What is the plastic moment (Mp) and how does it differ from the elastic moment (My)?

The plastic moment (Mp) is the bending moment at which the entire cross-section has yielded in tension and compression, creating a fully plastic stress distribution. This differs fundamentally from the elastic moment (My) where only the extreme fibers are at yield stress and the interior remains elastic. Elastic moment: My = Sx × Fy, where Sx = I/c is the elastic section modulus. The stress distribution is linear (triangular) — zero at neutral axis, Fy at extreme fibers. Plastic moment: Mp = Zx × Fy, where Zx is the plastic section modulus. The stress distribution is rectangular — Fy in compression above the plastic neutral axis (PNA), Fy in tension below. The PNA divides the cross-section into equal areas (C = T = Fy × A/2). The shape factor ξ = Zx/Sx = Mp/My represents the capacity gain from plastic design: W shapes (strong axis): ξ = 1.12-1.18, giving 12-18% more moment capacity than elastic — the most common practical range for structural design. HSS square: ξ = 1.12-1.15, similar to W shapes. HSS round: ξ = 1.27, higher because more material is concentrated away from the neutral axis. Solid circle: ξ = 1.70, substantial gain because the circular shape has more area far from the neutral axis. Solid rectangle: ξ = 1.50, because the rectangular stress block uses outer fibers fully. I-shapes gain less than solid shapes because the flanges already carry most of the moment elastically — the web contributes relatively little to Mp beyond My. LRFD design for compact sections already uses Zx for moment capacity (φMn = φ × Mp = 0.90 × Zx × Fy), so the 'plastic design' benefit is already partially captured in standard LRFD. Full plastic analysis goes further by allowing moment redistribution between sections.

What are the compact section requirements for plastic design?

Plastic design requires compact sections per AISC 360 Table B4.1b — sections that can develop the full plastic moment and sustain inelastic rotation without local buckling. The width-to-thickness limits for compression elements: W-shape flange (λ = bf/2tf): compact limit λp = 0.38√(E/Fy). For A992 (Fy = 50 ksi): λp = 0.38 × √(29,000/50) = 9.15. Most W shapes have bf/2tf = 5-9, satisfying the compact limit. W-shape web (λ = h/tw): compact limit λp = 3.76√(E/Fy). For Fy = 50 ksi: λp = 3.76 × √(580) = 90.6. Most W shapes have h/tw = 10-55, well within the compact limit. HSS flange (λ = b/t): compact limit λp = 1.12√(E/Fy). For Fy = 46 ksi (A500 Gr B): λp = 1.12 × √(630) = 28.1. HSS6×6×3/16 has b/t = 30.0 > λp → noncompact (not eligible for plastic design without Qs reduction). HSS6×6×1/4 has b/t = 22.0 < λp → compact. Round HSS (λ = D/t): compact limit λp = 0.07E/Fy. For Fy = 42 ksi: λp = 0.07 × 690 = 48.3. The rotation capacity R (ability to sustain plastic rotation before local buckling) is directly related to the slenderness ratio λ/λp — sections with λ < 0.5λp have high rotation capacity (suitable for SMF), while sections near λp have limited rotation capacity. AISC 341 for seismic design imposes stricter limits: λ ≤ λp (same as compact) but additionally requires demonstrated rotation capacity ≥ 0.04 rad for SMF connections. Most hot-rolled W shapes are compact for A992 steel. Some HSS shapes (thin-walled) are noncompact or slender, limiting their use in plastic design.

How does plastic hinge formation and collapse mechanism analysis work?

Plastic collapse analysis tracks the formation of plastic hinges until a mechanism (unstable configuration) forms. A plastic hinge is a cross-section where M = Mp and rotation can occur at constant moment — it behaves like a frictionless hinge with a constant resisting moment Mp. Collapse mechanism types: (1) Beam mechanism — hinges at supports and midspan. For a fixed-fixed beam under uniform load w: first hinges form at fixed supports (M = wL²/12 elastic → Mp at collapse), then a third hinge at midspan completes the mechanism. Collapse load: w_collapse = 16Mp/L². Elastic design would limit w to 12My/L². The ratio = 16Mp/12My = (4/3) × (Zx/Sx) ≈ 1.5× — plastic design allows ~50% more load for this configuration. (2) Sway mechanism — hinges at column tops and bottoms. Single-story portal frame with lateral load H: collapse load H_collapse = 4Mp/h (four hinges, one at each column end). (3) Combined mechanism — beam + sway, common in multi-story frames under combined gravity and lateral loads. (4) Panel mechanism — hinges around a panel in a Vierendeel frame. The collapse load is found by either: Virtual Work Method — equate external work to internal work: Σ(Pi × δi) = Σ(Mp × θj), where δi = virtual displacements and θj = hinge rotations. Assume a collapse mechanism with small virtual displacements, calculate work terms, solve for collapse load. Static (Equilibrium) Method — find the statically admissible moment distribution where Mp is reached at sufficient locations to form a mechanism, satisfying equilibrium everywhere and M ≤ Mp everywhere. The lowest collapse load from all possible mechanisms governs (lower bound theorem). Number of hinges required: for a statically indeterminate structure of degree r, at least r + 1 plastic hinges are needed to form a complete mechanism. A fixed-fixed beam (r = 3) requires 4 hinges, but 3 hinges actually form a mechanism because of the specific support conditions.

When is plastic design permitted and what are its limitations per AISC 360?

AISC 360-22 Section C1 and Appendix 1 govern plastic analysis and design. Plastic design is permitted when ALL of the following conditions are met: (1) Steel grade — Fy ≤ 65 ksi. Higher-strength steels (A514 Gr 100, Fy = 100 ksi) do not have sufficient ductility for plastic design. A992 (Fy = 50 ksi) is the standard plastic design steel. (2) Section type — all members where plastic hinges form must be compact (doubly symmetric). Singly-symmetric sections (channels, angles) are not eligible because the PNA does not coincide with the elastic neutral axis, complicating the hinge behavior. (3) Lateral bracing — adequate to prevent lateral-torsional buckling (LTB) before Mp is reached. Near plastic hinges, the unbraced length Lb ≤ Lp (plastic limit state per AISC F2). Between hinges, longer unbraced lengths may be acceptable with reduced capacity. (4) Load type — static loads only (not fatigue loading). Cyclic plastic deformation under fatigue loading leads to low-cycle fatigue failure. Plastic design is NOT permitted for crane runways, bridges, or machine supports. (5) Second-order effects — P-Δ and P-δ must be explicitly considered. Plastic analysis amplifies second-order effects because hinges reduce lateral stiffness. AISC Appendix 1 requires a second-order plastic analysis or the use of the amplified first-order method with stricter limits. (6) Connections — must develop the full plastic moment Mp of connected members. This is more demanding than elastic design connections (which only need to develop the design moment Mu < Mp). For EEP and BFP moment connections, the connection design moment = 1.1RyMp per AISC 358 for seismic, but for non-seismic plastic design, developing Mp is the minimum. (7) Serviceability — deflections at service load (not collapse load) must still satisfy limits. Plastic design increases the collapse load but service deflections are calculated elastically. Limitations: plastic design is most beneficial for highly indeterminate structures (continuous beams, frames) where redistribution is significant. For simply-supported beams or determinate structures, there is no redistribution benefit — Mp is used but no mechanism analysis is needed.

How much extra capacity does plastic design provide compared to elastic design?

The capacity gain from plastic design comes from two sources combined: (1) Section capacity gain (shape factor ξ = Zx/Sx): W shapes: 12-18% (ξ = 1.12-1.18), typical W24× beams have ξ ≈ 1.14 so Mp is 14% higher than My. HSS square: 12-15%. Round HSS: 27%. Solid rectangle: 50%. Solid circle: 70%. (2) Redistribution gain (mechanism vs first-hinge): This gain depends on the degree of static indeterminacy. Fixed-fixed beam (r = 3): elastic M_max = wL²/12 at supports, plastic collapse w = 16Mp/L². Elastic limit w_elastic = 12My/L². Collapse/elastic ratio = (16Mp)/(12My) = (4/3) × ξ. With ξ = 1.14: ratio = 1.52, or 52% more capacity. This is the combined section + redistribution benefit. Two-span continuous beam (r = 2): elastic M_support = 0.125wL², M_mid = 0.07wL². First hinge at support, then redistribution to midspan. Plastic Mp ≈ 0.086wL². Ratio ≈ 0.125/0.086 = 1.45 → 45% gain. Propped cantilever (r = 1): fixed end hinge + midspan hinge. Collapse/elastic ≈ 1.3 → 30% gain. Simply-supported beam (r = 0): determinate, no redistribution possible. Only section gain applies: 12-18% for W shapes. The practical benefit of plastic design is largest for: (1) continuous beams with 3+ spans (high redundancy), (2) portal frames where sway mechanism allows full redistribution, (3) fixed-base columns where base hinge formation redistributes moment to the beam. For typical steel building design: LRFD already captures the section capacity gain (Zx vs Sx) for compact sections. Full plastic analysis capturing the redistribution gain adds 15-35% more capacity for highly indeterminate structures, but is rarely used in day-to-day design because the additional analysis effort outweighs the material savings for typical structures. It is most commonly applied to portal frames, industrial structures, and continuous beams where the extra capacity justifies the analysis.

Plastic Design — AISC Plastic Analysis Method

Q: How does plastic hinge formation and collapse mechanism analysis work?

Plastic collapse analysis tracks the formation of plastic hinges until a mechanism (unstable configuration) forms. A plastic hinge is a cross-section where M = Mp and rotation can occur at constant moment — it behaves like a frictionless hinge with a constant resisting moment Mp. Collapse mechanism types: (1) Beam mechanism — hinges at supports and midspan. For a fixed-fixed beam under uniform load w: first hinges form at fixed supports (M = wL²/12 elastic → Mp at collapse), then a third hinge at midspan completes the mechanism. Collapse load: w_collapse = 16Mp/L². Elastic design would limit w to 12My/L². The ratio = 16Mp/12My = (4/3) × (Zx/Sx) ≈ 1.5× — plastic design allows ~50% more load for this configuration. (2) Sway mechanism — hinges at column tops and bottoms. Single-story portal frame with lateral load H: collapse load H_collapse = 4Mp/h (four hinges, one at each column end). (3) Combined mechanism — beam + sway, common in multi-story frames under combined gravity and lateral loads. (4) Panel mechanism — hinges around a panel in a Vierendeel frame. The collapse load is found by either: Virtual Work Method — equate external work to internal work: Σ(Pi × δi) = Σ(Mp × θj), where δi = virtual displacements and θj = hinge rotations. Assume a collapse mechanism with small virtual displacements, calculate work terms, solve for collapse load. Static (Equilibrium) Method — find the statically admissible moment distribution where Mp is reached at sufficient locations to form a mechanism, satisfying equilibrium everywhere and M ≤ Mp everywhere. The lowest collapse load from all possible mechanisms governs (lower bound theorem). Number of hinges required: for a statically indeterminate structure of degree r, at least r + 1 plastic hinges are needed to form a complete mechanism. A fixed-fixed beam (r = 3) requires 4 hinges, but 3 hinges actually form a mechanism because of the specific support conditions.

Q: How much extra capacity does plastic design provide compared to elastic design?

The capacity gain from plastic design comes from two sources combined: (1) Section capacity gain (shape factor ξ = Zx/Sx): W shapes: 12-18% (ξ = 1.12-1.18), typical W24× beams have ξ ≈ 1.14 so Mp is 14% higher than My. HSS square: 12-15%. Round HSS: 27%. Solid rectangle: 50%. Solid circle: 70%. (2) Redistribution gain (mechanism vs first-hinge): This gain depends on the degree of static indeterminacy. Fixed-fixed beam (r = 3): elastic M_max = wL²/12 at supports, plastic collapse w = 16Mp/L². Elastic limit w_elastic = 12My/L². Collapse/elastic ratio = (16Mp)/(12My) = (4/3) × ξ. With ξ = 1.14: ratio = 1.52, or 52% more capacity. This is the combined section + redistribution benefit. Two-span continuous beam (r = 2): elastic M_support = 0.125wL², M_mid = 0.07wL². First hinge at support, then redistribution to midspan. Plastic Mp ≈ 0.086wL². Ratio ≈ 0.125/0.086 = 1.45 → 45% gain. Propped cantilever (r = 1): fixed end hinge + midspan hinge. Collapse/elastic ≈ 1.3 → 30% gain. Simply-supported beam (r = 0): determinate, no redistribution possible. Only section gain applies: 12-18% for W shapes. The practical benefit of plastic design is largest for: (1) continuous beams with 3+ spans (high redundancy), (2) portal frames where sway mechanism allows full redistribution, (3) fixed-base columns where base hinge formation redistributes moment to the beam. For typical steel building design: LRFD already captures the section capacity gain (Zx vs Sx) for compact sections. Full plastic analysis capturing the redistribution gain adds 15-35% more capacity for highly indeterminate structures, but is rarely used in day-to-day design because the additional analysis effort outweighs the material savings for typical structures. It is most commonly applied to portal frames, industrial structures, and continuous beams where the extra capacity justifies the analysis.

Plastic design (also called limit state design or collapse design) uses the full plastic capacity of steel sections to determine member strength. Unlike elastic design, which limits stresses to the yield point, plastic design recognizes that steel can carry additional load after yielding through redistribution. This page covers the principles, methods, and AISC requirements for plastic design.

Elastic vs Plastic Design

Aspect	Elastic Design	Plastic Design
Stress limit	Fy (first yield)	Fy across full section
Moment capacity	Sx × Fy (elastic section modulus)	Zx × Fy (plastic section modulus)
Section modulus	Sx	Zx (always ≥ Sx)
Capacity gain	Baseline	15-30% more capacity
Redistribution	None	Yes, moment redistribution
Compact sections	Not always required	Required
Analysis	Elastic (linear)	Mechanism (nonlinear)
Application	All structures	Specific conditions per AISC

Plastic Moment Capacity

The plastic moment (Mp) is the moment capacity when the entire cross section has yielded:

Mp = Zx × Fy

where Zx = plastic section modulus, Fy = specified yield strength.

Shape Factor

The shape factor relates plastic and elastic section moduli:

Shape factor = Zx / Sx

Shape	Typical Shape Factor	Capacity Gain vs Elastic
W shape (strong axis)	1.12 - 1.18	12-18%
HSS (square)	1.12 - 1.15	12-15%
HSS (round)	1.27	27%
Solid circle	1.70	70%
Solid rectangle	1.50	50%
Angle	1.15 - 1.25	15-25%

W shapes gain 12-18% more moment capacity when designed plastically versus elastically.

Compact Section Requirements

Plastic design requires compact sections that can develop the full plastic moment without local buckling. AISC Table B4.1 defines the limits:

Width-to-Thickness Limits (Compression Elements)

Element	Compact Limit (λp)	Noncompact Limit (λr)
Flange of W (λ = bf/2tf)	0.38 × √(E/Fy)	1.0 × √(E/Fy)
Web of W (λ = h/tw)	3.76 × √(E/Fy)	5.70 × √(E/Fy)
Flange of HSS (λ = b/t)	1.12 × √(E/Fy)	1.40 × √(E/Fy)
Wall of round HSS (λ = D/t)	0.07 × E/Fy	0.31 × E/Fy

For A992 (Fy = 50 ksi), the compact limits are:

Element	λp (Compact)	λr (Noncompact)
W flange (bf/2tf)	9.19	24.1
W web (h/tw)	90.6	137.3
HSS wall (b/t)	27.0	33.7

Most standard W shapes are compact for A992 steel. Some HSS shapes are noncompact or slender, limiting their use in plastic design.

Plastic Hinge Formation

A plastic hinge forms when the moment reaches Mp at a section. The hinge allows rotation at constant moment (Mp), redistributing load to other parts of the structure.

Hinge Formation Sequence

For a fixed-fixed beam under uniform load:

First hinge: At the fixed supports (maximum negative moment)
Redistribution: Load transfers to mid-span as hinges rotate
Second hinge: At mid-span (positive moment)
Mechanism: Three hinges create a collapse mechanism

The collapse load is reached when enough hinges form to create a mechanism.

Collapse Mechanism Analysis

Types of Mechanisms

Mechanism Type	Configuration	Example
Beam mechanism	Hinges at supports + mid-span	Fixed-fixed beam
Sway mechanism	Hinges at column tops and bottoms	Single-story frame
Combined mechanism	Beam + sway combined	Multi-story frame
Panel mechanism	Hinges around a panel	Vierendeel frame

Virtual Work Method

The collapse load is found by equating external work to internal work:

External work = Internal work Σ (P_i × δ_i) = Σ (Mp × θ_j)

where P_i = applied loads, δ_i = displacements at load points, Mp = plastic moment capacity, θ_j = hinge rotations.

Static (Equilibrium) Method

An alternative approach: find the equilibrium moment distribution that satisfies the collapse condition:

Assume hinge locations (at points of maximum moment)
Draw the collapse moment diagram
Solve for the collapse load using equilibrium

Plastic Design per AISC 360

AISC 360 Chapter C and Appendix 1 address plastic analysis and design.

When Plastic Design Is Permitted

Condition	Requirement
Steel grade	Fy ≤ 65 ksi
Section type	Compact (doubly symmetric)
Lateral bracing	Adequate to prevent LTB before Mp
Load type	Static (not fatigue loading)
Second-order effects	Must be considered (P-δ, P-Δ)
Connections	Must develop Mp of connected members

Lateral Bracing Requirements for Plastic Design

Condition	Maximum Unbraced Length
Where Mp is developed	Per AISC Section F2.2 (for compact sections)
Near plastic hinges	Lp ≤ Lr (must prevent LTB)
Between plastic hinges	May be longer with reduced capacity
Last hinge to form	Elastic design limits apply

Plastic Design of Continuous Beams

Fixed-Fixed Beam

Elastic design: M_max = wL²/12 (at supports), M_mid = wL²/24

Plastic design: M_p = wL²/16 (all three hinges at Mp)

Capacity ratio: (wL²/16) / (wL²/12) = 0.75, meaning plastic design allows 33% more load (12/16 = 0.75, so plastic load / elastic load = 16/12 = 1.33).

Two-Span Continuous Beam

Elastic design: M_support = 0.125 wL², M_mid = 0.07 wL²

Plastic design: First hinge at interior support, then redistribution to mid-span. Mp = 0.086 wL².

Plastic Design of Frames

For portal frames and multi-story frames, plastic analysis finds the collapse mechanism:

Portal Frame

Beam mechanism: Mp develops at beam ends and mid-span

Sway mechanism: Mp develops at column tops and bases

Combined mechanism: The actual collapse mode is typically a combination.

The lowest collapse load from all possible mechanisms governs.

Moment Redistribution

AISC 360 allows up to 10% moment redistribution for elastic analysis of continuous beams. Full plastic design allows complete redistribution but requires compact sections and adequate bracing.

Frequently Asked Questions

What is the plastic moment? The plastic moment (Mp) is the maximum bending moment a cross section can resist when the entire section has yielded. Mp = Zx × Fy, where Zx is the plastic section modulus and Fy is the yield strength. For W shapes, Mp is about 12-18% higher than the elastic moment (My = Sx × Fy).

When can I use plastic design? Plastic design is permitted when: (1) all sections are compact, (2) Fy ≤ 65 ksi, (3) adequate lateral bracing is provided, (4) loads are static (not fatigue), and (5) connections can develop the full plastic moment. It is most beneficial for continuous beams and portal frames.

What is a plastic hinge? A plastic hinge is a cross section where the moment equals the plastic moment (Mp). The section yields and can rotate at constant moment, allowing load redistribution. A sufficient number of plastic hinges creates a collapse mechanism.

Why is plastic design not used more often? Plastic design requires compact sections, special bracing, and more complex analysis. Most building design uses LRFD (which already incorporates some plastic capacity through Zx for compact sections). Full plastic analysis is most beneficial for portal frames, continuous beams, and industrial structures where the extra capacity justifies the additional analysis effort.

Beam Capacity Calculator — Design a steel beam
Beam Formulas — Deflection and moment formulas
Section Properties — Sx and Zx data
Allowable Stress Design — ASD method
Portal Frame Design — Portal frame analysis

Disclaimer

This is a calculation tool, not a substitute for professional engineering certification. All results must be independently verified by a licensed Professional Engineer (PE) or Structural Engineer (SE) before use in construction, fabrication, or permit documents. The user is responsible for the accuracy of all inputs and the verification of all outputs.