<p> AOE/ESM 4084 - ENGINEERING DESIGN OPTIMIZATION Fall Semester, 1999</p><p>Homework Assignment 1 Due 2.00 PM, Tuesday, September 7</p><p>P</p><p> l(m) Section A-A</p><p>A cantilever beam is subjected to the point load P (kN), as shown above. The maximum bending moment in the beam is Pl (kN-m) and the maximum shear is P (kN). Formulate the minimum mass design problem using hollow circular cross-section. The material should not fail under bending stress or shear stress. The maximum bending stress is calculated as</p><p> = ( P l / I )  Ro</p><p>4 4 where I = (/4)  (Ro  Ri ) is moment of inertia of the cross-section. The maximum shearing stress is calculated as</p><p>2 2  = ( P / 3I )  (Ro + Ro Ri + Ri ) a) Transcribe the problem into the standard design optimization model (also</p><p> use Ro  20.0 cm, Ri  20.0 cm). Use the following data: P = 10 kN, l = 5.0 3 m, mass density,  = 7850 kg/m , allowable bending stress, a = 250 MPa,</p><p> allowable shear stress, a = 90 MPa. b) Solve the problem graphically to determining the active constraint(s) and approximate numeric answer for the optimal design variables and objective function. c) Determine the exact numerical result by solving the constraint equations. </p>