Question

Components of the stress tensor $\sigma$ at a point, with respect to the $(x, y, z)$ coordinate system, are given as $\begin{bmatrix} 18 & 24 & 0 \\ 24 & 32 & 0 \\ 0 & 0 & -20 \end{bmatrix}$ (MPa). a) Find the principal stresses $\sigma_i$ and the unit vectors $e_i$ along the principal directions. Arrange $\sigma_i$ such that $\sigma_1 \ge \sigma_2 \ge \sigma_3$. Further, choose the senses of $e_i$ such that $e_1 \cdot e_2 \times e_3 = +1$. b) Find the hydrostatic and deviatoric parts of $\sigma$. (c) Find the maximum shear stress $\sigma_{s\text{ max}}$ and the normal to the planes on which $\sigma_{s\text{ max}}$ acts. Express the normal in terms of the unit vectors $(i, j, k)$.

          Components of the stress tensor $\sigma$ at a point, with respect to the
$(x, y, z)$ coordinate system, are given as
$\begin{bmatrix} 18 & 24 & 0 \\ 24 & 32 & 0 \\ 0 & 0 & -20 \end{bmatrix}$ (MPa).
a) Find the principal stresses $\sigma_i$ and the unit vectors $e_i$ along the principal
directions. Arrange $\sigma_i$ such that $\sigma_1 \ge \sigma_2 \ge \sigma_3$. Further, choose the senses of $e_i$
such that $e_1 \cdot e_2 \times e_3 = +1$.
b) Find the hydrostatic and deviatoric parts of $\sigma$.
(c) Find the maximum shear stress $\sigma_{s\text{ max}}$ and the normal to the planes on
which $\sigma_{s\text{ max}}$ acts. Express the normal in terms of the unit vectors $(i, j, k)$.

Components of the stress tensor σ at a point, with respect to the
(x, y, z) coordinate system, are given as
< b m a t r i x > (MPa).
a) Find the principal stresses and the unit vectors ei along the principal
directions. Arrange such that σ1 ≥σ2 ≥σ3. Further, choose the senses of ei
such that e1 · e2 × e3 = +1.
b) Find the hydrostatic and deviatoric parts of σ.
(c) Find the maximum shear stress σs max and the normal to the planes on
which σs max acts. Express the normal in terms of the unit vectors (i, j, k).

Added by Peter H.

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