Select a shape tab, then enter the required section dimensions into the calculator to determine the plastic modulus of a cross-section.
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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 |
Z = BfTf(D - Tf) + \frac{Tw(D - 2Tf)^2}{4}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 (My = Fy × S) and full plasticity (Mp = Fy × 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:
M_p = F_y \times Z
In AISC LRFD (AISC 360), the design flexural strength of compact sections is φMn = 0.90 × Fy × 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 Zx Reference Table
Plastic moduli for common wide-flange sections (AISC SCM, 16th Ed.), with Mp at Fy = 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, Bf = 150 mm, Tf = 20 mm, Tw = 10 mm
Z = 150 \times 20 \times (300 - 20) + \frac{10 \times (300 - 2 \times 20)^2}{4} = 840{,}000 + 169{,}000 = 1{,}009{,}000 \text{ mm}^3For A992 (Fy = 345 MPa): Mp = 345 × 1,009,000 N·mm = 348 kN·m. Design capacity: φMn = 0.90 × 348 = 313 kN·m (compact section, no LTB).
