Showing papers on "Direct stiffness method published in 1986"

A First Course in the Finite Element Method

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

Stiffness Matrix for Geometric Nonlinear Analysis

Abstract: A new stiffness matrix for the analysis of thin walled beams is derived. Starting from the principle of virtual displacements, an updated Lagrangian procedure for nonlinear analysis is developed. Inclusion of nonuniform torsion is accomplished through adoption of the principle of sectorial areas for cross‐sectional displacements. This requires incorporation of a warping degree of freedom in addition to the conventional six degrees of freedom at each end of the element. Problems encountered in the use of this and similar matrices in three‐dimensional analysis are described.

Theory and methods of structural analysis

Abstract: Exact Theory of Plane Deformation Simplified Theories of the Structural Member Constitutive Properties Stiffness and Flexibility Properties of the Structural Member - Geometrically Linear Theory Stiffness and Flexibility Properties of the Structure Member - Geometrically Non-linear Theory Transformations Effective Member Equations Governing Equations of Structural Analysis Static-Kinematic Properties of Structures Displacement (Stiffness) Method - Linear Theory Force (Flexibility) Method - Linear Theory Non-linear Analysis Linearized Stability Analysis Space Structures General Non- linear Theory of the Structural Member Element Matrix for Non- linear Spatial Analysis Index.

Compound Strip Method for Analysis of Plate Systems

Abstract: A finite strip method (FSM) is developed for the analysis of linear elastic flat plate systems that are continuous over, deflecting supports. The approach presented incorporates the effect of the support elements in a direct stiffness methodology. The stiffness contribution of the support elements have been derived and are given in the form of strip matrices, which are directly added to the plate strip stiffness matrix at the element level. This summation of plate and support stiffness contributions constitutes a substructure, which is termed a compound strip. The validity of the compound strip method is demonstrated in several illustrative problems, which include single and multipanel plates continuous over flexible and rigid beams and columns. The FSM and finite element method (FEM) compare favorably for displacement and moment.

Element stiffness computation on CDC CYBER 205 computer

Abstract: An algorithm is presented for the evaluation of the linear stiffness matrices on the CDC CYBER 205 computer. Discussion is focused on the organization of the computation and the mode of storage of the different arrays to take advantage of CYBER 205 pipeline (streaming) capability. The algorithm is based on partitioning the element stiffness matrix by rows and columns into nodal submatrices and generating simultaneously the coefficients in all the nodal partitions for a group of distinct elements. An assessment is made of the performance of the proposed algorithm for generating the stiffness matrices of shear-flexible shallow shell elements having 5 degrees of freedom per node. The nomenclature used is given in Appendix II.

Compound Strip Method for Continuous Sector Plates

Abstract: A finite strip method is developed for the analysis of linear elastic curved plate systems that are continuous over non‐rigid supports. This approach incorporates the effect of the support elements in a direct stiffness methodology. The stiffness contribution of the support elements are derived and given as strip stiffness matrices which are combined with the plate strip matrices at the element level prior to assembly. This summation of plate and support stiffness contributions forms a substructure which is termed a compound strip. The compound strip methodology may be used readily to enhance computer programs based on traditional finite strip procedures. The validity of the compound strip method is demonstrated in several illustrative analyses. The compound strip methodology compares favorably to the finite element method for displacement and moment.

Stiffness Analysis of Thin‐Walled Girders

Abstract: For complex arrangements of thin-walled girders a generalized method of analysis is required that is suitable for rapid solution by computer. A method of elastic analysis based on a stiffness approach, is proposed which is equally applicable to straight and curved girders of thin-walled cross section. For straight girders, the member stiffness matrix is derived explicitly by inverting the appropriate member flexibility matrix and considering equilibrium of the member ends. The member stiffness matrix for curved girders is obtained by the same method but in numerical form due to the extreme complexity of the various terms. By transforming individual member stiffness matrices into system co-ordinates, a system stiffness matrix is established, whose inversion leads directly to a global solution for the structure. In this type of analysis uniformly distributed flexural and torsional loads must be applied as fixed-ends loads at the nodes. These are derived for both straight and curved girders. In addition, the distribution of bimoment is presented graphically for a variety of typical beam cross sections.

Numerical analysis of flexible retaining walls ---computer applications in geotechnical engineering. proceedings of a symposium, university of birmingham, england, april 1986

Abstract: A numerical method for analysing the behaviour of flexible retaining walls which allows soil-structure interaction to be modelled is presented. The method differs significantly from the traditional subgrade reaction approach in the ways that the soil stiffness and earth pressure limits are modelled. Three stiffness matrices are used in the analysis. One matrix represents the wall in bending while the other two represent the soil on each side of the wall. Each soil stiffness matrix is assembled using pre-calculated flexibility matrices obtained from finite element computations for elastic soil blocks. Earth pressure limits are determined from consideration of forces applied to the soil which allows the known effect of soil arching to be modelled. This occasionally permits pressures to locally exceed active and passive limits. The analysis has been incorporated into a computer program which is sufficiently economic and simple to be used as part of the general design process. Examples of its use are given.(a) for the covering abstract of the symposium see IRRD 815626.

Rotational Stiffness and Damping Coefficients of Fluid Film in a Finite Cylindrical Bearing

Abstract: The dynamic behavior of flexible rotors supported on fluid-film bearings are studied including the rotational stiffness and damping coefficients of the oil film. The rotational stiffness and damping coefficients have been evaluated for a finite cylindrical fluid-film bearing by solving the appropriate Reynolds equation for the oil film, using finite difference method. The resulting critical speeds and the unbalance response for a single-disk flexible-rotor system modelled by finite element method are compared with those which were obtained using a short bearing approximation. Presented at the 40th Annual Meeting in Las Vegas, Nevada May 6–9, 1985

In plane finite element analysis of arches

Abstract: A finite element procedure is described for the analysis of the in plane behaviour of frames composed of curved or straight members. A higher order quintic element is used to represent the element displacement fields, and its accuracy is compared with that of a cubic element. The virtual work equation is used to obtain the element stiffness matrix and the load vector. These are then assembled to form the structure stiffness equations, which are solved in the usual way. Structures with slope discontinuities are analysed by introducing a curved transition element at the discontinuity. The accuracy of this procedure is investigated (a).

A consistent formulation of trusses and non-warping beams in linearized finite displacements

Abstract: The explicit stiffness equations and the corresponding differential equations are formulated for a truss and a non-warping beam in the framework of the linearized finite displacement theory. The derivation is consistent with the theory of thin-walled members. One main objective is to show the exact correspondence between the stiffness equations and the differential equations with their boundary conditions. An alternative scheme of deriving the stiffness matrices is given as the direct modification of the already obtained matrix of thin-walled members.

Nonlinear analysis of thin-walled structures by a coupled finite element method

Abstract: This paper presents a finite element method which enables to analyze a large displacement behavior of elasto-plastic thin-walled structures which fail by overall, local or interactive instability. An incremental equilibrium equation for a thin-walled beam-column is derived in a stiffness matrix form by using a moving element coordinate system and an incremental variational principle. So called multipoint constraints technique is used to connect plate elements with a beam element at the coupling nodal point in the cross section. Numerical results of the present method are compared with the exact solutions and experimental results. Validity and efficiency of the present method are confirmed.

SOLVER — An Interactive Structures Analyzer for Microcomputers

Abstract: AN interactive, menu-driven computer program, SOLVER, for analyzing 2- or 3-dimensional skeletal structures such as continuous beams, trusses, frames and grids, was developed for use on microcomputers. Using the direct stiffness method in conjunction with recommendations of the 1982 National Design Specification of Wood Construction, the program determines stresses, node displacements, reactions, member deflections and combined stress interaction analysis of wood trusses and frames. The program offers several options for the analysis of member capacity as affected by strength property variations, loads, and joint fixity conditions. Options for tabulated output range from a summary report of structural response to detailed information of member forces, stresses, deflections and computational results from key steps in matrix structural analysis.

Analysis of thin-walled members by finite element-transfer matrix method

Abstract: The combined finite element-transfer matrix method is applied to the linear and nonlinear problems of thin-walled members. The transfer matrix is derived from the tangent stiffness matrix used in the FEM. To deal with complex structures the transfer matrix for the substructure, into which thin-walled members is devided, is introduced. In this method, procedures used in the FEM based on load increment are employed except for the estimation of approximate displacements for each specified increment load. Approximate displacement are estimated by the transfer matrix procedure. Some numerical examples presented in this paper show that for long thin-walled members, this method can be successfully applied to the linear and nonlinear problems by reducing the size of the matrix relative to less than that obtained by the FEM.

Development of computer program NAS3D using Vector processing for geometric nonlinear analysis of structures

Abstract: An algorithm for vectorized computation of stiffness matrices of an 8 noded isoparametric hexahedron element for geometric nonlinear analysis was developed. This was used in conjunction with the earlier 2-D program GAMNAS to develop the new program NAS3D for geometric nonlinear analysis. A conventional, modified Newton-Raphson process is used for the nonlinear analysis. New schemes for the computation of stiffness and strain energy release rates is presented. The organization the program is explained and some results on four sample problems are given. The study of CPU times showed that savings by a factor of 11 to 13 were achieved when vectorized computation was used for the stiffness instead of the conventional scalar one. Finally, the scheme of inputting data is explained.

Elastic-Plastic Constitutive Equation Using Non-Orthogonal Curvilinear Coordinates and its Application in Numerical Methods

Abstract: The elastic-plastic constitutive equations in solid mechanics are derived using arbitrary non-orthogonal curvilinear coordinates. These general equations, obtained in this paper, are applicable in both finite element and finite difference methods for dealing with irregular domains. Some examples are illustrated.