Abstract
Analytical solutions for free vibration analyses of a beam on elastic foundation are obtained for different support conditions. The analytical solutions are applied on three different axially loaded cases, namely; (1) one end clamped, the other end simply supported; (2) both ends clamped, and (3) both ends simply supported cases. Analytical solutions and frequency factors are evaluated for different ratios of axial load, <svg style="vertical-align:-0.1092pt;width:14.9875px;" id="M1" height="11.3125" version="1.1" viewBox="0 0 14.9875 11.3125" width="14.9875" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,11.113)"><path id="x1D441" d="M857 650l-6 -28q-44 -4 -61.5 -16.5t-29.5 -48.5q-11 -32 -37 -166l-78 -399h-29l-351 537h-4l-56 -276q-24 -120 -24 -164q0 -35 17.5 -46t75.5 -15l-6 -28h-245l7 28q41 2 62 14t31 44q10 30 41 171l53 245q8 44 6.5 60.5t-14.5 33.5q-10 15 -27 19.5t-64 6.5l6 28h153
l350 -516h5l48 257q25 131 25 171q0 34 -17.5 45t-77.5 15l7 28h240z" /></g> </svg>, acting on the beam to Euler buckling load, <svg style="vertical-align:-3.3907pt;width:20.625px;" id="M2" height="15.4" version="1.1" viewBox="0 0 20.625 15.4" width="20.625" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,11.112)"><use xlink:href="#x1D441"/></g> <g transform="matrix(.012,-0,0,-.012,14.925,15.187)"><path id="x1D45F" d="M393 379q-9 -16 -28 -29q-15 -10 -23 -2q-19 19 -36 19q-21 0 -52 -38q-57 -72 -82 -126l-40 -197q-23 -3 -75 -18l-7 7q49 196 74 335q7 43 -2 43q-7 0 -30 -14.5t-47 -37.5l-16 23q37 42 82 73t67 31q41 0 15 -113l-11 -50h4q41 71 85 117t77 46q29 0 45 -26
q13 -21 0 -43z" /></g> </svg>. The analytical solutions give results which are in excellent agreement with the variational iteration method (VIM) and homotopy perturbation method (HPM) results available in the literature for the particular problem considering all the cases provided in this study and the differential transform method (DTM) results available in the literature for the clamped-pinned case.
Highlights
The free vibration equation of an axially loaded beam on elastic foundation is a fourth-order partial differential equation
The analytical results are in excellent agreement with the results of the particular problem solved using the variational iteration method (VIM) method and the homotopy perturbation method (HPM) method available in the literature [8,9,10]
Analytical solution for free vibration of a one end clamped, and one end pinned beam on elastic foundation was previously given by Catal [7] with additional analyses by differential transform method (DTM)
Summary
The free vibration equation of an axially loaded beam on elastic foundation is a fourth-order partial differential equation For this particular engineering problem, the analytical solutions will be implemented in this study. Free vibration equations for one end clamped and other end supported beam on elastic foundation were solved by using the DTM for various axial loads acting on the beam [7] Both the variational iteration method (VIM) and homotopy perturbation method (HPM) were used to solve the free vibration equations of beam on elastic foundation for support conditions of one end clamped, and other end supported, both ends clamped and both ends supported considering various case studies [8,9,10,11]. The analytical results are in excellent agreement with the results of the particular problem solved using the VIM method and the HPM method available in the literature [8,9,10]
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