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# 2.1 Stress At A Point

## 2.1.1 Stresses on Rotated Planes – 3D

Suppose that we know the stress state of the soil on a particular plane, and we wish to determine the stress state on a different, rotated plane. This problem is equivalent to a transformation of the coordinate system. The components of the Cauchy stress tensor in the rotated coordinate space, $$\sigma_{ij}^{'}$$, can be determined from the stress components in the reference state, $$\sigma_{ij}$$, using the rotation matrix $$a$$ with components $$a_{ij}$$. Note that in soil mechanics, we often represent the shear stress components (i.e., the off-diagonal elements of the stress tensor) as $$\tau_{ij}$$ instead of $$\sigma_{ij}$$. Furthermore, the prime in $$\sigma'$$ usually denotes effective stress, but is used here to denote stresses in a rotated coordinate system.

$$\sigma' = a \sigma a^T$$

In matrix form:

$\left[ \begin{matrix} \sigma _{11}^{'} & \sigma _{12}^{'} & \sigma _{13}^{'} \\ \sigma _{21}^{'} & \sigma _{22}^{'} & \sigma _{23}^{'} \\ \sigma _{31}^{'} & \sigma _{32}^{'} & \sigma _{33}^{'} \\ \end{matrix} \right]=\left[ \begin{matrix} {{a}_{11}} & {{a}_{12}} & {{a}_{13}} \\ {{a}_{21}} & {{a}_{22}} & {{a}_{23}} \\ {{a}_{31}} & {{a}_{32}} & {{a}_{33}} \\ \end{matrix} \right]\left[ \begin{matrix} \sigma _{11}^{{}} & \sigma _{12}^{{}} & \sigma _{13}^{{}} \\ \sigma _{21}^{{}} & \sigma _{22}^{{}} & \sigma _{23}^{{}} \\ \sigma _{31}^{{}} & \sigma _{32}^{{}} & \sigma _{33}^{{}} \\ \end{matrix} \right]\left[ \begin{matrix} {{a}_{11}} & {{a}_{21}} & {{a}_{31}} \\ {{a}_{12}} & {{a}_{22}} & {{a}_{32}} \\ {{a}_{13}} & {{a}_{23}} & {{a}_{33}} \\ \end{matrix} \right]$

Figure 2.1.1. Stresses in a reference coordinate system (axes $$x_1$$, $$x_2$$, $$x_3$$), and rotated coordinate system (axes $$x_1'$$, $$x_2'$$, $$x_3'$$). https://upload.wikimedia.org/wikipedia/commons/thumb/7/76/Stress_transformation_3D.svg/2000px-Stress_transformation_3D.svg.png

## Try it yourself

Note: The $$a$$ matrix must be orthogonal (i.e., the axes in the rotated coordinate system must form 90 degree angles with each other)

$$\sigma$$
$$a$$
$$\sigma'$$
 2 0 0 0 1 0 0 0 1

## 2.1.2 Stresses on Rotated Planes – 2D

In soil mechanics, we often simplify three-dimensional stress conditions to two dimensions. The Mohr Circle is a very common tool used to interpret soil stress states, and it is inherently a two-dimensional representation of stresses. We should therefore consider rotation of stresses in two dimensions. Figure 2.1.2 defines the terms of the rotation tensor, $$a_{ij}$$, in a manner that is consistent with the three-dimensional procedure in Figure 1.

Figure 2.1.2. Two dimensional coordinate system rotation (note rotation is about the $$x_3$$ axis by angle $$\gamma$$).

The rotated stress state can therefore be defined as follows: $\left[ \begin{matrix} \sigma _{11}^{'} & \sigma _{12}^{'} \\ \sigma _{21}^{'} & \sigma _{22}^{'} \\ \end{matrix} \right]=\left[ \begin{matrix} \cos (\gamma ) & -\sin (\gamma ) \\ \sin (\gamma ) & \cos (\gamma ) \\ \end{matrix} \right]\left[ \begin{matrix} {{\sigma }_{11}} & {{\sigma }_{12}} \\ {{\sigma }_{21}} & {{\sigma }_{22}} \\ \end{matrix} \right]\left[ \begin{matrix} \cos (\gamma ) & \sin (\gamma ) \\ -\sin (\gamma ) & \cos (\gamma ) \\ \end{matrix} \right]$

Expanding out the matrix algebra results in the following expressions for the rotated stresses: \begin{align} & \sigma _{11}^{'}={{\sigma }_{11}}-{{\sigma }_{12}}\sin (2\gamma )-{{\sin }^{2}}(\gamma )\left( {{\sigma }_{11}}-{{\sigma }_{22}} \right) \\ & \sigma _{22}^{'}={{\sigma }_{22}}+{{\sigma }_{12}}\sin (2\gamma )+{{\sin }^{2}}(\gamma )\left( {{\sigma }_{11}}-{{\sigma }_{22}} \right) \\ & \sigma _{12}^{'}={{\sigma }_{12}}\cos (2\gamma )+\frac{\sin (2\gamma )}{2}\left( {{\sigma }_{11}}-{{\sigma }_{22}} \right) \\ \end{align}

## Try it yourself

$$\sigma$$
$$\gamma$$
deg
$$a$$
$$\sigma'$$
 2 0 0 1

## 2.1.3 Graphical Solution for 2D: Pole Method

A graphical solution to the two-dimensional stress state on a plane inclined at any angle can be solved using the Mohr Circle. The steps are as follows:

1. Plot stress state and draw Mohr Circle

Figure 2.1.3a. Graphical solution of stresses on inclined plane: Step 1.

2. Draw line from ($$\sigma_{22},\sigma_{12}$$) in a direction perpendicular to the line of action of $$\sigma_{22}$$. We assume here that $$\sigma_{22}$$ acts vertically as illustrated in Fig. 2.1.2.

Figure 2.1.3b. Graphical solution of stresses on inclined plane: Step 2.

3. Draw line from ($$\sigma_{11},\sigma_{12}$$) in a direction perpendicular to the line of action of $$\sigma_{11}$$.

Figure 2.1.3c. Graphical solution of stresses on inclined plane: Step 3.

4. The intersection of the lines from Step 2 and 3 is called the "pole". The pole is a unique point on the Mohr circle because the stress state along a plane inclined at any angle $$\theta$$ is defined by the intersection of the Mohr circle with a line drawn from the pole at an angle $$\theta$$.

Figure 2.1.3d. Graphical solution of stresses on inclined plane: Step 4.

5. Draw a line angled at $$\theta$$ from horizontal and $$\theta$$ from vertical. The intersection of these lines with the Mohr circle defines the state of stress.

Figure 2.1.3e. Graphical solution of stresses on inclined plane: Step 5.

You may notice that we are using $$\theta$$ to define the angle when we use the pole method, while we use $$\gamma$$ to define the angle when we used the rotation matrix in the previous section. There is a good reason for this. $$\theta$$ is an angle measured relative to the specific coordinate system under consideration, while $$\gamma$$ is a change of angle from a reference plane to a rotated plane. Say, for example, that we know the stress condition for an element on planes that lie at angles $$\theta_1$$ from horizontal and vertical, and we wish to find stresses on different planes rotated at angles $$\theta_2$$ from horizontal and vertical. In this case, $$\gamma$$ = $$\theta_2$$ - $$\theta_1$$. When finding the pole, it is important to draw the lines perpendicular to the line of action $$\sigma_{22}$$ and $$\sigma_{11}$$ in steps 2 and 3, respectively.

Figure 2.1.4. Mohr's circle for plane stress and plane strain conditions (Pole approach).