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Civil
Plastic section modulus of a symmetric I-section
First moment of the area on each side of the equal-area axis, added together, for a doubly symmetric I-section with equal flanges.
- Ref
- CIV-09
- Rev
- v1
- Field
- Civil
- Result
- Zp [m³]
The plastic section modulus is the sum of the first moments of the tension and compression areas about the equal-area axis. For a doubly symmetric I-section it is the flange term bf·tf·(H − tf) plus the web term tw(H − 2tf)²/4. Multiplied by the yield stress it gives the plastic moment Mp, which is larger than the first-yield moment by the shape factor.
First moment of the area on each side of the equal-area axis, added together, for a doubly symmetric I-section with equal flanges. It is a pure geometric property: multiplied by the yield stress it gives the plastic moment. Fillets are ignored, and local buckling can stop the plastic moment from being reached.
Calculator
- Flange contribution, bf·tf·(H − tf)0.00152 m³
- Web contribution, tw(H − 2tf)²/43.24×10−4 m³
- Elastic modulus I/(H/2) of the same I-section0.00164 m³
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- The shape factor Zp/S is 1.125, so full plasticity gives 12.46 % more moment than first yield.
- With a yield stress of 275 MPa the plastic moment is 507.1 kN·m, if the flanges and web are compact enough not to buckle first.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m³) | Local step | Change in result |
|---|---|---|---|---|---|---|
| H Overall depth | 0.05 to 1.5 m | 1.203e-4 | 0.01125 | 0.01113 | +10% | +12.8 % |
| tf Flange thickness | 0.002 to 0.19 m | 5.512e-4 | 0.007981 | 0.00743 | +10% | +7.38 % |
| bf Flange width | 0.05 to 0.6 m | 7.04e-4 | 0.004884 | 0.00418 | +10% | +8.24 % |
| tw Web thickness | 0.002 to 0.1 m | 0.001585 | 0.00476 | 0.003175 | +10% | +1.76 % |
Sweep of Flange thickness
| Flange thickness (m) | Plastic section modulus (m³) |
|---|---|
| 0.002 | 5.512e-4 |
| 0.0255 | 0.002214 |
| 0.049 | 0.003668 |
| 0.0725 | 0.004911 |
| 0.096 | 0.005945 |
| 0.1195 | 0.006769 |
| 0.143 | 0.007383 |
| 0.1665 | 0.007787 |
| 0.19 | 0.007981 |
Sweep one variable
Variables and how each one contributes
Plastic section modulus
What the equation gives you. Everything else feeds it.
Flange width
The width of each flange, in m. Wider flanges raise Zp in direct proportion to the flange term. Raising it by 10 % (to 0.22) moves the result by +8.24 %, from 0.001844 to 0.001996 m³.
Flange thickness
The thickness of each flange, in m. Thicker flanges add area far from the axis, but reduce the web depth. Raising it by 10 % (to 0.022) moves the result by +7.38 %, from 0.001844 to 0.00198 m³.
Overall depth
The overall depth of the section, in m. A deeper section raises Zp in both terms. Raising it by 10 % (to 0.44) moves the result by +12.8 %, from 0.001844 to 0.00208 m³.
Web thickness
The web thickness, in m. A thicker web raises the web term in direct proportion. Raising it by 10 % (to 0.011) moves the result by +1.76 %, from 0.001844 to 0.001876 m³.
Worked example with the values above
- Flanges: bf·tf·(H − tf) = 0.2 × 0.02 × 0.38 = 0.00152 m³.
- Web: tw(H − 2tf)²/4 = 0.01 × 0.36² / 4 = 3.24×10−4 m³.
- Zp = 0.001844 m³.
Assumptions and limits
- Linear elastic material, plane sections staying plane, and perfect bond between components.
- Sharp-cornered ideal geometry with no fillets, holes or local buckling.
Common mistakes
- Mixing millimetres with metres, which changes I by a factor of 10¹² when it is raised to the fourth power.
- Using the inertia about the wrong axis (width and depth swapped).
- Forgetting that composite properties depend on the modulus of each part, not only its area.
Statistics and uncertainty
These are exact geometric results for the stated shapes, so the uncertainty is in the dimensions. Because the second moment of area scales with depth cubed, a 2 % error in depth gives about a 6 % error in stiffness. Rolled and fabricated sections vary by a few per cent from nominal, and composite sections carry extra uncertainty from the modulus ratio.
Where it is used
- Calculating stiffness, stresses and capacity of beams and built-up members.
- Spacing connectors in composite and built-up sections.
Maths you need
- Second moment of area: I = ∫y² dA about the centroidal axis
- Parallel axis theorem: I = I₀ + Ad² about a parallel axis
- Neutral axis: the line of zero bending stress in an elastic section