How to Calculate Plastic Modulus
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.
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Instructions
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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.
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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.
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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 center of mass of each respective flange and the center 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.
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Tips & Warnings
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.
References
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