Mechanics problem

[Seismica]

I'm struggling with an engineering mechanics problem i'm doing as part of my revision. There is a gap in my notes and I can't seem to find anything useful online for my problem.

I need to find the Principal stresses at a point on a beam using Mohr's circle:

I know how to do this. The problem is, I don't know what σx and σy are, so I can't plot the circle as I have no scale for the horizontal axis (I have the maximum shear stress which represents the top of the circle).

From the given beam and cross section (Let's say it's an 'I' section beam for simplicity) I have created the shear force and bending moment diagrams, found the neutral axis and second moment of area/moment of inertia which has in turn allowed me to calculate the maximum shear stress.

Then i'm kind of stuck. I've used Mohr's circle to find the principal stresses before, but I don't know where σx and σy come from, they've always been given values in examples I have done before

How do I create a stress element diagram like the one below?

All I know is that they are normal stresses and the x and y parts show the direction, but I don't understand how to define them.

EDIT: Could it be as simple as the shear force at the point is the x-value and the moment force is the y-value?

 

[Exorcet]

Try using the stress invariants.

I2 = sigma_x*sigma_y + tau_xy

Set tau_xy to maximum shear, then solve for I2. Then do the zero shear case and you'll get an expression for sigma_x in terms of sigma_y or vice versa. You can then plug this into any formula you have relating maximum shear, sigma_x and sigma_y.

 

[Seismica]

Thankyou for the quick reply :-)

What does the I2 stand for? Would solving it at zero shear just give the values for the principle stresses or just give the values for sigma x and y for the circle?

I'm just struggling with how to apply the stress theory to an example. I thought it involved using My/I or P/A to find the normal stresses.

 

[Exorcet]

I2 is the second stress invariant. There are three invariants that come from the eigen values of the stress tensor. I1, I2, and I3 never change no matter how you transform your coordinates, so no matter where you are on Mohr's circle, they are constant. Solving for zero shear would give you the principle stresses.

As for what to apply, it depends on what you've been given. You said in the first post that you had max shear. Is this the only given info in the problem?

 

[Seismica]

I've been given an axial force acting down at a point on the beam, and a moment force acting across the beam. I've been asked to find the max shear stress and principal stresses at the top of the beam.

So i have been given a force and a moment. From these i have calculated the maximum shear stress and using My/I have found the normal/bending stress at the top and bottom of the cross section. That is where i'm stuck, i don't know how to bridge the gap between the two sets of notes i have.

 

[Exorcet]

OK, good you have more info than I thought you did. If the beam is symmetric and the forces act along axes of symmetry, you just use superposition because the axial force and moment are decoupled.

Now that you have the axial stress and shear, you can just plug into the equation for principle stress (or use Mohr's circle).

You have one normal stress (the axial stress) and the shear. You can plug in the axial stress for either sigma x or sigma y (depends on what coordinate system you're using).

 

[ECGadget]

^Using that for revision too ^

 

[Bram Turismo]

Man, I'm glad I'm past all the theoretical stuff about moments and shear, and off with practical uses! You just wait until you bump into metal plates upon which half I-beams, T-beams, are welded and you need to examine how shear and moment fields work upon this construction.

 

[Seismica]

OK, good you have more info than I thought you did. If the beam is symmetric and the forces act along axes of symmetry, you just use superposition because the axial force and moment are decoupled.

Now that you have the axial stress and shear, you can just plug into the equation for principle stress (or use Mohr's circle).

You have one normal stress (the axial stress) and the shear. You can plug in the axial stress for either sigma x or sigma y (depends on what coordinate system you're using).

I think i've got my head around it now, I'm beginning the understand how Mohr's circle is applied to an example.

But what if the cross section of the beam isn't symmetrical? I know it's uniform across the length of the beam, but it isn't an 'I' section beam. I'd rather not post exactly what it is, but is there any additional steps I would need to take for an asymmetric cross section?

 

[Exorcet]

I think i've got my head around it now, I'm beginning the understand how Mohr's circle is applied to an example.

But what if the cross section of the beam isn't symmetrical? I know it's uniform across the length of the beam, but it isn't an 'I' section beam. I'd rather not post exactly what it is, but is there any additional steps I would need to take for an asymmetric cross section?

At that point, when the loading does not line up with any symmetry, you have a coupled problem, which is bad. Your moment will induce other moments and so will your axial force. You need to solve for whole bunch of stiffnesses that just turn out to be zero in the symmetry case, superposition fails, and you get a system of differential equations.

This is something you'd probably want to put into a computer, so I'm hoping that the problem isn't really asking for you to solve this.

 

[Seismica]

At that point, when the loading does not line up with any symmetry, you have a coupled problem, which is bad. Your moment will induce other moments and so will your axial force. You need to solve for whole bunch of stiffnesses that just turn out to be zero in the symmetry case, superposition fails, and you get a system of differential equations.

This is something you'd probably want to put into a computer, so I'm hoping that the problem isn't really asking for you to solve this.

Ok, well in that case I think that it's a little advanced for what we are doing now, I doubt we will get such a problem in the exam (It's something we didn't even touch on in the lectures - I think that's something we will do at the start of next year).

You've been very helpful, thanks I think i'll attempt a couple of I-section beams, now that I know how to do them.

 

[Kylehnat]

I love engineering threads!