Mohr's Circle For A General Three-dimensional State of Stresses
To construct the Mohr's circle for a general three-dimensional case of stresses at a point, the values of the principal stresses and their principal directions must be first evaluated.
Considering the principal axes as the coordinate system, instead of the general, coordinate system, and assuming that, then the normal and shear components of the stress vector, for a given plane with unit vector, satisfy the following equations
Knowing that, we can solve for, using the Gauss elimination method which yields
Since, and is non-negative, the numerators from the these equations satisfy
- as the denominator and
- as the denominator and
- as the denominator and
These expressions can be rewritten as
which are the equations of the three Mohr's circles for stress, and, with radii, and, and their centres with coordinates, respectively.
These equations for the Mohr's circles show that all admissible stress points lie on these circles or within the shaded area enclosed by them (see Figure 3). Stress points satisfying the equation for circle lie on, or outside circle . Stress points satisfying the equation for circle lie on, or inside circle . And finally, stress points satisfying the equation for circle lie on, or outside circle .
Read more about this topic: Mohr's Circle
Famous quotes containing the words circle, general, state and/or stresses:
“To love someone is to isolate him from the world, wipe out every trace of him, dispossess him of his shadow, drag him into a murderous future. It is to circle around the other like a dead star and absorb him into a black light.”
—Jean Baudrillard (b. 1929)
“In former times and in less complex societies, children could find their way into the adult world by watching workers and perhaps giving them a hand; by lingering at the general store long enough to chat with, and overhear conversations of, adults...; by sharing and participating in the tasks of family and community that were necessary to survival. They were in, and of, the adult world while yet sensing themselves apart as children.”
—Dorothy H. Cohen (20th century)
“To motorists bound to or from the Jersey shore, Perth Amboy consists of five traffic lights that sometimes tie up week-end traffic for miles. While cars creep along or come to a prolonged halt, drivers lean out to discuss with each other this red menace to freedom of the road.”
—For the State of New Jersey, U.S. public relief program (1935-1943)
“American families, however, without exception, experience a double message in our society, one that claims a commitment to families and stresses the importance of raising bright, stable, productive citizens, yet remains so bound by an ideal of rugged individualism that parents receive little support in their task from the public or private sectors.”
—Bernice Weissbourd (20th century)