A compressed shell is stable while the membrane compression is small against its bending stiffness. Compression adds a negative term to the stiffness: the geometric stiffness K_G built from the membrane forces N of the elements. Buckling is the load level where the total stiffness becomes singular.
The reference state is the linear static solution of the load combination. K is factorised once, K = L·D·Lᵀ. With G = −K_G the problem becomes μ = 1/λ with C = D^-½ L⁻¹ G L⁻ᵀ D^-½, a symmetric matrix, solved by Lanczos with full reorthogonalisation. The largest positive μ gives the smallest positive λ. Negative λ (buckling if the loads were reversed) are reported separately. A load factor above 10⁶ is reported as no buckling. The Ritz residual tolerance is 10⁻⁵.
External pressure follows the wall as it deforms, so its load stiffness is added to K_G. Without it a long cylinder's buckling pressure comes out n²/(n² − 1) too high.
The combinations (Table 5.15 as coded)
- (1) P + D, and D alone when pressure is not acting.
- (2) 0.88 (P + D + L), 0.88 (P + D + T), the same with P not acting, and L reversed when it reverses.
- (4)/(5) 0.88 (P + D) ± k·(W or E), with k = 0.71 for wind or earthquake at strength level and 0.71/0.6 = 1.18 for a row already at allowable level; reversed, and with P not acting.
The factors 0.88 and k sit inside the combination, so λ is the margin on the combination as it stands. Each combination is a separate eigenvalue run.
From λ to a pass: 5.4.2 Method A
The critical location is the element with the largest destabilising energy density −φᵀK_G,e φ/area, not simply the node that moves most. There the capacity reduction factor βcr depends on the component:
- cylinder or cone, axial compression governing: βcr = 0.207 for Do/t ≥ 1247 (eq. 5.14), otherwise 338/(389 + Do/t) (eq. 5.15);
- cylinder or cone, hoop compression governing: βcr = 0.80 (eq. 5.16);
- formed head or sphere: βcr = 0.124 (eq. 5.17).
σm cancels from the uncapped test, so the check reduces to two conditions: βcr·λ ≥ 2, and σm ≤ 0.275·Sy once the cap applies. That is a reading of the coded test, not an extra rule in the code.
Capacity chain, illustrative skirt numbers (hand arithmetic)
| Skirt 3000 ID × 16 | Do/t = 3032/16 = 189.5; axial βcr = 338/(389 + 189.5) = 0.584 |
| Smallest λ for the uncapped test | 2 / 0.584 = 3.42 (hoop 2.5; formed head 2/0.124 = 16.1) |
| Assume λ = 24.7 (the dev log value) and σm = 30 MPa | σcrit = 0.584 · 24.7 · 30 = 433 MPa |
| Sy = 260 MPa | 0.55 Sy = 143 MPa < 433: capped, Sc = 143/2 = 71.5 MPa |
| Result | σm = 30 ≤ 71.5 MPa: passes; the cap sets the limit at σm ≤ 0.275 · 260 = 71.5 MPa |
λ = 24.7 is the value the dev log (2026-10-10) records for the opening in compression of an example skirt; σm = 30 MPa is assumed here and is not that run's stress.
Ends
With lines of support given, the model spans them: cut ends are held round (normal and circumferential translations in the end node's axes) and free to rotate, the support end also held along the meridian. Without them the support ring is held on the meridian and at three points round the circumference, so the shell can go oval and buckles as a long one, which is conservative. The result says when the critical element is next to a cut.
Limits
- Linear and perfect: no imperfection and no plasticity. All the knock-down is in βcr. Method B (5.4.3, elastic-plastic with imperfections) is not available; a critical location that is not a cylinder, cone, sphere or formed head is reported as not covered.
- The reference state is linear elastic, so a buckling load above yield is capped by the 0.55·Sy rule, not computed.
- A run that does not converge is not assessed, never a pass.
FE results are early access and under verification.