Mohr's circle method

The 3D Mohr circle is a graphical representation of the stress tensor and all its projections (or possibles values of normal stress $\sigma_n$ and shear stress $\tau$) on a given plane. Consider a horizontal plane in Fig. 5.18, the normal stress is the vertical stress $S_v$ and there is no shear stress. Consider a vertical plane with strike East-West in Fig. 5.18, you get the minimum principal stress $S_{hmin}$. Consider a vertical plane with strike North-South in Fig. 5.18, you get the maximum principal stress $S_{Hmax}$.

Likewise, non-trivial solutions of stress projection at an arbitrary plane angle include all the points delimited by the three Mohr circles. Let's consider solutions along each circle in Fig. 5.18.

• The blue circle (center C1) represents the possible state of stresses that result as a combination of $\sigma _v$ and $\sigma_{hmin}$. The shear stress $\tau$ and normal effective stress $\sigma_n$ of any plane in between the planes of $\sigma _v$ and $\sigma_{hmin}$ and colinear with $\sigma_{Hmax}$ can be found through the angle $\beta_1$ measured from $\sigma _v$.
• The green circle (center C2) represents the possible state of stresses that result as a combination of $\sigma_{v}$ and $\sigma_{Hmax}$. The shear stress $\tau$ and normal effective stress $\sigma_n$ of any plane in between the planes of $\sigma_{v}$ and $\sigma_{Hmax}$ and colinear with $\sigma_{hmin}$ can be found through the angle $\beta_2$ measured from $\sigma_{Hmax}$
• The red circle (center C3) represents the possible state of stresses that result as a combination of $\sigma_{Hmax}$ and $\sigma_{hmin}$. The shear stress $\tau$ and normal effective stress $\sigma_n$ of any plane in between the planes of $\sigma_{Hmax}$ and $\sigma_{hmin}$ and colinear with $\sigma_{v}$ can be found through the angle $\beta_3$ measured from $\sigma_{Hmax}$

Figure 5.18: The 3D Mohr circle

For this example (normal faulting, $S_{Hmax}$ azimuth E-W), a fault would occur with a strike E-W and dip 60$^{\circ }$ (assuming $\varphi = 30^{\circ}$). This is the orientation and point for maximum $\tau /\sigma _n$.

PROBLEM 5.3: Find the shear and normal effective stresses on a fault plane within the following state of stress and conditions:

• Fault: strike azimuth = 000$^{\circ }$, dip = 60$^{\circ }$.
• State of stress: $S_v =$ 23 MPa (principal), $S_{Hmax} =$ 20 MPa, $S_{hmin} =$ 13.8 MPa (azimuth: 090$^{\circ }$), and $P_p =$ 10 MPa.

SOLUTION

The effective stresses are: $\sigma_v =$ 13 MPa, $\sigma_{Hmax} =$ 10 MPa, $\sigma_{hmin} =$ 3.8 MPa. Based on the Mohr circle of $\sigma _v$ with $\sigma_{hmin}$ and trigonometry:

$$\sigma_n = \left( \frac{13 \text{ MPa} + 3.8 \text{ MPa}}{2} \rig... ... MPa} - 3.8 \text{ MPa}}{2} \right) \cos(2 \cdot 60^{\circ}) = 6.1 \text{ MPa}$$

$$\tau = \left( \frac{13 \text{ MPa} - 3.8 \text{ MPa}}{2} \right) \sin(2 \cdot 60^{\circ}) = 4.0 \text{ MPa} \: \: \blacksquare$$

PROBLEM 5.4: Find the shear and normal effective stresses on a fault plane within the following state of stress and conditions:

• Fault: strike azimuth = N60$^{\circ }$E, dip: 90$^{\circ }$.
• State of stress: $S_v =$ 30 MPa (principal), $S_{Hmax} =$ 45 MPa, $S_{hmin} =$ 25 MPa (azimuth: N30$^{\circ }$E), and $P_p =$ 15 MPa.

SOLUTION

The effective stresses are: $\sigma_v =$ 15 MPa, $\sigma_{Hmax} =$ 30 MPa, $\sigma_{hmin} =$ 10 MPa. Based on the Mohr circle of $\sigma_{Hmax}$ with $\sigma_{hmin}$ and trigonometry:

$$\sigma_n = \left( \frac{30 \text{ MPa} + 10 \text{ MPa}}{2} \righ... ...t{ MPa} - 10 \text{ MPa}}{2} \right) \cos(2 \cdot 30^{\circ}) = 25 \text{ MPa}$$

$$\tau = \left( \frac{30 \text{ MPa} - 10 \text{ MPa}}{2} \right) \sin(2 \cdot 30^{\circ}) = 8.7 \text{ MPa} \: \: \blacksquare$$