# Stress state in a beam

#### ladil123

Hello!

If I know the deflection (w) of a beam subjected to a point load in the middle, can I calculate the stress in that beam by calculating the shear force or moment ?
or use a 2d plane strain state ?

What is the best way to do it ?

Thanks

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#### skeleton

Yes. You need three equations:

deflection of beam as a function of (P,L,E,I)
bending stress of beam as a function of moment.
shear stress of beam as a function of shear.

Then you need to add the two stresses together, but they almost always don't add linearly. Bending stress is varies with the height, y, within the section; ie: high at both top and bottom fibers, and zero at the mid section. Shear stress is parabolic, with its maximum at the mid section. A bit of reading will reveal what shear stress engineers usually use at the extreme fibers (it is not zero).

This is all in most books on "Mechanics of Materials".

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#### CFDFEAGURU

Shear stress is linear and is zero at the mid plane. Bending stress is parabolic.

You had them backwards

#### ladil123

Ok.
So the bending stress sigma_B = M_max *Z/I , right? I=moment of inertia.

What about the stress due to shear then? With the equation for deflection I could integrate and get the shear force, but how do I get the shear stress ?

#### skeleton

Shear stress is linear and is zero at the mid plane. Bending stress is parabolic.

You had them backwards
I guess I wasn't clear. I was referring to the transverse stress level within a SECTION of the beam, and not referring along its length.

What you said is true but for longitudinal stress along the length.

#### dE_logics

I think I didn't get the question.

Yeah if you know the deflection, you can apply hooks law and figure out the stress.

But for that you also need to know 'sigma'; the constant.

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