Mohr's Circle

From the stress transformations, an equation in the form of a circle can be derived as follows:

$$ [\sigma _ \theta - \frac{\sigma _x + \sigma _y}{2}]^2 + \tau _ \theta ^2 = ( \frac {\sigma _x - \sigma _y}{2}) + \tau _{xy} ^2 $$

Where the centre of the circle is \( ( \frac{ \sigma _x + \sigma _y}{2}, \tau _ \theta ) \). The radius is \( \sqrt{(\frac{\sigma _x - \sigma _y}{2})^2 + \tau _{xy}^2 } \)

To plot Mohr's circle, a specific sign convention is used. If the shearing action on a face turns the element clockwise, it is deemed positive and plotted above the axis.

Since diametrically opposite points on the circle are stresses on perpendicular planes, two adjacent sides of the element can be chosen to be plotted.

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Where \( A \) and \( B \) are perpendicular sides of an element, \( (\sigma _{x'}, \tau _{x'y'}) \) can be determined using the \( \theta \) equations.

Principal Stresses

At \( \tau _ \theta = 0 \), the maximum and minimum pure normal stresses are called the principal stresses, calculated as follows:

$$ \sigma _{1,2} = \frac{\sigma _x + \sigma _y}{2} \pm \sqrt{(\frac{\sigma _x - \sigma _y}{2})^2 + \tau _{xy} ^2} $$

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