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Post-Buckling of Stringer-Stiffened Cylindrical Shell with Koiter Eigenmode Imperfection (Arc-Length)
Stringer- and ring-stiffened cylindrical shell (288 MITC4 quads + 336 3D beam elements) seeded with its first linear buckling eigenmode (`imperfection mode=1 scale=0.0006`) and traced into post-buckling via Crisfield arc-length continuation.
Axially compressed thin-walled aerospace cylinders are notoriously sensitive to initial geometric imperfections (Koiter's 1945 law), which trigger premature shell wall lobing and smooth out the sharp bifurcation point. In WebCAE, the `imperfection mode=1 scale=0.0006` command extracts the first buckling eigenvector φ₁ from `frame solve buckling`, normalizes it to unit maximum translational amplitude (max‖φ₁,i‖₂ = 1), and perturbs all 312 shell/beam nodes by w₀ = 0.6 mm (half the 1.2 mm wall thickness) before launching the nonlinear Crisfield arc-length solver.
1. Geometry & FEA Mesh
- •Cylindrical stiffened shell (R = 50 mm, H = 180 mm, skin thickness t = 1.2 mm): 288 MITC4 shell elements (24 circumferential sectors × 12 axial bays) + 288 longitudinal stringers (d = 2.5 mm) + 48 end-ring beam elements (d = 3.0 mm).
- •312 nodes (1,872 DOFs), composite/light-alloy modulus E = 10 GPa, ν = 0.33, ρ = 2,000 kg/m³.
- •Embedded modal (6 modes, f₁ = 393.95 Hz), linear buckling (4 modes, λ₁ = 1.0076), and nonlinear arc-length static results.
2. Loads & Boundary Conditions
- •Bottom ring (z = 0, 24 nodes) fully clamped across all 6 translational and rotational DOFs.
- •Top ring (z = 180 mm, 24 nodes) loaded by uniform axial compression F_z = −45.0 kN (−1,875 N/node).
- •Eigenmode 1 imperfection seeded with amplitude w₀ = 0.6 mm (`imperfection mode=1 scale=0.0006`).
3. Simulation Results & Findings
- ✓Linear bifurcation occurs at λ₁ = 1.00758 (critical axial load P_cr = 45.34 kN); fundamental vibration frequency is f₁ = 393.95 Hz.
- ✓With 0.6 mm mode-1 imperfection seeded, the nonlinear arc-length solver traces the post-buckling path in 5 arc steps (6 Newton iterations, 0 cutbacks).
- ✓Axial crown shortening reaches uz = −1.635 mm accompanied by radial buckling lobe growth ΔR = 0.191 mm.
WebCAE's `imperfection` command automates EN 1993-1-6 / NASA SP-8007 buckling-mode imperfection seeding and arc-length collapse analysis for shells, tanks, and lattice towers.
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