Engineers use the section modulus of the cross-section of a beam as one of the determinants of the beam's strength. In some cases, they employ the elastic modulus under the assumption that after a deforming force is removed, the beam returns to its original shape. In cases where plastic behavior is dominant, which means the deformation is permanent to some degree, they have to calculate the plastic modulus. This is a straightforward calculation when the beam has a symmetrical cross section and the beam material is uniform, but when the cross section or beam composition is irregular, it becomes necessary to divide the cross section into small rectangles, calculate the modulus for each rectangle and sum up the results.

## Rectangular Cross-Sectional Beams

When you apply stress to a point on a beam, it subjects part of the beam to a compressive force and the other part to a force of tension. The plastic neutral axis (PNA) is the line through the cross-section of the beam that separates the area under compression from that under tension. This line is parallel to the direction of the applied stress. One way to define the plastic modulus (Z) is as the first moment of area about this axis when the areas above and below the axis are equal.

If A_{C} and A_{T} are the areas of the cross section under compression and under tension respectively, and d_{C} and d_{T} are the distances from the centroids of the areas under compression and under tension from the PNA, the plastic modulus can be calculated with the following formula:

Z = A_{C} • d_{C} + A_{T} •d_{T}

For a uniform rectangular beam of height d and width b, this reduces to:

Z = bd^{2}/4

## Non-Uniform and Non-Symmetrical Beams

When a beam does not have a symmetrical cross section or the beam is composed of more than one material, the areas above and below the PNA can be different, depending on the moment of the applied stress. Locating the PNA and calculating the plastic modulus become multi-step processes that involve dividing the cross section area of the beam into polygons, each having equal areas undergoing compressive and tension forces. The plastic moment of the beam thus becomes a summation of the areas under compression, multiplied by the distance of each area to the centroid of compression and multiplied by the tensile strength of that section, which is then added to the same summation for the sections under tension.

The moment has a positive and negative component, depending on the direction of the stress, the axis and the combination of materials in the beam. The plastic modulus for the beam is thus the sum of the positive and negative moments divided by the material strength of the first polygon in the summation series for the plastic moment.

Tips

- 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.

About the Author

Chris Deziel holds a Bachelor's degree in physics and a Master's degree in Humanities, He has taught science, math and English at the university level, both in his native Canada and in Japan. He began writing online in 2010, offering information in scientific, cultural and practical topics. His writing covers science, math and home improvement and design, as well as religion and the oriental healing arts.