Moshe Eisenberger

Dynamic stiffness analysis of laminated beams using a first order shear deformation theory

In this paper the exact vibration frequencies of generally laminated beams are found using a new method, including the effect of rotary inertia and shear deformations. The effect of shear in laminated beams is more significant than in homogenous beams, due to the fact that the ratio of extensional stiffness to the transverse shear stiffness is high. The exact dynamic stiffness matrix is derived, and then any set of boundary conditions including elastic connections, and assembly of members, can be solved as in the classical direct stiffness method for framed structures. The natural frequencies of vibration of a structure are those values of frequency that cause the dynamic stiffness matrix to become singular, and one can find as many frequencies as needed from the same matrix. In the paper several examples are given, and compared with results from the literature.

Exact stiffness matrix for beams on elastic foundation

Abstract An exact stiffness matrix of a beam element on elastic foundation is formulated. A single element is required to exactly represent a continuous part of a beam on a Winkler foundation. Thus only a few elements are sufficient for a typical problem solution. The stiffness matrix is assembled in a computer program and some numerical examples are presented.

Buckling loads for variable cross-section members with variable axial forces

Abstract This work gives exact solutions for the buckling loads of variable cross-section columns, loaded by variable axial force, for several boundary conditions. Both the cross-section bending stiffness and the axial load can vary along the column as polynomial expressions. The proposed solution is based on a new method that enables one to get the stiffness matrix for the member including the effects of the axial loading. The buckling load is found as the load that makes the determinant of the stiffness matrix equal zero. Several examples are given and compared to published results to demonstrate the accuracy and flexibility of the method. New exact results are given for several other cases.

Vibrations and buckling of a beam on a variable winkler elastic foundation

Abstract Two methods for solving the eigenvalue problems of vibrations and stability of a beam on a variable Winkler elastic foundation are presented and compared. The first is based on using the exact stiffness, consistent mass, and geometric stiffness matrices for a beam on a variable Winkler elastic foundation. The second method is based on adding an element foundation stiffness matrix to the regular beam stiffness matrix, for vibrations and stability analysis. With these matrices, it is possible to find the natural frequencies and mode shapes of vibrations, and buckling load and mode shape, by using a small number of segments. It is concluded that the use of the element foundation stiffness approach yields better convergence at lower computation costs.

Buckling loads of variable thickness thin isotropic plates

This work presents accurate solutions for bifurcation buckling loads of rectangular thin plates with thickness that varies in the directions parallel to the two sides. The plates are subjected to biaxial compression and various combinations of boundary conditions are considered. The calculation of the critical loads was carried out by using the extended Kantorovich method. For the resulting ordinary differential equation, an exact method for the stability analysis of compressed members with variable flexural rigidity is used. The buckling load is found as the inplane load that makes the determinant of the stiffness matrix equal zero. New, exact results are given for many cases of uni-directional and bi-directional variation in thickness. The results are very accurate and exact for cases where an exact solution is available.

Exact longitudinal vibration frequencies of a variable cross-section rod

Abstract This paper presents the solution for the title problem. The solution is found using the exact element method, where the dynamic axial stiffness for the rod is found. The natural frequencies for the variable cross-section member are those for which the stiffness is equal zero. These values can be found up to any desired accuracy. This method is goof for any polynomial variation in the cross-sectional area and mass distribution along the member. The results of several examples are compared with results obtained from finite element analysis.

An exact high order beam element

Abstract This work gives the exact stiffness coefficients for an high order isotropic beam element. The terms are found directly from the solutions of the differential equations that describe the deformations of the cross-section according to the high order theory, which include cubic variation of the axial displacements over the cross-section of the beam. The model has six degrees of freedom at the two ends, one transverse displacement and two rotations, and the end forces are a shear force and two end moments. Also given are the equivalent end forces and moments for several cases of loading along the member. The components of the end moments are investigated, and are found for exact results. Comparison is made with the Bernoulli–Euler and Timoshenko beam models.

Exact Static and Dynamic Stiffness Matrices for General Variable Cross Section Members

This paper concerns the formulation of a new finite-element method for the solution of beams with variable cross section. Using only one element, it is possible to derive the exact static and dynamic stiffness matrices (up to the accuracy of the computer) for any polynomial variation of axial, torsional, and bending stiffnesses along the beam. Examples are given for the accuracy and efficiency of the method.

Vibrations and buckling of cross-ply nonsymmetric laminated composite beams

The exact element method was applied to calculate the natural frequencies, the buckling loads, and the influence of the axial load on the natural frequencies and mode shapes of nonsymmetric laminated composite beams. The theoretical model is using a first-order shear deformation theory and includes the effects of rotary inertia, shear deformation, and coupling between the longitudinal and transverse displacements. A parametric study was performed to investigate the influence of boundary conditions, materials and layup sequence on the buckling loads and natural frequencies of rectangular, cross-ply laminated composite beams.

Analysis of a beam column on elastic foundation

Abstract The analysis of beams on elastic Winkler foundation is very common in engineering. In many applications, transverse as well as axial forces exist. An exact analytical solution of a finite element beam column resting on a Winkler foundation is performed from which the exact stiffness terms are determined. The stiffness matrix is incorporated into a common beam program. Nodes are required only at points of discontinuity in stiffness, loading, or supports. Comparisons are made with case results appearing in literature.