**Section modulus** is a geometric (that is, shape-related) property of a beam used in structural engineering. Denoted *Z*, it is a direct measure of the strength of the beam. This kind of section modulus is one of two in engineering, and is specifically called the *elastic* section modulus. The other kind of elastic modulus is the *plastic* section modulus.

Pipes and other forms of tubing are as essential as stand-alone beams in the construction world, and their unique geometry implies that the calculation of the section modulus for this kind of material is different from that of other types. Determining the section modulus requires knowing various intrinsic, or built-in and unchangeable, properties of the material in question.

## Basis of the Section Modulus

Different beams made of different combinations of materials can have wide variations in the distribution of the smaller individual fibers in that section of the beam, pipe or other structural element under consideration. The "extreme fibers," or the ones at the ends of the sections, are forced to bear a greater fraction of whatever load the section is subjected to.

Determining the section modulus *Z* requires finding out the distance *y* from the *centroid* of the section, also called the *neutral axis*, to the extreme fibers.

## The Section Modulus Equation

The section modulus equation for an elastic object is given by *Z* = *I* / *y*, where *y* is the distance described above and *I* is the *second moment of area* of the section. (This parameter is sometimes called the *moment of inertia*, but as there are other applications of this term in physics, it is best to use "second moment of area.")

Because different beams have different shapes, the specific equations for different sections assume different forms. For example, that of a hollow tube such as a pipe is

## What is the "Second Moment of Area"?

The second moment of area *I* is an intrinsic property of the section and reflects the fact that the mass of the section may be distributed asymmetrically and affect how loads are handled.

Think of a solid steel door of a given size and mass and one of identical size and mass that has almost all of the mass on the outer edge while being very thin in the middle. Intuition and experience probably tells you that the latter door would respond less readily to an attempt to push it open close to the hinge than the door with a uniform construction and therefore more mass situated closer to the hinge.

## Section Modulus of Pipe

The equation for the section modulus of a pipe or hollow tube is given by

The derivation of this equation is not important, but because the cross-sections of pipes are circular (or are treated as such for computational purposes if they are close to circular), you would expect to see a π constant, because this pops up when computing areas of circles.

Noting that *I* = *Zy*, the second moment of area *I* for a pipe is

Which means that in this form of the section modulus equation, *y* = *R*.

## Section Modulus of Other Shapes

You may be asked to find the section modulus of a triangle, rectangle or other geometric structure. For example, the equation of a hollow rectangular section has the form:

where *b* is the width of the cross-section and *h* is the height.

## Online Section Modulus Calculator

While it is easy track down online section modulus calculators for all sorts of shapes, it's good to have a firm handle on the equations and why the variables are what they are and why they appear where they do in the formulas. One such calculator is provided in the Resources.

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