In a system, two connected rigid bars $$AC$$ and $$BC$$ are of identical length $$L$$ with pin supports at $$A$$ and $$B.$$ The bars are interconnected at $$C$$ by a frictionless hinge. The rotation of the hinge is restrained by a rotational spring of stiffness, $$k.$$ The system initially assumes a straight line configuration, $$ACB.$$ Assuming both the bars as weightless, the rotation at supports, $$A$$ and $$B,$$ due to a transverse load, $$P$$ applied at $$C$$ is:

A steel strip of length, $$L = 200$$ $$mm$$ is fixed at end $$A$$ and rests at $$B$$ on a vertical spring of stiffness, $$k = 2$$ $$N/mm.$$ The steel strip is $$5$$ $$mm$$ wide and $$10$$ $$mm$$ thick. $$A$$ vertical load, $$P = 50$$ $$N$$ is applied at $$B,$$ as shown in the figure. Considering $$E = 200$$ $$GPa,$$ the force (in $$N$$) developed in the spring is _______________

A simply supported beam $$AB$$ of span, $$L = 24$$ $$m$$ is subjected to two wheel loads acting at a distance, $$d = 5$$ $$m$$ apart as shown in the figure below. Each wheel transmits a load, $$P = 3$$ $$kN$$ and may occupy any position along the beam. If the beam is an $$I$$-section having section modulus, $$S = 16.2\,c{m^3},$$ the maximum bending stress (in $$GPa$$) due to the wheel loads is __________________

A horizontal beam $$ABC$$ is loaded as shown in the figure below. The distance of the point of contraflexure from end $$A$$ (in $$m$$) is