plastic-analysis

name	plastic-analysis
description	Plastic structural analysis — plastic hinge theory, limit load (upper/lower bound theorems), plastic moment (Mp, shape factor), collapse mechanisms (beams/frames/slabs), Johansen yield line theory, interaction formulas (M-N-V), shakedown analysis (Melan/Koiter), plastic collapse in pressure vessels (ASME VIII DBA), elastoplastic FEA (bilinear hardening), and plastic collapse assessment (API 579 FAD).
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Plastic Structural Analysis — Complete Skill

Plastic Hinge Theory

Plastic Moment (M_p)

Elastic limit (M_y): M_y = σ_y × S [S = elastic section modulus = I/c; first yield moment]

Plastic moment (M_p): Full plastification of cross-section; rectangular stress block (σ = +σ_y above NA; σ = -σ_y below NA) M_p = σ_y × Z_p [Z_p = plastic section modulus = first moment of area about plastic neutral axis (PNA)]

Shape factor (f): f = M_p / M_y = Z_p / S [always ≥ 1] Rectangular section: Z_p = bh²/4; S = bh²/6 → f = 1.5 Wide flange (W-shape): f ≈ 1.12–1.18 (most W-shapes); flanges carry most of moment Circular solid: f = 4/(3π) × D³ / (π D³/32) = 16/(3π) = 1.70 Circular hollow: f ≈ 1.27 (thin-wall limit)

Plastic neutral axis (PNA): Divides cross-section into equal areas for axial load = 0 A_above × ȳ_above - A_below × ȳ_below = Z_p [first moment of each half]

Plastic Hinge

Definition: When M = M_p at a cross-section; that section rotates freely (unlimited rotation at constant moment M_p) Physical: zone of plasticity spreads from outermost fibers inward; full plastification when M = M_p Hinge length: L_p ≈ 0.5h to 1.0h in length (real hinge has finite length; idealized as point)

Number of hinges for collapse: Statically determinate structure: one hinge → collapse Statically indeterminate (degree of indeterminacy = n): n+1 hinges needed for collapse

Collapse Mechanisms

Simply supported beam (concentrated load P at midspan): One hinge forms at midspan (moment = M_p): P_c = 4M_p / L [collapse load] Shape factor advantage: P_c = f × P_elastic_max = 1.12–1.5 × P_first_yield (for W-shapes)

Fixed-fixed beam (midspan point load): Two hinges at ends (where M = M_p); one at midspan → total 3 hinges for 2 degrees redundancy Virtual work: P × L/2 × δ/L = 2 × M_p × θ + M_p × 2θ → P_c × δ/2 = 4M_p × θ With δ = L/2 × θ: P_c = 16M_p / L

Propped cantilever (point load at midspan): Hinge at fixed end + hinge at load point = 2 hinges for 1 degree redundancy Virtual work: M_p × (θ + 2θ) = P × L/2 × θ → P_c = 6M_p / L

Upper and Lower Bound Theorems

Lower bound (static) theorem: Any stress distribution satisfying equilibrium (internal forces = external loads) + material strength (moment ≤ M_p everywhere) → that load is ≤ P_collapse "Safe load" → never overestimates capacity

Upper bound (kinematic) theorem: Any collapse mechanism (virtual work) gives load ≥ P_collapse "Optimistic" → never underestimates capacity True collapse load: both bounds coincide → P_lower = P_upper = P_collapse

Engineering use: Both bounds rapidly evaluate collapse without full elastoplastic analysis If bounds are close (< 10%): good estimate of P_collapse from either

Plastic Analysis of Frames

Method of Sections (Static)

For statically indeterminate frames:

1. Identify possible hinge locations (loaded sections, joints, regions of maximum moment)
2. Assume collapse mechanism (which sections form hinges)
3. Write equilibrium equations with M = M_p at hinge locations
4. Solve for P_c
5. Verify: M ≤ M_p at all non-hinge cross-sections (lower bound check)

Independent mechanisms: N possible mechanisms for a frame = n + 1 critical sections (at max potential moments) Identify beam mechanisms + sway mechanisms + combined mechanism Combined mechanism: P_c lower than individual → governs

Example (two-story frame under lateral load H): Sway mechanism: four hinges at column bases and tops of first-story columns Virtual work: H × h_story × θ = 4 × M_p × θ → H_c = 4M_p / h

Interaction with Axial Load (M-N Interaction)

Plastic moment with axial load N: For rectangular section: M_pN = M_p × [1 - (N/N_p)²] [N_p = σ_y × A = squash load] Reduced M_p under axial load N

For W-shapes (approximate): M_pN = M_p for N ≤ 0.15 N_p (axial load effect negligible below 15% squash) M_pN = 1.18 M_p × [1 - (N/N_p)] for N > 0.15 N_p (AISC LRFD interaction surface)

Shear interaction: Combined V-M interaction: M_pV = M_p × √(1 - (V/V_p)²) [V_p = plastic shear = 0.6 σ_y × A_web] V interaction significant only when V > 0.6 V_p

Yield Line Theory (Slabs)

Johansen Yield Line Method

Two-way slab collapse analysis: Yield lines: lines along which M = M_p (moment capacity of slab strip) Rigid plate segments rotate about axes along supported edges (axes of rotation)

Isotropic slab (same M_p in both directions): Square slab (simply supported, uniform load w): Yield lines: from corners to center (X-pattern); w_c = 24 M_p / (L²) [L = side length]

Rectangular slab (L × B): w_c from virtual work over assumed yield line pattern: w × Σ (V_i × A_i) = Σ (M_p × l_i × θ_i_relative) [virtual work; V_i = virtual displacement; A_i = segment area; l_i = yield line length]

Orthotropic slab (M_px ≠ M_py): Affine transformation: replace orthotropic with equivalent isotropic by scaling geometry

Nodal forces: At corners of slab and free edges: nodal force contribution to virtual work Complex for unsupported corners; use equilibrium at yield line junction

Pressure Vessel Plastic Analysis

ASME VIII Division 2 Plastic Analysis

Limit load analysis (ASME VIII App. 5): Apply load amplification factor: load multiplier λ_L = P_applied × (some factor) Determine λ_L at which plastic collapse occurs (incremental FEA or analytical limit load) ASME requirement: λ_L (collapse) ≥ 2 × P_design [safety factor 2 for pressure vessel] Or directly: P_max_elastic / φ where φ from ASME limit load factor

Plastic collapse criterion (ASME VIII Div. 2, Part 5): Twice-elastic-slope criterion (TES): load vs. displacement; draw line with twice initial elastic slope; intersection with actual curve = collapse load Alternative: plastic strain limit method (P_c when ε_p > 5% anywhere in wall) Limit analysis: analytical or FEA; most accurate for complex geometries

Net Section Collapse (NSC) for flawed vessels (API 579): FAD (Failure Assessment Diagram) approach at collapse limit L_r = P / P_yield [ratio of applied load to yield load] K_r = K_I / K_Ic [ratio of stress intensity to fracture toughness] Failure if: K_r²/(1-L_r^1.4f(L_r)) > 1 [combined fracture + collapse; API 579 Level 2]

Shakedown Analysis

Melan's Theorem (Lower Bound Shakedown)

Shakedown (elastic shakedown): After initial plastic deformation, structure responds elastically forever Useful for variable loading (pressure cycling, thermal transients)

Melan's theorem: If a time-invariant residual stress field ρ(x) exists such that: σ_elastic(x,t) + ρ(x) is within yield surface for ALL load combinations → shakedown occurs Elastic action at steady state → plastic deformation bounded

Shakedown load P_s: P_s ≥ P_y (first yield); P_s ≤ 2 × P_y (Bleich factor for combined thermal+mechanical) Bauschinger: P_s ≤ 2 × P_y for kinematic hardening; = 2 P_y for elastic-perfectly plastic

Ratcheting (no shakedown): Net plastic strain accumulates each cycle → progressive collapse or fatigue Occurs when: σ_elastic range > 2 × σ_y (no residual stress can contain) ASME III: 3S_m limit for primary+secondary to prevent ratcheting

Koiter's Theorem (Upper Bound Shakedown)

Kinematic shakedown: Any combination of kinematically admissible plastic strain rates ε̇_p gives: P_s ≤ P_calculated Provides upper bound; true P_s bracketed between Melan and Koiter bounds

FEA Plastic Analysis

Material Models

Elastic-perfectly plastic: σ = E × ε for ε < σ_y/E σ = σ_y for ε ≥ σ_y/E (no hardening) Conservative; simple; used for limit analysis

Bilinear kinematic hardening: Hardening slope E_t after yield; σ = σ_y + E_t × ε_p E_t = 0.01–0.05 × E typical (strain hardening)

Multilinear isotropic hardening: True stress-strain curve from coupon test; most accurate for real material

Implementation (ABAQUS/Ansys): *PLASTIC, HARDENING=ISOTROPIC (or KINEMATIC) σ₁, ε_p1; σ₂, ε_p2; ... [tabulated true stress-plastic strain pairs]

Arc-Length Method (Riks)

Required for snap-through and limit load tracing: Standard Newton-Raphson fails at limit point (singular stiffness) Arc-length: traces load-displacement path including post-collapse Ansys: NLGEOM, ON; Arc-length control in Solution Controls ABAQUS: *STATIC, RIKS

Interpretation: P vs. displacement curve: limit load where load reaches maximum (before decrease) Limit load ≈ load at first local maximum of curve

Standards and References

Standard	Scope
ASME VIII Div. 2 Appendix 5	Plastic and limit load analysis for pressure vessels
AISC 360-22 Ch. C/H	Inelastic analysis for steel frames
Eurocode 3 EN 1993-1-1	Plastic design of steel structures
API 579-1/ASME FFS-1 App. A	Failure assessment (plastic collapse component)
Johansen "Yield Line Theory" (1962)	Original yield line reference

Output

Provide: structure type (beam/frame/slab/pressure vessel), material (S_y [MPa]; hardening model), cross-section (Z_p [cm³]; S [cm³]; shape factor f; M_p [kN·m]), mechanism type and assumed hinge locations (with virtual work equation written out), collapse load P_c [kN] or distributed load w_c [kPa] (from upper bound calculation), lower bound check (M ≤ M_p at all non-hinge sections: pass/fail), M-N interaction check if axial load (M_pN [kN·m]; N/N_p ratio), shakedown analysis (σ_elastic_range vs. 2σ_y; shakedown or ratcheting?), ASME VIII limit load (λ_L ≥ 2? or TES method applied), API 579 FAD point (K_r, L_r) if flaw present, FEA material model (E_t [GPa]; convergence), collapse load from FEA [kN] vs. analytical bound, and applicable standard (AISC 360, ASME VIII Div. 2 App. 5, API 579, Eurocode 3).