A question about cutting the section to compute the shearing tension

[uts]

Hello Dears! I've just registered here. It sounds nice!

As I've seen in different solution manuals here in my country, there are several realizations of how to cut the section we've got, to compute the resultant shearing tension of a shear loading.

For instance, as you see in the following section we have a structure that is consist of three parts; 2 lateral parts and a thin medial part, that are joint with screw and nuts. The problem is to compute the shearing tension in the a-a position.

[PLAIN]http://upload1.imgdl.ir/images/711Untitled_1.jpg

The most acceptable solution I've found, is that cutting the medial thin part is just enough to detaching the hatchet part. So in the tension formula (Tension = V . Q / I . t ) the "t" will be the medial part diameter and the "Q" will be computed for the hatched part. That was the first approach.

The second one is cutting the whole section across the a-a line. So that the "t" and "Q" parameters will be computed for the section show in the following figure;

[PLAIN]http://upload1.imgdl.ir/images/4062.jpg

Which one is correct?!
Sorry for my probable English grammatical and vocab mistakes. I'll appreciate that if you share your knowledge and proficiency about this problem with me.

 

[nvn]

I think you can do it either way you prefer. It depends on whether you want to (a) find the shear stress on the entire section a-a, or (b) find the shear stress on only the medial part.

 

[AlephZero]

I think you need to find it from first principles. Look up the way that the shear force distribution is calculated for Timoshenko beam theory. (See Timoshenko's "Theory of Elasticity", or a similar textbook). I mean the equations that give the parabolic distribution of shear in a rectangular beam, and the approximately constant shear in an I beam. The shear stress depends on an integral of the axial stresses (or forces) through the depth of the beam. Then use the same method for your beam cross section.

Your beam profile has 3 flanges not 2, so the standard "Tension = V . Q / I . t" formula for an I beam will give the wrong answers.

There are two different assumptions that you could make:

(1) The bolted joint is tight enough so the beam acts like one solid piece of metal, and the shear stress is the same in the three parts of the web

(2) The shear stress in the outside and the middle parts is different, and the shear force is only transferred from the outside to the middle by the bolt.

 

[Studiot]

there are several realizations of how to cut the section we've got, to compute the resultant shearing tension of a shear loading.

It's not at all clear to me what is being discussed here.

The OP seems to be talking about shear stresses, due to shear loading

The responses appear to be about shear stresses that arise as a result of bending of a beam.

 

[AlephZero]

It's not at all clear to me what is being discussed here.

The OP seems to be talking about shear stresses, due to shear loading

The responses appear to be about shear stresses that arise as a result of bending of a beam.

If you apply a shear load to the end of a cantilever, the beam bends.

There is no debate about the total shear force at any section of the beam. That is statically determinate. The question is how the shear force (or shear stress) is distributed over the cross section.

Euler-Bernouilli beam theory (a.k.a. engineer's beam bending theory) doesn't answer that question at all.

Timoshenko's beam theory gives an approximate answer by considering the equilibrium of the axial forces (caused by bending) and the shear forces, through the depth of the beam. This is not entirely straight forward, because the bending behaviour doesn't depend on the precise geometric shape of the cross section, only on its I value, but the shear force distribution does depend on the details of the shape.

For example the shear stress distribution in I-beam, and the shear stress in rectangular section beam with the same I value, are very different.

However Timoshenko makes some (unavoidable) assumptions about the shear stress distribution which for a general shaped beam (e.g a circular section) are incompatible with the stress boundary conditions that relate to the geometry. At this point the arguments about "the right way to do it" can easily generate more heat than light, and if you really "want the right answer" then perhaps you should give up trying to assume the structure is a beam and just make a 3-D model. However (IMO) that is not relevant to the OP's question, and Timoshenko beam theory would give a sensible results.

You need to remember that even a "full 3D" FE model is only an approximation to reality. For the OP's structure, do you really want to model the full details of the clamping loads from the bolt, including the clearance between the bolt and the holes in the beam? I think not, for most "real life" engineering situations.

Timoshenko beam theory also includes an extra flexibility term for the beam in shear. Unlike Euler beam theory, sections through the beam are not assumed to remain normal to the neutral axis. The shear flexibility is also affected by the shear force/stress distribution over the beam section, though the OP was not asking about that. (And for slender beams, it is a small effect on the beam stiffness in any case, and the shear force/stress distribution is a small correction to that small effect)