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.

中文翻译：

---

地基非线性对双特征值附近可剪杆屈曲后行为的影响

摘要 研究了非线性弹性基础上受压、简支、可剪切弹性杆的后屈曲行为。对于辅助参数的临界值，特征值是双倍的，因此 Liapunov-Schmidt 方法导致两个分岔方程。这些方程描述了临界值附近辅助参数值的屈曲后行为。研究了二次分岔的发生，这取决于地基和抗剪刚度的非线性。结果表明，这些效应对杆的后屈曲行为有显着影响。确定了初级和次级分支的分岔类型、稳定性、模式相互作用和渐近扩展。结果表明，对于非线性硬化弹性地基，二次分支是稳定的。