A homogeneous, prismatic, linearly elastic steel bar fixed at both the ends has a slenderness ratio (l/r) of 105, where l is the bar length and r is the radius of gyration. The coefficient of thermal expansion of steel is $12 \times 10^{-6}$ $^{\circ}$C. Consider the effective length of the steel bar as $0.5l$ and neglect the self-weight of the bar.

The differential increase in temperature (rounded off to the nearest integer) at which the bar buckles is

For the 6m long horizontal cantilever beam PQR shown in the figure, Q is the midpoint. Segment PQ of the beam has flexural rigidity $EI = 2 \times 10^5$ kN. $m^2$ whereas the segment QR has infinite flexural rigidity. Segment QR is subjected to uniformly distributed, vertically downward load of 5 kN/m.

The magnitude of the vertical displacement (in mm) at point Q is _______ (rounded off to 3 decimal places).

The structural design method that DOES NOT take into account the safety factors on the design loads is

The horizontal beam PQRS shown in the figure has a fixed support at point P, an internal hinge at point Q, and a pin support at point R. A concentrated vertically downward load (V) of 10 kN can act at any point over the entire length of the beam.

The maximum magnitude of the moment reaction (in kN.m) that can act at the support P due to V is ________________ (in integer).