simple beam question for the engineers

[gtc]

Help me settle an argument I'm about to have.

Suppose you have two identical beams, carrying identical loads at their center. One beam is simply supported at each end, and the other beam is fixed at each end.
Which beam has a higher maximum stress?

[Porsche-O-Phile]

Typically a restrained beam will because it cannot (as with a simply supported beam) distribute the load over the entire span via deflection. I'll have to dig out my structures books to verify with equations (it's been a while) but simply supported beams can deflect over the entire span whereas a restrained end beam must develop resistive moments at the ends and would deflect some distance along the span.

I'll double-check but that's what I recall offhand.

[TimT]

Which stress? bending? shear?

[TimT]

Fixed at both ends max M - Pl/8 (fixed at both ends)

Max deflection= (Pl^3)/(192EI)

[TimT]

Simple beam without fixed ends..

Max M = (Pl/4)

deflection= ((Pl^3)/(48EI))

just plug the numbers in

[TimT]

By inspection, you can see the answer

[cgarr]

How big is your Wife? Around here we just did visual inspections. We needed fire access to the attic in the shop so what the city did was bring down the fattest Fireman and if he fit thru the hole it would pass!

[Porsche-O-Phile]

Just double-checked, Tim is correct - Mmax will be greater for the simple span although it's worth noting that there is probably more overall stress in the fixed-beam situation since Mmax occurs at three different points (each end plus the center) whereas in the simple span it "simply" builds from zero at each end to Mmax at mid-span. In a restrained beam the Mmax is less, but there are more moment forces and changes in their direction going on, arguably more stress placed on the beam as a whole). Obviously a restrained beam also requires much greater attention to the connections due to the moments that are built up there.

In a fixed-beam situation you also have to account for negative bending near the ends - this can be a problem especially with concrete similar to what you get when you get continuous beams over columns or bearing walls or whatever (you need to reinforce the top or you can fail it in the negative direction).

Thanks to Tim for digging out those equations - it's good to look at this stuff again.

EDIT: another thing you need to consider with restrained ends is thermal expansion - because the ends are restrained, it is easier to get buckling along the weak axis and/or torsional stresses introduced, but that's another discussion I suppose...