Question

2. For the box beam with loading shown, (a) determine the state of stress in the x-y plane at points A and B and indicate on square differential elements oriented with x and y; (b) represent the state of stress on Mohr's circle for both points and identify the principal stresses and maximum in- plane shear stress; (c) show the principal stress orientation of the reference frame by rotating the square differential element for both points. 1200 lb 800 lb 6 in. B 8 in 8 AH 6 in. in. 3 ft 2.5 ft-1-2.5 ft-_5 ft ft-+2.5 ft-5 ft

Answer 1

As given by R_1 + R_2 = 2000: Sigma m_s_2 = 0 = 800 times 13 + 1200 times 5 = R_1 times 10 R_1 = 80 times 13 + 5 times 120 = 1640 lb R_2 = 360 lb \r\n I = (8^4/12 - 64/12) = 283.33 in^4 Bending moment M_AB = 1640 times 2.5 - 800 times 5.5 = 300 lb ft = 3600 lb ln \r\n T_b = M_by/I = T = V/1 (A^3) sigma_b /A = 3600 times 4/233.33 = 61.715 lb/in^2 (Tension) sigma_b/B = 3600 times 3/233.33 = 46.2834 ln/in^2 (compression) t/A = 0 T/A = 840/233.33 times 8 times (8 times 2 times 35) = 12.6 lb/in^2 (clock wise) rightarrow share force at action AB V_A,B = 840 lb rightarrow Element A: \r\n Principle stress sigma_1 = 61.715 lb/in tau_max = 30.86 lb/in^2 Element B: Taken clock wise shear to positive \r\n 2 theta = tan^- 1(12.6/23.14) = 28.59 theta = 14.28 sigma_1 = -23.14 - 26.35 = -49.49 lb/in^2 sigma_2 = -23.14 + 26.35 = + 3.21 lb/in^2