Question 4.5: Determination of Angle of Twist of a Rod with Fixed Ends A c...
Determination of Angle of Twist of a Rod with Fixed Ends
A circular brass rod (Figure 4.4a) is fixed at each end and loaded by a torque T at point D. Find the maximum angle of twist.
Given: a = 20 in., b = 40 in., d = 1 in., T = 500 lb·in., and G = 5.6 \times 10^{6} psi
The reactions at the end are designated by T_{A} and T_{B}.
Statics: The only available equation of equilibrium for the free-body diagram of Figure 4.4b yields
T{A} + T_{B} = T
Therefore, the problem is statically indeterminate to the first degree.
Deformations: The angle of twist at section D for the left and right segments of the bar are
\phi_{A D}=\frac{T_A a}{G J}, \quad \phi_{B D}=\frac{T_B b}{G J} (a)
Geometry: The continuity of the bar at section D requires that
\phi_{A D}=\phi_{B D} \quad \text { or } \quad T_A a=T_B b (b)
Equations (a) and (b) can be solved simultaneously to obtain
T_A=\frac{T b}{L}, \quad T_B=\frac{T a}{L} (4.13)
Substituting the given numerical values into this equation, we have
\phi_{\max }=\frac{500(20) 40}{5.6\left(10^6\right) \frac{\pi}{32}(1)^4(60)}=0.012 rad =0.7^{\circ}