Structures, Beams: Proof of formulas for a beam

[hmvv]

Ok I'm struggling with something here. I am rusty so have probably just misunderstood something simple mathematically. There are equations to create the Bending moment diagram for each different type of beam subject to a UDL; cantilever, simply supported, fully fixed ends etc. However I just don't understand how they got them.

A simply supported beam with a UDL of q and length L will have bending moments at each end of (qL^2)/2. That I get and can show proof yet its the centre moment I cannot reproduce, (qL^2)/8

Or cantilever at free end, (qL^4)/8EI {I do understand the EI bit btw}

and fully fixed beams atall, (qL^2)/12 at ends and (qL^2)/24 in the centre. {Doesn't make sense to me as the reaction forces are still the same as a simply suppoted beam: qL/2}

People write these on their beams when doing other structural analysis like its obvious yet I just can't see it. Clearly I've forgotten some basics about beams and I'm baffled, anyone care to help?

thanks in advance :)

 

[timthereaper]

This, plus some assumptions and calculus:

http://en.wikipedia.org/wiki/Euler–Bernoulli_beam_theory#Static_beam_equation

 

[SteamKing]

Ok I'm struggling with something here. I am rusty so have probably just misunderstood something simple mathematically. There are equations to create the Bending moment diagram for each different type of beam subject to a UDL; cantilever, simply supported, fully fixed ends etc. However I just don't understand how they got them.

Most of the beam formulas can be derived using simple statics. An introductory text on strength of materials will probably derive some of these formulas as examples.

A simply supported beam with a UDL of q and length L will have bending moments at each end of (qL^2)/2. That I get and can show proof yet its the centre moment I cannot reproduce, (qL^2)/8

If a beam is simply-supported at both ends, the bending moments there will both be equal to zero, not qL²/2. A moment would develop at the ends of the beam only if the ends of the beam were restrained from rotating, which they can't be if there is only a simple support. The central bending moment for a simply supported beam with a UDL is indeed qL²/8.

Or cantilever at free end, (qL^4)/8EI {I do understand the EI bit btw}

The deflection of a cantilever is equal to qL⁴/8EI, not the bending moment, so you have to be clear about what you are discussing here.

and fully fixed beams atall, (qL^2)/12 at ends and (qL^2)/24 in the centre. {Doesn't make sense to me as the reaction forces are still the same as a simply suppoted beam: qL/2}

The reactions are going to be the same for a beam of the same length and total loading because the two beams still must each be in static equilibrium. (That is, the sum of the reactions on the beam is equal to and opposite of the total applied load.)

However, because the ends of the beams may be restrained differently (fixed versus simple supports), the bending moments may not be distributed in the same way along the length of each beam.

People write these on their beams when doing other structural analysis like its obvious yet I just can't see it. Clearly I've forgotten some basics about beams and I'm baffled, anyone care to help?

thanks in advance :)

People write these equations because they have learned them and probably use them frequently. They don't need to derive them every time their use is required. That's why beams of different combinations of support conditions and loadings are compiled: you can use the tables to calculate reactions, moments, and deflections quickly without having to go thru a lengthy derivation each time.

Here is a table of some common beam formulas:

http://www.awc.org/pdf/DA6-BeamFormulas.pdf