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Analysis of axially loaded tapered beams with general end restraints on two-parameter foundation

Taha M. H., Nassar M.

Journal of Theoretical and Applied Mechanics

2014

Vol. 52 nr 1

215--225

The stability and free vibration of axially-loaded tapered beams with elastic end restraints resting on two-parameter foundations are studied using the differential quadrature method (DQM). The governing differential equation is discretized at sampling points, and then the boundary conditions due to elastic end restraints are implemented and substituted into the governing differential equation yielding a system of homogeneous algebraic equations. The equivalent two-parameter eigenvalue problem is obtained and solved for critical loads in the static case and for natural frequencies in the dynamic case. The obtained solutions are found compatible with those obtained from other techniques. The influences of different parameters on the critical loads and natural frequencies are investigated.

Free vibrations of a stepped beam with two uniform and/or tapered parts

El-Din S. S., Mahmoud A. A., Nassar M. A.

Mechanics and Mechanical Engineering

2000

Vol. 4, nr 2

165--181

Closed form solutions for free vibrations of a stepped beam of two parts are given. Each part is of rectangular cross-sectional area. The parts may be of uniform cross-section and/or tapered with both equal and different tapered ratio in the horizontal and vertical planes. General constraints at the ends are possible at the three ends of the beam. The equations of motion of the beam are given in terms of trigonometric functions, hyperbolic functions, and the well known Bessel functions. Various special cases are deduced from the present solution and showed complete agreement with previous closed form solutions for these special cases.