Calculator Input
Choose a section type, enter dimensions, and calculate plastic modulus, elastic modulus, shape factor, and plastic moment.
Example Data Table
These sample cases show how plastic modulus and moment capacity change with geometry.
| Section | Input Example | Plastic Modulus (mm³) | Shape Factor | Plastic Moment (kN·m) |
|---|---|---|---|---|
| Rectangle | b = 150 mm, h = 300 mm, fy = 250 MPa | 3,375,000 | 1.5 | 843.75 |
| Hollow Rectangle | B = 200 mm, H = 300 mm, t = 10 mm, fy = 250 MPa | 972,000 | 1.2078 | 243 |
| Solid Circle | D = 200 mm, fy = 250 MPa | 1,333,333.333 | 1.6977 | 333.333 |
| Symmetric I-Section | bf = 200 mm, tf = 12 mm, tw = 8 mm, h = 300 mm, fy = 250 MPa | 843,552 | 1.1138 | 210.888 |
Formula Used
Plastic section modulus is the first moment of fully yielded area about the plastic neutral axis.
Rectangle
Zp = b × h² / 4
Ze = b × h² / 6
Mp = fy × Zp
Hollow Rectangle
bi = B − 2t, hi = H − 2t
Zp = (B × H² − bi × hi²) / 4
I = (B × H³ − bi × hi³) / 12, Ze = I / (H / 2)
Solid Circle
Zp = D³ / 6
I = π × D⁴ / 64
Ze = π × D³ / 32
Hollow Circle / Tube
Di = Do − 2t
Zp = (Do³ − Di³) / 6
I = π × (Do⁴ − Di⁴) / 64, Ze = I / (Do / 2)
Symmetric I-Section
hw = h − 2tf
Zp = bf × tf × (h − tf) + tw × hw² / 4
I = 2[(bf × tf³ / 12) + (bf × tf × (h/2 − tf/2)²)] + tw × hw³ / 12
Shape factor = Zp / Ze. Larger values indicate more bending reserve beyond first yield.
How to Use This Calculator
- Select the section shape that matches your geometry.
- Choose the working length unit: mm, cm, m, or in.
- Enter section dimensions and the material yield stress.
- Press Calculate to place the result block above the form.
- Read the plastic modulus, elastic modulus, shape factor, area, and plastic moment.
- Use the graph for quick comparison between plastic and elastic section modulus.
- Use the export buttons to save the result as CSV or PDF.
- For unusual sections or code-based design checks, treat this as a fast screening tool.
FAQs
1) What is plastic section modulus?
Plastic section modulus measures the first moment of fully yielded compression and tension areas about the plastic neutral axis. It estimates ultimate bending capacity once yielding spreads across the whole section depth.
2) How is plastic section modulus different from elastic section modulus?
Elastic modulus predicts first yield under linear stress distribution. Plastic modulus predicts full plastic resistance. Their ratio gives the shape factor, which shows the reserve strength available after the extreme fibers first yield.
3) Why is shape factor useful?
Shape factor compares plastic capacity with elastic capacity. Values above one mean the section can resist additional moment after first yield. It is useful when comparing section efficiency and bending reserve.
4) Which sections does this calculator cover?
This version covers solid rectangles, hollow rectangles, solid circles, tubes, and symmetric I-sections about the strong axis. Unsymmetric, tapered, or built-up sections need a separate plastic analysis.
5) Do units affect the final answer?
Units matter only for consistency. The calculator converts dimensions internally, then returns plastic modulus in the chosen cubic unit and plastic moment in kilonewton-metres for easy interpretation.
6) Why is yield stress included?
Yield stress converts plastic section modulus into plastic moment using Mp = fy × Zp. Higher yield stress increases bending capacity, assuming local buckling and design-code limits do not control first.
7) Is this enough for final structural design?
No. Final design must also check local buckling, lateral torsional buckling, shear interaction, code resistance factors, and serviceability. This tool works best for learning, comparison, and preliminary assessment.
8) What assumptions are used here?
The formulas assume homogeneous material, full yielding, and the stated section dimensions. I-section results assume symmetry about the major axis, with the plastic neutral axis at mid-depth.