Determine the temperature rise necessary to induce buckling in a 1 m long circular rod of diameter 40 mm shown in the figure below. Assume the rod to be pinned at its ends and the coefficient of thermal expansion as 20 $$ \times $$ 10^-6/^oC. Assume uniform heating of the bar.

At a point in a stressed body the state of stress on two planes $$45$$⁰ apart is as shown below. Determine the two principal stresses

A concentrated load $$P$$ acts on a simply supported beam of span $$L$$ at a distance $${L \over 3}$$ from the left support. The bending moment at the point of application of the load is given by

A circular rod of diameter $$d$$ and length $$3d$$ is subjected to a compressive force $$F$$ acting at the top point as shown below. Calculate the stress at the bottom-most support point A.