Question

4. The beam AC shown in Figure 4 is supported by two columns AE and BD, and carries a load P (P-50 kN). The columns have the same square cross section hxh, Young's modulus E 2x105 MPa. Determine the minimum dimension h of the cross section such that both columns do not fail in elastic buckling. Use the factor of safety of 1.2 against buckling. Pin connections are used for ends E, A, and B as shown in Figure 4. (similar to 2018 exam question) 2 m 50 kN 2 m 2 m Figure 4

Answer 1

F.B.D of ABC Taking moment about B. rightarrow F_A5 times 3 = 50 times 2 F_AE = 100/2 = 33.33 kN sigma f_y = 0 rightarrow R_B - F_AE + 50 R_B = 33.33 + 50 R_B = 83.33 kN FBD of AE Because P_r = z^2 EI/Less^2 Less_AE = L = 2M Because both end pinned downarrow F.O.S 1.2 times 33.33 times 10^3 = P_r = Z^2 times 2 times 10^5 times 1/12 times h h^3/(2000)^2 h = 31.4 mm FBD of BD Less_BD = Max^m{BF = 2 FD times 0.7 = 0.7 times 3 = 2.1 Less BD = 2.1 m = 2100 mm Because p_r = Z^2 elementof L/Less^2 downarrow F.O.S 1.2 times 83.33 times 10^3 = Z^2 times 2 times 10^5 times 1/12 times h^4/2.1 times 10^3)^2 h = 40.46 mm Hence the minimum dimension required to present = h_max = 40.46 mm