible and carries several rigid axi‐symmetric discs. Viscous damping is assumed to act in all parts of the system and gravitational effects are ignored.
A theory of the solution of steady state network problems with special reference to mine ventilation networks
Authors: Yevdokimov A. G.
Published: August 2005

When calculating complex interconnected systems (electrical, power, water supply, gas systems, etc.) by digital computer, a number of common problems arise, connected both with the mathematically correct formulation of the problem under investigation, and with the algorithms of the solution, the programming of which makes it possible to reduce the computing time, the store capacity and the volume of input data required. The present article is devoted to the solution of these problems. The procedure is illustrated by formulating and manipulating the equations relating to complex interconnected mine ventilation networks.
Finite element method for piezoelectric vibration
Authors: Allik Henno , Hughes Thomas J. R.
Published: August 2005

A finite element formulation which includes the piezoelectric or electroelastic effect is given. A strong analogy is exhibited between electric and elastic variables, and a ‘stiffness’ finite element method is deduced. The dynamical matrix equation of electroelasticity is formulated and found to be reducible in form to the well‐known equation of structural dynamics, A tetrahedral finite element is presented, implementing the theorem for application to problems of three‐dimensional electroelasticity.
On the numerical solution of a class of problems in a linear first strain–gradient theory of elasticity
Authors: Oden J. T. , Rigsby D. M. , Cornett D.
Published: August 2005

This paper is concerned with the development of general discrete models for the analysis of boundary‐value problems in the first strain–gradient theory of elasticity. Extensions of the finite element method are constructed for this purpose, and general equations of motion are derived for finite elements of a class of micro‐polar materials which are characterized by strain energy functions involving strains and second gradients of strains or displacements. The notion of generalized nodal doublets is introduced. The problem of a composite consisting of a strain–gradient sensitive microlayer embedded between semi‐infinite bodies is examined as an example problem. Some of the results are compared with available exact solutions.
A multidata method of approximate Laplace transform inversion
Authors: Cost Thomas L. , Becker Eric B.
Published: August 2005

A newly developed numerical Laplace transform inversion technique is described. A derivation of the method, termed the ‘multidata method’, is presented along with a description of a similar collocation method. Similarities and differences between the two numerical methods are described and discussed. Results of parameter studies of both methods are presented which demonstrate the sensitivity to error displayed by the collocation method and the magnitude of the improvement in accuracy obtainable with the multidata method as compared with the collocation method, especially when errors exist in the function to be inverted.
An application of the finite element technique to boundary value problems of potential flow
Authors: Doctors L. J.
Published: August 2005

This report summarizes an experimental application of the finite element approach to two‐dimensional inviscid fluid flow.

The method results in a matrix equation relating the vector of velocities at the nodal points with the vector of singularities. The singularities, which are concentrated at the nodes, consist of a source and a vortex.

On solution of this equation (defining the flow in the region), boundary conditions must te stipulated. This usually involves setting the internal singularities to zero and making certain modifications ro those lying on the boundary of the region.

Some results of the application to a particular problem are included in the paper.
A conforming triangular finite element plate bending solution
Authors: Anderheggen Edoardo
Published: August 2005

A conforming plate bending solution using simple polynomial deflection functions of third‐degree inside each triangular element is presented. In order to avoid normal slope discontinuities along the sides of the elements, the plate displacement parameters are subjected to ‘slope continuity conditions’ acting as constraints to the minimum potential energy problem. This is then solved by the classical method of Lagrange introducing multipliers as new auxiliary variables. If a special variational formulation of the problem is used, it can be shown that the Lagrangean multipliers are generalized stress parameters. The suggested solution is therefore basically a ‘mixed’ solution, the unknown variables of the problem being both displacement and stress parameters. Several numerical results are presented.
Calculations of plate frequencies from complementary energy formulation
Authors: Sakaguchi R. L. , Tabarrok B.
Published: August 2005

The complementary variational principle has been used to derive the differential equations and the associated boundary conditions of the vibrating plate in terms of bending moments. It is shown that in this formulation, the plate possesses an infinite number of zero frequency modes in which the plate remains in a state of constant strain under a set of self‐equilibrating bending moments. In applying the Rayleigh Ritz procedure for the non‐zero frequency modes of the plate, it is shown that it the assumed functions are orthogonal to only a finite number of zero frequency modes, then one may obtain frequencies which are lower than the true frequencies of the plate. An iliustrative example is given in the paper.
The numerical solution of non‐linear consolidation problems
Authors: Poskitt T. J.
Published: August 2005

It has long been recognized that permeability, compressibility and viscosity vary during the consolidation of soil. However, the introduction of these variations into consolidation theory leads to formidable difficulties due to their rendering the equations highly non‐linear. Numerical methods are given herein which enable the governing equation to be solved for a variety of different non‐linear laws.
Non‐axial‐symmetric thermal stresses in circular discs or cylinders
Authors: Kettleborough C. F.
Published: August 2005

A solution is presented for the computation of the transient thermoelastic stresses in a hollow cylinder with temperature boundary conditions given as a circumferential variation of surface heat transfer coefficient. The temperature distribution is solved explicitly. The problem is set up using the Airy stress function which leads to the biharmonic equation. This approach requires the satisfaction of three Michell integrals at the inner boundary in order to ensure single‐valued displacements and rotation. An iterative method is described in which these integrals are all simultaneously satisfied and thus provide the necessary non‐zero boundary conditions for the solution of the biharmonic equation which is rapidly solved by Gaussian elimination. Results are presented for the general case where the temperature is a function of r and θ. The computer program is checked by assuming a constant value of the surface heat transfer coefficients. In this case a closed form solution is obtained.
Note on dynamic relaxation
Authors: Wood Winifred L.
Published: August 2005

The Dynamic Relaxation (DR) method of solving a set of simultaneous linear equations requires an estimate of the spectral radius of the matrix. Dividing each equation by the corresponding row sum of moduli of the elements of the matrix gives a convenient upper bound of unity to this. This note shows that the DR method then gives a faster asymptotic rate of the convergence than the degenerate Chebyshev method which it closely resembles.
Use of optimization methods to solve heat convection problems with unknown wall boundary conditions
Authors: Gray Michael , Smith Russell A.
Published: August 2005

In convection heat transfer problems, it is convenient to solve the governing differential equations by using straightforward marching techniques for numerical integration. However, the boundary layer equations in these heat transfer problems are not of the initial value type even though they are parabolic equations. Consequently marching techniques are not successful until correct initial values are known. A means of determining these initial values is presented herein. Selected optimization methods are utilized in conjunction with marching integration techniques to solve third‐ and fourth‐order ordinary differential equations which have three unknown initial boundary conditions. Two optimization methods, one deterministic and the other non‐deterministic in nature, are used both independently and in combination as determined by the particular circumstances. Two applications are used to demonstrate the technique and comparison is made with existing solutions.
A simple algorithm for the stiffness matrix of triangular plate bending elements
Authors: Kratochvíl J. , Ženíšek A. , Zlámal M.
Published: August 2005

For a hierarchy of polynomials on the triangle there is derived an algorithm for computing the stiffness matrix of the plate bending element. The algorithm is easy to program and means a considerable saving of the computing time. The same approach can be used for any elliptic equation with constant coefficients.
Global‐local finite element
Authors: Mote C. D.
Published: August 2005

Finite element procedures usually require more degrees of freedom for a specified accuracy than does a classical Ritz procedure if suitable coordinate functions are available. This paper develops a combined global and local dependent variable representation which couples the conventional and finite element Ritz methods. This hybrid method preserves much of the flexibility of the finite element method while increasing the solution accuracy for a specified system order. The method is illustrated by examination of a beam and a plate vibration problem.
On completeness of shape functions for finite element analysis
Authors: Taylor Robert L.
Published: August 2005

Some elements commonly used for analysis are examined for examined for completeness of polynomial interpolation and computational efficiency. Extensions to n‐dimensional space are shown to be natural consequences of the interpolation, thus all elements considered here allow for finite element approximation in higher than three‐dimensional spaces (e.g. space–time interpolations). From the study it is concluded that ‘serendipity’ class elements from the most efficient elements up to third‐degree polynomial approximations. The method used here to develop the serendipity shape functions allows for different orders of interpolation along each edge. Thus, in zones where high accuracy is required meshes can now be easily changed from linear to quadratic or higher‐order elements. Computations on some simple problems have demonstrated this to be a superior method than using large numbers of low ordered elements.
Finite element analysis of plates with curved edges
Authors: Chernuka M. W. , Cowper G. R. , Lindberg G. M. , Olson M. D.
Published: August 2005

The application of the high precision triangular plate bending element to problems with curved boundaries is considered. Appropriate edge conditions for nodal points on these boundaries are derived. The error inherent in representing the shape of a curved boundary by a series of straight segments is found to be the limiting factor on accuracy, while the effect of approximations in the actual boundary conditions is minor. To overcome the first type of error, the high precision element is modified to include one curved edge. Substantial improvements in accuracy are obtained, as demonstrated in example calculations for circular and elliptical plates.
On the stability of explicit methods for the numerical integration of the equations of motion in finite element methods
Authors: Fu C. C.
Published: August 2005

This paper deals with the stability of the numerical solutions of a dynamic finite element analysis. The solutions are obtained through a stepwise integration of the equations of motion. Upper bounds on the steplength of the integration are obtained from a stability analysis of using a simple finite difference approximation for the equations of motion, and are shown to depend strongly on the particular element is use and on how the mass of the element is distributed at its nodes.

As an example, the two‐dimensional wave propagation in a semi‐infinite plate subjected to a suddenly applied moment along its edge is studies. Through the example, we show that the bound on the steplength, obtained from the simple analysis, can provide a useful guide on choosing the steplength in other higher order integration methods. In particular, we show that, for stability considerations, the upper bound on the steplength should also hold for a fourth order explicit method. In order to achieve an acceptable accuracy of the solution, we show that the steplength should be approximately one half of the bound for the higher order explicit method as well as a higher order implicit method.

Solution of the example has been compared with that of the Timoshenko theory.
Viscoelastic stress analysis
Authors: Carpenter William Calvin
Published: August 2005

This paper formulates the viscoelastic problem so as to allow the use of higher‐order differential equation solution techniques. Viscoelastic models describing the functional relationships of stress and strain for the material are used. However, the procedure outline is valid for more general relationships. The advantages of using Runge–Kutta integration formulae are indicated and an example demonstrates the excellent results obtainable. The computational efficiency of the method is emphasized and error prediction is discussed.
Solution of eigenvalue problems by sturm sequence method
Authors: Gupta K. K.
Published: August 2005

A generalized eigenvalue algorithm is presented herein along with the complete listing of the associated computer program, which may be conveniently utilized for the efficient solution of certain broad classes of eigenvalue problems. Extensive applications of the procedure are envisaged in the analysis of many important engineering problems, such as stability and natural frequency analysis of practical discrete structural systems, idealized by the finite element technique. The procedure based on the Sturm sequence method is accurate and fast, possessing several significant advantages over other known methods of such analysis. Numerical results are also presented for two representative structural engineering problems.
Tetrahedral polynomial finite elements for the Helmholtz equation
Authors: Silvester P.
Published: August 2005

It is shown that the Helmholtz equation in three dimension leads to finite element approximations on tetrahedral elements that closely resemble the corresponding two‐dimensional treatment on triangle. For each polynomial order, there exist two numeric universal matrices independent to tetrahedron size and shape; the element matrices are always given as linear combinations of row and column permutations of these. Numeric matrices are given up to third‐order, and the permutation schemes are shown in detail. Experimental computer programs using these elements have shown fast matrix assembly times; convergence rates are essentially similar to those obtained with the corresponding triangular elements.
Effect of shear deformation on the elastica with axial strain
Authors: Huddleston John V.
Published: August 2005

The problem of the elastica is generalized to include the effect if shear as well as axial strain. This leads to a non‐linear two‐point boundary‐value problem composed of five first‐order differential equations. At first the problem is linearized to study the way in which the bucking loads are modified by the combination of effects. For a simple column, the eigenvalues are found to be zeros of a third‐order polynomial, and these are evaluated by a straightforward Newton–Raphson technique using a digital computer. The postbucking behaviour is then studied by using the computer and a ‘shooting’ method to solve the non‐linear problem. For the special case of the prismatic column, graphs plotted by the computer are displayed to show the variation of load with both lateral and axial deflection for various combinations of axial compressibility and shear deformability. For one particular case, the deflected center line at various load levels is shown.
On the unity of the constant strain/constant moment finite element methods
Authors: Hellan Kåre
Published: August 2005

The paper is an attempt to unify the finite element methods of ‘first’ order in plate stretching and plate bending, i.e. those assuming in each element constant strains in stretching or constant moments in bending. Reference is made to the stretching–bending analogy between the method of Turner, Clough, Martin and Topp and the method of Elias. By another analogy property a plate stretching method which is the dual to the constant moment method of Herrmann (and Hellan) is derived. The transformation of variables which carries the last pair of dual methods identically into the first pair is finally demonstrated.
A mixed finite element method for bending of plates
Authors: Poceski Apostol
Published: August 2005

A simple mixed finite element method is developed. The finite element is a rectangular triangle and rectangle. In the element the deflections are assumed to be simple four‐element polynomials, bending moments, Mx and My with a partially linear distribution, and a constant, Mxy, expressed in terms of the node deflections. The element matrix is of the order of 8 × 8. It is derived in a common engineering way. The unknowns are the deflections at the nodes and mid‐diagonal, the two moments at the end of the diagonal and the two moments on the cathetus. The results obtained by this method show good convergence and an improvement in the accuracy of the moments as well as in the deflections, compared with results obtained by similar methods, such as those of Herrmann.
Numerical stability in quasi‐static elasto/visco‐plasticity
Authors: Cormeau I.
Published: August 2005

The numerical stability of simple marching schemes used in elasto/visco‐plasticity is investigated. An assumption of convexity renders possible the derivation of a theoretical stability criterion based on the identification of the numerical process with the integration of a non‐linear, first order, system of ordinary differential equations. Explicit stability criteria are obtained for commonly used visco‐plastic laws. Selected examples illustrate the necessity and effectiveness of the proposed stability criteria in actual computations.
Solution of generalized geometric programs
Authors: Avriel M. , Dembo R. , Passy U.
Published: August 2005

A cutting plane algorithm for the solution of generalized geometric programs with bounded variables is described and then illustrated by the detailed solution of a small numerical example. Convergence of this algorithm to a Kuhn–Tucker point of the program is assured if an initial feasible solution is available to initiate the algorithm. An algorithm for determining a feasible solution to a set of generalized posynomial inequalities which may be used to find a global minimum to the program as well as test for consistency of the constraint set, is also presented. Finally an application in optimal engineering design with seven variables and fourteen non‐linear inequality constraints is formulated and solved.
Use of the over‐relaxation technique in the simulation of large groundwater basins by the finite element method
Authors: Gambolati Giuseppe
Published: August 2005

The over‐relaxation technique has been used in conjunction with the finite element method in a regional time‐dependent simulation of the subsidence of Venice. One year of computer experiments have shown that in basins with a ratio h/R ⩽ 10−2 between the vertical and the horizontal dimension, the over‐relaxation technique can lead to unsatisfactory results. In single precision and for relatively large time steps, the solution of the final linear system can be inaccurate if the optimum over‐relaxation factor ω is not correctly assessed. In transient analyses, instability can also occur. The latter may be avoided by properly reducing the time step or by switching to double precision. Steady state simulations can also require double precision to provide accurate results even when the best ω is used. Instability and inaccuracy disappear in basins for which h/R ⩾ 10−1. In addition the processor time diminishes significantly as in this case the number of iterations necessary to obtain a good solution is considerably smaller than the order of the matrix.