Calculating Principal Stress in Combined Loading of a Beam

[P0zzn]

Consider a beam under combined loading. Axial force, bending moment and torsion.
I'm interested in determining the principle stress in any stress element on surface of beam.

Well as per rule I've to show my attempt so:
normal stress=axial load/beam crosssection
shear stress=torsion*radius/polar MOI
bending stress=moment * radius/ MoI about NA

normal stress and bending stress have same line of action so undergo vector addition.
Now we have a normal stress and shear stress. We got principal stress and orientation from Mohr's circle.

Sounds good... But it didn't work. Where did i go wrong?

 

[FredGarvin]

How about posting exactly what you did. There's no way we can tell you where you went wrong without seeing what you did.

 

[P0zzn]

Sorry sir. But i clearly mentioned my approach to that problem. As far calculations and Mohr's circle is involved, I'm quiet sure, that's not a problem.

I'd highly appreciate your effort if you could list the stresses acting on the stress element under specified loading.

 

[CarlAK]

You need to think about which stress acts along each of the 3 axes.
Along the beam (call this x), you get a combination of axial stress + bending stress (positive or negative depending on relation to neutral axis)
-shear, as would be obtained from a shear diagram, acts along y axis.
-in the z axis, a normally loaded beam would have zero I believe, but if you have torsion then that is probably the third principal stress

I may have missed something, if so I hope a real structural/materials engineer will chime in.

 

[FredGarvin]

How are we supposed to know if you made a simple sign error?

 

[P0zzn]

CarlAK your effort is apprecable.
I don't differ from you upto the computation of stress along X axis as you said, but there is no direct shear involved. So torsìon actually produces shear stress in y axis. And being honest i have no idea about stress in Z axis.
Anyways Thanks.

Fred, i am afraid sign isn't the problem.

 

[ank_gl]

Consider a beam under combined loading. Axial force, bending moment and torsion.
I'm interested in determining the principle stress in any stress element on surface of beam.

Well as per rule I've to show my attempt so:
normal stress=axial load/beam crosssection
shear stress=torsion*radius/polar MOI
bending stress=moment * radius/ MoI about NA

normal stress and bending stress have same line of action so undergo vector addition.
Now we have a normal stress and shear stress. We got principal stress and orientation from Mohr's circle.

Sounds good... But it didn't work. Where did i go wrong?

So.. what didnt work? Where did you go wrong?

This is the way i list them(according to the loadings in OP)

$$\sigma$$$$_{xx}$$ = My/I + F/A

$$\tau$$$$_{xy}$$ = VQ/I

$$\tau$$$$_{yz}$$ = 16T/(pi)d^3

 

[P0zzn]

I'm sorry but are you sure of that ank_gl??
I don't think there is direct shear involved. So possibly 2nd eqn isn't needed.
Anyways, thank you.

 

[ank_gl]

You mentioned combined loading, so i assumed you also included direct shear. If its pure bending, then yes, 2nd equation won't apply.