scispace - formally typeset
Search or ask a question

Showing papers on "Direct stiffness method published in 1972"



Journal ArticleDOI
TL;DR: In this paper, a direct computer-oriented approach is presented enabling the treatment of coupled combinations of three possible types of modifications, namely, removal of freedoms, addition of freedoms and modification of elements.
Abstract: In order to perform modifications of structures analyzed by the Matrix Displacement Method, a direct computer-oriented approach is presented enabling the treatment of coupled combinations of three possible types of modifications, namely, removal of freedoms, addition of freedoms, and modification of elements. The formulation is extended to cover modifications of elements in substructures together with the corresponding modification to the main structure. The method is derived from the laws of partitioned matrices and Boolean transformation of freedoms within the structural stiffness matrix, and fully exploits symmetry and positive-definiteness. The presented form of the modification equations allows the use of an alternative existing approach of updating the triangularized factor of the structural stiffness matrix due to the modification of elements. The method is extended to provide a recursive hypermatrix Cholesky algorithm. All formulations are accompanied by operation counts to enable rapid determination of break-even points for reanalysis or possible iterative solutions.

39 citations


Journal ArticleDOI
TL;DR: In this article, two new procedures for a direct solution without generating the inverse are developed, based on the Sherman and Morris algorithm and the second on a solution of a reduced set of equations in which the independent unknowns are computed directly.
Abstract: This study presents ways to implement the algorithm of Sherman and Morrison when modifications occur in a small part of the structure. For such limited modifications, a corresponding small number of elements in the original stiffness matrix are affected, and the inverse of the new stiffness matrix can be computed efficiently by this method. However, for matrices of high order, this method also becomes time consuming. Therefore, two new procedures for a direct solution without generating the inverse are developed. The first procedure is based on the Sherman and Morris algorithm and the second on a solution of a reduced set of equations in which the independent unknowns are computed directly. Numerical difficulties due to matrix singularity and ways to overcome them are analyzed. A comparative study demonstrates the efficiency of the proposed methods in terms of the number of arithmetic operations in reanalysis of limited structural design.

22 citations


Journal ArticleDOI
TL;DR: In this article, a finite element method is presented for free vibration analysis of thin plates subjected to complex force systems applied in the middle-plane of the plate, where the equation of motion is characterized by the basic stiffness, consistent mass, and incremental stiffness matrices.

22 citations


Journal ArticleDOI
TL;DR: In this paper, the equations for the free undamped vibration of a structure in an ideal incompressible fluid medium and their finite element formulation are briefly reviewed and the relevant matrices (stiffness and loading) for two prismatic fluid elements are given explicitly and some numerical results are presented.
Abstract: The equations for the free undamped vibration of a structure in an ideal incompressible fluid medium and their finite element formulation are briefly reviewed. The relevant matrices (stiffness and loading) for two prismatic fluid elements are given explicitly and some numerical results are presented.

18 citations


Journal ArticleDOI
TL;DR: In this paper, a method of analysis and computer program is presented for cellular structures of constant depth with arbitrary geometry in plan view. The development is based on the finite element method, which is used to capture both the membrane and plate bending behavior of the deck and web components.
Abstract: A method of analysis and computer program are presented for cellular structures of constant depth with arbitrary geometry in plan view. The development is based on the finite element method. Special elements are developed to capture both the membrane and plate bending behavior of the deck and web components. The well established direct stiffness method is used for the element assembly. The structure may be subjected to a variety of force and displacement boundary conditions such as distributed dead and live loads in addition to concentrated nodal loads and prescribed nodal displacements. After solving for the unknown nodal displacements, the reactions are computed in addition to the internal forces which are output at points selected by the user.

12 citations


01 May 1972
TL;DR: In this paper, the solution of problems involving static equilibrium, natural vibrations, and stability of arbitrary, stiffened, shells of revolution, subjected to symmetric static loading is reviewed using a numerical integration technique combined with the direct stiffness method, and compared to solutions employing other numerical techniques, such as finite differences and finite elements.
Abstract: : The solution of problems involving static equilibrium, natural vibrations, and stability of arbitrary, stiffened, shells of revolution, subjected to symmetric static loading is reviewed using a numerical integration technique combined with the direct stiffness method, and compared to solutions employing other numerical techniques, such as finite differences and finite elements The numerical integration technique is extended to linear and nonlinear analysis of static equilibrium, stability, and vibrations under the influence of initial prestress, of arbitrary shells of revolution subjected to general unsymmetric (no line of symmetry) loadings The coupling of the Fourier harmonics of the various shell functions in the case of nonlinear analysis are presented for a spherical cap subjected to unsymmetric static loading Moreover, the classical buckling load is established for a prolate spheroid subject to hydrostatic pressure, and for an example shell of revolution of complex geometry subjected to a complex loading system (Author)

7 citations


Book
01 Jun 1972

7 citations


Proceedings ArticleDOI
01 Apr 1972
TL;DR: In this article, the authors present an assessment of the solution procedures available for the analysis of inelastic and/or large deflection structural behavior, and compare and evaluate each with respect to computational accuracy, economy, and efficiency.
Abstract: This paper presents an assessment of the solution procedures available for the analysis of inelastic and/or large deflection structural behavior. A literature survey is given which summarized the contribution of other researchers in the analysis of structural problems exhibiting material nonlinearities and combined geometric-material nonlinearities. Attention is focused at evaluating the available computation and solution techniques. Each of the solution techniques is developed from a common equation of equilibrium in terms of pseudo forces. The solution procedures are applied to circular plates and shells of revolution in an attempt to compare and evaluate each with respect to computational accuracy, economy, and efficiency. Based on the numerical studies, observations and comments are made with regard to the accuracy and economy of each solution technique.

6 citations


Journal ArticleDOI
TL;DR: A three-dimensional finite element method for general anisotropy materials is presented that uses a parallelepiped element and includes the directional properties of the materials, making it easily applicable to problems in wood mechanics.
Abstract: A three-dimensional finite element method for general anisotropy materials is presented. The method uses a parallelepiped element and includes the directional properties of the materials, making it easily applicable to problems in wood mechanics. By varying the stiffness and allowing some elements to assume zero stiffness, a variety of shapes can be studied, including those with cracks and voids. The limitations of the method are primarily related to the available computer storage and to the fitting of rectangular-sided elements to curved boundaries. Examples are presented to indicate the types of problems this technique is capable of solving. These examples include wood mechanics problems of compression, both parallel and at angles to the ring and grain, tension of a member with holes, and torsion.

5 citations


Proceedings ArticleDOI
01 Jan 1972
TL;DR: In this paper, an approach has been developed to analyze the dynamic response of the structure with continuously distributed mass based on the stiffness matrix method satisfying the equations of motion at any point in the structure.
Abstract: An approach has been developed to analyze the dynamic response of the structure with continuously distributed mass based on the stiffness matrix method satisfying the equations of motion at any point of the structure. The effect of the soil medium on the vibration of the structure has been taken into account, the structure and the pile-foundation have been considered as a monolytic system. An optimum column spacing length can be obtained by analyzing the structure for few different column spacing lengths.

Journal ArticleDOI
TL;DR: In this article, a direct stiffness method for analyzing structural plates which possess nonlinear material properties is developed to predict the ultimate load of certain plates subjected to lateral loading, where the continuous plate is replaced by interconnected beam elements whose physical properties are determined by forcing the moment-rotation behavior to be identical with that of the plate.
Abstract: A direct stiffness method for analyzing structural plates which possess nonlinear material properties is developed to predict the ultimate load of certain plates subjected to lateral loading. Because of the mathematical complexity of the problem, a numerical technique is suggested whereby the continuous plate is replaced by interconnected beam elements whose physical properties are determined by forcing the moment-rotation behavior to be identical with that of the plate. This technique yields a model that assumes concentrated plasticity and distributed elasticity. The success of the beam model is made possible by approaching the problem from a differential point of view and using a three-parameter moment-rotation equation. The solutions are then obtained using a fourth-order Runge-Kutta integration technique.

Journal ArticleDOI
TL;DR: In this paper, a technique for finding the finite-element equations (including geometric nonlinearity) for a curved shell using Marguerre's strain-displacement relations is described.
Abstract: A technique is described for finding the finite-element equations (including geometric nonlinearity) for a curved shell using Marguerre’s strain-displacement relations. Both the displacements and the shell geometry are interpolated by arbitrary plate functions. Strain-energy tensors are developed which yield the stiffness equations through application of an energy principle. An automated method for explicitly generating the stiffness terms is presented, and applied to give several third and fifth order triangular elements of nonconstant curvature. Applicability is analyzed for both shallow and deep shells, and numerical results are given for a shallow conoid problem.

Book ChapterDOI
K.I. Majid1
01 Jan 1972
TL;DR: In this paper, the authors focus on the matrix methods of structural analysis and show that matrix methods are a powerful tool for dealing with large and complex structures and become especially useful when a computer is used to perform the numerical operations.
Abstract: This chapter focuses on the matrix methods of structural analysis. Matrix methods invariably appear to be rather cumbersome, particularly in dealing with simple structural problems. In reality, however, they are straightforward and form a powerful tool for dealing with large and complex structures. They become especially useful when a computer is used to perform the numerical operations. The first method, which is used more widely than the second, is the matrix displacement method, which expresses the internal member forces in terms of the joint displacements and then proceeds to solve a set of joint equilibrium equations to determine the unknown displacements. This method is popular because it does not involve the concept of redundancies, and it deals with statically determinate and indeterminate structures with equal efficiency. The second method is the matrix force method. This method expresses the conditions of equilibrium between the externally applied loads and the resulting internal forces, and solves the compatibility equations for the unknown redundant.

01 May 1972
TL;DR: A DISCRETE ELEMENT ANALYSIS as discussed by the authors, which is based on an ITERATIVE PROCEDURE called the TANGENT STIFFness METHOD, is used to evaluate the performance of a general discemble.
Abstract: A DISCRETE ELEMENT ANALYSIS WHICH CONSIDERS GEOMETRIC, MATERIAL, AND SUPPORT NONLINEARITIES OF STATICALLY LOADED PLANE FRAMES IS DEVELOPED. A COMPUTER PROGRAM HAS BEEN WRITTEN TO IMPLEMENT AND VERIFY THE ANALYSIS. FRAME GEOMETRY, LOADS, CROSS SECTIONS, AND SUPPORTS (NONLINEAR CONCENTRATED AND DISTRIBUTED SPRINGS) CAN BE SUFFICIENTLY GENERAL TO WORK PRACTICAL FRAME PROBLEMS. THE METHOD OF ANALYSIS IS BASED ON AN ITERATIVE PROCEDURE CALLED THE TANGENT STIFFNESS METHOD. UNBALANCED NODAL POINT FORCES ARE APPLIED TO A TEMPORARILY LINEAR STRUCTURE WHOSE POSITION DEPENDENT STIFFNESS MATRIX IS THE TANGENT STIFFNESS MATRIX OF THE STRUCTURE. THE FRAME MEMBERS ARE DIVIDED INTO A NUMBER OF DISCRETE ELEMENTS. LOAD-DISPLACEMENT EQUATIONS FOR AN INDIVIDUAL DISCRETE ELEMENT ARE DERIVED WHICH ARE VALID FOR LARGE DISPLACEMENTS. A NUMERICAL TECHNIQUE IS USED TO DETERMINE THE FORCE-DEFORMATION RESPONSE OF A CROSS SECTION WITH NONLINEAR STRESS-STRAIN CURVES. LOADS AND NONLINEAR SUPPORTS ARE INPUT IN NORMAL ENGINEERING TERMS AND CAN BE REFERENCED EITHER TO THE STRUCTURE OR TO THE MEMBER AXES. WHEN NECESSARY, THE LOADS AND NONLINEAR SUPPORTS ARE INTERNALLY TRANSFORMED TO MEMBER COORDINTES AND DISCRETIZED TO CONCENTRATED VALUES AT THE NODAL POINTS. CASTIGLIANO'S FIRST THEOREM IS APPLIED TO DEVELOP MATRIX EXPRESSIONS FOR THE STIFFNESS MATRIX OF A GENERAL DISCRETE ELEMENT AND THESE EXPRESSIONS ARE USED TO OBTAIN THE STIFFNESS MATRIX FOR THE SPECIFIC DISCRETE ELEMENT USED IN THE FRAME SOLUTIONS. A NUMBER OF PROBLEMS ARE WORKED AND COMPARED WITH EXISTING ANALYTICAL OR EXPERIMENTAL SOLUTIONS.

01 Aug 1972
TL;DR: In this paper, an incremental variational principle is developed to relate the strain tensor of the actual imperfect shell to the first and second order incremental Green strain tensors of the perfect shell.
Abstract: : For an elastic thin cylindrical shell with arbitrary geometric imperfections a refined finite element stiffness matrix is precisely formulated in terms of Lagrangian variables. A general incremental variational principle is developed to relate the strain tensor of the actual imperfect shell to the first and second order incremental Green strain tensors of the perfect shell. Applying to the variational principle a complete cubic polynomial coordinate function for the nodal normal displacement, and a linear function for the nodal in-plane displacements, the incremental stiffness matrix of a triangular curved shell element is obtained. For a shell with arbitrary geometry, the element stiffness matrix is evaluated by numerical integration. (Author)

Journal ArticleDOI
01 Jun 1972
TL;DR: In this article, a method of an analysis of box-girder and multicell-branch-style bridge-decks is proposed, based on the MATRIX progress-procedure technique.
Abstract: A METHOD OF ANALYSIS OF BOX-GIRDER AND MULTICELLULAR BRIDGE DECKS IS PROPOSED. ALTHOUGH SIMILAR TO THE DIRECT STIFFNESS TECHNIQUE, IT INCORPORATES THE MATRIX PROGRESSION CONCEPT FOR OBTAINING THE STIFFNESS MATRICES FOR THE INDIVIDUAL PLATE ELEMENTS: RESULTS FROM EXPERIMENTS WITH TWO PERSPEX MODELS ARE SHOWN TO AGREE SATISFACTORILY WITH THEORETICAL CALCULATIONS. THE METHOD OF ANALYSIS IS SYSTEMATIC AND CONVENIENT FOR COMPUTATION. COMPUTER STORE REQUIRED FOR A MULTI-CELL BOX IS SMALL COMPARED TO OTHER NUMERICAL METHODS SINCE THE PLATES BETWEEN JUNCTIONS CAN BE CONSIDERED AS INDIVIDUAL ELEMENTS AND FURTHER SUBDIVISIONS ARE UNECESSARY. CONVERGENCE OF THE MOMENTS AND FORCE QUANTITIES IS SLOWER THAN THAT OF THE DISPLACEMENTS. HOWEVER, FOR THE EXAMPLES CITED SUMMATION OF 19 HARMONICS WAS CONSIDERED TO BE SUFFICIENT. THE GENERAL FLEXIBILITY OF THE METHOD MAKES IT POSSIBLE TO REPRESENT PRECISELY ANY LOAD CONDITION, WHETHER IT IS NORMAL OR IN-PLANE, BY USING SUITABLE FOURIER REPRESENTATION. ADDITIONAL FORCES APPLIED THROUGH PRESTRESSING CAN ALSO BE INCORPORATED IN THE LOAD VECTOR WHEN DIVIDED INTO CORRESPONDING COMPONENTS IN THE VERTICAL AND HORIZONTAL DIRECTIONS. THE PROBLEM OF INTERMEDIATE SUPPORTS CAN BE ANALYZED USING AN INFLUENCE TECHNIQUE IN WHICH THE INTERNAL REACTIONS ARE FIRST CALCULATED FROM THE CONDITIONS THAT THE RESULTANT DEFLECTIONS AT THE SUPPORTED POINTS ARE ZERO. THE FINAL SOLUTION IS THAT OF A SINGLE-SPAN STRUCTURE UNDER THE COMBINED ACTION OF THE APPLIED LOAD AND THE REACTIONS. THE ADVANTAGE OF THE METHOD IS ITS APPLICABILITY TO ELEMENTS OTHER THAN THE ISOTROPIC RECTANGULAR PLATE ELEMENTS. A SMALL CHANGE OF THE GOVERNING MATRICES CAN MAKE THE SOLUTION SUITABLE FOR ORTHOTROPIC RIGHT OR FAN-SHAPED PLATES. FURTHERMORE THE STIFFNESS MATRIX FOR A SHELL ELEMENT CAN BE OBTAINED BY USING THE SAME MATRIX PROGRESSION TECHNIQUE. THESE EXTENSIONS ARE LIKELY TO MAKE THIS METHOD SUITABLE FOR A NUMBER OF COMPLEX BRANCH PLATE PROBLEMS INCLUDING THE RIGHT OR CURVED BOX-TYPE BRIDGE DECKS. /TRRL/


Book ChapterDOI
01 Jan 1972
TL;DR: In this article, the structural stiffness matrix from each element stiffness matrix is constructed from known displacement in each nodal point of an element, and the equations of equilibrium are introduced to calculate displacement boundary conditions.
Abstract: Finite element analysis is an application of the direct stiffness method of structural analysis; the steps of the method are as follows: 1. Conceive of a prototype embracing the important feature of actual structure. Subdivide it into nodal points and elements. 2. Form the matrix connecting displacements and forces at each nodal point of element (element stiffness matrices). 3. Assemble the structural stiffness matrix from each element stiffness matrix. 4. Introduce applied forces and write the equations of equilibrium. 5. Introduce displacement boundary conditions. 6. Solve for displacements at nodal points. 7. From known displacement in each nodal point of an element, find element stresses. 8. Recycle for non-linear properties