Plastic Collapse Analysis

Plastic Collapse Analysis (structures/fem/plastic-collapse-analysis)

Use when you must find the plastic collapse (limit) load of a beam or a simple frame. This leaf analyzes the limit state of rigid-perfectly- plastic bending: the fully plastic moment Mp = sigma_y*Zp built from the plastic section modulus Zp, the shape factor nu = Zp/Z as the section reserve between first yield and full plasticity, and the plastic hinge mechanisms of statically indeterminate beams and frames whose collapse loads come from the kinematic (virtual work) and static (equilibrium plus yield) theorems. It pairs with the elastic side of the pack: beam-frame-analysis solves the same rigid-jointed frames elastically and stops at the member end actions, while this leaf carries the member into the plastic mechanism state, so the two bound the strength check from below (first yield) and above (plastic collapse).

Domain quick reference

• Fully plastic moment: Mp = sigma_yZp with Zp the first moment of area about the equal-area axis (plastic neutral axis). Closed forms: rectangle Zp = bh2/4; solid circle Zp = d3/6; doubly symmetric I-beam with the plastic neutral axis in the web Zp = bt_f(d - t_f)
  • t_w*(d - 2*t_f)**2/4, the flanges at full stress at their lever arms about the mid-depth plus the two half-webs.
• Elastic section modulus: rectangle Z = bh**2/6; circle Z = pid3/32; I-beam I = (b*d3 - (b - t_w)(d - 2t_f)*3)/12 and Z = 2I/d.
• Shape factor: nu = Zp/Z is exactly 3/2 for the rectangle, exactly 16/(3pi) = 1.69765272631 for the circle, and the closed-form quotient for the I-beam (about 1.10 to 1.25 for rolled shapes). First-yield moment My = sigma_yZ = Mp/nu: the moment at which the extreme fibre reaches sigma_y. Flanged sections put the material far from the neutral axis, so first yield and full plasticity nearly coincide and the reserve is smallest.
• Hinge count rule: a collapse mechanism needs r + 1 plastic hinges for r-fold static indeterminacy: 1 (simply supported, r = 0), 2 (propped cantilever, r = 1), 3 (fixed-fixed beam, r = 2), 2 (pinned-base frame sway, r = 1), 4 (fixed-base frame sway, r = 3).
• Kinematic theorem per case (virtual work Wdelta = sum of Mptheta_i with delta = thetaL/2 under the central load): simply supported Wc = 4Mp/L; propped cantilever Wc = 6Mp/L; fixed-fixed beam Wc = 8Mp/L. Portal sway under a top lateral load H on columns of height h (delta = thetah): pinned bases Hc = 2Mp/h, fixed bases Hc = 4*Mp/h.
• Static theorem: at the collapse load the equilibrium moment diagram of the mechanism state has |M| <= Mp everywhere with |M| = Mp exactly at the hinge stations.
• Elastic first-yield context loads: W_y = 4My/L (simply supported, peak WL/4), W_y = 16My/(3L) (propped cantilever, hogging peak 3WL/16 at the fixed end), W_y = 8My/L (fixed-fixed, peaks WL/8 at both ends). Collapse-to-yield ratios Wc/Wy = nu, 9*nu/8 and nu.
• FAR 25.303 context: ultimate load is 1.5 times the limit load, so the plastic collapse load factor lambda = collapse load / limit load must reach 1.5 and the ultimate margin lambda/1.5 clears 1 exactly when the collapse load clears the 1.5-factor ultimate load.
• Units are SI throughout (m, N, Pa). Scope: rigid-perfectly-plastic bending, small-deflection mechanisms, constant Mp along each member, no axial-moment interaction, no distributed-load collapse, no combined mechanisms beyond the sway mechanism, no strain hardening.

Workflow

1. Establish the section reserves of the cross section: run plastic_section_modulus, elastic_section_modulus and shape_factor on the shape dims (rectangle (b, h); circle (d); I-beam (d, b, t_f, t_w)) and read the shape factor as the reserve between first yield and full plasticity.
2. Establish the yield capacity of the member: fully_plastic_moment with the yield stress and the plastic section modulus for Mp, and divide by the shape factor for the first-yield moment My = Mp/nu.
3. Form the plastic hinge mechanism of the single-span beam and compute the plastic collapse load under the central point load with collapse_load_beam, the kinematic (virtual work) theorem applied to the r + 1 hinge mechanism.
4. Cross-check the mechanism state with the static theorem: sample the collapse-state equilibrium moment diagram (reactions Wc/2 each end, or 4Mp/L at the fixed end and 2Mp/L at the pin of the propped cantilever) and confirm |M| <= Mp with equality only at the hinge stations.
5. Get the elastic first-yield context loads W_y from My and the collapse-to-yield ratios Wc/Wy (nu, 9*nu/8, nu) for the reserve report.
6. Get the sway mechanism collapse loads of a simple rectangular frame under a top lateral load with portal_sway_collapse_load on the column height, the fully plastic moment and the base condition.
7. Express the reserve against the applied limit load: collapse_load_factor for lambda and ultimate_margin for the FAR 25.303 1.5 ultimate-factor verdict, adequate when the collapse load clears 1.5 times the limit load.
8. Confirm the deterministic checks with the contract test scripts/test_plastic_collapse_analysis.py.

Worked example

sigma_y = 250 MPa; rectangle 50 x 100 mm (b = 0.05, h = 0.1), circle d = 0.1 m, I-beam 400 x 200 x 12 x 8 mm (d = 0.4, b = 0.2, t_f = 0.012, t_w = 0.008). All values below are the real module outputs.

• Section reserves: rectangle Zp = 0.000125 m^3 (125 cm^3), Z = 8.33333333333e-05 m^3, nu = 1.5 exactly, Mp = 31250 N m, My = 20833.3333333 N m. Circle Zp = 0.000166666666667 m^3 (166.7 cm^3), Z = 9.81747704247e-05 m^3, nu = 16/(3*pi) = 1.69765272631, Mp = 41666.6666667 N m, My = 24543.6926062 N m. I-beam Zp = 0.001213952 m^3, Z = 0.00108074325333 m^3, nu = 1.12325660721, Mp = 303488 N m, My = 270185.813333 N m. The shape-factor ladder 1.6977 (circle) > 1.5 (rectangle) > 1.1233 (I-beam) is the reserve ordering: the I-beam material sits in the flanges, far from the neutral axis.
• Single-span beams at rectangle Mp = 31250 N m, L = 3 m, one central point load: Wc = 4Mp/L = 41666.6666667 N (1 hinge at midspan) simply supported, Wc = 6Mp/L = 62500 N (2 hinges, fixed end and under the load) propped cantilever, Wc = 8*Mp/L = 83333.3333333 N (3 hinges, both supports and midspan) fixed-fixed.
• Static theorem check at collapse: sampling the collapse-state moment diagram at 2001 stations gives max |M| = 31250 N m = Mp with overshoot above Mp exactly 0.0 for all three cases; the fixed-fixed collapse state runs linearly from -Mp at x = 0 through +Mp under the load to -Mp at x = L with reactions Wc/2 = 41666.6667 N.
• Elastic first-yield context loads: W_y = 27777.7777778 N, 37037.037037 N and 55555.5555556 N, giving Wc/Wy = 1.5 (= nu), 1.6875 (= 9*nu/8) and 1.5 (= nu): the propped cantilever gains the extra 12.5% because only its fixed-end peak yields first.
• Portal frame with I-beam columns, Mp = 303488 N m, h = 4 m, top lateral load: pinned bases Hc = 2Mp/h = 151744 N (2 top-corner hinges), fixed bases Hc = 4Mp/h = 303488 N (4 corner hinges), exactly double.
• Margin logic at a 60000 N limit: pinned portal lambda = 2.52906666667, ultimate load required 90000 N, margin 1.68604444444, adequate; fixed portal lambda = 5.05813333333, margin 3.37208888889, adequate. The pinned-base limit load that fails at ultimate is Hc/1.5 = 101162.666667 N. Fixed-fixed beam at a 40000 N limit: lambda = 2.08333333333, margin 1.38888888889, adequate; at a 60000 N limit: lambda = 1.38888888889, margin 0.925925925926, inadequate: the plastic mechanism forms below the FAR 25.303 ultimate load.

Verification

• Confirm plastic_section_modulus("rectangle", 0.05, 0.1) returns 0.000125 and the closed forms equal b*h2/4, d3/6 and the I-beam expression by construction.
• Confirm the shape factors 1.5, 1.69765272631 and 1.12325660721 with the ordering 1.6977 > 1.5 > 1.1233, and the nu = Mp/My identity for every section (anchor residual 2.22044604925e-16).
• Confirm the collapse loads 41666.6666667, 62500 and 83333.3333333 N equal the closed forms 4Mp/L, 6Mp/L and 8*Mp/L with the r + 1 hinge counts and station texts.
• Confirm max |M| over the 2001-station sampled collapse-state diagram equals Mp = 31250 with overshoot no greater than 1e-9 relative for all three beam cases (anchor 0.0) and equality only at the hinge stations.
• Confirm the portal sway values 151744 N (pinned) and 303488 N (fixed), the fixed-base sway exactly double the pinned-base value.
• Confirm ultimate margins 1.38888888889 (adequate) and 0.925925925926 (inadequate) with the verdict flip exactly where the collapse load crosses 1.5 times the limit load.
• Confirm every non-physical input raises ValueError: unknown shape, wrong dims arity, nonpositive dims, I-beam with d <= 2*t_f or t_w >= b, sigma_y <= 0, unknown beam case, span = 0, mp <= 0, unknown frame base, height = 0, nonpositive limit load and ultimate_factor = 0 (13 anchor cases).
• Run the contract test offline: python3 scripts/test_plastic_collapse_analysis.py (34 tests, deterministic).

Related leaves

• structures/fem/beam-frame-analysis: the elastic complement in this pack, the stiffness-method frame solution that stops at the member end actions with no yield stress and no hinge.
• structures/materials/ramberg-osgood: the material stress-strain curve at a point, the input material model context for the yield stress.
• structures/materials/multiaxial-yield-criteria: pointwise yield margins of a stress state, the first-yield side of the check.
• structures/fem/buckling-analysis: elastic instability of slender compression members, the alternative ultimate failure mode.
• structures/fem/cylindrical-shell-buckling: elastic ovalization collapse of curved shell sections, a different geometry and mechanism.

Pitfalls

• Reading the shape factor as a strength margin: nu = Zp/Z is a section reserve, not a factor on the applied load. The fixed-fixed beam collapses at nu times its first-yield load, but the propped cantilever reaches 9*nu/8 because only its single hogging peak yields first, so the load reserve depends on the elastic moment diagram, not on the section alone.
• Treating the collapse load as an ultimate load: collapse at the plastic mechanism is itself failure. The FAR 25.303 check is lambda = collapse load / limit load >= 1.5, so a mechanism that forms below 1.5 times the limit load (margin below 1.0, verdict inadequate) fails the ultimate requirement even though the elastic stresses at the limit load look acceptable.
• Assuming a 50/50 reaction split at collapse: the propped cantilever collapse state carries 4Mp/L at the fixed end against 2Mp/L at the pin, and only that asymmetric diagram satisfies the static theorem with |M| = Mp at both hinge stations.
• Using the elastic section modulus for the fully plastic moment: Mp needs the plastic section modulus Zp about the equal-area axis, not the extreme-fibre Z; on the worked rectangle the two differ by the shape factor 1.5, which is exactly the reserve the plastic check exploits.
• Counting hinges instead of mechanisms: a set of r + 1 hinges is only a collapse mechanism when it forms a kinematically admissible mechanism at a load that satisfies equilibrium with |M| <= Mp, which is why the kinematic and static theorems are applied together.
• Applying I-beam closed forms outside the web: the Zp expression and its shape-factor range hold only for a doubly symmetric I-beam with the plastic neutral axis in the web (d > 2*t_f and t_w < b); the module rejects proportions outside that scope with ValueError.

Behavior contract (gate 3)

Run the deterministic contract test (stdlib unittest, offline):

python3 scripts/test_plastic_collapse_analysis.py

The test covers the section-reserve pass (plastic and elastic section moduli and shape factors of the three worked sections), the yield capacity pass (fully plastic moments and first-yield moments), the plastic hinge mechanisms and collapse loads of the three single-span beam cases by the kinematic theorem, the static theorem cross-check on the 2001-station collapse-state moment diagram, the elastic first-yield context loads and collapse-to-yield ratios, the pinned and fixed base portal sway mechanisms, the collapse load factor and the FAR 25.303 ultimate margin verdicts, the 13 ValueError rejection cases and the determinism check.

Compliance

• Standards referenced, not reproduced: FAR 25.303 and CS 25.303 (structure must withstand 1.5 times the limit loads without failure) frame the ultimate-factor context of the margin; the relations above are standard engineering methodology, summary-only per standards-map.yaml.
• compliance: STANDARDS-REF, gated: false.