= Euler's critical load
{c}
{title2=$P_{\rm cr}=\pi^2EI_A/(KL)^2$}
{wiki}
= Euler buckling
{c}
{synonym}
A straight slender <elastic beam> under axial compression loses stability when bending can decrease its total <potential energy>. With pinned ends a distance $L$ apart, its quadratic potential is $\tfrac12\int_0^L[EI_A(y'')^2-P(y')^2]dx$. Stationarity gives $EI_Ay''''+Py''=0$, with $y=y''=0$ at each end. The first nontrivial mode is $\sin(\pi x/L)$, at $P_{\rm cr}=\pi^2EI_A/L^2$. More generally $P_{\rm cr}=\pi^2EI_A/(K L)^2$, with an effective-length factor $K$ determined by endpoint <constraints>: $K=1$ for pinned-pinned, $2$ for fixed-free and $1/2$ for fixed-fixed. Real imperfections remove the perfectly sharp onset, and short columns can yield before this elastic instability.
Back to article page
