# Deflection at any point formula

Good Afternoon Everyone,

I am need of a little assistance. I am working on determing the precise stress at any point along a beam and would like a little help. I know the deflection at every point and would like an easy way to relate this back to stress. Any help?

[PLAIN]http://sphotos.ak.fbcdn.net/hphotos-ak-snc3/hs358.snc3/29503_523051459059_53800516_30961326_1066764_n.jpg [Broken]

RMX

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Hey RMXByker,

It is a good question and Studiot provided you a good link talking about the relationship between deflection and internal moment. With this equation, however, you are assuming small angle deflection, so be sure you remember that everything I am about to say only applies for small angle deflection.

Also, to determine all stress at every point in the beam would be almost impossible. I am going to interrupt that you are looking for the bending stress distribution (ignoring all shear stresses). Please correct me on what you are looking for if this is wrong.

In the attached document (cause I can't seem to make the build in TeX to work), I have made up a little sheet for you. I hope it helps.

#### Attachments

• PFhelp0001.pdf
27.3 KB · Views: 167
I solve a very similar problem in this thread:

I can't get it to work with PF

Mech_Engineer
Gold Member
I can't get it to work with PF

The last attachment I posted is basically a fully symbolic derviation of the beam bending formula using the integration procedure. MathCAD did all the heavy lifting for me in terms of symbolic manipulation, but it can at least give you an idea of what you're in for (a lot of work).

You'll end up having to split the beam into three sections, integrate three times for each section, and then solve a system of 9 equations with 9 unknowns at the end. It's not pretty. A pdf of the MathCAD sheet you'll be most interested in is attached to the following post:

https://www.physicsforums.com/showpost.php?p=1600375&postcount=19

The integration tree you'll need is as follows:

$$\nu''''=\frac{q(x)}{EI}$$

$$\nu'''=\frac{V(x)}{EI}$$

$$\nu''=\frac{M(x)}{EI}$$

$$\nu'=\theta(x)$$

$$\nu=\delta(x)$$

Thank you, I know how to solve a beam.

At present I use MathType, but this has no word processor or picture (graph) capabilities.

Mech_Engineer
Gold Member
One thing that will make your problem more difficult is you won't be able to make the assumption that your max deflection & bending moment will always be in the middle. You'll also have to think about what you want to do about cases where a is near one side of the beam or the other.

Mech_Engineer