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 .

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