limit-analysis

name	limit-analysis
description	Plastic limit analysis — lower and upper bound theorems, yield line theory (slabs), collapse mechanisms, virtual work method, interaction diagrams (M-N-V), shakedown analysis, progressive collapse, ASME VIII plastic design rules, Drucker-Prager yield criterion, FEA limit load analysis.
metadata	{"priority":7,"promptSignals":{"phrases":["limit analysis","plastic collapse","limit load","yield line analysis","lower bound theorem","upper bound plasticity"],"minScore":3}}

Plastic Limit Analysis — Complete Skill

Fundamental Theorems of Plasticity

Lower Bound Theorem

Statement: Any statically admissible stress field that satisfies equilibrium and does not violate yield anywhere provides a lower bound on the true collapse load. P_collapse_true ≥ P_lower_bound [any safe equilibrium solution is conservative]

Requirements for lower bound:

1. Stress field must satisfy equilibrium (∂σ_ij/∂x_j + b_i = 0)
2. Stress must satisfy yield criterion (e.g., von Mises: σ_eff ≤ σ_y) everywhere
3. Boundary conditions on tractions must be satisfied
4. Does NOT require stress to be the actual stress distribution

Approach: assume any statically admissible stress distribution → compute max load → P_collapse ≥ this value

Upper Bound Theorem

Statement: Any kinematically admissible mechanism that satisfies compatibility and velocity boundary conditions, when equated with external work rate = internal plastic dissipation rate, gives an upper bound on collapse load. P_collapse_true ≤ P_upper_bound [any mechanism overestimates collapse load]

Virtual work principle: P × δ = Σ M_p × θ [for plastic hinge mechanisms; δ = virtual displacement; θ = rotation at plastic hinge; M_p = plastic moment]

Requirements for upper bound:

1. Mechanism must be kinematically compatible (satisfy velocity BC)
2. Internal dissipation calculated from correct flow rule (associated plasticity)
3. Does NOT require equilibrium at non-hinge points

True collapse load lies between: P_lower ≤ P_true ≤ P_upper

Plastic Moment Capacity

Fully plastic moment (rectangular cross section): M_p = σ_y × b × t² / 4 [b = width; t = thickness; shape factor f = M_p / M_e = 1.5]

For W (wide flange) sections: f ≈ 1.12–1.18 (shape factor; ratio of plastic to elastic section modulus) M_p = σ_y × Z_p [Z_p = plastic section modulus; tabulated in steel design codes]

Plastic section modulus: Z_p = ∫ |y| dA [integration over full cross section; first moment of area about plastic neutral axis] For rectangular: Z_p = b × t²/4; Z_e (elastic) = b × t²/6; Z_p/Z_e = 1.5

Beam and Frame Collapse Analysis

Single-Span Beam (Propped Cantilever)

Propped cantilever, uniform load w per unit length, span L: Mechanism 1 (interior hinge at x₀ + end hinge at fixed end): External work: w × L × δ_avg = w × L × δ/2 [δ = mid-span virtual displacement] Internal work: M_p × θ_A + M_p × θ_x₀ Equate: w_collapse = 11.66 M_p / L² [from optimal hinge location x₀ = 0.414L]

Fixed-fixed beam, central point load P: Two hinges (both ends) + one at midspan = mechanism (degree of indeterminacy = 2) P × δ = M_p × θ + M_p × 2θ + M_p × θ = 4 M_p × θ; δ = L/2 × θ P_collapse = 8 M_p / L

Simply supported beam (single span, central load): One plastic hinge at midspan: P × δ = M_p × 2θ; δ = L/2 × θ → P_collapse = 4 M_p / L (Statically determinate → lower bound = upper bound; exact)

Portal Frame Collapse

Rectangular portal frame (span L, height H, fixed bases): Three mechanisms:

1. Beam mechanism: hinges in beam (at beam-column joints and beam center)
2. Sway mechanism: hinges at column bases + beam-column joints
3. Combined (critical): combination of 1 and 2

Beam mechanism (horizontal load H_f, vertical load V_f): P_beam = 2 M_p × (1/L + 2/L) = 4 M_p / L [for vertical load on beam] P_sway = 2 M_p × 2 / H = 4 M_p / H [for horizontal load on frame; sway mechanism] Combined: typically gives lower collapse load → controls design

Rank of mechanisms: Number of possible mechanisms = degree of static indeterminacy + 1 Frame with 3 redundants → 4 possible mechanisms → test all; minimum P_upper controls

Yield Line Theory (RC Slabs / Thin Plates)

Application to Reinforced Concrete Slabs

Yield line: line of maximum moment where plastic rotation concentrates; separates slab into rigid regions At yield line: M = M_n (nominal moment capacity per unit width)

Typical yield line patterns: Simply supported rectangular slab: diagonal yield lines from corners to center Two-way slab with all edges fixed: yield lines along diagonals + parallel lines near edge Circular slab, central load: radial yield lines (fan pattern)

Virtual work method: External work: W_ext = Σ(loads × virtual displacement of their application point) Internal work: W_int = Σ(M_n × yield_line_length × θ_n) [θ_n = component of rotation perpendicular to yield line]

For isotropic square slab (side a, uniform load w, simply supported): W_ext = w × a² × δ/3 [average displacement = δ/3 for triangular shape factor] W_int = M_n × 4 × a × θ_y = 4 M_n × a × 2δ/a = 8 M_n × δ w_collapse = 24 M_n / a²

Orthotropic slabs (M_x ≠ M_y): Johansen's stepped yield criterion: M_x cos²α + M_y sin²α = M_n (along yield line at angle α) Affine transformation: scale coordinates to convert to isotropic problem → solve → scale back

Shakedown Analysis

Melan's Shakedown Theorem (Lower Bound)

Shakedown: structure may yield in first load cycles but eventually reaches elastic steady state Shakedown load P_SD: max load where structure shakes down (no progressive incremental collapse)

Melan's theorem: if any time-independent residual stress ρ_ij can be found such that (σ_ij^elastic + ρ_ij) does not violate yield for all loads in load space → structure will shake down

Shakedown limit vs. collapse limit: P_SD ≤ P_collapse (always) For non-cyclic loading: P_SD = P_collapse For cyclic load range ΔP: P_SD < P_collapse (shakedown governs)

ASME VIII-2 (Pressure Vessel) shakedown rule: Allowable load = 2/3 of limit load (collapse load); elastic stress range ≤ 2 S_y (Bree diagram) For thermal + pressure: use Bree diagram → region of shakedown vs. ratcheting

Ratcheting (Progressive Deformation)

Ratcheting: incremental plastic strain each cycle → eventual failure or excessive deformation Bree diagram: boundary between elastic shakedown, reverse plasticity, and ratcheting Load parameter: X = σ_p / S_y (pressure stress); Y = σ_T / S_y (thermal stress) Shakedown boundary: X + Y/2 ≤ 1 (simplified); Bree diagram gives exact boundary

M-N-V Interaction (Combined Loading)

Beam Cross Section Interaction Diagram

Combined axial force N and moment M: For rectangular section: N/N_p + (M/M_p)² = 1 [approximate; N_p = σ_y × b × t; M_p = σ_y × b × t²/4]

Exact rectangular interaction: M/M_p = 1 - (N/N_p)² [complete plastic capacity with axial]

For W-sections (simplified): M/M_p = 1 - (N/N_p)^1.18 [approximate; from finite element calibration]

Including shear V: τ_y = σ_y/√3 (Tresca/Mises); reduced moment capacity with shear: M_pv = M_p × √(1 - (V/V_p)²) × (1 - V²/(3 V_p²)) [approximate Timoshenko interaction]

Drucker-Prager yield criterion (for soils and rocks): f = αI₁ + √J₂ - k = 0 [I₁ = first stress invariant; J₂ = second deviatoric invariant; α, k = material constants] α = 2sinφ / (√3 × (3-sinφ)); k = 6c cosφ / (√3 × (3-sinφ)) [c = cohesion; φ = friction angle] Used in foundation limit analysis → bearing capacity

ASME Plastic Design Rules

ASME VIII Division 2 — Limit Load Analysis

Elastic-perfectly plastic (EPP) model: σ_y taken at temperature; no strain hardening; most conservative Limit load: structure cannot sustain any additional load increment (collapse)

Lower bound limit load procedure (ASME KD-230):

1. Run elastic-plastic FEA with EPP material; monotonically increase load
2. Limit load = load when solution does not converge or displacement accelerates unboundedly
3. Factor of safety: design load = limit load / 2.4 (ASME VIII-2 standard safety factor = 1.5 for primary membrane stress limit)

Pseudo-elastic methods (ASME Annex B): Twice-yield method: run elastic analysis; if max stress > 2σ_y → go to plastic analysis Elastic compensation method (ECM): iteratively redistribute over-stressed element stiffness → converges to lower bound

Progressive Collapse (AISC DoD Criteria)

Progressive collapse: local failure triggers redistribution → cascading global failure (Ronan Point, 2001 WTC) GSA/DoD criteria: remove any single column → alternate load path must carry load without collapse Demand-Capacity Ratio (DCR): DCR = Q_UD / Q_CE ≤ 2.0 [Q_UD = unfactored demand; Q_CE = expected capacity]

Tie force method (DoD UFC 4-023-03): Horizontal tie force: F_t = minimum(beam capacity or required tie force) Vertical tie: column design to carry load from column above without support below Peripheral tie: perimeter beam tie around floor

FEA Limit Load Analysis

Method: elastic-perfectly plastic material; gradually increase load (arc-length method in ANSYS/Abaqus) GMNPA (Geometric and Material Nonlinear Plasticity Analysis): Includes large deformation (updated Lagrange); captures snap-through and post-buckling Load-displacement curve: rises to limit load → load drops or displacement accelerates → plastic collapse

Tangent stiffness method: Monitor global stiffness K_T: limit load when K_T → 0 (stiffness vanishes at collapse) Sensitivity to mesh: elements in yield zone must capture strain gradient; min 4 elements across expected yield zone

Standards

Standard	Scope
ASME VIII Division 2 Annex B	Plastic collapse and limit analysis for pressure vessels
AISC 360 (Plastic Design)	Chapter I: composite sections; Appendix 1: inelastic analysis
DoD UFC 4-023-03	Design of buildings to resist progressive collapse
EN 1993-1-1 (Eurocode 3)	Steel structures including plastic design
ACI 318	RC structures (yield line theory for slabs in commentary)
ASME KD-230	Pressure vessel limit analysis procedures

Output

Provide: structure type (beam/frame/slab/pressure vessel), loading (P [kN], w [kN/m], moment [kN·m]), plastic moment capacity M_p [kN·m] (cross section dimensions, σ_y), collapse mechanism type (beam/sway/combined/yield-line), upper bound load P_upper [kN] from virtual work (show work), lower bound P_lower [kN] from equilibrium, true P_collapse estimate (average or confirmed exact), shape factor f = M_p/M_e, interaction diagram check (N/N_p + (M/M_p)² ≤ 1), shakedown assessment (Bree diagram if cyclic), FEA limit load (EPP material, load factor at collapse), safety margin vs. design load, progressive collapse check (DCR < 2.0), and applicable standard (ASME VIII-2 Annex B, AISC 360, DoD UFC 4-023-03).