Gere & Goodno — Stresses in Beams (Advanced Topics): Problem 6.2-6

6.2-6 A round titanium tube of outside diameter \( d_2 \) and a copper core of diameter \( d_1 \) are bonded to form a composite beam, as shown in the figure. (a) Derive formulas for the allowable bending moment \( M \) that can be carried by the beam based upon an allowable stress \( \sigma_{Ti} \) in the titanium and an allowable stress \( \sigma_{Cu} \) in the copper. (Assume that the moduli of elasticity for the titanium and copper are \( E_{Ti} \) and \( E_{Cu} \), respectively.) (b) If \( d_2 = 40 \text{ mm} \), \( d_1 = 36 \text{ mm} \), \( E_{Ti} = 120 \text{ GPa} \), \( E_{Cu} = 110 \text{ GPa} \), \( \sigma_{Ti} = 840 \text{ MPa} \), and \( \sigma_{Cu} = 700 \text{ MPa} \), what is the maximum bending moment \( M \)? (c) What new value of copper diameter \( d_1 \) will result in a balanced design? (i.e., a balanced design is that in which titanium and copper reach allowable stress values at the same time).