Question

(30 pts) The bar ACB having two different cross-sectional areas Ai and Az js held between rigid supports at A and B-See figure. A load P acts at point C. which is distance b, from end A and b2 from end B. Modulus of elasticity (E) is the same for both sections Az A: 8 b: b2 a) Obtain formulas for the reactions Ra and Re at supports A and B due to load P b) Obtain a formula for the displacement 6c of point C. c) what is the ratio of the stresses σ1 in the region AC to σ2 in region CB

Answer 1

(a) balance all force acting in (x - alpha) direction R_A = P + R_B rightarrow (1) Therefore It is fixed at A and B Therefore S_AC + S_BC = 0 Rightarrow R_A times B_1/E times A_1 + R_B times b_2/E times A_2 = 0 Rightarrow R_A times b_1/E times A_1 + R_B times b_2/E times A_2 = 0 Rightarrow R_A times b_1/A_1 + R_B b_2/A_2 = 0rightarrow (ii) put value of R_A in (ii) (P + R_B) b_1/A_1 + R_B b_2/A_2 = 0 Rightarrow p b_1/A_1 + R_B b_1/A_1 + R_B b_2/A_2 = 0 Rightarrow R_B [b_1/A_1 + b_2/A_2] = -p b_1/A_1] Rightarrow R_B = -p b_1 A_2/[b_1 A_2 + b_2 A_1] rightarrow (iii) put value of (iii) in (i) R_A = P + R_B = p - pb_1 A_2/[b_1 A_2 + b_2 A_1]