Problem C$-2[2$D State of Stress and Mohr's Circle$]$

Strength of Materials ME $2050,$ Fall $2024,$ Exam$-2$

A structural member is subjected to a plane stress loading condition. The state of stress at a point of the member is shown in the figure.

$(1)$ Draw the Mohr's circle for this $2$D state of stress;

$(2)$ Determine the principal stresses, and the maximum inplane shear stresses and associated normal stresses developed at this point;

$(3)$ Represent the obtained critical stresses in $(2)$ on new $2$D stress elements, and indicate the principal stress orientation $({}_P)$ and the maximum in$-$plane shear orientation $({}_S)$ on the new stress elements with respect to the original stress elements.

Note: directly plug in the given values into the plane stress transformation equations will not validate the problem.