(Solved): A torsion-bar spring consists of a prismatic bar, usually of round cross section, that is twisted a ...

A torsion-bar spring consists of a prismatic bar, usually of round cross section, that is twisted at one end and held fast at the other to form a stiff spring. An engineer needs a stiffer one than usual. So, the engineer builds in both ends and applies the torque in the central portion of the span, as shown in the figure. This effectively creates two springs in parallel. If the load is not applied at the center and the spring has a uniform diameter, one leg of the span will be underutilized. For optimal design, the diameter of each leg should be selected such that the maximum shear stress in each leg is the same. Using $x=0.35L,L=12\mathrm{in},T=1100\mathrm{lbf}$-in, and $G=11.5\mathrm{Mpsi}$, design a torsion-bar spring such that the maximum shear stresses in each leg are equal and the spring has the same spring rate (angle change) as a uniform-diameter part with $x=$ $0.35\mathrm{L}$ and $d=d_1=d_2=0.4\mathrm{in}$. Specify $d_1,d_2$, the spring rate $k$, and the torque and maximum shear stress in each leg. The left diameter $(d)$ of the torsion bar is The right diameter $\left(d_2\right)$ of the torsion bar is The spring rate $(k)$ of the torsion bar is The torque $\left(T_i\right)$ in the torsion bar is The torque $\left(T_2\right)$ in the torsion bar is The maximum shear stress $(??)$ in the torsion bar is The maximum shear stress $\left({?}_2\right)$ in the torsion bar is in. in. lbf-in/rad. Ibf-in. lbf-in. psi. psi.