Example data table
| Shape | Key dimensions | Axis | Zp (mm³) |
|---|---|---|---|
| Rectangle | b=200, h=300 | Major | 4,500,000 |
| Solid circle | d=200 | Any | 1,333,333.333 |
| Hollow circle | D=250, d=200 | Any | 2,604,166.667 |
| Rectangular tube | B=200, H=300, t=10 | Major | 2,885,333.333 |
| I-section | bf=200, tf=20, tw=12, h=300 | Major | 3,161,760 |
Formulas used
- Definition: Plastic section modulus Zp = ∫ |y| dA about the plastic neutral axis.
- Rectangle: Major Zp = b·h²/4, Minor Zp = h·b²/4.
- Solid circle: Zp = d³/6.
- Hollow circle: Zp = (D³ − d³)/6.
- Rectangular tube: Outer minus inner rectangle, with bᵢ=B−2t and hᵢ=H−2t.
- I-section (major): PNA location is solved from equal areas above and below.
- I-section (minor): Centerline PNA, sum of rectangle contributions Zp = Σ(hᵢ·bᵢ²/4).
How to use this calculator
- Select a cross-section shape from the shape list.
- Choose the bending axis: major or minor.
- Select the input unit, then enter the dimensions.
- Click Calculate to show results above the form.
- Use Download CSV or Download PDF to export.
Plastic section modulus guide
1) What the value represents
The plastic section modulus (Zp) measures how much fully yielded material is available to resist bending. Unlike elastic modulus (S), Zp assumes the entire compression block and tension block reach yield stress. Engineers use it to estimate plastic moment capacity, where Mp = Fy · Zp.
2) Units and scaling behavior
Zp has units of length cubed, such as mm³ or in³. If you scale every dimension of a shape by 2, Zp increases by 2³ = 8. This cubic scaling is why small thickness changes in tubes or webs can noticeably change bending capacity in plastic design.
3) Rectangle and tube trends
For rectangles about the strong axis, Zp = b·h²/4, so depth h dominates. For rectangular tubes, subtracting the inner void reduces Zp, but keeping material far from the neutral axis is efficient. A taller tube generally outperforms a wider tube for strong-axis bending.
4) Circles and hollow circles
A solid circle uses Zp = d³/6 and is identical about any diameter. Hollow circles follow (D³ − d³)/6, so increasing outer diameter is powerful, while a larger inner diameter removes capacity. Thin-walled tubes can still be excellent when diameter is large.
5) Major vs minor axis for I-sections
I-sections are strong about the major axis because flanges place area far from the plastic neutral axis. About the minor axis, flange width matters more than overall depth. This calculator reports both axes; use the axis that matches your bending direction in the member and connection layout.
6) Plastic neutral axis behavior
The plastic neutral axis (PNA) splits the cross-section into equal areas in compression and tension. For symmetric shapes, it lies at mid-depth. For unsymmetric I-sections, the PNA shifts toward the larger flange. The tool computes the PNA location for major-axis I-section calculations.
7) From Zp to capacity
To estimate plastic moment, multiply Zp by yield strength. Example: if Zp = 3.0×10⁶ mm³ and Fy = 250 MPa, then Mp ≈ 750 kN·mm, which is 0.75 kN·m. Always apply your code’s resistance and stability factors.
8) Practical checks before using results
Confirm that dimensions match the real profile and that thickness is not larger than geometry permits. For thin elements, local buckling can prevent full plasticity, reducing usable capacity. If your standard requires class/compactness limits, verify them before relying on Mp derived from Zp.
FAQs
1) What is the difference between Zp and elastic section modulus?
Zp assumes the whole section yields, while elastic section modulus (S) is based on linear stress distribution up to first yield. Zp is always larger than S for the same shape and axis.
2) Which axis should I select?
Select the axis about which the member bends. Major (strong) axis typically corresponds to the larger depth direction. Minor (weak) axis corresponds to bending about the narrower dimension or the web centerline.
3) Why does the circle axis not matter?
A circle is rotationally symmetric, so its area distribution is the same about any diameter. Therefore, both elastic and plastic section properties are identical for all bending directions through the center.
4) Can I use Zp directly for allowable stress design?
Use Zp only if your design method allows plastic behavior. Some allowable-stress approaches focus on elastic limits, making S more appropriate. Always follow your governing code and load combinations.
5) Why might real capacity be lower than Mp = Fy·Zp?
Local buckling, lateral-torsional buckling, residual stresses, holes, weld access, and connection details can reduce usable strength. Codes also apply resistance factors and section class limits before allowing full plastic capacity.
6) What units should I use for Fy with Zp?
Keep units consistent. If Zp is in mm³ and Fy is in MPa (N/mm²), Mp will be N·mm. Convert to kN·m or other units as needed after computing.
7) Does this tool cover tapered or curved sections?
This version covers common prismatic shapes built from rectangles and circles. Tapered, curved, or perforated sections require section-by-section integration or finite element methods to locate the PNA and compute Zp accurately.