The influence of foundation nonlinearity on the post-buckling behavior of a shearable rod near double eigenvalues. (15th October 2020) - Glavardanov, V.B.; Grahovac, N.M.; Berecki, A.D.; Zigic, M.M.
Highlights: Post-buckling behavior of a shearable rod on a nonlinear foundation is studied. Secondary bifurcations are determined near the double eigenvalues. For sufficiently large foundation nonlinearity bifurcations do not depend on shear. There are eleven post-buckling behaviors for every double eigenvalue. The asymptotic expansions of post-buckling states are constructed. Abstract: The post-buckling behavior of a compressed, simply supported, shearable elastic rod on a nonlinear elastic foundation is studied. For the critical values of auxiliary parameter the eigenvalues are double so that the Liapunov-Schmidt method leads to two bifurcation equations. These equations describe the post-buckling behavior for the values of auxiliary parameter near the critical ones. The occurrence of secondary bifurcations, depending on the nonlinearity of foundation and shear rigidity, is investigated. It is shown that these effects have a significant influence on the post-buckling behavior of the rod. The type of bifurcations, stability, mode interactions and asymptotic expansions of primary and secondary branches are determined. It is shown that the secondary branches can be stable for the nonlinear hardening elastic foundation. The complex bifurcation analysis reveals that for every critical value of the auxiliary parameter, there are eleven different post-buckling behaviors, depending on the nonlinearity of foundation and shear rigidity.