This article aims to evaluate the critical weight of flexible pipe subjected to applied end moments at fixed support locations. The pipe is hinged at one end, while the other end is free to slide over a frictionless support. The horizontal distance between the two supports is fixed. The model formulation is developed by the variational approach, and the finite element method is employed to obtain the numerical solutions. The critical weights are evaluated for various values of end moments and the proportional parameter of the end moments.
<abstract><p>This article explores the equilibrium configurations of a cantilever beam described by the minimizer of a generalized total energy functional. We reformulate the problem as a boundary value problem using the Euler-Lagrange condition and investigate the existence and uniqueness of minimizers. Furthermore, we discuss the dependence of solutions on the parameters of the boundary value problems. In addition, the Adomian decomposition method is derived for approximating the solution in terms of series. Finally, numerical results for the equilibrium configurations of cantilever beams are presented to support our theoretical analysis.</p></abstract>
In this paper, we investigate the existence and uniqueness of minimizers of a fractional variational problem generalized from the energy functional associated with a cantilever beam under a uniformly distributed load. We apply the fractional Euler–Lagrange condition to formulate the minimization problem as a boundary value problem and obtain existence and uniqueness results in both L2 and L∞ settings. Additionally, we characterize the continuous dependence of the minimizers on varying loads in the energy functional. Moreover, an approximate solution is derived via the homotopy perturbation method, which is numerically demonstrated in various examples. The results show that the deformations are larger for smaller orders of the fractional derivative.
In this paper, the linearization of third-order ordinary differential equations, which are the transformed equations of the quintic nonlinear beam equation, is presented. First of all, a third-order ordinary differential equation can be linearized if its coefficients satisfied the conditions of the linearization theorem. So, the conditions for linearization are investigated. After that, in each case, the linearizing transformation is defined, and finally the linear third-order equation is obtained. Moreover, after calculating the solutions to linear third-order equations and substituting the original variables to the solutions, the exact solutions to the equation of motion are obtained.
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