Mohr's Circle - Mohr's Circle For A General Three-dimensional State of Stresses | Mohr Circle General Three-dimensional State Stresses

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

$\begin{align} \left( T^{(n)} \right)^2 &= \sigma_{ij}\sigma_{ik}n_jn_k \\ \sigma_\mathrm{n}^2 + \tau_\mathrm{n}^2 &= \sigma_1^2 n_1^2 + \sigma_2^2 n_2^2 + \sigma_3^2 n_3^2 \end{align}\,\!$

Knowing that, we can solve for, using the Gauss elimination method which yields

$\begin{align} n_1^2 &= \frac{\tau_\mathrm{n}^2+(\sigma_\mathrm{n} - \sigma_2)(\sigma_\mathrm{n} - \sigma_3)}{(\sigma_1 - \sigma_2)(\sigma_1 - \sigma_3)} \ge 0\\ n_2^2 &= \frac{\tau_\mathrm{n}^2+(\sigma_\mathrm{n} - \sigma_3)(\sigma_\mathrm{n} - \sigma_1)}{(\sigma_2 - \sigma_3)(\sigma_2 - \sigma_1)} \ge 0\\ n_3^2 &= \frac{\tau_\mathrm{n}^2+(\sigma_\mathrm{n} - \sigma_1)(\sigma_\mathrm{n} - \sigma_2)}{(\sigma_3 - \sigma_1)(\sigma_3 - \sigma_2)} \ge 0 \end{align}\,\!$

Since, and is non-negative, the numerators from the these equations satisfy

as the denominator and

These expressions can be rewritten as

$\begin{align} \tau_\mathrm{n}^2 + \left^2 \ge \left( \tfrac{1}{2}(\sigma_2 - \sigma_3) \right)^2 \\ \tau_\mathrm{n}^2 + \left^2 \le \left( \tfrac{1}{2}(\sigma_1 - \sigma_3) \right)^2 \\ \tau_\mathrm{n}^2 + \left^2 \ge \left( \tfrac{1}{2}(\sigma_1 - \sigma_2) \right)^2 \\ \end{align}\,\!$

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:

“It was my heaven’s extremest sphere,
The pale which held that lovely deer;
My joy, my grief, my hope, my love,
Did all within this circle move!”
—Edmund Waller (1606–1687)

“In the drawing room [of the Queen’s palace] hung a Venus and Cupid by Michaelangelo, in which, instead of a bit of drapery, the painter has placed Cupid’s foot between Venus’s thighs. Queen Caroline asked General Guise, an old connoisseur, if it was not a very fine piece? He replied “Madam, the painter was a fool, for he has placed the foot where the hand should be.””
—Horace Walpole (1717–1797)

“A perfect personality ... is only possible in a state of society where man is free to choose the mode of work, the conditions of work, and the freedom to work. One to whom the making of a table, the building of a house, or the tilling of the soil, is what the painting is to the artist and the discovery to the scientist,—the result of inspiration, of intense longing, and deep interest in work as a creative force.”
—Emma Goldman (1869–1940)

“The goal in raising one’s child is to enable him, first, to discover who he wants to be, and then to become a person who can be satisfied with himself and his way of life. Eventually he ought to be able to do in his life whatever seems important, desirable, and worthwhile to him to do; to develop relations with other people that are constructive, satisfying, mutually enriching; and to bear up well under the stresses and hardships he will unavoidably encounter during his life.”
—Bruno Bettelheim (20th century)