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# Direct stiffness method

About: Direct stiffness method is a(n) research topic. Over the lifetime, 2584 publication(s) have been published within this topic receiving 53131 citation(s).

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01 Jan 1986

TL;DR: In this paper, Galerkin's Stiffness matrix is used to measure the stiffness of a bar in a 3D-dimensional space using a 3-dimensional truss transformation matrix.

Abstract: 1 INTRODUCTION Brief History Introduction to Matrix Notation Role of the Computer General Steps of the Finite Element Method Applications of the Finite Element Method Advantages of the Finite Element Method Computer Programs for the Finite Element Method 2 INTRODUCTION TO THE STIFFNESS (DISPLACEMENT) METHOD Definition of the Stiffness Matrix Derivation of the Stiffness Matrix for a Spring Element Example of a Spring Assemblage Assembling the Total Stiffness Matrix by Superposition (Direct Stiffness Method) Boundary Conditions Potential Energy Approach to Derive Spring Element Equations 3 DEVELOPMENT OF TRUSS EQUATIONS Derivation of the Stiffness Matrix for a Bar Element in Local Coordinates Selecting Approximation Functions for Displacements Transformation of Vectors in Two Dimensions Global Stiffness Matrix for Bar Arbitrarily Oriented in the Plane Computation of Stress for a Bar in the x-y Plane Solution of a Plane Truss Transformation Matrix and Stiffness Matrix for a Bar in Three-Dimensional Space Use of Symmetry in Structure Inclined, or Skewed, Supports Potential Energy Approach to Derive Bar Element Equations Comparison of Finite Element Solution to Exact Solution for Bar Galerkin's Residual Method and Its Use to Derive the One-Dimensional Bar Element Equations Other Residual Methods and Their Application to a One-Dimensional Bar Problem Flowchart for Solutions of Three-Dimensional Truss Problems Computer Program Assisted Step-by-Step Solution for Truss Problem 4 DEVELOPMENT OF BEAM EQUATIONS Beam Stiffness Example of Assemblage of Beam Stiffness Matrices Examples of Beam Analysis Using the Direct Stiffness Method Distribution Loading Comparison of the Finite Element Solution to the Exact Solution for a Beam Beam Element with Nodal Hinge Potential Energy Approach to Derive Beam Element Equations Galerkin's Method for Deriving Beam Element Equations 5 FRAME AND GRID EQUATIONS Two-Dimensional Arbitrarily Oriented Beam Element Rigid Plane Frame Examples Inclined or Skewed Supports - Frame Element Grid Equations Beam Element Arbitrarily Oriented in Space Concept of Substructure Analysis 6 DEVELOPMENT OF THE PLANE STRESS AND STRAIN STIFFNESS EQUATIONS Basic Concepts of Plane Stress and Plane Strain Derivation of the Constant-Strain Triangular Element Stiffness Matrix and Equations Treatment of Body and Surface Forces Explicit Expression for the Constant-Strain Triangle Stiffness Matrix Finite Element Solution of a Plane Stress Problem Rectangular Plane Element (Bilinear Rectangle, Q4) 7 PRACTICAL CONSIDERATIONS IN MODELING: INTERPRETING RESULTS AND EXAMPELS OF PLANE STRESS/STRAIN ANALYSIS Finite Element Modeling Equilibrium and Compatibility of Finite Element Results Convergence of Solution Interpretation of Stresses Static Condensation Flowchart for the Solution of Plane Stress-Strain Problems Computer Program Assisted Step-by-Step Solution, Other Models, and Results for Plane Stress-Strain Problems 8 DEVELOPMENT OF THE LINEAR-STRAIN TRAINGLE EQUATIONS Derivation of the Linear-Strain Triangular Element Stiffness Matrix and Equations Example of LST Stiffness Determination Comparison of Elements 9 AXISYMMETRIC ELEMENTS Derivation of the Stiffness Matrix Solution of an Axisymmetric Pressure Vessel Applications of Axisymmetric Elements 10 ISOPARAMETRIC FORMULATION Isoparametric Formulation of the Bar Element Stiffness Matrix Isoparametric Formulation of the Okabe Quadrilateral Element Stiffness Matrix Newton-Cotes and Gaussian Quadrature Evaluation of the Stiffness Matrix and Stress Matrix by Gaussian Quadrature Higher-Order Shape Functions 11 THREE-DIMENSIONAL STRESS ANALYSIS Three-Dimensional Stress and Strain Tetrahedral Element Isoparametric Formulation 12 PLATE BENDING ELEMENT Basic Concepts of Plate Bending Derivation of a Plate Bending Element Stiffness Matrix and Equations Some Plate Element Numerical Comparisons Computer Solutions for Plate Bending Problems 13 HEAT TRANSFER AND MASS TRANSPORT Derivation of the Basic Differential Equation Heat Transfer with Convection Typical Units Thermal Conductivities K and Heat-Transfer Coefficients, h One-Dimensional Finite Element Formulation Using a Variational Method Two-Dimensional Finite Element Formulation Line or Point Sources Three-Dimensional Heat Transfer by the Finite Element Method One-Dimensional Heat Transfer with Mass Transport Finite Element Formulation of Heat Transfer with Mass Transport by Galerkin's Method Flowchart and Examples of a Heat-Transfer Program 14 FLUID FLOW IN POROUS MEDIA AND THROUGH HYDRAULIC NETWORKS AND ELECTRICAL NETWORKS AND ELECTROSTATICS Derivation of the Basic Differential Equations One-Dimensional Finite Element Formulation Two-Dimensional Finite Element Formulation Flowchart and Example of a Fluid-Flow Program Electrical Networks Electrostatics 15 THERMAL STRESS Formulation of the Thermal Stress Problem and Examples 16 STRUCTURAL DYNAMICS AND TIME-DEPENDENT HEAT TRANSFER Dynamics of a Spring-Mass System Direct Derivation of the Bar Element Equations Numerical Integration in Time Natural Frequencies of a One-Dimensional Bar Time-Dependent One-Dimensional Bar Analysis Beam Element Mass Matrices and Natural Frequencies Truss, Plane Frame, Plane Stress, Plane Strain, Axisymmetric, and Solid Element Mass Matrices Time-Dependent Heat-Transfer Computer Program Example Solutions for Structural Dynamics APPENDIX A - MATRIX ALGEBRA Definition of a Matrix Matrix Operations Cofactor of Adjoint Method to Determine the Inverse of a Matrix Inverse of a Matrix by Row Reduction Properties of Stiffness Matrices APPENDIX B - METHODS FOR SOLUTION OF SIMULTANEOUS LINEAR EQUATIONS Introduction General Form of the Equations Uniqueness, Nonuniqueness, and Nonexistence of Solution Methods for Solving Linear Algebraic Equations Banded-Symmetric Matrices, Bandwidth, Skyline, and Wavefront Methods APPENDIX C - EQUATIONS FOR ELASTICITY THEORY Introduction Differential Equations of Equilibrium Strain/Displacement and Compatibility Equations Stress-Strain Relationships APPENDIX D - EQUIVALENT NODAL FORCES APPENDIX E - PRINCIPLE OF VIRTUAL WORK APPENDIX F - PROPERTIES OF STRUCTURAL STEEL AND ALUMINUM SHAPES ANSWERS TO SELECTED PROBLEMS INDEX

921 citations

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TL;DR: In this article, the Haskell-Thompson transfer matrix method is used to derive layer stiffness matrices which may be interpreted and applied in the same way as stiffness matrix in conventional structural analysis, and the exact expressions are given for the matrices, as well as approximations for thin layers.

Abstract: The Haskell-Thompson transfer matrix method is used to derive layer stiffness matrices which may be interpreted and applied in the same way as stiffness matrices in conventional structural analysis These layer stiffness matrices have several advantages over the more usual transfer matrices: (1) they are symmetric; (2) fewer operations are required for analysis; (3) there is an easier treatment of multiple loadings; (4) substructuring techniques are readily applicable; and (5) asymptotic expressions follow naturally from the expressions (very thick layers; high frequencies, etc) While the technique presented is not more powerful than the original Haskell-Thompson scheme, it is nevertheless an elegant complement to it The exact expressions are given for the matrices, as well as approximations for thin layers Also, simple examples of application are presented to illustrate the use of the method

660 citations

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TL;DR: In this paper, the structural mechanics of assemblies of bars and pinjoints, particularly where they are simultaneously statically and kinematically indeterminate, are investigated, and an algorithm is set up which determines the rank of the matrix and the bases for the four subspaces.

Abstract: The paper is concerned with the structural mechanics of assemblies of bars and pinjoints, particularly where they are simultaneously statically and kinematically indeterminate. The physical significance of the four linear-algebraic vector subspaces of the equilibrium matrix is examined, and an algorithm is set up which determines the rank of the matrix and the bases for the four subspaces. In particular, this algorithm gives full details of any states of self-stress and modes of inextensional deformation which an assembly may possess. A scheme is devised for the segregation of inextensional modes into rigid-body modes (up to six of these may be allowed by the foundation constraints) and “internal” mechanisms. In some circumstances a state of self-stress may impart first-order stiffness to an inextensional mode. A matrix method for detecting this effect is devised, and it is shown that if there is no state of self-stress which imparts first-order stiffness to a given mode, then that mode can undergo rather large distortion which involves either zero change in length of the bars or, possibly, changes in length of third or higher order in the displacements. The significance of negative stiffness, as indicated by the matrix method, is discussed. The paper contains simple examples which illustrate all of the main points of the work.

603 citations

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TL;DR: In this article, the generalized finite element method (GFEM) was used to solve complex, 3D structural mechanics problems and the performance of the GFEM and FEM in the solution of a 3D elasticity problem was compared.

Abstract: The present paper summarizes the generalized finite element method formulation and demonstrates some of its advantages over traditional finite element methods to solve complex, three-dimensional (3D) structural mechanics problems. The structure of the stiffness matrix in the GFEM is compared to the corresponding FEM matrix. The performance of the GFEM and FEM in the solution of a 3D elasticity problem is also compared. The construction of p-orthotropic approximations on tetrahedral meshes and the use of a-priori knowledge about the solution of elasticity equations in three-dimensions are also presented.

532 citations