## Principal Stress

In the element below, calculate the max tensile principal stress.

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__**Hint**

**Hint**

For the special case of a two-dimensional stress state, the equation for principal stress reduces to:

$$$\sigma _a,\sigma _b=\frac{\sigma_x+\sigma_y}{2}\pm \sqrt{(\frac{\sigma _{x}-\sigma_{y}}{2})^{2}+\tau_{xy}^{2}}$$$

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__**Hint 2**

**Hint 2**

For the special case of a two-dimensional stress state, the equation for principal stress reduces to:

$$$\sigma _a,\sigma _b=\frac{\sigma_x+\sigma_y}{2}\pm \sqrt{(\frac{\sigma _{x}-\sigma_{y}}{2})^{2}+\tau_{xy}^{2}}$$$

Comparing the problem's figure to the above equation format,
$$\sigma_y$$
is -140 MPa due to the downward direction the arrow is pointed in. Thus,

$$$\sigma _a,\sigma _b=\frac{275+-140}{2}\pm \sqrt{(\frac{275-(-140)}{2})^{2}+70^{2}}$$$

$$$\sigma _a,\sigma _b=67.5\pm \sqrt{43,056.25+4,900}=67.5\pm219$$$

$$$\sigma _a=67.5+219=286.5\:MPa$$$

$$$\sigma _b=67.5-219=-151.5\:MPa$$$

The max principal stress is 286.5 MPa.

286.5 MPa