Shear Deformation - Point Moment 7

[Celt83]

This is probably something that is very easy and I am way overthinking it.

I've done the unit force method which shows that the shear deformation from a point moment is 0 at all locations on a simple span. I've done the same thing to show that the shear component of the slope at any location on the beam is constant = k M / A G L. Here is where I get hung up if the slope is constant and non-zero why is the deflection 0 from a math stand point, the mechanics make sense to me but I'm lost in the math on this one.

I have a great mechanics of materials book by Timoshenko and Young which for me so far has the best break down of shear deformations I've been able to find and in there they present that the slope due to shear is simply dy/dx = tau,max / G = k V_x / A G, which aligns with my unit method solution of k M / A G L.

integrating that once yields: y,shear = k M x / G A L + C1

Initial conditions are x=0, y=0 and x=L, y=0 the first condition yields C1 = 0, however the second condition yields C1 = k M / A G ... so I get that something is missing here which would should result in y,shear = 0 but I am at a loss.

would greatly appreciate any insight on this one.

Open Source Structural Applications:

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[BAretired]

Maybe we can also agree that, irrespective of its orientation, a straight prismatic beam with constant shear along its length will experience deformation parallel to the axis of the beam and zero deformation normal to the beam.

BA

 

[cal91]

Maybe we can also agree that, irrespective of its orientation, a straight prismatic beam with constant shear along its length will experience deformation parallel to the axis of the beam and zero deformation normal to the beam.

Depends on boundary conditions.

Cantilever with transverse point load at the end has constant shear but transverse shear deflection.

Simply supported beam with point moment at the end has constant shear but longitudinal shear deflection.

 

[BAretired]

Depends on boundary conditions.

Cantilever with transverse point load at the end has constant shear but transverse shear deflection.

Simply supported beam with point moment at the end has constant shear but longitudinal shear deflection.

Agreed. I should have specified that the beam is simply supported.

BA

 

[KootK]

Maybe we can also agree that, irrespective of its orientation, a straight prismatic beam with constant shear along its length will experience deformation parallel to the axis of the beam and zero deformation normal to the beam.

I can agree to that with one caveat: said beam has to span between two points of support, each providing restraint in the direction normal to the beam.

Logically, I think that hearkens back to IDS's argument that, effectively, a straight thing can only connect two points by way of a straight line (my paraphrase). And a beam with a constant shear diagram is a straight thing from the perspective of shear deformation.

I know, me and the definitions...

...I certainly gained a lot from this.

Ditto. If there's an afterlife, I suspect that Timoshenko is out there somewhere monitoring this conversation with interest and regretting his inability to take part.

 

[IDS]

Logically, I think that hearkens back to IDS's argument that, effectively, a straight thing can only connect two points by way of a straight line (my paraphrase). And a beam with a constant shear diagram is a straight thing from the perspective of shear deformation.

Nice paraphrase.

I'm tempted to ask what the principles of energy conservation have to say about whether Timoshenko is or is not spinning in his grave, but maybe we should stay off that one.

Doug Jenkins
Interactive Design Services

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