Non-linear forced vibration analysis of higher-order shear-deformable functionally graded material beam in thermal environment subjected to harmonic excitation and resting on non-linear elastic foundation

Abstract

Geometrically non-linear forced vibration analysis of higher-order shear-deformable functionally graded material beam under harmonic excitation and supported on three-parameter non-linear elastic foundation is presented. The beam is immovably clamped and is considered to be under static thermal loading due to uniform temperature rise. Reddy's third-order shear-deformable beam theory in conjunction with von Kármán geometric non-linearity is considered to derive the governing equations employing Hamilton's principle, and Ritz method is followed for approximating the displacement and rotation fields. A numerical algorithm based on iterative substitution method and Broyden's method is proposed to predict the stable regions of frequency-response behavior. The frequency-response curves are presented in normalized plane for variations of load-amplitude, elastic foundation parameters, temperature rise, gradation index and functionally graded material composition, and their effects are discussed in detail. It is found that the load-amplitude, elastic foundation parameters, thermal loading and some of the functionally graded material compositions significantly affect the frequency response; whereas, the effect of gradation index is found to be relatively small. A comparative frequency-response curve between Voigt model and Mori-Tanaka scheme of functionally graded material modeling is presented, and it shows negligible difference between these two models. The present problem under thermal environment is studied for the first time through this work, and the proposed model and the numerical algorithm provide a simplified approach to study the non-linear frequency-response behavior.

A new higher-order beam theory for buckling and free vibration responses of laminated composite and functionally graded porous beams

“…Although the elasticity theory describes the beam's responses precisely, it is computationally complex and difficult to apply to arbitrary boundary conditions or complicated geometries. As a result, several alternative methods have been proposed, including layer-wise theories, 10,11 equivalent single-layer theories (ESLT), [12][13][14][15] zig-zag theories, [16][17][18][19] and Carrera's Unified Formulation. 20,21 Among these theories, the ESLT is widely utilised due to its formulation simplicity and programming ease.…”

Free and Forced Vibration Characteristics Analysis of a Multispan Timoshenko Beam Based on the Ritz Method

“…Wu and Chen [20] dealt with the free and forced vibration behaviors of laminated composite beams based on higher-order zig-zag theory, where the transverse shear stress is introduced into the equation of motion by using Hamilton's principle. More studies corresponding to the vibration characteristics of single-span beam can be found in [21][22][23][24][25].…”

Effect of porosity and skew edges on transient response of functionally graded sandwich plates

“…Dozio 3 used Carrera unified formulation to derive exact solutions for dynamic analysis of single-layered FGM plates by considering two dimensional higher-order theories. Paul and Das 4 performed forced vibration analysis of FGM beam resting on an elastic foundation using a higher-order shear deformation theory (HSDT). Jha et al 5 presented a higher-order shear and normal deformation theory for analysing the free vibration behaviour of FGM plates.…”