What Are the Derivations and Uses of Beam Deflection Formulas?

[...harry...]

i have two different beam deflection formulas, one for rigid supports and one for simple supports - but i know there are many different ones out there.

i was wondering if anyone could tell me how the formulas were created, oh and also what exactly does moment of inertia have to do with delfection??

 

[FredGarvin]

The easy part first. The area moment of inertia is a numerical value that describes a geometry's resistance to bending. So it the larger the value of the area moment of inertia, the greater resistance to bending something will be. It is analogous to the moment of inertia in dynamics where it is a measure of the resistance to angular acceleration.

Beam equations are derived from differential equations that govern the behavior. In what is known as "classical beam theory" there is one major assumption made that simplifies the analysis. That assumption is that the cross section taken through any part of the beam will always remain perpendicular to the neutral axis and will remain in its original shape as well.

I would suggest doing a search for "beam equations derivations" or similar to find there are a ton of pages that will talk about where the "plug and play" equations come from. Here's an example:
http://en.wikipedia.org/wiki/Euler-Bernoulli_beam_equation

 

[...harry...]

what are plug and play equations? is this where you just put in the numbers and get an answer?
are the derivations of the equations all complex math of is there a simpler explanation... even if its not mathematical?

 

[Mech_Engineer]

Basically what Fred is talking about are pre-solved algebraic equations that describe a beam's deflection or angle. Here's a link to another thread about deriving a plug-n-play equation for an irregularly loaded beam:

https://www.physicsforums.com/showthread.php?t=209319&highlight=harder+beam+equation

If you're interested in how beam equations are derived, you could buy a mechanics of materials or beam theory textbook, which would take you through different methods of approximating beam bending. Roark's Formulas for Stress and Strain is an engineer's bible for pre-solved beam bending equations, and briefly describes the methods used for solving them.

 

[FredGarvin]

what are plug and play equations? is this where you just put in the numbers and get an answer?
are the derivations of the equations all complex math of is there a simpler explanation... even if its not mathematical?

The derivations are, like I mentioned, based on differential equations. I don't really see any way to simplify them. That is why, for most people, the pre solved equations are so nice. Complex beam problems can usually be solved through combinations of the pre-solved equations. That is called superposition. It too has some underlying assumptions that dictate its use.

Here are some pages with some plug and play equations I mentioned:
http://www.engineersedge.com/beam_bending/beam_bending10.htm
http://www.neng.usu.edu/mae/faculty/stevef/info/beam_eq.htm
https://ecourses.ou.edu/cgi-bin/ebook.cgi?doc=&topic=me&chap_sec=&page=&appendix=beams