A limit-load analysis asks one question: does the structure still find an equilibrium under the factored loads when its material is elastic and perfectly plastic? It does not follow strains or the shape change. It follows VIII-2 5.2.3: a material that yields at 1.5S and does not harden, small displacements, loads that rise in proportion until equilibrium is lost.
Material point
Each through-thickness layer point of a shell element is in plane stress with von Mises yield and associated flow. The code writes the yield function with the projection matrix P = ⅓·[[2, −1, 0], [−1, 2, 0], [0, 0, 6]]:
The element is the layered MITC4 shell. Through the thickness, Simpson with 5 points puts the mid-surface on a panel boundary, which makes the fully plastic bending moment exact, Mp = σy·t²/4 per unit width. Transverse shear and drilling stay elastic and do not enter the yield function.
Load control and what stops it
Increments are cut when Newton fails and grown again when it is easy, down to Δλ_min = 10⁻⁴. Newton uses the consistent tangent; above 1500 elements a quasi-Newton (L-BFGS on the start factor) is used, and a quasi-Newton failure is confirmed by full Newton before it counts. The step ends as a limit when the tangent goes singular, or on a step guard (the plateau as Δλ_min defines it). A step-guard stop is a lower bound, never a bracketed margin.
Combinations and the verdict
Table 5.4 is applied to FEXLA's rows, each its own load path from zero (no superposition in plasticity): 1.5 (P + D); 1.3 (P + D) + 1.7 L; 1.3 (P + D) + 1.1 L or 0.54 W; with wind 1.3 (P + D) + 1.1 W + 1.1 L; with earthquake 1.1 E; and the test condition (1/βT)(PT + D) with βT = 0.95 hydrostatic, 0.80 pneumatic. Pressure not acting, L not acting and reversed occasional loads are run too. The thermal state and displacement-controlled loads are left out (they do not change an elastic-perfectly plastic limit). A combination that is a multiple s ≤ λ′ of a run that converged to λ′ is covered, with no run of its own.
PASS: the path converged at λ = 1, or the combination is covered. FAIL: no equilibrium below λ = 1. Not assessed: budget, cancel or solver error, never a pass. After a pass the run continues to λ = 3 to find the collapse factor, reported as a margin only if it is a tangent stop with a bracket under 0.2 %.
Closed cylinder, FEXLA load model (hand arithmetic from cylinderLimit)
| Ri 500, t 10, Rm = 505 | a = Rm = 505; b = Ri²/(2·Rm) = 247.52 |
| Denominator | √(a² − a·b + b²) = √(255 025 − 125 000 + 61 268) = 437.37 |
| Validation (σy = 250 MPa) | p_L = 250 · 10 / 437.37 = 5.716 MPa; FEXLA 5.7229 (+0.12 %, the faceting 1/cos(π/64) = 1.0012) |
| Job (S = 138 MPa, σy = 1.5 S = 207) | p_L = 207 · 10 / 437.37 = 4.733 MPa |
| Pass criterion LL1 = 1.5·P | 1.5·Pd ≤ 4.733, so Pd ≤ 3.155 MPa for a clean shell |
The last two rows apply the coded closed-form target to a design stress; they are an illustration, not a run. A real job also sees nozzles, pads and junctions where the limit is lower.
What passing means, and what it does not
- Passing shows no collapse of the small-displacement perfectly plastic model under the factored loads. It says nothing about strains (5.3.3), about stiffness loss from shape change (5.2.4), buckling (5.4), fatigue or ratcheting.
- A passing set supersedes only the elastic primary rows (5.2.1.4), in a run with both.
- Warnings are raised for walls with Rm/t at or below 4.5 (a shell model drops the through-wall stress; B7 measured −1.1 % at Ro/Ri 1.25 and −3.4 % at 1.5, conservative) and for parts loaded mainly in bending (5.2.1.3.2(b)).
- The job is on the VIII-2 route only (S must be the Division 2 value), refuses above 15 000 elements and warns above 5000. Snow and wind during the test are not modelled.
FE results are early access and under verification.