# How to Calculate Plastic Modulus

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The plastic modulus (also known as the "plastic section modulus") is a theoretical tool used in structural engineering to quantify the strength of beams and how those beams deform under stress. It is based strictly on two-dimensional beam cross sections.

The "plastic" in the name refers to the type of deformation to which the beams in question are prone -- in this case, deformation through irreversible ("plastic") processes. Different beam geometries exhibit different characteristic plastic modulus formulas. The higher the plastic modulus, the more reserve strength the beam has after stress-induced deformation has begun.

Determine the geometry of the beam whose plastic modulus you are interested in. Common geometries you will often encounter include rectangular (including square), solid circular, hollow circular and I-beam.

Measure the dimensions of your beam using a tape measure. Alternatively, you can look up these values if you have the beam's documentation handy. The most important values for calculating the plastic modulus of a beam are generally the width and height of the beam's cross section. If your beam has any unusual geometry (such as flanges in an I-beam), though, measure those dimensions as well.

Apply one of the following formulas for the plastic modulus Z based on the geometry of the beam you're dealing with:

Rectangular: Z = (b x h^2)/4

where b is the width (or base) of the beam cross section and h is its height

Solid circular: Z = (d^3)/6

where d is the diameter of the beam cross section

Hollow circular: Z = (d_2^3 - d_1^3)/6

where d_2 is the outer diameter of the beam cross section and d_1 is the inner diameter

I-beam: Z = (b_1 x t_1 x y_1) + (b_2 x t_2 x y_2)

where b is the width (or base) of each respective flange in the beam cross section, t is the thickness of each respective flange, and y is the distance between the centre of mass of each respective flange and the centre of mass of the beam as a whole

Plug the indicated values into the appropriate formula, and your plastic modulus will be given as Z.

- You are only interested in the two-dimensional cross section of the beam (as if you were looking at the beam head on). Length is not a useful variable in calculating the plastic modulus.

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