## Which beam is stronger?

This is for a deck:

Which beam is stronger?

A) a beam with cross-section dimensions 4"x10" and 8' long
B) a beam with cross-section dimensions 4"x12" and 12' long

Both beams are same material and are both simply supported at the ends. Same load type (either concentrated or distributed doesn't really matter) and same load location.

Here are my thoughts:
Moment of inertia for beams: A: 333.33 in^4; B: 576 in^4
Lengths of beams: A: 96 in; B: 144 in

Using the equation for a concentrated load at the center: P = (deflection*48*E*I)/(L^3)
The deflection of beam A is about half the deflection of B. Does this mean A is stronger with the given loads?
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 A good way to compare is to use what is known as the section modulus (usually given the symbol Z or S) Values of Z are often tabulated in structural tables, but can easily be computed as the ratio of the moment of intertia to the distance from the neutral axis to the edge of the beam (c). $$Z = \frac{I}{c}$$ In the case of a rectangular beam c=h/2 where h is the height of the beam. If M is the maximum moment on the beam ( computed directly from the loads and geometry of the beam) and f is the allowable maximum stress in the material, a particularly simple equation then results. $$Z = \frac{M}{f}$$ does this help?

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 Quote by walt88 Does this mean beam A is stronger with the given loads?
walt88: Yes, that is correct. If both beams have exactly the same material strength, beam A is 1.953 times stiffer than beam B with respect to (w.r.t.) deflection, and 1.042 times stronger than beam B w.r.t. stress.

## Which beam is stronger?

Don't confuse member strength and stiffness with performance under load. The larger section is both stronger and stiffer, but because it is on a longer span, it may be more highly stressed and may deflect more under the same load.

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