For the beam and loading shown in the figure, the second derivative of the deflection curve of the beam at the mid-point of $A C$ is given by $\frac{\alpha M_0}{8 E I}$. The value of $\alpha$ is _________ (rounded off to the nearest integer).
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 linearly elastic planar structure shown in the figure is acted upon by two vertical concentrated forces. The horizontal beams UV and WX are connected with the help of the vertical linear spring with spring constant k = 20 kN/m. The fixed supports are provided at U and X. It is given that flexural rigidity EI = 105 kN-m2, P = 100 kN, and a = 5 m. Force Q is applied at the center of beam WX such that the force in the spring VW becomes zero.
The magnitude of force Q (in kN) is __________. (round off to the nearest integer)
If the mid-span deflections of these beams are denoted by $${\delta _1}$$ and $${\delta _2}$$ (as indicated in the figures), the correct option is