Elastic Rod — Euler Buckling
Kirchhoff rod: θ''(s) + (P/EI)sin θ = 0 — exact elastica post-buckled shapes
Rod Configuration (Elastica)
Bifurcation Diagram P vs Deflection
Parameters
Load P/P_cr:
1.20
Mode n:
1
EI (flex. rigidity):
1.00
Boundary condition:
Pinned–Pinned
Cantilever
Fixed–Fixed
P_cr (mode 1):
—
EI/L²
P/P_cr:
—
Max deflection:
—
L
End shortening:
—
L
Mode:
—
Euler loads: P_n = n²π²EI/L² (pinned-pinned)
Post-buckled shape is an
elastica
— solution via elliptic integrals K(k).
At P=P_cr: δ ~ √(P−P_cr) — supercritical pitchfork bifurcation.