Free Section Modulus Calculator

The elastic section modulus is a fundamental geometric property in the design of beams subjected to bending. This free online section modulus calculator serves as a comprehensive beam section analysis tool, delivering not only the section modulus but also the neutral axis location and the area moment of inertia for a wide array of standard structural profiles. Whether you are sizing a simple rectangular beam or analyzing a complex channel or I‑shaped member, this structural beam calculator quickly provides the key beam section properties needed for stress and deflection checks.

Relationship between bending stress and section modulus

The maximum normal stress in a beam cross‑section due to an applied bending moment can be calculated as:

$$\sigma_m = \frac{M \, c}{I}$$

where $M$ is the bending moment at the section, $c$ is the farthest distance from the neutral axis to the outer fiber, and $I$ is the second moment of area about the neutral axis. Because the ratio $I/c$ depends solely on the geometry, it is convenient to define the elastic section modulus $S$:

$$S = \frac{I}{c}$$

Thus the stress simplifies to:

$$\sigma_m = \frac{M}{S}$$

This is analogous to the direct stress formula $\sigma = F/A$, with the bending moment acting like axial force and the section modulus acting like cross‑sectional area. For any given cross‑section, once the moment of inertia and the extreme‑fiber distance are known, the section modulus follows directly by division.

Considerations in practice

The bending moment $M$ comes from a structural analysis of the beam. For standard steel shapes, tables list the elastic section modulus directly. However, when dealing with custom or non‑standard geometries, using a dedicated section modulus calculator (such as this one) is the most efficient and error‑free method. In a prismatic beam, the maximum bending stress will occur at the point of maximum bending moment. It is important to note that the equations above assume linearly elastic material behavior (Hooke’s law). When the stresses exceed the yield strength, the plastic section modulus must be used.

Plastic section modulus

When a beam section is loaded beyond its elastic limit, the stress distribution becomes nonlinear. The plastic section modulus $Z$ is defined to relate the plastic moment $M_p$ (the moment required to cause yielding across the entire section) to the yield strength $\sigma_Y$:

$$M_p = Z \, \sigma_Y$$

As the bending moment increases, some fibers first reach the yield stress while others remain elastic. Only when the entire section has reached the yield stress does full plastification occur; the moment at that state is the plastic moment. The elastic formulas are no longer applicable once any portion of the section undergoes plastic deformation.

Elastic section modulus formulas for common cross‑sections

The following table lists the distances to the centroidal axes ($y_c$ and $x_c$), the second moments of area ($I_x$ and $I_y$), and the elastic section moduli ($S_x = I_x / y_c$ and $S_y = I_y / x_c$) for typical structural shapes. All formulas assume the section is oriented with the principal axes horizontal and vertical.

Shape	$y_c$	$x_c$	$I_x$	$I_y$	$S_x$	$S_y$
Square (side $a$)	$\dfrac{a}{2}$	$\dfrac{a}{2}$	$\dfrac{a^{4}}{12}$	$\dfrac{a^{4}}{12}$	$\dfrac{a^{3}}{6}$	$\dfrac{a^{3}}{6}$
Rectangle (width $b$, depth $d$)	$\dfrac{d}{2}$	$\dfrac{b}{2}$	$\dfrac{b d^{3}}{12}$	$\dfrac{d b^{3}}{12}$	$\dfrac{b d^{2}}{6}$	$\dfrac{d b^{2}}{6}$
Hollow rectangle (outer $b\times d$, inner $b_i\times d_i$)	$\dfrac{d}{2}$	$\dfrac{b}{2}$	$\dfrac{b d^{3} - b_{i} d_{i}^{3}}{12}$	$\dfrac{d b^{3} - d_{i} b_{i}^{3}}{12}$	$\dfrac{I_x}{y_c}$	$\dfrac{I_y}{x_c}$
Tee section (flange width $b$, thickness $t$, web depth $d$, web thickness $t_w$)	$\dfrac{b t^{2} + t_{w} d (2t+d)}{2(t b + t_{w} d)}$	$\dfrac{b}{2}$	$\dfrac{b (d+t)^{3} - d^{3} (b - t_{w})}{3} - A (d+t - y_{c})^{2}$	$\dfrac{t b^{3} + d t_{w}^{3}}{12}$	$\dfrac{I_x}{d+t - y_{c}}$	$\dfrac{I_y}{x_{c}}$
Channel section (flange width $b$, thickness $t$, web depth $d$, web thickness $t_w$)	$\dfrac{b t^{2} + 2 t_{w} d (2t+d)}{2(t b + 2 t_{w} d)}$	$\dfrac{b}{2}$	$\dfrac{b (d+t)^{3} - d^{3} (b - 2 t_{w})}{3} - A (d+t - y_{c})^{2}$	$\dfrac{(d+t) b^{3} - d (b - 2 t_{w})^{3}}{12}$	$\dfrac{I_x}{d+t - y_{c}}$	$\dfrac{I_y}{x_{c}}$
Wide‑flange beam (equal flanges, width $b$, flange thickness $t$, web depth $d$, web thickness $t_w$)	$\dfrac{d}{2} + t$	$\dfrac{b}{2}$	$\dfrac{b (d+2t)^{3} - (b - t_{w}) d^{3}}{12}$	$\dfrac{b^{3} t}{6} + \dfrac{t_{w}^{3} d}{12}$	$\dfrac{I_x}{y_{c}}$	$\dfrac{I_y}{x_{c}}$
Angle (leg lengths $b$ and $d$, thickness $t$)	$\dfrac{d^{2} + b t - t^{2}}{2(b+d-t)}$	$\dfrac{b^{2} + d t - t^{2}}{2(b+d-t)}$	$\dfrac{b d^{3} - (b-t)(d-t)^{3}}{3} - A(d - y_{c})^{2}$	$\dfrac{d b^{3} - (d-t)(b-t)^{3}}{3} - A(b - x_{c})^{2}$	$\dfrac{I_x}{d - y_{c}}$	$\dfrac{I_y}{b - x_{c}}$
Circle (radius $R$)	$R$	$R$	$\dfrac{\pi}{4} R^{4}$	$\dfrac{\pi}{4} R^{4}$	$\dfrac{\pi}{4} R^{3}$	$\dfrac{\pi}{4} R^{3}$
Hollow circle / pipe (outer radius $R$, inner radius $R_i$)	$R$	$R$	$\dfrac{\pi}{4} (R^{4} - R_{i}^{4})$	$\dfrac{\pi}{4} (R^{4} - R_{i}^{4})$	$\dfrac{I_x}{R}$	$\dfrac{I_y}{R}$

For the Tee, Channel, and Angle shapes, $A$ is the cross‑sectional area: $A = t b + t_{w} d$ (Tee), $A = t b + 2 t_{w} d$ (Channel), and $A = t (b+d-t)$ (Angle).

Units

The second moment of area $I$ has dimensions of length to the fourth power: $\text{mm}^{4}$ or $\text{m}^{4}$ in the International System, and $\text{in}^{4}$ in US customary units. The elastic section modulus $S$ carries dimensions of length cubed: $\text{mm}^{3}$, $\text{m}^{3}$, or $\text{in}^{3}$. Interestingly, these are the same units as volume, reflecting the geometric nature of the section modulus. If a full deflection analysis is needed, a dedicated beam deflection calculator can be used alongside this section modulus tool.