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\subsection*{Shear Stress Resultant}
\[ T = \int_A{p \* \tau \* dA} = \int_A{p \* \left(G \* p \* \dfrac{d\phi}{dx}\right) \* dA} \qquad \text{Resultant Torque} \]
\[ \text{If $G$ is constant,} \]
\[ T = G \* \dfrac{d\phi}{dx} \* \int_A{p^2 \* dA} \]
\[ T = G \* \dfrac{d\phi}{dx} \* I_p \qquad \text{where $I_p$ is the polar moment of inertia} \]
\[ \dfrac{d\phi}{dx} = \dfrac{T}{G \* I_p} \qquad \text{Torque-Twist Equation} \]
\[ \dfrac{d\phi}{dx} = \dfrac{T}{G \* I_p} = \dfrac{\tau}{G \* p} \]
\[ \tau = \dfrac{T_p}{I_p} \qquad \text{Torsional Shear Stress Equation} \]
\[ \sigma = \dfrac{M \* c}{I} \]
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