A beam of solid circular cross-section and length 1.0 m is simply supported at i

Question

A beam of solid circular cross-section and length 1.0 m is simply supported at its ends. The beam is required to support a lateral load of 6.0 kN uniformly distributed along its length and a torque of 1.5 kNm, with a safety factor of 2.0 against yielding. You have a choice of two materials from which to make the beam:- AISI 4142 steel: ? = 7900 kg/m3, S_Y = 1584 MPa 2024-T4 aluminum: ? = 2700 kg/m3, S_Y = 303 MPa, where ?, S_Y are the density and uniaxial yield stress respectively. What is the mass of the lightest beam satisfying these conditions? HELP PLEASE!!!!!!!!

Explanation / Answer

Max bending moment = wl^2/8
w = 6 kN/m
M = 750000 Nmm
T = 1500000 Nmm

= 32M/d3

= 16T/d3

von mises srtess = (2 + 32)0.5 = 13253928.3/d3 MPa

Case 1

Allowable stress = 1584/2 = 792 Mpa

d3 = 16734.757 d = 25.578 mm

Mass = 4.0594 kg ( volume = area of cross section * length)

Case 2

Allowbale stress = 303/2 = 151.5 MPa

d^3 = 87484.675 d = 44.392 mm

Mass = 4.17902 kg

Lightest beam is stee of mass 4.0594 kg

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