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
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".
Shear stress is linear and is zero at the mid plane. Bending stress is parabolic. You had them backwards
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 ?
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.
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.