Showing papers on "Direct stiffness method published in 1991"

VICONOPT: Program for Exact Vibration and Buckling Analysis or Design of Prismatic Plate Assemblies

Abstract: ASUMMARY is given of key features of the computer program VICONOPT,1 which covers prismatic assemblies of anisotropic plates exactly for buckling and vibration analysis and also for design subject to buckling constraints. Contents VICONOPT (VIpasa with CONstraints and OPTimization) is a 23,000 line Fortran 77 computer program that incorporates the earlier programs VIPASA2 and VICON.3 It covers any prismatic assembly of anisotropic plates and Fig. 1 shows typical cross sections. Each plate can carry any combination of NL, NT9 and Ns, the longitudinally invariant in-plane forces per unit length of plate edge shown in Fig. 1. VICONOPT performs analysis or optimum design. The analysis includes calculation of critical buckling load factors, or undamped natural frequencies, and mode shapes. VIPASA uses the stiffness matrix method based on exact flat plate theory with Winkler foundations. It also uses an algorithm that guarantees convergence on all required eigenvalues and permits the user to employ nested (to any level) substructuring very concisely and flexibly to reduce solution times, data preparation, and computer memory usage. The mode of buckling or vibration is assumed to vary sinusoidally in the longitudinal direction x> with the displacement amplitudes u, v, w, and ^ shown in Fig. 1 and with computations being repeated for a user specified set of half-waveleng ths X. Plate bending and membrane behaviors are uncoupled and the bending and in-plane stiffness matrices D and A are respectively fully populated and orthotropic, which treats balanced symmetric laminates. The global stiffness matrix becomes complex when anisotropy or shear loading are present, thus, increasing solution time. The nodal lines of zero displacement are straight and in the y direction if all plates are orthotropic with Ns = 0, and so satisfy simply supported end conditions. Otherwise, solutions only approximate such end conditions and become excessively conservative as A approaches L Dead load values of NL, NT9 and Ns are permitted for both buckling and vibration problems. In the former case they are additional to live load values that are factored until buckling occurs. Plate loadings may be given as data, although NL is usually calculated from the total longitudinal load on the panel or

Identification of a Nonproportional Damping Matrix from Incomplete Modal Information

Abstract: This work provides a method of modeling the damping matrix of a structure from incomplete experimental data combined with a reasonable representation of the mass and stiffness matrices developed by finite element methods and reduced by standard model reduction techniques. The proposed technique uses the reduced mass and stiffness matrices and the experimentally obtained eigenvalues and eigenvectors in a weighted least squares or a speudo-inverse scheme

Computation of Member Stiffness in Bolted Connections

Abstract: The member stiffness in a bolted connection has a direct influence upon safe design with regard to both static and fatigue loading, as well as in the prevention of separation in the connection This work provides a simple technique for computing the member stiffness in many types of bolted connections Finite element analyses are performed for joints having a range of materials and geometries, and the results are generalized by nondimensionalization An exponential expression for the stiffness is determined, and the results are compared with those of some of the techniques currently used

New displacement-based methods for optimal truss topology design

Abstract: Two alternate methods for maximum stiffness truss topology design are presented. The ground structure approach is used, and the problem is formulated in terms of displacements and bar areas. This large, nonconvex optimization problem can be solved by a simultaneous analysis and design approach. Alternatively, an equivalent, unconstrained, and convex problem in the displacements only can be formulated, and this problem can be solved by a nonsmooth, steepest descent algorithm. In both methods, the explicit solving of the equilibrium equations and the assembly of the global stiffness matrix are circumvented. A large number of examples have been studied, showing the attractive features of topology design as well as exposing interesting features of optimal topologies.

Consistent Frame Buckling Analysis by Finite Element Method

Abstract: Conventional element assembly process based on the initial configuration of structures does not necessarily imply the satisfaction of compatibility conditions in the deformed state. This may result in a violation to the rule that all physical relations should be specified for deformed structures in a buckling analysis. Based on an exact consideration of the interelement compatibility for structural members, one can derive a joint matrix for internally generated moments and a moment matrix for externally applied moments, in addition to the elastic and geometric stiffness matrices commonly used in nonlinear analysis. To highlight the linking between the fundamental mechanics equations and their finite element counterparts, the present derivation will be made only for planar frames that are allowed to buckle both in and out of plane. The validity of the present procedure is confirmed in the numerical examples.

Statically Indeterminate Structures

Abstract: Computation of Deflections. Work and Energy Theorems. Analysis of Statically Indeterminate Structures by the Force or Flexibility Method. Analysis of Statically Indeterminate Structures by the Displacement or Stiffness Method. Continuous Girders and Frames with Variable Moment of Inertia. Approximate and Practicable Methods. Elastic Arches, Members with Closed Ends, and Frames with Curved Members. Flexible Members. Appendix. General References. Index.

Stress analysis of flat plates with shear using explicit stiffness matrix

Abstract: An explicit expression for the stiffness matrix is worked out for a triangular plate bending element considering the effect of transverse shear deformation. The element has twelve nodes on the sides and four nodes internal to it. The formulation is displacement type and the use of area co-ordinates makes it possible to obtain the shape functions explicitly. Separate polynomials are assumed for transverse displacement and rotations. To obtain the element stiffness matrix no matrix inversion or numerical integration need be carried out and only a few matrix multiplications of low order are necessary. The element, which is initially of thirty five degrees of freedom, can be reduced to a thirty degrees of freedom one by condensation of the internal nodes. An interesting feature of the element developed is that the values of nodal moments computed at a node point, considering different elements surrounding the node, do not vary significantly. Thus the nodal moments can be obtained directly at node points. Also, the element does not give rise to any inconvenience like locking, even for very thin plates. The straightforward approach in formation of the element stiffness will cut down the storage space considerably and will also call for less CPU time, thus making the use of the element well suited to low capacity computers. A number of plate bending problems have been worked out using the present element for different thickness to side ratios and a comparison has been made with the available results. Good accuracy has been observed in all cases, even for a small number of elements.

A Hybrid Method for the Vibration Analysis of Rotor-Bearing Systems:

Abstract: The transfer matrix method together with a digital computer form the foundation of the dynamic analysis of rotor-bearing systems. The properties of each segment of the rotating shaft are expressed in simple matrix form and the overall dynamic behaviour is then obtained by successive multiplication of the element matrices. The main drawback associated with this method is the numerical instability in calculating natural frequencies for complex systems.The finite element method, on the other hand, uses the element stiffness and mass matrices to form the global equation of motion for the complete system. This avoids the numerical problems of the transfer matrix method at the expense of the computer memory requirements.The new method described in this paper combines the transfer matrix and finite element techniques to form a powerful algorithm for vibration analysis of rotor-bearing systems. It is shown that the accuracy improves significantly when the transfer matrix for each shaft segment is obtained from fi...

Implementation of radiation boundary conditions in the finite element analysis of electromagnetic wave propagation

Abstract: Radiation boundary conditions are formulated which permit the simulation of two-dimensional electromagnetic wave phenomena with the finite-element method using conventional elements over a bounded domain. Implementation of such boundary conditions preserves the symmetry of the global stiffness matrix with all the advantages that this implies, including economy of storage and solution. A number of wire-antenna systems have been modeled with this technique in a finite-element computer program called FEAST. The results demonstrate good agreement with published reference data. >

The dynamic stiffness matrix method in the analysis of rotating systems

Abstract: The dynamic stiffness method has been applied to the evaluation of the natural frequencies of rotating systems. To this purpose, a rotating “Rayleigh beam,” defined by adding the effect of the rotary inertia and the gyroscopic effects to the Bernoulli-Eider beam, has been formulated and its dynamic stiffness matrix is presented in this paper. The effects due to the presence of concentrated disks, as well as of elastic, isotropic supports, have been included in the formulation. The usual matrix assembly procedure is used in order to obtain the global dynamic stiffness matrix of the system. The natural frequencies of the system are determined by utilizing an iterative root searching technique. Numerical results, obtained for a rotor system taken from the literature on this subject, are presented. Presented at the 45th Annual Meeting in Denver, Colorado May 7–10, 1990

Vector finite-element analysis using IBM 3090–600e VF

Abstract: A finite-element computer program consists of four basic modules, namely, computation of element stiffness matrices, assembly of the global stiffness matrix, solution of the system of linear simultaneous equations, and calculation of stresses. Vector algorithms consisting of direct vector Fortran code and the Engineering and Scientific Subroutine Library (ESSL) routines are presented for these modules. Numerical investigations were conducted using the IBM 3090–600E VF computer. For element stiffness matrix calculation, using a vector length equal to the number of elements produced speed-up in the range of 3·2 to 3·7 over the corresponding scalar code. The vectorized global element stiffness matrix assembly was accomplished with the help of an INDEX array and the ESSL routine DAXPYI that resulted in the speed-up of 2·6 to 4·3. The ESSL routines DPBF and DPBS gave speed-up of 7·7 to 53·6 over a scalar Gaussian solver. The scalar-to-vector speed-up for the stress calculation ranges from 2·2 to 5·33. The speed-up of the totally vectorized program over the scalar program is from 7·4 to 51·9. As would be expected, the equation solution takes up the most computational effort in a finite-element analysis. For the models we analysed the equation solution consumed 93 per cent to almost 100 per cent of the total CPU time in the scalar computations. Vectorizing the equation solver only reduces the percentage of its share to the range of 63 per cent to 88 per cent. In a totally vectorized program, the share of the equation solver effort is 85 per cent to 97 per cent. For the models analysed, the additional speed-up accomplished by vectorizing the whole program over the one with a vectorized equation solver only was from 10 per cent to 40 per cent.

Anisoparametric Interpolation in Hybrid‐Stress Timoshenko Beam Element

Abstract: A hybrid‐stress Timoshenko beam element with consistent mass and geometric stiffness matrices is presented. It is demonstrated that the use of the hybrid‐stress model removes the necessity for a finite element shear‐correction factor, which can be determined through a deflection‐matching analysis at the formulation stage of two‐node anisoparametric displacement model elements. The element is assessed by static, dynamic, and stability test problems.

Identification of structural system parameters from time domain data : direct identification of mass, stiffness and damping parameters of a structure

Abstract: A method is presented for identifying the discrete linear dynamic model expressed by mass, stiffness and damping parameters of a mechanical structure. First, the characteristic matrix of the structure system is determined from the identified modal parameters. Secondary, the mass parameters are identified from the autocorrelation function of the measured responses and the ones calculated from modal parameters in the case of one sine-wave excitation. Consequently the stiffness and damping parameters are obtained from the identified mass parameters and the characteristic matrix. Finally, a numerical simulation example and an experimental example which is a cantilevered pipe structure are demonstrated in detail. It is shown that the present method is applicable for the identification problems of practical structures since the test data can be obtained without additional complications.

Modern Structural Analysis: The Matrix Method Approach

Abstract: Preliminary Considerations Nodal Variables - Stiffness and Flexibility Matrices of a Structure Element Variables - Total Stiffness Matrix of an Element Direct Stiffness Method Supplementary Procedures to the Stiffness Method Basic Element Variables The Equations of Equilibrium and the Equations of Compatibility of a Structure Computation of the Displacements of Statically determinate Framed Structures The Flexibility Method The Displacement or Stiffness Method Appendix A. The International System of Units Appendix B. End Actions in Fixed-At-Both-Ends Elements Appendix C. Vector Quantities Appendix D. Computation of the Total Stiffness Matrix of an Element and the Equivalent Actions to be Placed at the Nodes of a Structure Using Shape Functions.

Equivalent systems for inelastic analysis of prismatic and nonprismatic members

Abstract: This paper deals with the analysis of prismatic and nonprismatic members of any arbitrary variation in moment of inertia, and where the material is permitted to be stressed well beyond its elastic limit, thus causing the modulus of elasticity to vary along their length. The stress and deflection characteristics of such members are determined by the method of the equivalent systems, which permits to replace the original member of variable stiffness with one of uniform stiffness, whose elastic line is identical to the one of the original variable stiffness member. It is proven mathematically that the inelastic analysis of members with continuously varying moment of inertia and modulus of elasticity can be carried out by using equivalent linear systems of constant stiffness and applying known handbook formulas or methods of linear elementary mechanics. The member can be analyzed for both elastic and inelastic ranges, all the way to failure, thus permitting observation of progressive deterioration of the memb...

Stiffness Formulations of Planar Kinematics

Abstract: A simple theory using the stiffness concept is presented for linkage-motion analysis. The nonlinear incremental mechanism equations are established and their solution procedure is proposed. The kinematic system is first regarded as an instable elastic structure. The tangent stiffness equations are updated for current mechanism configurations. The obtained singular stiffness matrix is then used to predict incremental nodal displacements. The stress-associated deformation is designated as numerical error. The proposed stiffness method follows successively the mechanism motion in an incremental-iterative manner. Computed numerical examples show that the elastic deformation can easily be removed with a few iterations. The proposed stiffness approach to kinematics may be incorporated into existing finite-element computer program systems and can be applied to general three-dimensional mechanisms.

Including bolt bending stiffness in finite-element analysis of flanged connections

Abstract: The stiffness matrix of a spring constraint is derived that can restrict only rotation in a mesh of finite elements having no rotation as a degree of freedom It is connected to two nodes of the structure and does not increase the total number of degrees of freedom To validate the stiffness matrix, a simple 2-D numerical example is presented which is in excellent agreement with theoretical results Integral grey cast iron flanges (BS4504: 1969) are analysed including and not including the bending stiffness of the bolts The analysis shows that neglecting the bending stiffness of the bolts results in up to 35% error of overestimating the maximum stress in large-diameter flanges

Minimum Required Lateral Stiffness of Structures

Abstract: In this paper an approximate method is presented to calculate the minimum lateral stiffness required to insure global structural stability. The technique is based on setting the critical stability load of a structure equal to the critical buckling load of the primary member. It uses an approximation for the change in potential energy due to lateral displacement of the column and requires that the stiffness yield a positive energy. Sixteen examples of simple structures are analyzed, four of which have analytical solutions.

A Microcomputer-aided Three-dimensional Finite Element Method. A Tandem-type Substructuring Method Using LDU Decomposition Method of Stiffness Submatrices.

Abstract: This paper presents a tandem-type substructuring method using a LDU decomposition method of the small-size stiffness submatrices for developing an efficient calculation method of a microcomputer/personal computer-aided finite element method (FEM). This method needs to utilize the disk memory storage unit such as floppy and hard disks in order to store the data of the stiffness submatrices and/or their decomposed ones. Especially, the stiffness submatrices of the storage unit are divided from a substructure stiffness matrix within the limit of the machine memory capacity to be used. The data of the LDU submatrices are placed in the data set of the stiffness submatrices in order to save storage space in the unit. The FEM using the method is applied to an efficient analysis of a three-dimensional elastic block with semi-cylinder notches under tension. Morover, the FEM is available for an analysis of structures of a large-size matrix required for a memory capacity larger than that of the machine itself. This method also contributes to the decrease of memory capacity and central processing unit time in the conventional FEM.

Analysis of steel frame structures in fire.

Abstract: The main aim of the present research is to develop a method of analysis for structural frames exposed to fire including the effects of material and geometric non-linearities. A matrix stiffness method based on a secant stiffness approach is used providing a full temperature deformation history. The approach has previously been used for the analysis of continuous beams and is extended in the present work to include axial forces. These not only affect the longitudinal displacement, but also reduce the member stiffness and create secondary moments due to the p-delta effect. The influence of material unloading on the moment-axial force-curvature relationship is studied by examining a cross-section subjected to different combinations of bending moment and axial force at both ambient temperature and in fire. A computer program, based on the method is used to conduct a limited parametric study. This includes the influence of slenderness ratio, the magnitude of axial load and moment, the size of cross-section and grade of steel. Both uniform and non-uniform temperature profiles are considered for isolated beams, columns and simple portal frame. The importance of the p-delta effect is also investigated.

Inelastic Analysis of Structures Based on Stress Redistribution

Abstract: An important characteristic of structures that exhibit material nonlinearity is that the internal stresses are redistributed when the stiffness properties of the members change. Based on this physical characteristic of structures, an efficient analytical method for the nonlinear analysis of structures is presented here. In his analytical method, the overall response of the structure is determined by considering: (1) The elastic response of the structure to external loads; and (2) the stress redistribution that results from the member yield excursions. One advantage of this method is that the stress redistribution may be represented in terms of the elastic stiffness matrix of the structure and not the tangent stiffness matrix. Another advantage of this method is that the only parameters that have to be evaluated and updated throughout the analysis are the member stresses and the member deformations. Hence, there is no direct evaluation of the structure displacements or the tangent stiffness matrices.