Rigidly jointed frame-critical load-circular frequencies

Question

Question 1: The rigidly jointed frame shown in Figure has bending stiffnesses such that EI_AB = 1000kNm²and EI_BC = EICD = 1600kNm².

A) Determine the joint rotations at B and C.
B) Hence, plot a bending moment diagram for the frame stating principal values.
C) Sketch a displaced shape for the frame.

Question 2: Determine the critical load (W_crit) to cause elastic instability of the rigidly jointed  frame shown in figure below. The frame members have constant bending stiffnesses (EI) as indicated on the figure, where E is the modulus of elasticity and I is the second moment of area.

Question 3. For the collapse loads indicated in figure below, find the required plastic moment of resistance (M_p). The frame members have  varying  plastic moments of resistance as indicated on the figure.

Question 4: The two-storey  frame shown in Figure below is such  that  the self-mass may be considered to be located at the beam levels and the beams may be presumed to be rigid. The self-mass (m) at each beam level and column stifnesses (EI) are  indicated on the figure, where E  is the modulus of elasticity and I is the second  moment of area. Determine the natural circular frequencies and natural modes of vibrations of the frame and provide sketches of these.