Problem about Proof A simply supported beam

[davidpotter]

Hello everyone!
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 :)
author: internet việt nam
have a nice day.

 

[Simon Bridge]

The different equations are the result of physics and probably some approximation.
You should consult the source for a reference.

 

[SteamKing]

Hello everyone!
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.

This is not correct. If the beam is simply supported at both ends, the bending moment at the supports is zero. That's inherent in what a simple support is: it's just a resting point which cannot resist the rotation of the beam under load; hence, no moment can develop at these locations.

That I get and can show proof yet its the centre moment I cannot reproduce, (qL^2)/8

This is because you have not analyzed the statics of a simply supported beam properly.

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

Now, you've moved on from bending moments to talking about deflections.

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}

Beams fixed at both ends are not statically determinate, like the cantilever or the simply supported beam. In order to determine the reactions and bending moments at the ends, a different kind of analysis must be performed.

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 :)
author: internet việt nam
have a nice day.

People do this because they understand the methods used to analyze these situations. They don't need to re-derive everything from first principles every time a new analysis is required. Many books on structural design carry tables for reference for determining reactions and bending moments for a handful of simple (and not so simple) cases. Saves time.

I don't know what books on strength of materials or mechanics you can refer to, but all of the above is basic stuff.

These links may provide you with an introduction to beam analysis:

http://www.learnengineering.org/2013/08/shear-force-bending-moment-diagram.html

http://www.mhhe.com/engcs/engmech/beerjohnston/mom/samplechap.pdf

There's many other examples on the web which can be uncovered by a simple search.