In this paper, the static and dynamic buckling loads of a viscously damped imperfect finite column lying on an elastic foundation with cubic – quintic nonlinearity but trapped by a step load (in the dynamic case) is investigated analytically. The main objective is to determine analytically both the static and dynamic buckling loads by means of perturbation and asymptotic procedures and relate both buckling loads in one single formula. The formulation contains small perturbations particularly in the viscous damping and imperfection amplitude. Multi – timing perturbation techniques and asymptotics are easily utilized in analyzing the problem. The results, which are nontrivially obtained, are implicit in nature and are valid as long as the magnitudes of the small perturbations become asymptotically small compared to unity.

No abstract

No abstract

In this investigation, regular perturbation procedures in asymptotic expansions of the relevant variables are employed to discuss the static buckling analysis of a finite deterministically imperfect but viscously damped column resting on some quadratic–cubic nonlinear elastic foundations, but struck by a step load. The governing equation for the system under discussion is fully nonlinear, so that a closed form and easy solution to the problem is not possible. An approximate analytical solution to the problem is obtained using asymptotic and perturbation techniques and numerical results obtained show that increase in imperfection factors lower the static buckling loads of the column.

The exact and asymptotic analyses of the buckling of a quadratic-cubic model structure subjected to static loading are discussed. The governing equation is first solved using the phase plane method and next, using the method of asymptotics. In the asymptotic method, we discuss the possibilities of using regular perturbation method in asymptotic expansions of the relevant variables to get an approximate analytical solution to the problem. Finally, the two results are compared using numerical results obtained with the aid of Q-Basic codes. In the two methods discussed in this work, it is clearly seen that the static buckling loads decrease as the imperfection parameters increase. It is also observed that the static buckling loads obtained using the exact method are higher than those obtained using the method of asymptotics.