Abstract
The natural frequencies of non-uniform beams resting on two layer elastic foundations are numerically obtained using the Generalized Differential Quadrature (GDQ) method. The Differential Quadrature (DQ) method is a numerical approach effective for solving partial differential equations. A new combination of GDQM and Newton’s method is introduced to obtain the approximate solution of the governing differential equation. The GDQ procedure was used to convert the partial differential equations of non-uniform beam vibration problems into a discrete eigenvalues problem. We consider a homogeneous isotropic beam with various end conditions. The beam density, the beam inertia, the beam length, the linear (k<sub>1</sub>) and nonlinear (k<sub>2</sub>) Winkler (normal) parameters and the linear (k<sub>3</sub>) Pasternak (shear) foundation parameter are considered as parameters. The results for various types of boundary conditions were compared with the results obtained by exact solution in case of uniform beam supported on elastic support.
Highlights
Beams resting on linear and non-linear elastic foundations have many practical engineering applications as railroad tracks, highway pavement, buried pipelines and foundation beams
Forced vibration of Euler–Bernoulli of nonuniform beams resting on two layer elastic foundations under axial and transverse load is analyzed
In order to discuss the stability and accuracy of the GDQM, uniform beams are solved using the present approach for implementing the boundary conditions and the results are compared with the exact results available in the literature
Summary
Beams resting on linear and non-linear elastic foundations have many practical engineering applications as railroad tracks, highway pavement, buried pipelines and foundation beams. Bagheri et al [27] studied the nonlinear responses of clamped–clamped buckled beam They used two efficient mathematical techniques called variational approach and Laplace iteration method in order to obtain the responses of the beam vibrations. Ramzy et al [30] presented a new technique of GDQM for determining the deflection of a non-uniform beam resting on a non-linear elastic foundation, subjected to axial and transverse distributed force. Ramzy et al [32] studied free vibration of uniform and non-uniform beams resting on fluid layer under axial force using the GDQM. The main goal of this study, to present a new combination of a GDQM and Newton’s method to obtain the fundamental frequencies and the corresponding modal shapes of non-uniform beams resting on two layer elastic foundations under appropriate boundary conditions
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