Koiter's imperfection sensitivity theory

Definition

Consider a conservative elastic structure with load parameter $\lambda$ that buckles at $\lambda_c > 0$ with a single buckling mode $u_1$. Let $\xi$ be the amplitude of the mode. The initial post-buckling path of the perfect structure has the expansion $$\frac{\lambda}{\lambda_c} = 1 + a \xi + b \xi^2 + \cdots.$$ The post-buckling coefficients $a$ and $b$ are computed from the third and fourth variations of the potential energy. Let the structure carry an initial imperfection of amplitude $\varepsilon$ in the shape of $u_1$, and let $\lambda_s(\varepsilon)$ be its maximum load, when it exists. Koiter's imperfection sensitivity theory is the asymptotic analysis of this situation for small $\abs{\varepsilon}$. Its main conclusions are:

1. If $a \ne 0$ (asymmetric bifurcation), then for imperfections of one sign $1 - \lambda_s / \lambda_c$ is of order $\abs{\varepsilon}^{1/2}$.
2. If $a = 0$ and $b < 0$ (unstable symmetric bifurcation), then $1 - \lambda_s / \lambda_c$ is of order $\abs{\varepsilon}^{2/3}$.
3. If $a = 0$ and $b > 0$ (stable symmetric bifurcation), then the imperfect structure has no maximum load near $\lambda_c$ and the structure is imperfection-insensitive.