A linearly elastic beam of length 2l with flexural rigidity EI has negligible mass. A massless spring with a spring constant k and a rigid block of mass m are attached to the beam as shown in the figure.

The natural frequency of this system is

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