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4.2 Stress At A Point - 3D

4.2.1 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, $\hat{\sigma}_{ij}$ , 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 $\hat{\sigma}$ usually denotes effective stress, but is used here to denote stresses in a rotated coordinate system.

\hat{σ} = a σ a^{T}

$\hat{\sigma} = a \sigma a^T$

In matrix form:

[\begin{matrix} {\hat{σ}}_{11} & {\hat{σ}}_{12} & {\hat{σ}}_{13} \\ {\hat{σ}}_{21} & {\hat{σ}}_{22} & {\hat{σ}}_{23} \\ {\hat{σ}}_{31} & {\hat{σ}}_{32} & {\hat{σ}}_{33} \end{matrix}] = [\begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{matrix}] [\begin{matrix} σ_{11}^{} & σ_{12}^{} & σ_{13}^{} \\ σ_{21}^{} & σ_{22}^{} & σ_{23}^{} \\ σ_{31}^{} & σ_{32}^{} & σ_{33}^{} \end{matrix}] [\begin{matrix} a_{11} & a_{21} & a_{31} \\ a_{12} & a_{22} & a_{32} \\ a_{13} & a_{23} & a_{33} \end{matrix}]

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

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)

σ

$\sigma$

a

$a$

\hat{σ}

$\hat{\sigma}$

2	0	0
0	1	0
0	0	1

Directly computing components of the

a

$a$ matrix is not always convenient. If you prefer, you may specify the rotation angles about the

x_{1}

$x_1$ ,

x_{2}

$x_2$ , and

x_{3}

$x_3$ axes (

α

$\alpha$ ,

β

$\beta$ , and

γ

$\gamma$ , respectively) below to set the

a

$a$ matrix.

α

$\alpha$ deg

β

$\beta$ deg

γ

$\gamma$ deg