# Law of physics for beam sizes

1. May 19, 2012

### rodsika

Supposed in builiding contruction, the beam span is 12 meters compared to 6 meters. Would it make the beam size (and column size) twice in the longer 12 meters span given the same loading? Is there a law of physics that says the sizes are proportional to the span? Or is it algorithmic or nonlinear?

Don't worry. I'm not going to construct the builiding myself, of course. Just wondering how the beam depth would increase if the span requirement is greater. If you have any formula to give an idea what is the relationship, do let me know.. thanks a lot.

2. May 21, 2012

### scutterbob

The load type will make a difference, point load vs uniform loads. The beam depth but more specificly the section modulus and moments of inertia determine the strength and stiffness of the beam, and of course the material strength.

P = point load
L = Span Length
W = uniform load (kip/ft or kN/m)

Moment from point load = P*L / 4 assuming (simple supports)
Moment from uniform load = W L2 / 8 assuming (simple supports)

As you can see for uniform loads (most typical) the length term is squared.

b = beam width
h = beam height
I = moment of inertia (rectangluar beam = b*h3 / 12)
For simplicty (and normally only in simple timber construction we assumed a rectangular shape) you see the height of the beam is a cubed term hence it is very significant.
We would in steel normally use a roughly "I" shaped beam because it efficently provides big moments of inertia by concentraiting the larger areas at the higher stressed locations. The web is then assumed to only carry shears.

y = half of beam height for this case.
σ(stress) = M *y / I = M/S

It is always necessary to account for the selfweight of the beam and other supporting elements in addition to any supported live, dead or environmental loads.

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