# Section modulus

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Section modulus

In structural engineering, the section modulus of a beam is the ratio of a cross section's second moment of area to its greatest distance from the neutral axis.

The section modulus is directly related to the strength of a corresponding beam. It is expressed in units of volume, e.g., mm3.

Elastic Section Modulus

For the common cross sections, these are the equations:

* Rectangular - $S = \left(1/6\right)bh^2$
* I section (Strong Axis)- $S = \left(bf\right)\left(tf\right)^3\left(d-tf\right) + 0.25\left(tw\right)\left(d-2tf\right)^2$
* I section - $S = 0.5\left(bf\right)^2\left(tf\right)+ 0.25\left(d-2tf\right)\left(tw\right)^2$
* Solid Circle - $S = pi*d^3/32$
* Pipe - $S = d^3/6\left(1-\left(1-2t/d\right)^3\right)$ - $S = pi*\left(Do^4-Di^4\right)/\left(32*Do\right)$
* Hollow Rect - $S = bh^2/4\left(1-\left(1-2w/b\right)\left(1-2t/h\right)^2\right)$
* Diamond - $S = bh^2/12$
* Channel - $S = bt\left(h-t\right) + 0.25\left(h-2t\right)^2$

Plastic Section Modulus

The plastic Section Modulus is the sum of the areas on each side of the neutral axis times the distance from the centroid of the section to the centroid of the areas on either side of the centroid.

For the two flanges of an I-Beam with the web excluded:

* $Z = bty+b1t1y1$ [American Institute of Steel Construction LRFD 3rd Ed. pg. 17-34]

where:b=widtht=thickness

y and y1 are the distances to the centroids of the flanges respectively.

* Solid Circle - $S = d^3/6$for a hollow Circle $S = \left(d2^3-d1^3\right)/6$rajiv A k

References

* [1] American Institute of Steel Construction: Load and Resistance Factor Design, 3rd Edition, pg. 17-34
* http://www.groupsrv.com/science/post-865320.html

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