Combined bending and shear and twisting stresses in beams

Hi,
I will discuss here with you the way to analyse a beam for the amount of maximum normal and shear stresses developed at a point in a beam, which is subjected to combined stresses.

Beams subjected to combined stresses:

Beam subjected to combined normal, bending, shear and twisting stresses

I assume you are familiar with the normal stresses, shear stresses, bending stresses and shear stresses due to Twisting.
Also you might be familiar with the Principal stresses and maximum shear stresses at a point.
If you know all these terms then you have sufficient tools to analyze a beam who is subjected to combined stresses.
 Any point in a beam can be subjected a following stresses at the same time:

(1) Normal stress = Applied axial load/ Cross Sectional Area = P/A
(2) Bending stress = (M/I).y
(3) Shear Stress due to transverse load = (V.Q)/(I.b)
(4) Shear stress due to twist/Torque = (T/J).r

Combine all stresses and convert along the normal stress and
shear stress

If combination of two or more than two of the above stresses arises at a point in a beam, you have find the resultant of all of them along the normal and tangential direction along a face.

Generally normal loading is only in axial direction(x-axis) so the resultants will be only an axial load, and shear stress.

So when you have found the resultants you can easily apply the formula for the principal stress at an element when this is subjected to a normal stress accompanied by a shear stress.

(Reference: An online video lecture on youtube from IITK)