Two pipe columns (AB, FC) are pin connected to a rigid beam (BCD), as shown in the figure. Each pipe column has a modulus of E, but heights (L₁ or L₂) and outer diameters (d₁ or d₂) are different for each column. Assume the inner diameter of each column is 3/4 of the outer diameter. Uniformly distributed downward load q = 2P/L is applied over a distance of 3L/4 along BC, and concentrated load P/4 is applied downward at D.

(a) Derive a formula for the displacement at point D in terms of P and column flexibilities f₁ and f₂.

(b) If d₁ = (9/8) d₂, find the L1₁ or L₂ ratio so that beam BCD displaces downward to a horizontal position under the load system in part (a).

(c) If L₁ = 2L₂, find the d₁/d₂ ratio so that beam BCD displaces downward to a horizontal position under the load system in part (a).

(d) If d₁ = (9/8)d₂ and L₁/L₂ 51.5, at what horizontal distance x from B should load P/4 be placed so that beam BCD displaces downward to a horizontal position under the load system in part (a)?