The response of a finite beam on a tensionless Pasternak foundation subjected to a harmonic load

Nonlinear thermal dynamic buckling of FGM beams

“…The virtual total potential energy of the FGM beam is equal to the sum of the virtual energy of external applied loads which is absent in this study, the virtual strain energy of the beam and the virtual energy of elastic foundation [45,46]. Thus, the virtual potential energy of the beam per unit width is equal to…”

Nonlinear dynamic analysis of Timoshenko beam-columns partially supported on tensionless Winkler foundation

“…Explicit expressions for the natural frequencies and the associated amplitude ratios of a double-beam system and the analytical solution of its critical buckling were also derived by Zhang et al [6]. Moreover, semi-analytical closed form solutions [7] and numerical methods such as the method of power series expansion of displacement components employing Hamilton's principle [8], the Galerkin method [9], the differential quadrature element method [10], a combination between the state space and the differential quadrature methods [11], double Fourier transforms [12] and the finite element technique [13][14][15] have also been used for the vibration and buckling analysis of beam-columns on one or two-parameter linear or nonlinear (tensionless) elastic foundations taking into account or ignoring shear deformation effect.…”

Nonlinear resonances in infinitely long 1D continua on a tensionless substrate

“…In the past, several authors have investigated various aspects of the problem, both in a general framework [8][9][10][11][12][13][14] or by considering specific details, such as finite [14][15][16] or infinite (or semi-infinite) domains [17][18][19], harmonic in time and fixed in space [19,20] or fixed in time and moving in space [21][22][23] loads, cables [17,18,23] or beams and plates [14,16,19,22,[24][25][26] mechanical models, numerical [19,26,27] or analytical approximate [16][17][18] solutions. We refer to the Introduction in [18] for a detailed description of the literature.…”