Showing papers on "Direct stiffness method published in 1984"

Finite layer analysis of layered elastic materials using a flexibility approach. Part 1—Strip loadings

Abstract: It is well known that the analysis of a horizontally layered elastic material can be considerably simplified by the introduction of a Fourier or Hankel transform and the application of the finite layer approach. The conventional finite layer (and finite element) stiffness approach breaks down when applied to incompressible materials. In this paper these difficulties are overcome by the introduction of an exact finite layer flexibility matrix. This flexibility matrix can be assembled in much the same way as the stiffness matrix and does not suffer from the disadvantage of becoming infinite for an incompressible material. The method is illustrated by a series of examples drawn from the geotechnical area, where it is observed that many natural and man-made deposits are horizontally layered and where it is necessary to consider incompressible behaviour for undrained conditions. For abstract of part 2 see TRIS no. 378330. (Author/TRRL)

Sparse Orthogonal Schemes for Structural Optimization Using the Force Method

Abstract: Historically there are two principal methods of matrix structural analysis, the displacement (or stiffness) method and the force (or flexibility) method. In recent times the force method has been used relatively little because the displacement method has been deemed easier to implement on digital computers, especially for large sparse systems. The force method has theoretical advantages, however, for multiple redesign problems or nonlinear elastic analysis because it allows the solution of modified problems without restarting the computation from the beginning. In this paper we give an implementation of the force method which is numerically stable and preserves sparsity. Although it is motivated by earlier elimination schemes, in our approach each of the two main phases of the force method is carried out using orthogonal factorizaton techniques recently developed for linear least squares problems.

A procedure for analysing space frames with partial warping restraint

Abstract: Stiffness matrices for three-dimensional beam elements that include the effect of warping restraint on elastic torsional response have been derived by various investigators. Using one of the available stiffness matrices and assuming that the warping boundary conditions can be specified on a member-by-member basis, an elastic ‘warping’ support is introduced to represent conditions of partial warping restraint at the member ends. The concept of a ‘warping indicator’ is then introduced to facilitate use of warping springs. Following this, static condensation is used to eliminate the restrained warping degrees-of-freedom. The condensed stiffness matrices for the elements can then be assembled to yield a global stiffness matrix. In the global matrix, continuous warping degrees-of-freedom, that is, those internal to a member represented by several elements, are expressed in local co-ordinates. The remainder are expressed in global co-ordinates. In the force recovery phase, it is shown that an ‘indirect’ method yields most accurate results for the bimoment and warping torsion when the twist function is represented by a cubic polynomial. Solutions to examples of linear elastic analysis are compared with well-known analytical solutions to demonstrate the application of the method.

Beam on Elastic Foundation Finite Element

Abstract: A stiffness matrix for a beam on elastic foundation finite element and element load vectors due to concentrated forces, concentrated moments, and linearly distributed forces are developed for plane frame analysis. This element stiffness matrix can be readily adopted for the conventional displacement method. For the force method, an element flexibility matrix and element displacement vectors due to the aforementioned loads are also presented. Whereas most other analyses of a beam on elastic foundation finite element approximate the foundation by discrete springs or by cubic hermitian polynomials, the present stiffness and flexibility matrices are derived from the exact solution of the differential equation. Thus, results of this finite element analysis are exact for Navier and Winkler assumptions. Numerical examples are given to demonstrate the efficiency and simplicity of the element.

Generalized Method for Estimating Drift in High‐Rise Structures

Abstract: An approximate method for estimating the drift of multi-bent structures is presented. Structures that are singly or doubly symmetrical in plan and comprising any combination of shear walls, coupled walls, rigid frames and braced frames, can be considered. Results for structures that are uniform with height compare closely with results from stiffness matrix computer analyses. The method is developed from coupled-wall deflection theory which is expressed in terms of two non-dimensional structural parameters. The parameters involve three structural properties: the individual bending stiffness of the walls, the overall bending stiffness related to axial deformations of the walls and racking stiffness caused by reverse bending of the beams. Similar properties are calculated for rigid frames, braced frames and shear walls and then combined to determine values of the two parameters for the total structure. These values are then substituted into a generalized equation to obtain the deflection profile. This method accounts for axial deflection of the vertical components and is, therefore, more accurate for very tall structures. The method provides a rapid estimate of the drift in a high-rise structure as well as allowing an easy means of comparing the suitability of different structural solutions for a tall building. The method also illustrates the fundamental dependence of the behavior of continuous type cantilevers on two characteristic parameters.

Dynamic-stiffness matrix of soil by the boundary-element method: Conceptual aspects

Abstract: Starting from a weighted-residual formulation, the various boundary-element methods, i.e. the weighted-residual technique, the indirect boundary-element method and the direct boundary-element method, are systematically developed for the calculation of the dynamic-stiffness matrix of an embedded foundation. In all three methods, loads whose analytical response in the unbounded domain can be determined are introduced acting on the continuous soil towards the region to be excavated. In the weighted-residual technique and in the indirect boundary-element method, a weighting function is used; in the latter case, it is selected as the Green's function for the surface traction. In the direct boundary-element method, the surface traction along the structure-soil interface is interpolated. The same type of boundary matrices which have a clear physical interpretation are identified in the three formulations, each of which is illustrated with a simple static example. The indirect boundary-element method leads to the most accurate results. The guaranteed symmetry and the fact that the displacement arising from the applied loads can easily be calculated and compared to the prescribed displacement makes the indirect boundary-element method especially attractive for calculating the dynamic-stiffness matrix of the soil. Instead of calculating the dynamic-stiffness matrix of the embedded foundation with the boundary-element method, it can be determined as the difference of those of the regular free field and of the excavated part. The calculation of the former does not require the Green's function for the surface traction. The dynamic stiffness of the excavated part can be calculated by the finite-element method.

Approximate Stiffness Matrix for Tapered Beams

Abstract: A method for finding a modified bending stiffness matrix for a member of varying section is presented. The assumption is made that the displacement function for a uniform beam may be used as an approximation to the correct displacement function, thus leading to greater simplicity in the computation, while providing sufficient accuracy for most purposes. It is shown that for two practical applications of the approximate stiffness matrix for tapered beams, economy of effort can be achieved without any great loss of accuracy. For large tapers, the accuracy of the method is acceptable. Even greater accuracy is to be expected for smaller tapers.

Identification of large flexible structures mass/stiffness and damping from on-orbit experiments

Abstract: Two methods for identifying the mass, damping and stiffness matrices of a linear vibrating system are presented. Both methods require the measurement of acceleration, velocity and displacement at various locations of the system. In the first method, the response of the system subjected to known forces is used while the second method employs the free vibration data. The unknown parameters are recovered through the standard least squares procedure. Numerical results are presented for several examples.

Shells of revolution with local deviations

Abstract: A finite element model that is suitable for the analysis of shells of revolution with arbitrary local deviations is presented. The model employs three types of shell elements: rotational, general and transitional. The rotational shell elements, which are most efficient, are used in the region where the shell is axisymmetric. The general shell elements, which can simulate almost any shell geometry, are used in the local region of the deviation. The transitional shell elements connect these two distinctively different types of elements and make it possible to combine them in a single analysis. The form of the global stiffness matrix is somewhat unique in the new model. Non-zero terms are not confined to a narrow band along the diagonal, but occur throughout the matrix. This is due to the following: (1) two different types of nodes, ring nodes and point nodes, are combined in a single analysis; and (2) a locally non-axisymmetric geometry creates a coupling of the Fourier harmonic coefficients of the rotational elements. Yet, the matrix still contains many scattered zero terms that should be considered for numerical efficiency. In this paper an efficient solution procedure that is effective for this situation is developed. The steps include the use of a substructuring technique and separate partial harmonic analysis. A numerical example is presented and compared with existing solutions to demonstrate the capabilities and the efficiency of the new model.

Structural design sensitivity analysis with generalized global stiffness and mass matrices

Abstract: First- and second-order derivatives of measures of structural response with respect to design variables are calculated using the generalized global stiffness and mass matrix formulation of structural equations. The adjoint variable method is extended for first-order design sensitivity calculation and a mixed direct differentiationadjoint variable method is employed to calculate second derivatives of static structural response measures. The variational or virtual work form of the structural equations is shown to be well suited for design sensitivity analysis with generalized global stiffness and mass matrices, avoiding the requirement for explicit reduction of the matrices. The method is illustrated using a simple structure with a multipoint constraint.

Structural Analysis of Suspension Bridges

Abstract: A practical structural analysis of suspension bridges by the stiffness matrix method is presented. This analysis is based on deflection theory. In this method all live loads are applied at arbitrary points and locations along a stiffening girder or truss because of the inclusion of the load terms derived by means of the Laplace transformation. Therefore, it is possible to reduce the number of division elements. Horizontal cable tension can be determined by integral of Green's function with respect to deflection. Irrespective of the number of division elements, the magnitude of horizontal cable tension can be obtained precisely. The method is applicable to an analysis of suspension bridges with any number of spans and variable cross sections. The validity of the stiffness matrix method proposed herein is examined and checked by numerical calculations. This stiffness matrix method can be developed and extended to a three‐dimensional analysis of suspension bridges.

Thin-Walled Core Element for Multistory Buildings

Abstract: The stiffness matrix of a thin‐walled open‐section element including the effect of shear strains due to warping is presented Closed form expressions for the stiffness coefficients are obtained for the case of negligible uniform torsion stiffness; an iterative procedure, based on a compatibilization scheme, is presented for the case when both uniform torsion and warping shear deformations are considered The proposed element can be used for modeling core shear walls in multistory buildings and is suitable for implementation in a computer program based on the direct stiffness method Two simple examples of multistory core wall structures are presented to assess the effect of uniform torsion stiffness and warping shear deformation on dynamic torsional behavior

Evolution of assumed stress hybrid finite element

Abstract: Early versions of the assumed stress hybrid finite elements were based on the a priori satisifaction of stress equilibrium conditions. In the new version such conditions are relaxed but are introduced through additional internal displacement functions as Lagrange multipliers. A rational procedure is to choose the displacement terms such that the resulting strains are now of complete polynomials up to the same degree as that of the assumed stresses. Several example problems indicate that optimal element properties are resulted by this method.

An algorithm for assembly of stiffness matrices into a compacted data structure

Abstract: A data structure is described that stores only the non‐zero terms of the assembled stiffness matrix. This storage scheme results in considerable reduction in memory demand during the assembly phase of a finite element program. Therefore, larger matrices can be formed in the main memory of the computer. When secondary store must be used this approach reduces the I/O cost during the assembly stage. An algorithm is derived that starts with the element connectivity information and generates the compacted data structure. The element matrices are then assembled to form the stiffness matrix with this storage scheme. The assembly algorithm is described and a FORTRAN listing of the routines is presented. The reduction in storage is demonstrated with the aid of numerical examples.

Treatment of contact stiffness in structural analysis (1st report: mathematical model of contact stiffness and its applications)

Abstract: This paper describes a treatment of machined surfaces in contact in the structural analysis. In order to collect data on the contact stiffness, the relationship between the interface pressure and the surface approach of components subjected to normal forces was experimentally investigated, and then the obtained data were arranged for an easy computational utilization. Using those data, the influence of the contact stiffness on the resulting deformations of structures having joints was investigated by means of a finite element method. From the analysis, it was found that the contact stiffness has great effect of the deformation analysis of components subjected to both normal and tangential forces.

Constrained Stiffness Matrix Adjustment Using Mode Data.

Abstract: : A procedure is introduced that uses, in addition to mode data, structural connectivity information to optimally adjust deficient stiffness matrices. The adjustments are performed such that the percentage change to each stiffness coefficient is minimized. The physical configuration of the analytical model is preserved, and the adjusted model will reproduce exactly the modes used in the identification. The theoretical development is presented, and the procedure is demonstrated by numerical simulation of a test problem. (Author).

Direct Methods for Solving Finite Element Equations

Abstract: The application of the finite element method to a boundary value problem leads to a system of equations K α = G, where the stiffness matrix K is often large, sparse, and positive definite. This chapter reviews the solution of such systems by Gaussian elimination and the closely related Cholesky method. These so-called direct, that is, non-iterative, methods are the most widely used methods for finite element computation today. Apart from being important in their own right, the direct methods provide the basis for the incomplete factorization technique used to precondition the conjugate gradient method. The chapter discusses node-ordering strategies aimed at making K a matrix with a small band or envelope and presents storage schemes appropriate for such matrices. The mathematics of Gaussian elimination and the Cholesky method is basic regardless of how the nodes are ordered. It then examines some special techniques for solving the stiffness equations. Each essentially corresponds to a node-ordering strategy that differs from the band-minimizing type and the implementation of Gaussian elimination differs accordingly. The chapter also explains the main types of errors that arise in constructing and solving finite element equations.

The inclusion of flexible joints in the stiffness method

Abstract: The joint area between beam type elements may be regarded as a sub-structure, the behaviour of which may be defined by the stiffness or flexibility matrix. The flexibility of such a sub-structure may be determined experimentally by the measurement of displacements under a variety of loading conditions, and the report describes the method of obtaining the stiffness matrix from this data. Alternatively, the joint area may be analysed by a detailed finite element calculation and the report describes two methods of obtaining the stiffness matrix, one using applied forces, the other by prescribing displacements. The method of inputting the derived stiffness matrix into the PAFEC 75 finite element program is fully described with examples. (Author/TRRL)

Orthogonal Schemes for Structural Optimization

Abstract: : Historically there are two principal methods of matrix structural analysis, the displacement (or stiffness) the force method has been used relatively little because the displacement method has been deemed easier to implement on digital computers, especially for large sparse systems. The force method has theoretical advantages, however, for multiple redesign problems or nonlinear elastic analysis because it allows the solution of modified problems without restarting the computation from the beginning. In this paper we give an implementation of the first phase of the force method which is numerically stable and preserves sparsity. A primary feature of our work is the development of an efficient algorithm for computing a banded basis for the null space by orthogonal decomposition. Numerical test comparisons for several practical structural analysis problems are provided. (Author)

A finite element method for stress analysis or elastoplastic body with polygonal line strain——hardening

Abstract: In this paper, the stress-strain curve of material is fitted by polygonal line composed of three lines. According to the theory of proportional loading in elastoplasticity, we simplify the complete stress-strain relations, which are given by the increment theory of elastoplasticity. Thus, the finite element equation with the solution of displacement is derived. The assemblage elastoplastic stiffness matrix can be obtained by adding something to the elastic matrix, hence it will shorten the computing time. The determination of every loading increment follows the von Mises yield criteria. The iterative method is used in computation. It omits the redecomposition of the assemblage stiffness matrix and it will step further to shorten the computing time. Illustrations are given to the high-order element application departure from proportional loading, the computation of unloading fitting to the curve and the problem of load estimation.

Dynamic Analysis of Elastic-Plastic Structures by Mode-Superposition and Equivalent Plastic Loads

Abstract: A method is shown for the dynamic analysis of elastic-plastic structures by mode superposition and treating plastic strain as additional applied loads. This method reduces the elastic-plastic analysis to elastic analysis of the same structure with additional forces. In the elastic analysis, the modes of vibration and stiffness matrix are elastic and remain the same in all time steps. This eliminates the calculation of a new stiffness matrix for each time step and hence greatly reduces the amount of numerical computation. Dynamic elastic-plastic responses of an unsymmetrical portal frame were calculated as an illustration of this method. The elastic modes and stiffness matrix were calculated by the finite element method.