Two-dimensional elasticity solutions for functionally graded beams resting on elastic foundations

Deflection profile analysis of beams on two-parameter elastic subgrade

“…According to this technique, the M th term approximate solution of (2), (8) is sought in the form (9) where ( ) i Q t are coordinates in modal space and ( ) i P x are the normal modes of vibration written as (10) No difficulty arises at all to show that for a beam with simply supported end conditions, taking into account equation (10), equation (9) can be written as (11) Substituting equation (11) into the governing equation (8), one obtains (12 ) which after some simplifications and rearrangements yields (13) To determine the expression for ( ) i Q t , the expression on the LHS of equation (13) is required to be orthogonal to the function . Thus, multiplying equation (13) by and integrating with respect to x from x=0 to x=L, leads to (14) where , , (16) and considering only the i th particle of the system, equation (14) can then be written as (17) where , and (18) To obtain the solution of the equation (17), it is subjected to a Laplace transform defined as (19) where s is the Laplace parameter. Applying the initial conditions (3), one obtains the simple algebraic equation given as In what follows, we seek to find the Laplace inversion of equation (21).…”

“…Beams on elastic foundation and under the actions of the moving loads have received a considerable attention in literature; see for example references [10][11][12][13][14][15][16][17][18]. However, most of these works employed the simplest mechanical model which was developed by Winkler and generally referred to as a one-parameter model.…”

Free Vibration Analysis of Functionally Graded Nanocomposite Beams on Elastic Foundation Using a Mesh-Free Method

“…The first group used analytical and semi-analytical methods to study the bending and vibrations of FG beams resting on elastic foundations. Ying et al (2008) presented exact solutions for functionally graded simply supported beams resting on a Winkler-Pasternak elastic foundation based on the two-dimensional theory of elasticity. Fallah and Aghdam (2011) studied the free vibration and post-buckling of FG beams on nonlinear elastic foundations subject to axial loads.…”

An Analytical Solution for Free Vibration of Elastically Restrained Timoshenko Beam on an Arbitrary Variable Winkler Foundation and Under Axial Load

“…The displacement of the beam is determined as the linear combination of a Fourier series and an auxiliary polynomial function. Ying et al investigated the precise solutions for free vibration and bending of functionally graded beams on a Winkler-Pasternak elastic foundation (Ying et al, 2008). The beam is considered as orthotropic at any point, while material properties varying exponentially along the thickness direction.…”