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Shallow Schwedler Space Dome: Nonlinear Snap-Through (Arc-Length)
Geometrically nonlinear corotational beam-shell snap-through analysis of a shallow 12 m Schwedler reticulated dome using Crisfield arc-length continuation.
Shallow reticulated domes with low rise-to-span ratios (H/D = 0.06) exhibit strong geometric softening under apex downward loading before transitioning into post-buckling tensile membrane action. Linear eigenvalue buckling (λ₁ = 1.783) overestimates the stiffness along the equilibrium path. Using corotational beam and MITC4 shell elements with Crisfield spherical arc-length continuation (frame solve static nlgeom=on method=arc), WebCAE traces the full nonlinear load-displacement path up to w_max = 358.5 mm (~50% of the initial dome rise).
1. Geometry & FEA Mesh
- •12 meridional sectors and 3 ring tiers (r = 2.0, 4.0, 6.0 m) + apex crown node (37 nodes, 150 active DOFs).
- •96 tubular steel members (Ø90 × 5 mm pipe, E = 210 GPa, ν = 0.30, ρ = 7850 kg/m³): 36 meridional ribs, 36 ring purlins, and 24 Schwedler diagonals.
- •24 four-node glazing/cladding shell panels (t = 4 mm) in outer tiers.
- •Pinned perimeter supports around the r = 6.0 m base ring; concentrated downward loads at the apex and inner ring.
2. Loads & Boundary Conditions
- •Pinned perimeter supports (Ux = Uy = Uz = 0) at all 12 nodes of the outer base ring (r = 6.0 m, z = 0).
- •Concentrated downward vertical load at the apex node #1 and distributed downward nodal loads around the inner ring (r = 2.0 m).
- •Crisfield spherical arc-length continuation (method=arc, steps=20, tol=1e-5) combined with linear modal and buckling analyses.
3. Simulation Results & Findings
- ✓Full arc-length convergence across 8 continuation steps (23 Newton iterations, 0 cutbacks) to λ = 1.000.
- ✓Apex vertical deflection reaches 358.49 mm, demonstrating pronounced geometric softening between λ = 0.48 (u = 54 mm) and λ = 0.74 (u = 198 mm) followed by tension stiffening.
- ✓Fundamental vibration frequency f₁ = 27.69 Hz and first linear buckling factor λ₁ = 1.783 stored in the same bundle.
Structural engineers designing long-span roofs and reticulated domes can verify nonlinear snap-through stability alongside linear buckling and modal frequencies in a single model.
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