Mohr's Circle - Mohr's Circle For A General Three-dimensional State of 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
as the denominator and
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 is a good lesson—though it may often be a hard one—for a man who has dreamed of literary fame, and of making for himself a rank among the world’s dignitaries by such means, to step aside out of the narrow circle in which his claims are recognized, and to find how utterly devoid of all significance, beyond that circle, is all that he achieves, and all he aims at.”
    —Nathaniel Hawthorne (1804–1864)

    “The general Mistake among us in the Educating of our Children, is, That in our Daughters we take Care of their Persons and neglect their Minds; in our Sons, we are so intent upon adorning their Minds, that we wholly neglect their Bodies.”
    —Richard Steele (1672–1729)

    “The present war having so long cut off all communication with Great-Britain, we are not able to make a fair estimate of the state of science in that country. The spirit in which she wages war is the only sample before our eyes, and that does not seem the legitimate offspring either of science or of civilization.”
    —Thomas Jefferson (1743–1826)

    “Europe has a press that stresses opinions; America a press, radio, and television that emphasize news.”
    —James Reston (b. 1909)