A further study on the postbuckling of extensible elastic rods

Thermal Buckling and Postbuckling of Euler-Bernoulli Beams Supported on Nonlinear Elastic Foundations

“…Tiersten [30] has recently revisited the Euler-Bernoulli beam theory and delineated various approximations made to simplify the problem. Several investigators [5,9,31] have set 1 in Eq. (8b) while studying postbuckling deformations of beams.…”

“…Lestari and Hanagud [2] have studied the nonlinear vibrations of buckled beams with elastic end constraints, and the beam subjected simultaneously to axial and lateral loads. Other works [3][4][5] have analyzed either analytically or numerically nonlinear ordinary differential equations associated with the postbuckling behavior of initially straight rods subjected to compressive loads and different boundary conditions. Kreider and Nayfeh [6] have studied experimentally and theoretically the nonlinear vibrations of a clamped-clamped buckled beam.…”

Time Integration of Non-Linear Dynamic Equations by Means of a Direct Variational Method

“…Equation (16) is the well-known Du$ng model without damping. (15) The change of variable (1) and the introduction of a function as suggested in equation (8) and in particular in equation (11) lead to the transformed equation and its BC:…”

“…The values of ¹/n, in all the analyzed cases, are larger than the usual t of the most popular numerical integration schemes (e.g., method of central di!erence, Newmark method, Wilsonmethod, etc.). Let us show a numerical example by setting kM "100 (softening spring) in equation (15) and the initial conditions v "4 and vR "0. This oscillator is known to have a period of…”

“…This vibration problem was also solved before using WEM but only regarding the space domain [15]. Here, instead, the two involved variables*space and time*are included in the WEM approach.…”