beam deflection

In the case of a loaded beam, what is the more likely mode of failure due to overload - bending or fracture?

Were the beam made of cast iron the more likely failure mode might be fracture, not bending, in which case the shear stress would be of primary interest.

This bridge shows that it is quite possible to construct bridges in cast iron. In this particular example, nearly every beam is in compression, which is the best way to use the material.

I don't know why you don't do shear, but I almost always do, though usually I just do combined stress and don't bother sorting it all out. So, the question really is: why don't you do shear stress? You must have learned to draw shear stress diagrams in Statics and Strengths of Materials.

I send you the answer yesterday but I see it was not accepted so that I send it again.

Consider a beam length "L" rectangular section bxh

If it is a cantilever (for simplicity) the maximal bending moment will be M= F*L

The maximal bending stress will be σ = M/W where W= b*h^2/6 thus σ = 6*F*L/(b*h^2)

The maximal shear stress will be τ = 1.5*F/(b*h)

Now let us make the ration σ/τ

σ/τ= 4*L/h.

If the ratio L/h is >1/4 σ >τ. Almost all beams have a L/h ratio >> 1 so always σ >τ!

As you see since the highest stress has to be checked it will be the bending one.

I do not believe anyone from CR4 has answered your question properly. The answer is that deflection due to bending is usually much larger than shear deformation. For many years, particularly when engineers were doing hand calculations, shear deformation has been ignored because, for most situations, it is small compared with bending. In the old days, axial deformation for frames was also ignored because it was small and difficult to incorporate in hand calculations. When an engineer cannot find an exact solution, what does he do? He approximates.

When you have a short beam with heavy point loads, shear deformation cannot be ignored as it may be in the same order of magnitude as the bending deformation. In those cases, it must be considered in the analysis.

Happily, most finite element analysis (FEA) programs include shear deformation in the stiffness matrix for the various elements. And they also include axial deformations. In the end, deformation may or may not be the deciding factor in your design, but it may be comforting to know that all aspects have been considered in the analysis.

But there are a great deal more imponderables in the field of structural analysis than these. Even the most comprehensive structural analysis is only as good as the information entered into the program. Complete reliance on a computer program is a huge mistake. You must temper the output with common sense and engineering "gut feel".