Longitudinal shear in built-up beams 2

[MusicEngineer]

Hello all,

Hope everyone is well today. I am in the process of solving a conceptual question and I would appreciate everyone's input on this.

Please see the sketch attached. This is not the exact question I'm trying to solve but it is similar enough such that the concept can be carried over.

The goal is to find out the Longitudinal Shear at the connecting interface of the two beams shown and to figure out which weld size works best. I want to use the shear flow equation to compute the required shear force per unit length but I'm having some difficulty understanding how this equation is applied.

The equation is taken from a mechanics of materials text book. q=VQ/I; where q is the required shear (lb/ft), V is the shear force at the cross-section, Q is the moment of the area taken about the neutral axis, and I is the moment of intertia.

I have been able to calculate all the terms in equation except the 'Q' term based on this scenario. Any help would be greatly appreciated! Also, if there are any other methods to calculate the longitudinal shear at the connection interface of the two members, please feel free to share.

 

[JAE]

That is one heavy T section.

To calculate Q, draw a horizontal plane through the point where the welds occur. The area of the shape on one side of that plane has a centroid. Take the distance from that centroid of that area to the NA of the total shape. That is your d value. The area of the shape on one side of the plane is your A. Q = Ad.

 

[MusicEngineer]

Thank you JAE, that answer was spot on, concise and well explained. I appreciate your input. And yes, that is one HEAVY T section. The reason I drew that T section like that is because I wanted to make sure that the NA was located on the side of the T section.

 

[MusicEngineer]

Follow up question: Is the moment of area above the horizontal plane equal to the moment of area below the HP? In other words, is it necessary that the area above the horizontal plane muliplied by the distance from the centroid of this area to the NA is equal to the area below the HP multiplied by the distance from the centroid of the lower area to the NA?

If not, does this mean that we take the larger of the two 'Q' values?

 

[BAretired]

1. I agree with the explanation of JAE and it is the method most engineers would use including me. But you asked for other methods.

2. You could calculate the flexural stresses at top and bottom of the existing W beam at a certain point in the span, say p[sub]1[/sub]. Since the beam is symmetrical, the average stress in the beam is the average of those two values, say f[sub]1[/sub]. You could do the same for another point, say p[sub]2[/sub] and find f[sub]2[/sub].

The axial force in the beam at p[sub]1[/sub] is A.f[sub]1[/sub]. The axial force in the beam at p[sub]2[/sub] is A.f[sub]2[/sub]. The difference is the force in the weld, F[sub]w[/sub]. If p[sub]1[/sub] and p[sub]2[/sub] are separated by distance L, the average force in the weld is F[sub]w[/sub]/L or A(f[sub]2[/sub]-f[sub]1[/sub])/L.

3. You could calculate the flexural stress at the centroid of the Tee section at two points in the span, then determine the total force in the Tee at two points and repeat the same process as in 2. above.

This is, in fact how the VQ/I formula was derived. The change in axial force in a portion of the beam must be delivered by a horizontal shear in the weld.

BA

 

[BAretired]

I was still writing when you wrote your follow-up question. All three methods give exactly the same result.

BA

 

[JAE]

On your follow up question why don't you take your section and try it out to see if the two sides Q values equal out.

 

[MusicEngineer]

The Q values do equal out. It makes sense now. Thanks.