IJRRAS 14 (1) ● January 2013 www.arpapress.com/Volumes/Vol14Issue1/IJRRAS_14_1_ 20.pdf 176 STATIC AND DYNAMIC BEHAVIOR OF TAPERED BEAMS ON TWO-PARAMETER FOUNDATION Mohamed Taha Hassan & Mohamed Nassar Dept. of Eng. Math and Physics, Faculty of Eng., Cairo University, Giza. ABSTRACT The static and dynamic behaviors of tapered beams resting on two-parameter foundations are studied using the differential quadrature method (DQM). The governing differential equations are derived and discretized; then the appropriate boundary conditions are discretized and substituted into the governing differential equations yielding a system of homogeneous algebraic equations. The equivalent two-parameter eigenvalue problem is obtained and solved for critical loads in the static case (=0) and natural frequencies in the dynamic case with a prescribed value of the axial load (P o P cr ). The obtained solutions are found compatible with those obtained from other techniques. A parametric study is performed to investigate the significance of different parameters. Ke ywor ds: Tapered beams, two-parameter foundation, differential quadrature, critical load and natural frequencies. 1. INTRODUCTION Nonprismatic elements are commonly used in many practical applications to optimize weight or materials. The static and dynamic behavior of such elements need design criteria to identify the optimal configurations. The analytical treatments of such elements are intractable due to the complicated governing equations whereas the numerical techniques offer tractable alternatives. Different configurations are studied by many researchers to obtain stability and/or vibration behaviors of such structural elements. Closed forms and analytical solutions for simple cases of prismatic and non-prismatic elements are found in literature. Taha and Abohadima [1-2] studied the free vibration of non-uniform beam resting on elastic foundation using Bessel functions. Taha [3] investigated the nonlinear vibration of initially stressed beam resting on elastic foundation by employing the elliptic integrals. Ruta [4] used the Chebychev series to obtain solutions for non-prismatic beam vibration. Asymptotic perturbation has been used by Maccari [5] to analyze the nonlinear dynamics of continuous systems. Sato [6] reported the transverse vibration of linearly tapered beams using Ritz method. He studied the effect of end restraints and axial force on the vibration frequencies. Numerical methods such as the FEM [7-9], the differential transform methods [10, 11] and the differential quadrature method [12-14] are used to study certain configurations of such elements. The free vibration of tapered beams with nonlinear elastic restraints was studied by Naidu [9] using the FEM, and the effect of tapering ratio and end restraints were analyzed. The behavior of non-prismatic beams resting on elastic foundations had received a little attention in literature due to the complexity in its mathematical treatment and most researches in that area were carried out to investigate special cases. In the present work, the stability and vibration behavior of axially -loaded tapered beams resting on a two-parameter foundation will be investigated using the DQM. The present work differs than Naidu work [9] in implementing the two-parameter foundation and axial compression load. The governing equations are formulated in dimensionless form, discretized over the studied domain; and the boundary conditions are discretized and substituted into governing equations yielding a system of homogeneous algebraic equations. Using eigenvalue analysis yields the critical loads in static case ( =0) and natural frequencies for a prescribed axial load value (P o P cr ). The obtained solutions will be verified and the effects of different parameters related to the studied model on the stability and frequency parameters will be illustrated. 2. FORMULATION OF THE PROBLEM 2.1 Vibration equation The free vibration equation of a non-prismatic beam axially-loaded by P o and resting on a two-parameter foundation shown in Fig.(1) is given as: 2 2 2 2 2 1 2 2 2 2 ( ) ( ) ( ) 0 o Y Y Y EI X P k AX kY X X X X t (1) where I(X) is the moment of inertia of the beam cross section at X; is the mass density per unit volume; E is modulus of elasticity; A(X) is the area of cross section at X; Y(X, t) is the lateral displacement; P o is the axial load
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STATIC AND DYNAMIC BEHAVIOR OF TAPERED BEAMS ON TWO-PARAMETER FOUNDATION · 2018-09-14 · In the present work, the stability and vibration behavior of axially-loaded tapered beams
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IJRRAS 14 (1) ● January 2013 www.arpapress.com/Volumes/Vol14Issue1/IJRRAS_14_1_ 20.pdf
176
STATIC AND DYNAMIC BEHAVIOR OF TAPERED BEAMS ON
TWO-PARAMETER FOUNDATION
Mohamed Taha Hassan & Mohamed Nassar
Dept. of Eng. Math and Physics, Faculty of Eng., Cairo University, Giza.
ABSTRACT
The static and dynamic behaviors of tapered beams resting on two-parameter foundations are studied using the
differential quadrature method (DQM). The governing differential equations are derived and discretized; then the
appropriate boundary conditions are discretized and substituted into the governing differential equations yielding a
system of homogeneous algebraic equations. The equivalent two-parameter eigenvalue problem is obtained and
solved for critical loads in the static case (=0) and natural frequencies in the dynamic case with a prescribed value
of the axial load (PoPcr). The obtained solutions are found compatible with those obtained from other techniques. A
parametric study is performed to investigate the significance of d ifferent parameters.