# Buckling A part experiencing uniaxial compressive [[Stress|stress]] will buckle when load exceeds its ***critical load*** $P_\text{crit}$. Characteristics of buckling: - sudden deformation - large displacements ## Euler's Buckling Formula > $ > P_\text{crit}=\frac{\pi^2EI}{L_e^2} > $ > > -- Leonhard Euler (1707-1783, Switzerland) $E$ - Young's modulus $I$ - [[Second Moment of Area|area moment of inertia]] $L_e$ - effective length - distance between inflection points - column pinned at both ends: $L_e=L$ - fixed at one end, free on the other: $L_e=2L$ - fixed at one end, roller on the other: $L_e=0.5L$ - pinned at one end, roller on the other: $L_e=0.7L$ Can divide by area to achieve critical stress: $ \sigma_\text{crit}=\frac{\pi^2EI}{AL_e^2} $ Introducing the radius of gyration $r=\sqrt{I/A}$: $ \sigma_\text{crit}=\frac{\pi^2EI}{(L_e/r)^2} $ $L_e/r$ - effective slenderness ratio Maximum compressive stress calculated using the ***secant formula***: $ \sigma_\text{max}=\frac{P}{A}\left[1+ \frac{ec}{r^2}\sec\left(\frac{L}{2r}\sqrt{\frac{P}{EA}}\right)\right] $