Natural Frequency and Mode Shapes of Exponential Tapered AFG Beams on Elastic Foundation

Abstract

Abstract.A displacement based semi-analytical method is utilized to study non-linear free vibration and mode shapes of an exponential tapered axially functionally graded (AFG) beam resting on an elastic foundation. In the present study geometric nonlinearity induced through large displacement is taken care of by non-linear strain-displacement relations. The beam is considered to be slender to neglect the rotary inertia and shear deformation effects. In the present paper at first the static problem is solved through an iterative scheme using a relaxation parameter and later on the subsequent dynamic analysis is carried out as a standard eigen value problem. Energy principles are used for the formulation of both the problems. The static problem is solved by using minimum potential energy principle whereas in case of dynamic problem Hamilton's principle is employed. The free vibrational frequencies are tabulated for exponential taper profile subject to various boundary conditions and foundation stiffness. The dynamic behaviour of the system is presented in the form of backbone curves in dimensionless frequency-amplitude plane and in some particular case the mode shape results are furnished.

Nonlinear response of axially functionally graded Timoshenko beams on elastic foundation under harmonic excitation

“…Only a few articles are available in literature relating to elastic foundation supported AFG thin beams, where Euler-Bernoulli theory is utilised for mathematical formulation of the problem. Huang and Luo [42] presented a new and simple method to calculate the critical buckling loads of beams with axial inhomogeneity on elastic foundation, whereas, Lohar et al [43][44][45] investigated the dynamic behaviour of AFG beam on elastic foundation. Research work in the field of AFG Timoshenko beam is extremely rare.…”