Showing papers on "Direct stiffness method published in 1983"

Coupled vibrations of beams—an exact dynamic element stiffness matrix

Abstract: A uniform linearly elastic beam element with non-coinciding centres of geometry, shear and mass is studied under stationary harmonic end excitation. The Euler-Bernoulli-Saint Venant theory is applied. Thus the effect of warping is not taken into account. The frequency-dependent 12 × 12 element stiffness matrix is established by use of an exact method. The translational and rotational displacement functions are represented as sums (real) of complex exponential terms where the complex exponents are numerically found. Built-up structures containing beam elements of the described type can be analysed with ease and certainty using existing library subroutines.

A new approach to the dynamic analysis of structures using fixed frequency dynamic stiffness matrices

Abstract: The dynamic analysis of structures by the standard finite element method introduces additional inaccuracies into the solution which are not present when the method is used for static analyses. These inaccuracies can arise from two sources: (i) the element formulation and (ii) the reduction of the size of the matrices by a static condensation (i.e. using the Guyan method1,2). The errors in both cases are caused by neglecting frequency-dependent terms in the functions relating the displacements at any point in the structure to the displacements at certain fixed points (i.e. nodes in the element formulation and ‘masters’ in the condensation). A new method of solution is proposed in this paper in which the frequency-dependent terms are retained implicitly by using dynamic stiffness matrices defined at a number of fixed frequencies. The dynamic stiffness matrices may be condensed efficiently to a relatively small number of master degrees-of-freedom using a front solution algorithm. The final stage in the solution uses these matrices to synthesize a high-order eigenvalue problem. A method of solving such an eigenvalue problem, of arbitrary order, is described in a separate paper.3 Numerical examples are given to show the accuracy and efficiency of the proposed method compared with conventional methods of solution.

Elastic Critical Loads on Multistory Rigid Frames

Abstract: A direct matrix method for the calculation of the elastic critical loads on rigid frames is presented. The effect of the secondary axial forces in columns is readily included. The procedure leads to a linear eigenvalue problem where the smallest root from a product matrix represents the critical load. The computational aspects of the method are considered, and the analysis is reduced to the setting up of two main matrixes. The first matrix is directly evaluated from the geometrical and loading properties, and the second is the linear elastic stiffness matrix under a fictitious system of transverse forces. Economy in the setting up of the stiffness matrix is achieved by treating individual columns as substructures. The method is applied to three example frames, and the numerical results are compared with those obtained from stability-function solutions.

Finite Element Solutions of Deep Beams

Abstract: The finite element solutions of deep beams are presented. Two types of deep beam elements are introduced. The first element simulates the exact analytical solution of the deep beam of rectangular cross section. The element stiffness matrix is developed by solving the plane stress elasticity equations with the selection of the proper boundary conditions. The second element simulates the approximate solution of the deep beam of arbitrary cross section as it assumes a cubic normal strain distribution along the beam cross section. The element stiffness matrix is derived by applying the principle of virtual displacements of the system. The two elements are found to yield similar results for beam aspect ratios which are greater than 1.0. The deep beam element solutions were compared to the conventional engineering theory of beam solutions for simple (cantilever) structures and complex (turbine generator pedestal) structures; the results of the two methods of solution were found to differ significantly.

Number of non‐zero blocks in the stiffness matrix of the finite element network

Abstract: In Reference 1 a series of general formulae for evaluating the total number of non-zero blocks in the stiffness matrix, both of two-dimensional and three-dimensional (arbitrary polygonal/polyhedral) finite element networks, was presented. In this paper, an alternative, more direct and efficient method for deriving such formulae is presented. In addition, this new method can be used for more complex situations. In fact, it may be used for almost all kinds of finite element networks.

Research on the control of large space structures

Abstract: The research effort on the control of large space structures at the University of Houston has concentrated on the mathematical theory of finite-element models; identification of the mass, damping, and stiffness matrix; assignment of damping to structures; and decoupling of structure dynamics. The objective of the work has been and will continue to be the development of efficient numerical algorithms for analysis, control, and identification of large space structures. The major consideration in the development of the algorithms has been the large number of equations that must be handled by the algorithm as well as sensitivity of the algorithms to numerical errors.

Effective method of storing global stiffness matrices built around regular nodal grids

Abstract: All fundamental stages of the computer process, which accomplishes stress--strain-state computations by the finite-element method (FEM), are related to the formation and processing of a global stiffness matrix (GSM). Solution of practical problems reduces to GSM of extremely high orders; the method employed to store the GSM in the memory of a computer determines to a significant degree the efficiency of the corresponding software. At the present time, the method of storing GSM in the form of the upper or lower symmetric portion of the band has come into the most widespread use [1-7].

On the Use of Certain Properties of Stiffness Matrices in Finite Element Methods

Abstract: The stiffness matrices encountered in the analysis of elastic systems using finite element methods possess certain properties the use of which in the solution procedures reduces the computer time and increases the accuracy of the results. The presentation places emphasis on the application of the theory to skeletal systems using one dimensional elements. Reference, however, has been made to two and three dimensional finite elements.