Showing papers on "Direct stiffness method published in 1968"

Finite Strip Method Analysis of Elastic Slabs

Abstract: Rectangular elastic isotropic and orthotropic slabs for a variety of plate loadings and boundary conditions are analyzed by the finite strip method, in which a combination of trigonometric and hyperbolic series satisfying the boundary conditions in the longitudinal direction is used together with a simple polynomial function in the transverse direction. By using variational principles, the stiffness and load matrices of each strip are readily derived. The stiffness and load matrices for the whole slab are then assembled and solved to give the displacements, from which the moments can be calculated. Excellent accuracy has been achieved for various numerical examples using only a relatively small number of discrete computational variables. Beam-slab structures such as bridges should be readily amenable to the application of this method.

Matrix bandwidth minimization

Abstract: In the analysis of a structural problem by the finite element method, a large order stiffness matrix is created which describes mathematically the inter-connectivity of the system The structure is defined in three dimensional space by discrete points called nodes Each node is represented by its coordinates in the space The nodes are then connected by the various finite elements that the particular computer program may utilize (ie, bar members, rectangular or triangular panels, three dimensional tetrahedrons, etc)

Solution of nonlinear problems of elastoplasticity by finite elementmethod.

Abstract: By using the finite element and direct stiffness method, piecewise linear load deflection relationship, and step-by-step computational algorithm, a general procedure for the solution of problems of elastoplasticity is developed. The change of material properties at each step is expressed in the general differential form of Prandtl-Reuss equations of the theory of plastic deformations, with special treatment of the plane strain case. The effect of initial stresses is also considered when the rotations at each step are not negligible, although the strains are assumed small compared to unity. Application of the principle of virtual work to define the equilibrium of an element subject to initial and additional stresses yields a geometrical stiffness matrix which satisfies macroscopic equilibrium requirements. Computational difficulties arising from the repeated solution of large numbers of linear simultaneous equations are overcome by using an iteration method with an over-relaxation factor. The accuracy of the propagated solution is improved by the application of half-step procedure, which is a special case of the Runge-Kutta method.

Curved Beam Stiffness Coefficients

Abstract: A three-dimensional stiffness coefficient matrix for constant cross section curved beams is derived. This matrix relates the beam end point rotations and translations to the internal moments, torques, shears, and axial forces. The stiffness matrix is transformed into a three-dimensional spatial coordinate system by means of a coordinate transformation matrix. The transformation matrix is derived by rotating the orthogonal coordinate axis through three consecutive rotations. Finally an example problem, simulating a spherical dome is worked to demonstrate the application of the curved beam stiffness matrix coefficients.

Stability analysis of shells of revolution by the finite-element method.

Abstract: An extension of the finite-element displacement method to the analysis of linear bifurcation buckling of general shells of revolution under static axisymmetric loading is presented. A systematic procedure for the formulation of the problem is based upon the criterion that the condition for neutral stability of a system is the vanishing of the second variation of the total potential energy from the stable equilibrium state to the perturbed bifurcation state; this results in an eigenvalue problem. For solution, the shell is discretized into either a series of conical frusta or of frusta with meridional curvature. The prebuckling equilibrium solution is axisymmetric, but the perturbation-displacement field within each element is represented by Fourier circumferential components of the generalized displacements which are defined at the nodal circles. The present formulation is applied to a number of shells of revolution with arid without meridional curvature, and comparisons are made with other theoretical and available experimental results.

Dynamic Analysis of Shells Using Doubly-Curved Finite Elements

Abstract: : This paper discusses the theoretical development of a computer program for the static and dynamic analysis of general shell structures. The theory is based on the finite element concept and uses both the generalized finite element method and the direct stiffness method to form the pertinent equations. The treatment of shell surface geometry, the displacement functions and elemental degrees of freedom, and the modification of the generalized stiffness method required for the implementation of the triangular element are described. The correlation of both theoretical and experimental results with those obtained by the present method are shown along with idealizations required for accurate results. Static and dynamic solution results are compared,

Explicit expressions for triangular torus element stiffness matrix.

Abstract: Explicit expressions for stiffness matrix of triangular torus elements associated with linear displacement fields and generalized Hookian material

Use of Fourier series in the finite element method.

Abstract: P IAN1'2 has demonstrated the use of the energy method in deriving element stiffness matrices in connection with the displacement method of structural analysis. His procedure is based on the representation of an element displacement function in terms of m undetermined coefficients where the number m may be larger than the number of generalized displacements n.2 When m is larger than n, it is possible to satisfy not only stress equilibrium and boundary displacement continuity but also to maintain slope continuity along the normal directions of the element edges. Having the element displacement function, the total energy of the element! may be evaluated, and then the condition of the minimum potential energy yields the stiffness matrix. The aforementioned procedure is illustrated in Ref. 2 by determining the stiffness matrix for a rectangular plate under plane stress conditions. The displacement function is chosen in the form of a power series as follows : u = ai +

Structural analysis by matrix decomposition

Abstract: A physical interpretation is derived for the Choleski decomposition method as applied to structural analysis. Both the stiffness (displacement) and flexibility (force) methods of structural analysis are treated. The interpretation is shown to be valuable in providing physically meaningful upper and lower bounds on the elements of the decomposed diagonal matrix. These bounds are useful in error analysis of structural computations on a digital computer.

Stability of Spaceframes

Abstract: Nonlinear equations governing the behavior of spaceframes whose members undergo finite deflections and moderate chord rotations are presented. The beam-column effect including the effect of flexural shortening is taken into account, while twist due to torsion is assumed to be small. The method of solution is based upon the Newton method, and the corresponding tangent stiffness matrix is given. The solution for a particular loading is obtained by applying successive corrections to the approximate solution by means of the tangent stiffness matrix. A tangent stiffness matrix which is not positive, definitely implies that the potential energy of the corresponding equilibrium configuration is not at a minimum, and thus the configuration is unstable. Therefore, the lowest value of the load for which the determinant of the tangent stiffness matrix vanishes gives the critical buckling load for the structure. The formulation may be used for predicting the buckling loads and the load-deformation history for arches or shell-like spaceframes. Buckling of a shallow arch is investigated as an illustration. The first two branching points on the load-deflection curve, indicating the asymmetric modes of buckling, are determined.

A new finite element model for solid continua

Abstract: : The proposed finite element model is based on separate assumptions of interior and interelement displacements and on the assumed boundary tractions of each individual element. The associated variational functional for this model is presented. This method has the same merits of the assumed stress method in that a compatibility displacement function at the interelement boundary can be easily constructed. In the plate bending problems, the matrix to be inverted to obtain the element stiffness matrix by the present method is, in general, of smaller order than that of the assumed stress method when the same order of approximation is used in both methods. (Author)

Linear and Nonlinear Analysis of Shells of Revolution with Asymmetrical Stiffness Properties

Abstract: : In this paper, the authors apply an extension of the direct stiffness method of structural analysis to the linear and nonlinear analysis of shells of revolution under arbitrary static loading with variable thickness properties in the circumferential direction and the meridional direction. The primary difference between this research and previously published research is that all the Fourier harmonics in the circumferential direction are now coupled. The thickness variation in the circumferential direction yields an 8N x 8N element stiffness matrix in which N is the number of harmonics. The resulting computer program is used to conduct a linear, nonlinear, and stability analysis of the Apollo aft heat shield.

Embankment analysis and field correlation

Abstract: EXPERIMENTAL RESULTS FROM A PROGRAM TO DEVELOP ACCURATE AND RELIABLE INSTRUMENTATIONS FOR STRESS AND DEFORMATION MEASUREMENTS IN HIGHER EMBANKMENTS ARE NUMERICALLY ANALYZED FOR A PARTICULAR EMBANKMENT TO COMPARE THE RESULTS WITH THE FIELD MEASUREMENTS. THIS COMPARISON PERMITS AN ASSESSMENT OF THE ACCURACY OF THE MATHEMATICAL MODEL AND PREDICTION TECHNIQUE. THE ANALYSIS IS BASED ON A PLANE STRAIN SYSTEM WITH LINEARLY ELASTIC ISOTROPIC MATERIALS THAT IS CONSTRUCTED AND LOADED INCREMENTALLY. THE FINITE ELEMENT METHOD IS EMPLOYED TO OBTAIN STRESSES AND DISPLACEMENTS IN THIS TYPICAL CROSS-SECTION. THE CONTINUOUS SYSTEM IS IDEALIZED AS AN ASSEMBLAGE OF SMALLER, BUT FINITE, ELEMENTS CONNECTED AT A DISCREET NUMBER OF POINTS CALLED NODES. THE BASIS FOR THE METHOD IS A CONSISTENTLY DERIVED STIFFNESS FOR EACH OF THESE ELEMENTS, THE STIFFNESS BEING A RELATIONSHIP BETWEEN THE GENERALIZED FORCES AND THE DISPLACEMENTS AT THE NODES OF EACH ELEMENT. STIFFNESS OF THE ELEMENTS USED IS PREDICTED ON A MATERIAL POSSESSING LINEAR ELASTIC PROPERTIES. EXAMINATION OF THE ANALYTICAL RESULTS AND COMPARISON WITH THE FIELD DATA INDICATE THAT THE MATHEMATICAL MODEL CAN LEAD TO QUALITATIVE PREDICTIONS. THE WEAKEST FEATURE OF THE MATHEMATICAL MODEL IS THE MATERIAL CHARACTERIZATION. IT IS RECOMMENDED THAT: (1) ESTIMATES FOR STRESS AND DISPLACEMENT FIELDS SHOULD BE OBTAINED PRIOR TO PLACEMENT OF INSTRUMENTATION, (2) LATERAL PRESSURES SHOULD BE MEASURED AS WELL AS VERTICAL STRESSES, (3) VERTICAL PRESSURE CELLS MIGHT BE USED TO CALIBRATE THE CELLS MEASURING HORIZONTAL STRESSES, AND (4) AN ACCURATE DETERMINATION OF THE MATERIAL PROPERTIES OF THE EARTH FILL MUST BE MADE WHEN INSTRUMENTING AN EMBANKMENT.

Matrix Methods of Aerospace Structural Analysis. Invited Paper

Abstract: : The future development of key areas in the field of matrix structural analysis is considered. The relationship of these areas to the overall design problem is established, and sources of past difficulties within the areas are identified. Avenues of approach to the difficulties are suggested and illustrated by reference to recently conducted research. In the area of modelling, the triangular membrane element and the bar-panel idealization are evaluated. A method of improving results obtained from the analysis of models composed of triangular membrane elements is presented. The loss of numerical accuracy in linear structural analysis is considered and methods of improving accuracy are described. Problems in the field of nonlinear structural analysis are reviewed. The possibility of an interactive computational approach to the synthesis of complex structures is suggested, and an interactive graphics approach to the synthesis of a structure subjected to steady-state vibration is demonstrated.

Elastic-Plastic Analysis of Plane Framed Structure by Means of the Matrix Methods

Abstract: In the present paper, the matrix methods are developed for elastic-plastic analysis of plane framed structures subjected to proportional loading on the basis of the plastic hinge concept : After the bending moment at a joint reaches the fully plastic moment, the joint behaves as a hinge for the further increase of load. In the case of the force method, the number of redundant forces decreases corresponding to the formation of plastic hinges, and the flexibility matrix for the structure should be modified. The structure collapses when any redundant force cannot be eliminated for the new formation of a plastic hinge. In the case of the displacement method, it is easy to modify the stiffness matrix of the structure by the use of the member stiffness matrix corresponding to the formation of plastic hinges. The structure collapses when the structure stiffness matrix ceases to be positive definite.The present method can easily be modified to be effective for structures which may yield by axial forces. For a structure subjected to a prescribed loading program, the stress analysis may be conducted by the present method by taking unloading process of plasticity into account.

Structural analysis by matrix methods for forces not at the coordinates

Abstract: The analysis of structures with forces not at the coordinates is developed and demonstrated for both the flexibility and stiffness methods. In the flexibility method a “free coordinate state” is defined in a superposition of element forces. The free coordinate state element displacements δ i 0 are computed from Betti's Law using the displacement configurations associated with the columns of the element flexibility matrices. In the stiffness method a “fixed coordinate state” is defined in a superposition of element displacements. The fixed coordinate state element forces P i 0 are computed from Betti's Law using the displacement configurations associated with the columns of the element stiffness matrices. It is shown that the problems of “thermal expansion” and “lack of fit” are treated by the same procedure as that used for forces not at the coordinates.