Module 8: Principal Stresses and Mohr's Circle - Examples & Applications

A stressed element is subjected to normal stresses

\sigma_x = 50 MPa

\sigma_y = -10 MPa

, and shear stress

\tau_{xy} = 40 MPa

. Calculate the principal stresses analytically.

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STEP-BY-STEP

Steps to Construct Mohr's Circle:

Establish a Cartesian coordinate system with normal stress ( $\sigma$ ) on the x-axis (positive tension to the right) and shear stress ( $\tau$ ) on the y-axis (positive downward is standard convention in mechanics of materials to match physical rotation).
Plot the known state of stress on the right vertical face of the physical element as Point A: $(\sigma_x, \tau_{xy})$ .
Plot the known state of stress on the top horizontal face of the element as Point B: $(\sigma_y, -\tau_{xy})$ .
Connect points A and B with a straight line. Where this line intersects the horizontal $\sigma$ -axis is the Center ( $C$ ) of the circle.
Draw a circle using $C$ as the center, passing exactly through points A and B. The distance from C to A is the radius $R$ .
The rightmost intersection point of the circle on the $\sigma$ -axis is the maximum principal stress ( $\sigma_1$ ). The leftmost intersection point is the minimum principal stress ( $\sigma_2$ ). The highest (or lowest) vertical point of the circle is the maximum shear stress ( $\tau_{max}$ ).

A point on a steel bracket is subjected to the following state of plane stress:

\sigma_x = 80 MPa

(tension),

\sigma_y = -40 MPa

(compression), and

\tau_{xy} = 25 MPa

. Determine the principal stresses and maximum shear stress.

0 of 3 Steps Completed