Plastic Modulus Calculator

Calculate the plastic section modulus Z for rectangular, round, or I-beam sections from your dimensions in mm, cm, m, in, or ft units.

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Plastic Modulus Formulas

Plastic modulus (Z) equals the first moment of each half-section's area about the plastic neutral axis. Formulas by shape match the calculator tabs above:

Shape	Plastic Modulus (Z)	Elastic Modulus (S)
I-Section	BfTf(D − Tf) + Tw(D − 2Tf)² / 4	I / (D/2)
Rectangle	bh² / 4	bh² / 6
Solid Circle	D³ / 6	πD³ / 32

Shape Factor (f = Z / S)

The shape factor is the ratio of plastic to elastic section modulus. It represents reserve moment capacity between first yield (M_y = F_y × S) and full plasticity (M_p = F_y × Z). A shape factor of 1.5 means 50% additional moment capacity beyond first yield.

Shape	Shape Factor (f)	Engineering Implication
Rectangle	1.50	50% reserve above first yield
Solid Circle	≈ 1.70	70% reserve; inefficient for bending
Diamond (vertex up)	2.00	Double the elastic moment capacity
Typical W-Section	1.10 – 1.15	Low reserve; material concentrated at flanges far from NA

Wide-flange sections intentionally place material far from the neutral axis for elastic efficiency, which compresses the shape factor toward 1.0. This is the key trade-off: I-sections maximize elastic stiffness but offer limited plastic reserve compared to solid rectangular sections.

Plastic Moment Capacity

The plastic moment is the theoretical maximum moment a fully yielded cross-section can sustain:

In AISC LRFD (AISC 360), the design flexural strength of compact sections is φM_n = 0.90 × F_y × Z. Sections must satisfy compactness limits (Table B4.1b) for Z to govern; non-compact and slender sections use reduced capacities per the lateral-torsional buckling and local buckling provisions.

Common Steel Grades: Yield Strength Reference

Grade	Fy (ksi)	Fy (MPa)	Typical Application
A36	36	248	Plates, angles, misc. steel
A572 Gr.50	50	345	General W-shapes, HSS
A992	50	345	W-shapes preferred; limits H/t for seismic
EN S355	51.6	355	European structural steel
Aluminum 6061-T6	35	241	Lightweight structural members

AISC W-Section Z_x Reference Table

Plastic moduli for common wide-flange sections (AISC SCM, 16th Ed.), with M_p at F_y = 50 ksi:

Section	Zx (in³)	Sx (in³)	Shape Factor	Mp at 50 ksi (kip-ft)
W8×31	30.4	27.4	1.11	127
W10×45	54.9	49.1	1.12	229
W12×50	71.9	64.2	1.12	300
W14×82	139	123	1.13	579
W18×35	66.5	57.6	1.15	277
W21×44	95.4	81.6	1.17	397
W24×55	114	94.4	1.21	475

Deeper, lighter W-sections (e.g., W24×55 vs. W14×82) show higher shape factors because the web constitutes a larger share of total area, increasing Z relative to S.

What is Plastic Modulus?

The plastic modulus (Z) is a cross-section geometric property representing the first moment of area of each half-section about the plastic neutral axis (PNA). The PNA divides the section into two equal areas, ensuring equilibrium between tensile and compressive yield forces. For doubly symmetric sections (I-beams, rectangles, circles), the PNA coincides with the centroidal axis. For asymmetric sections (T-sections, unequal angles), the PNA shifts toward the larger area.

Elastic section modulus (S = I/c) governs at the onset of yielding in the extreme fiber; plastic modulus (Z) governs after yielding spreads through the full section depth. Z is always greater than or equal to S, and their ratio defines the shape factor.

Worked Example

I-section: D = 300 mm, B_f = 150 mm, T_f = 20 mm, T_w = 10 mm

For A992 (F_y = 345 MPa): M_p = 345 × 1,009,000 N·mm = 348 kN·m. Design capacity: φM_n = 0.90 × 348 = 313 kN·m (compact section, no LTB).