Plastic Section Modulus and Shape Factor Calculator

Elastic and plastic section modulus, shape factor, first-yield moment and plastic moment for common sections.

Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.

How this calculator works

When a beam is loaded, the outermost fibre reaches yield first. That is the first-yield moment, My = fy × Ze, using the elastic section modulus. But the beam does not collapse then: the yielded zone spreads inward until the whole section has yielded and a plastic hinge forms. That is the plastic moment, Mp = fy × Sp, using the plastic section modulus.

The ratio Sp / Ze is the shape factor. It is 1.5 for a solid rectangle, about 1.7 for a solid circle, and 1.1 to 1.2 for a typical I-section, because most of the material is already out at the flanges. Plastic design is only valid for compact sections braced against buckling, and design capacity uses a strength reduction factor from the standard: that is a check for an engineer.

Formulas used

  • Z_e = I / y_{max}
  • S_p = \text{first moment of area of each half about the equal-area axis}
  • \xi = S_p / Z_e
  • M_p = f_y S_p

Worked example

With the default numbers in the calculator above:

  1. Elastic section modulus: 493.8 ×10³ mm³
  2. Plastic section modulus: 552.6 ×10³ mm³
  3. Shape factor: 1.119
  4. First-yield moment: 123.5 kN m
  5. Plastic moment: 138.2 kN m

Change the numbers in the calculator to see the full working, every step, for your own case.

Common questions

What is a plastic hinge?

A section that has fully yielded. It keeps carrying its plastic moment while it rotates, so the beam behaves as if it had a hinge there.

Why is the shape factor higher for a rectangle than an I-section?

A rectangle has a lot of material near the neutral axis that is still elastic at first yield. An I-section has most of its material in the flanges, which yield almost together.

Can I use this for AS 4100 design?

No. It gives the textbook section properties. Design capacity involves compactness limits, the strength reduction factor and bracing, which an engineer must check.

Related calculators

Further reading: Hibbeler, Mechanics of Materials; Gere and Goodno, Mechanics of Materials; Shigley, Mechanical Engineering Design. Plastic analysis texts such as Neal, The Plastic Methods of Structural Analysis.

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