Nonlinear analysis of Euler beams resting on a tensionless soil with arbitrary configurations

Background The nonlinear interaction between an elastic Euler beam and a tensionless soil foundation is studied. The exact analytical solutions of the nonlinear problem are rather complicated. The main difficulty is imposing compatibility conditions at lift-off points. These points are determined as a part of the solution, although being needed to get the solution itself. In the current work, semi-analytical solutions are derived using the Rayleigh–Ritz method. The principle of vanishing variation of potential energy is adopted. The solution is approximated using a set of suitable trial functions. Accurate high-order approximate analytical solutions are obtained using MAXIMA symbolic manipulator. Lift-off points are identified through an iterative procedure and compatibility conditions are satisfied automatically. The methodology is designed to accommodate arbitrary configurations for the load distribution and the beam properties. Results Exact solutions are revised briefly to verify the semi-analytical solutions in terms of deflection, bending moment, and shear. Semi-analytical solutions for constant beam properties including various support conditions and load distributions are verified. Convergence of high-order semi-analytical solutions is illustrated for cases including one and two contact points. A parametric study is provided to illustrate the effect of soil stiffness on the contact length. The case of a finite beam with free ends is considered. The semi-analytical solutions for variable beam moment of inertia are provided and verified. Conclusions Highly accurate semi-analytical solutions can be obtained for the problem considered using the Rayleigh–Ritz method along with a symbolic manipulator. Arbitrary load and support configurations can be modeled, and the locations of lift-off points are well predicted. The semi-analytical solutions are extremely valuable for cases of variable moment inertia since exact solutions are rather rare.