Section Modulus Equations and Formulas

What is a section modulus?

Section modulus is a crucial geometric property. It is used in designing beams or flexural members. It interacts with other properties like tension area, compression radius of gyration, and stiffness moment of inertia. Relationships among these properties vary with shape. Common shape-specific section modulus equations are provided below, with distinctions between elastic (S) and plastic (Z) section moduli.

The elastic section modulus (S) is defined as S = I / y, where I is the second moment of area, and y is the distance from the neutral axis to any fiber. Typically, it’s reported with y = c, where c is the distance to the most extreme fiber, as shown in the table below. It’s frequently employed to calculate the yield moment (My) using My = S × σy, where σy is the material’s yield strength.

Cross SectionEquation of Section Modulus (Zₑ)
Rectangular Sectionbd²/6
I Section[BD³ – (B – b)d³]/6D
Circle(π/32) × d³
Hollow Circleπ/32 × [(D⁴ – d⁴)/D]
Hollow Rectangle Section(BD³ – bd³)/6D
Diamond SectionBH²/24
C Section[BD³ – (B – b)d³]/6D

Section Modulus of Different Geometries

Section Modulus of Different Geometries
Section Modulus of Geometrical shapes

Plastic Section Modulus (PNA)

The Plastic section modulus is utilized for materials dominated by irreversible plastic behavior, which is not intentionally encountered in most designs. It relies on the plastic neutral axis (PNA), defined as the axis dividing the cross-section so that compression force equals tension force.

For sections with constant yielding stress, the areas above and below the PNA are equal, but for composite sections, this may differ. The plastic section modulus is the sum of the cross-sectional areas on each side of the PNA multiplied by their distance from the local centroids to the PNA.

Plastic Section Modulus (PNA)

Zp = AcYc + AT YT
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