Plastic Moment (Mp) — Fy × Z, Full Section Yielding

The plastic moment (Mp) is the flexural capacity a steel cross-section achieves when every fiber has yielded — tension on one side of the plastic neutral axis, compression on the other, all at the yield stress Fy. It is the maximum possible moment the section can carry before local buckling, lateral-torsional buckling, or strain limits intervene.

Where the yield moment My = Fy × S represents first yielding (extreme fiber reaches Fy), the plastic moment Mp = Fy × Z represents full plastification. The ratio Z/S (shape factor) quantifies how much additional moment the section can carry between first yield and the plastic hinge.

PRELIMINARY — NOT FOR CONSTRUCTION. All content is for educational and reference use only. Must be independently verified by a licensed Professional Engineer (PE) or Structural Engineer (SE) before use in any project.

Derivation — From Stress Blocks to Mp

When a compact beam section is loaded beyond My, yielding progresses inward from the extreme fibers. At the fully plastic state:

     Compression (all fibers at Fy)
    ┌──────────────────────────────┐
    │  C = Fy × Ac                │ ← area above PNA
    ├──────────────────────────────┤ ← Plastic Neutral Axis (PNA)
    │  T = Fy × At                │ ← area below PNA
    └──────────────────────────────┘
     Tension (all fibers at Fy)

For equilibrium: C = T → Ac = At = A/2 (equal areas above and below the PNA)

The moment is the couple formed by C and T:

Mp = C × ȳc + T × ȳt = Fy × (Ac × ȳc + At × ȳt) = Fy × Z

Where:

Ac, At = areas above and below PNA (= A/2 for doubly-symmetric sections)
ȳc, ȳt = distances from PNA to centroids of Ac, At
Z = Ac × ȳc + At × ȳt = plastic section modulus

Worked Example: Rectangular Section (b × h)

A = b × h
Plastic neutral axis: h/2 from either face (by symmetry)
Ac = At = (b × h)/2
ȳc = ȳt = h/4  (centroid of each half from PNA)

Z = 2 × [(b × h/2) × h/4] = b × h² / 4

Compare: S = b × h² / 6

Shape factor = Z/S = (bh²/4) / (bh²/6) = 1.50

A rectangular section carries 50% more moment at full plasticity than at first yield.

Worked Example: W-Shape (Doubly-Symmetric I-Shape)

For the strong axis of a W-shape:

Zx = bf × tf × (d − tf) + tw × (d − 2tf)² / 4

Where the first term is the flange contribution and the second is the web contribution. This formula assumes the PNA is at mid-depth (doubly symmetric).

For W14x48 (A992, Fy = 50 ksi):

Zx = 78.4 in³ (from AISC Manual)
Mp = 50 × 78.4 / 12 = 327 ft-kips
Sx = 70.3 in³ (elastic)
My = 50 × 70.3 / 12 = 293 ft-kips
Shape factor = 78.4/70.3 = 1.115

Plastic Neutral Axis (PNA)

For doubly-symmetric sections, the PNA coincides with the geometric centroid. For singly-symmetric sections (e.g., T-shapes, channels), the PNA shifts from the centroid to divide the area equally:

T-shape example: A tee has a wide flange (table) and a narrow stem. The PNA is usually in the flange because the flange has more area — it moves toward the heavier side to satisfy Ac = At.

When the PNA ≠ centroid (singly symmetric), the neutral axis under elastic bending also differs from the PNA, and the moment-curvature relationship is asymmetric (different yield moments in positive and negative bending).

Shape Factors for Common Sections

Section	Z/S (Shape Factor)	Comments
W-shape (strong axis)	1.10 – 1.18	Flanges carry most Mp; web contributes little
W-shape (weak axis)	1.50 – 1.55	Rectangular-like behavior of flanges
Channel (strong axis)	1.16 – 1.22	Similar to W-shape
Rectangular HSS	1.10 – 1.30	Depends on wall thickness (B/t ratio)
Round HSS	1.27	Constant for thin-walled tubes
Solid round (bar)	1.70	Very high — significant plastic reserve
Solid rectangular	1.50	Classical solution
Tee (stem in compression)	1.60 – 2.20	Highly asymmetric — PNA shifts markedly
Angle (equal leg)	1.50 – 2.00	Principal axis bending

The shape factor reflects how efficiently the section uses material. Higher shape factors mean more material near the neutral axis (less efficient for Mp/weight). W-shapes are efficient (Z/S ≈ 1.12) because most material is concentrated in flanges far from the neutral axis — they reach Mp with only modest additional rotation beyond My.

Requirements to Achieve Mp

A section can only reach Mp if ALL the following conditions are met:

Compact section: λ ≤ λp for all elements (AISC 360 Table B4.1b) — no local buckling
Adequate lateral bracing: Lb ≤ Lp — no lateral-torsional buckling before Mp
Adequate shear strength: Shear does not reduce flexural capacity per AISC 360 G2
No axial load interaction: P/Py < 0.15 (or reduced Mp per interaction equations)

If any condition is violated, the available moment is less than Mp. AISC 360 Chapter F provides the reduced capacity formulas.

Mp in Plastic Design vs. Elastic Design

	Elastic Design (ASD)	Plastic Design (LRFD)
Capacity used	My = Fy × S (or reduced per LTB)	Mp = Fy × Z (if compact + braced)
Analysis method	Elastic (first-order or second-order)	Plastic (mechanism-based, plastic hinges)
When allowed	Always	Only with AISC 360 Appendix 1 requirements
Rotation demand	Low — no hinge formation	High — hinges rotate to redistribute moments
Common applications	Beams, columns, general design	Moment frames, continuous beams, portal frames

Plastic design exploits the ability of compact sections to form stable plastic hinges that rotate while maintaining Mp. This allows moment redistribution — the structure can carry more load than the first-hinge elastic analysis predicts.

Mp Reduction for Combined Loading

Axial Force Interaction (AISC 360 H1)

When a beam-column has significant axial load (Pr/Pc ≥ 0.2), the available plastic moment is reduced:

Mrx = Mpx × (1 − Pr/Pc) / (1 − 0.5 × Pr/Pc)    [for strong-axis bending]

Shear Interaction (AISC 360 G2)

When Vu > 0.5φVn, the web's ability to carry moment is reduced. For compact I-shapes:

Reduced yield stress for shear zone: Fy_reduced = (1 − ρ)Fy where ρ = (2Vu/φVn − 1)²
This reduces Mp proportionally

Frequently Asked Questions

What is the difference between Mp and My?

My = Fy × S is the moment at which the extreme fiber first reaches Fy. Beyond My, yielding progresses inward. Mp = Fy × Z is the moment when the entire section has yielded. My is used for elastic design; Mp is used for plastic design when the section is compact and adequately braced. The ratio Mp/My = Z/S is the shape factor, typically 1.10-1.18 for W-shapes.

Can a non-compact section reach Mp?

No. By definition, a non-compact section has one or more elements with λ > λp, meaning local buckling occurs before full plastification can be achieved. The section's capacity is somewhere between My and Mp, determined by linear interpolation (AISC 360 F3 or F4). A slender section (λ > λr) cannot even reach My — elastic local buckling governs.

How does AISC 360 use Mp in beam design?

For compact I-shaped members with Lb ≤ Lp, Mn = Mp = Fy × Zx. This is the base case — full plastic moment, no reduction. For Lp < Lb ≤ Lr, Mn transitions from Mp at Lp down to 0.7FySx at Lr (LTB reduction). For non-compact flanges, Mn < Mp regardless of Lb because the section itself cannot develop full plasticity.

What happens to Mp after the plastic hinge forms?

Once a plastic hinge forms at Mp, the section continues to rotate while maintaining approximately constant moment (strain-hardening may cause a slight increase at very large rotations). The rotation capacity — how much rotation the hinge can sustain before local buckling or fracture — depends on the compactness ratio λ/λp and the loading type. Compact sections (λ << λp) have the highest rotation capacity.

Related Terms and Pages

Educational reference only. Plastic moment capacity must be verified per AISC 360 Chapter F by a licensed Professional Engineer, including checks for compactness, unbraced length, shear, and combined loading for all construction applications.

Disclaimer: This content is for educational purposes only. Results must be verified by a licensed professional engineer. Steel Calculator provides preliminary design tools — NOT a substitute for professional engineering judgment.