Moshe F. Rubinstein

Creep analysis of axisymmetric bodies using finite elements

Abstract An analysis and a numerical computer program are developed for calculating the creep strains in an arbitrary axisymmetric body of revolution subjected to axisymmetric loads. The method of solution is an extension of the direct stiffness method. The body is replaced by a system of discrete triangular cross section ring elements interconnected along circumferential nodal circles. The equation of equilibrium for the body are derived from the principal of minimum potential energy. The creep behavior of the body is obtained by the use of an incremental approach. The method involves starting with the elastic solution of the problem and calculating the creep strains for a small time increment. These creep strains are treated as initial strains to determine the new stress distribution at the end of the time increment. The procedure is repeated until either the final time is reached or until the stress distribution is not changed, i.e., a steady-state condition is reached. The method is quite general and is independent of the type of creep law used. Results are presented for a plane-strain cylinder solution and a pressure vessel with spherical, elliptical, and flat end closures.

Structural reanalysis by iteration

Abstract Reanalysis based on an iteration process is expressed as an infinite series. It is shown that for large changes in the design, convergence is poor or not possible. A method to improve convergence is presented, based on an expression of the matrix of changes in stiffness as a linear combination of two matrices. The method is then modified to further improve convergence separately for each component of the unknown displacement vector. The advantage of the proposed procedure is demonstrated by numerical examples.

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.

On the effect of rotatory inertia and shear in beam vibration

Abstract An energy approach is used to account for rotatory inertia and shear in formulating beam vibration problems.

Error analysis in structural computations

Abstract The errors in structural computations are assessed and bounds are established on their magnitude. The matrix decomposition approach is shown to be most useful in error analysis. Both positive definite and positive semi-definite matrices are treated. The discrete finite-digit representation of numbers in a digital computer and the nature of digital computer calculations are considered in the development.

Stability analysis by matrix decomposition

Abstract A method is presented for the stability analysis of structures using a combination of matrix iteration and matrix decomposition. The method is extented to stability analysis of structures in which two critical loads are present whose absolute values are relatively close or identical.

A simple interpretation of generalized forces using Betti's Law

Abstract Bettis Law is used to compute the generalized forces at the coordinates of a structure It is shown that the displacement configurations assiciated with the columns of the stiffness matrix of the structure provide the necessary information for the calculations.

Analysis of large complex frames on a small computer

Abstract The stiffness matrix method is combined with an iteration technique to analyze frames in which the members are nonprismatic and have shapes which are complex for purposes of analysis, or cannot be expressed analytically. The complex nonprismatic members are approximated by a combination of prismatic and tapered segments, and the matrix method is used to compute their characteristics as well as the fixed end moments and shears required in the iteration process. The proposed method of analysis is suitable for analyzing large complex frames on a small digital computer. The analysis includes gravity loads as well as lateral loads that may result from wind, blast or earthquake.

Creative problem solving combining man and machine: Problem solving in the computer age must make use of the unique abilities of humans and machines

What kinds of tools for human imaginative thinking are most suitable for the era of computers? How can we prepare to be more creative and productive in professional work, and in dealing with personal, everyday-life situations? How can we creatively anticipate and act on change?

Dynamic stability analysis by the conjugate gradient method

Abstract A method is presented to determine the regions of dynamic instability of a structural system. The finite element method is employed in problem formulation, and the conjugate gradient method is used to compute the boundaries of dynamic instability by minimizing appropriate Rayleigh quotients. The solution of the governing equation is reduced to a problem of finding the frequencies which bound the region of dynamic instability The conjugate gradient method used in the computations has the following advantages: (1) it is not necessary to explicitly invert the stiffness matrix; (2) the computations may be carried out directly from the element stiffness and mass matrices, and therefore, it is not necessary to synthesize the corresponding system matrices; and (3) computer storage required in the computations is less than that required in conventional techniques. An algorithm is presented for computing the boundary frequencies for the first mode of dynamic instability. Higher modes of dynamic instability are computed from a modified algorithm based on revised Rayleigh quotients which sweep out the modes previously calculated. An example, in which the dynamic stability of a plate is established, illustrates the method.