Section Modulus Equations and Calculators Common Shapes

Strength of Materials | Beam Deflection and Stress

Section modulus is a geometric property for a given cross-section used in the design of beams or flexural members. Other geometric properties used in design include area for tension, radius of gyration for compression, and moment of inertia for stiffness. Any relationship between these properties is highly dependent on the shape in question. Equations for the section moduli of common shapes are given below. There are two types of section moduli, the elastic section modulus (S) and the plastic section modulus (Z).

For general design, the elastic section modulus is used, applying up to the yield point for most metals and other common materials.

The elastic section modulus is defined as S = I / y, where I is the second moment of area (or moment of inertia) and y is the distance from the neutral axis to any given fiber. It is often reported using y = c, where c is the distance from the neutral axis to the most extreme fiber , as seen in the table below. It is also often used to determine the yield moment (My) such that My = S × σy, where σy is the yield strength of the material.

Cross section Shape Equation Comment
Rectangle

Section Modulus Rectangle

Calculator:

Section Modulus Rectangle Calculator

Solid arrow represents neutral axis
Section Modulus I-Beam

Section Modulus I-Beam

Calculator:

Section Modulus I Beam Universal Calculator

NA indicates neutral axis
I-Beam section Modulus

I-Beam Section Modulus

Calculator:

Section Modulus I Beam Center Neutral Axis Calculator

NA indicates neutral axis
Section Modulus Circle

Section Modulus Equation

Calculator:

Section Modulus Circle Round Center Neutral Axis Calculator

Solid arrow represents neutral axis
Pipe Section Modulus

Pipe Section Modulus

Calculator:

Section Modulus Hollow Round Center Neutral Axis Calculator

NA indicates neutral axis
Rectangular Tube Section Modulus

Rectangaular Pipe Section Modulus Equation

Calculator:

Section Modulus Hollow Rectangle Square Center Neutral Axis Calculator

NA indicates neutral axis
Section Modulus Diamond

Section Modulus Diamond Equation

Calculator:

Section Modulus Diamond Shape Center Neutral Axis Calculator

NA indicates neutral axis
Section Modulus C-Channel

Section Modulus Equation C-Channel

Calculator:

Section Modulus Channel Shape Center Neutral Axis Calculator

NA indicates neutral axis

 

Plastic Section Modulus (PNA)

The Plastic section modulus is used for materials where (irreversible) plastic behavior is dominant. The majority of designs do not intentionally encounter this behavior.

The plastic section modulus depends on the location of the plastic neutral axis (PNA). The PNA is defined as the axis that splits the cross section such that the compression force from the area in compression equals the tension force from the area in tension. So, for sections with constant yielding stress, the area above and below the PNA will be equal, but for composite sections, this is not necessarily the case.

The plastic section modulus is then the sum of the areas of the cross section on each side of the PNA (which may or may not be equal) multiplied by the distance from the local centroids of the two areas to the PNA:

Plastic Section Modulus

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