In an effort to gain the insight necessary to solve the problems myself, and to show those damned civil engineers that I can indeed grasp their useless abstractions (useless because there are a great number of higher-level mathematical techniques which could be used to solve these problems in a much more straight forward manner) I will now derive and explain Mohr's circle in relation to analyzing plane stresses.
Examination
of the normal (O) and the shear stresses (T) acting upon a cross-section
of a larger body (figure to the left) allows us to analyze the stresses
and strains on sections inclined with respect to this one (figure to the
right.) By equilibrium we find that, via sign conventions,
which will simplify things as the derivation progresses. The stresses
acting on the inclined X1, Y1 element (right) can be expressed in terms
of the stresses on the X, Y element (left) by using Newtonian equations
of equilibrium. Given that the force acting on an area is equal to the
stress multiplied by the area, or
Equations (1) and (2) give the normal and shear stresses acting on the inclined plane in terms of the angle, theta, and the original stresses acting on the segment. For special cases when the angle is zero, we note that Ox_{1} = Ox and Tx_{1}y_{1} = Txy, as we would expect.
Civil engineers claim that this can be "expressed in a more convenient form" by using certain trigonometric identities, but, honestly, convenience is a matter of opinion. Now, this is where I would have simply stopped. Right here is plenty of information I need to solve just about any plane stress problem thrown my way. It may be ugly, and it certainly could be simplified using some sort of vector or matrix notation, however, provided you can handle dealing with squared sines and cosines, and don't mind the derivation of and/or memorizing these equation, it's not too bad. BUT NOOOOOO! Some dumb-ass civil engineer was afraid of manipulating squared sines and cosines, and used the following identities:
Now, again, here is where I would have stopped. I would have a set of equations (slightly more complicated than (1) & (2) in my opinion) that is still within reason that I could derive on the fly and/or memorize. And some civil engineers were obviously fine with stopping here because they went ahead and named these equations the transformation equations for plane stress, because they transform the stress components from one set of axis to another.
Also, some slightly interesting things pop out of these equations. For
one, when the original stresses are zero, except for Ox, then the element
is said to be in uniaxial stress, meaning that it only is experiencing
stress along one axis. In the uniaxial stress state, equations (3) and
(4) reduce to
To find the maximum and minimum normal stresses throughout the entire range of angles, one can easily take the first derivative of (3) with respect to theta, set it to zero, and solve for the angle. This will give what is called the principal plane on which the principal stresses act. If this all sounds overly complicated... you're right! Why not just use the tried and true terminology "maximizing and minimizing the function" instead of inventing these two new terms with unrelated and unclear meaning? Well.... that's civil engineers for you.
Now is where we begin to get into the unnecessary jargon. All the excess
baggage some engineer created to make it so that utilizing these relationships
would not require higher math. This (and many other examples of engineer
idioticy) most likely stems from the fact that most engineers slept through
their higher level math classes, and suffer from acute mathematical insecurities
(and probably rightly so.) It's these abstract constructions which attempt
to simplify the work, yet ultimately make it more difficult for those of
us more mathematically inclined, that really piss me off. If
you represent equation (5) geometrically with a 90^{o} triangle,
(left), we can obtain general formulas for the principal stresses. First,
we note that the hypotenuse of the triangle is,
The equations of Mohr's circle can be derived from the transformation
equations (3) and (4). By simply rearranging the first equation, we find
that the two expressions comprise the equation of a circle in parametric
form.
However, by resubstitution of equation (6) and by recognizing that the average stress value between the X and Y axis, Oave, is,
With Ox, Oy, and Txy known, the procedure for constructing Mohr's circle
is as follows:
O_{1,2}, representing the maximum and minimum normal stresses and their respective angles away from point A (where theta = 0^{o}) can be found by simply looking at the O values when T = 0. In the drawing above, O_{1} represents the maximum, and O_{2} the minimum. Furthermore, T_{max/min}, representing the maximum and minimum shear stresses and their respected angles can be found by locating the T values when O = Oave. At this point, T is simply equal to the radius, R, or equation (6).
In addition to these helpful points, all other possible points for the shear and normal stresses can be found on this circle. In order to find another value of Ox, Oy for a given rotation, one must simply start at the A and B points (A representing the Ox value and B, the Oy value), and rotate in a positive theta direction (by the orientation shown above, this is in a counterclockwise direction, in keeping with the right hand rule) for 2theta (from equations (3) and (4) above). The resulting points, D and D', will yield the Ox, Txy, and Oy, Txy (respectively) for that rotation.
As I have likely mentioned before (likely because, I can't really recall) to me this seems all very abstract and difficult to use. However, the aforementioned bastiches will be requiring this on my upcoming test, so I felt a need to more fully understand it. Granted, I still don't understand it as fully as I would hope, but it ought to be enough to get me through this one, insignificant little test.
P.S.: I apologize for my editorializing and opinionated presentation of this topic. I rarely do this when I analyze problems I don't understand (even when I do not like the method, such as the Lewis Dot structure). This time, however, I have some very strong feelings about my predicament. Also, in all fairness, if you were given the problem where O1 = O2 and Tmax = 0, i.e. the Mohr's circle was simply a little dot with R = 0, using the Mohr's circle method would arrive you at any and all answers much quicker than using equations (3) and (4). However, I don't think this extreme simplification of one special case warrants the abstraction being a required bit of knowledge for civil engineers. -->