Mohr's Circle and understanding its maximums and minimums

[Adder_Noir]

Hi!

I've been looking at Mohr's circle and by and large I understand what's going on. I can see how if the perpendicular stresses sigma x and sigma y are known and the plane is tilted to an angle from the horizontal trig can be used to evaluate the nominal compressive force and nominal shear force on that plane. I can also understand the use of the tan function to produce two values for the angle theta which are 90 degrees apart which I assume is due to the period of the tan function.

In short basically I've done quite a bit of homework on this already myself. One thing puzzles me though. I notice that tau (the shear stress) is a maximum at 45 degrees orientation to the horizontal. I was curious originally as to why it wasn't a maximum when at 90 degrees to the horizontal. I got a bit confused thinking surely that would mean maximum shear stress but then realized the compressive force sigma is acting on a point it's not two forces either side of the middle of the object so I concluded that shear is not a maximum here as the sigma force is acting along the plane of shear - I hope you understand what I mean.

Two big questions remain though:

1)Considering what happens to the magnitude of the shear force when the rotation moves from 45 to 90 degrees, why doesn't it start to grow very large as its angle *approaches* 90 degrees? I would have thought at an angle of say for example 80 degrees most of the compressive sigma force would be trying to shear apart the two parts of the object separated by the plane very strongly thus making the shear force here greater than at 45 degrees not less. I don't understand why shear stress is so small here.

2)At an angle of 135 degrees rotation shear stress is considered a minimum. Does this mean it is a magnitude minimum or is this more of a vector type minimum which means it's of equal magnitude to that at 45 degrees just pointing the opposite way?

I'll post a link to a great pic on wikipedia:

http://en.wikipedia.org/wiki/File:Principal_stresses_2D.svg"

Thanks in advance

 

[kyiydnlm]

On the pic you provided, ignore the bottom element and let's say index 1 to 4 from left to right. 1 and 3 are on exactly same plane; they are same thing. 2 and 4 on same plane, they are identical. The stresses of each of the pairs are in same direction.

2 and 4 are on principle plane; maximum normal stress, zero shear. 45 degree from principle plane, 1 and 3, maximum shear and minimum normal stress.

When approching 2 or 4, normal stresses increase, and shears decrease; when approching 1 or 3, shear stress increase, normal ones decrease.

do not confuse yourself, 1 and 3 are same thing, 2 and 4 are same thing.

if you put them in the center of the circle and rotate them, 1-3 (or 2-4) will overlap.

Remember, the elements are not something flying around a center; they are representatives of plane.

 

[Adder_Noir]

I haven't had chance to digest the full meaning of your post yet but I can tell just from the way it's written it will make a whole lot of sense. Thanks, I'll post again when I've digested it all. I've also found a new angle on a concept problem I had with this thing I'll post details of that too cheers!