Question

A cantilever beam AB with a circular cross section and length L = 750 mm supports a load P = 800 N acting at the free end (see figure). The beam is made of steel with an allowable bending stress of l20 MPa. Determine the required diameter d_min (figure part a) of the beam, considering the effect of the beam's own weight. Repeat part (a) if the beam is hollow with wall thickness t = d/8 (figure part b); compare the cross-sectional areas of the two designs.

Answer 1

P = 800 N, L = 750 m (or) 0.75 m bending moment at point A M_A = PL + w times L/a straightline 1 weight of the beam, w = pi times d^2/4 times L times e A = diameter, e = density of steel w = pi/4 times d^2/4 times 0.75 times 77,000 N/m^2 w = 45,356.85 d^2 N straightline 2 from eq 2 M_A = 800 times 0.75 + 45356.85 d^2 times 0.75/2 M_A = 600 + 17,016 d^2 straightline 3 maximum bending moment m_max = sigma_a s = sigma_a (pi d^3/3^2) straightline 4 from eq 3 & 4 600 + 17016 d^2 = sigma_a (pi d^3/32) sigma_a = 120 mpa (or) 120 times 10^6 pa 600 + 17016 d^2 = 120 times 10^6(3.1416 times d^3/32) 11.781 times 10^6 d^3 - 17016 d^2 - 600 = 0 d_min = 37.6 mm b) t = d/8 calculate the weight of the beam, w = rho L[pi/4(d^2 - (3d/40^2]} w = 77000 times 0.75 lambda