$3\mathrm{.}11.$ Two two$-$element shear gages $($Fig$.2\mathrm{.}3$ m or n $)$ are mounted on a cylindrical piece at diametrically opposite locations, as shown in Fig. P$3\mathrm{.}11.$

$($a$)$ Position the four elements in a half Wheatstone bridge in order to have the bridge output proportional to shear strain $($or torque $T)$ and independent of bending moment $M$ and axial load $P.$

$($b$)$ Determine the bridge output $\frac{E}{V}$ and prove that it is independent of $M$ and $P.$ The cylindrical piece has a circular cross section with radius of $R$ and a shear modulus of $G.$

$3\mathrm{.}11.$ Two two$-$element shear gages $($Fig$.2\mathrm{.}3$ m or n $)$ are mounted on a cylindrical piece at diametrically opposite locations, as shown in Fig. P$3\mathrm{.}11.$

$($a$)$ Position the four elements in a half Wheatstone bridge in order to have the bridge output proportional to shear strain $($or torque $T)$ and independent of bending moment $M$ and axial load $P.$

$($b$)$ Determine the bridge output $\frac{E}{V}$ and prove that it is independent of $M$ and $P.$ The cylindrical piece has a circular cross section with radius of $R$ and a shear modulus of $G.$