Question

PROBLEM 3.130 Shafts \( A \) and \( B \) are made of the same material and have the same crosssectional area, but \( A \) has a circular cross section and \( B \) has a square cross section. Determine the ratio of the maximum torques \( T_{A} \) and \( T_{B} \) when the two shafts are subjected to the same maximum shearing stress \( \left(\tau_{A}=\tau_{B}\right) \). Assume both deformations to be elastic. SOLUTION Let \( c= \) radius of circular section \( A \) and \( b= \) side of square section \( B \). For equal areas \( \pi c^{2}=b^{2} \) \[ c=\frac{b}{\sqrt{\pi}} \] Circle: \[ \tau_{A}=\frac{T_{A} c}{J}=\frac{2 T_{A}}{\pi c^{3}} \quad \therefore \quad T_{A}=\frac{\pi}{2} c^{3} \tau_{A} \] Square: From Table 3.1, Ratio: \[ \begin{aligned} c_{1} & =0.208 \\ \tau_{B} & =\frac{T_{A}}{c_{1} a b^{2}}=\frac{T_{B}}{c_{1} b^{3}} \quad \therefore \quad T_{B}=c_{1} b^{3} \tau_{B} \\ \frac{T_{A}}{T_{B}} & =\frac{\frac{\pi}{2} c^{3} \tau_{B}}{c_{1} b^{3} \tau_{B}}=\frac{\frac{\pi}{2} \cdot \frac{b^{3}}{\pi^{3 / 2}} \tau_{B}}{c_{1} b^{3} \tau_{B}}=\frac{1}{2 c_{1} \sqrt{\pi}} \frac{\tau_{A}}{\tau_{B}} \end{aligned} \] For the same stresses, \[ \tau_{B}=\tau_{A} \quad \therefore \frac{T_{A}}{T_{B}}=\frac{1}{(2)(0.208) \sqrt{\pi}} \quad \frac{T_{A}}{T_{B}}=1.356 \]

          PROBLEM 3.130
Shafts \( A \) and \( B \) are made of the same material and have the same crosssectional area, but \( A \) has a circular cross section and \( B \) has a square cross section. Determine the ratio of the maximum torques \( T_{A} \) and \( T_{B} \) when the two shafts are subjected to the same maximum shearing stress \( \left(\tau_{A}=\tau_{B}\right) \). Assume both deformations to be elastic.
SOLUTION
Let \( c= \) radius of circular section \( A \) and \( b= \) side of square section \( B \).
For equal areas \( \pi c^{2}=b^{2} \)
\[
c=\frac{b}{\sqrt{\pi}}
\]
Circle:
\[
\tau_{A}=\frac{T_{A} c}{J}=\frac{2 T_{A}}{\pi c^{3}} \quad \therefore \quad T_{A}=\frac{\pi}{2} c^{3} \tau_{A}
\]
Square:
From Table 3.1,
Ratio:
\[
\begin{aligned}
c_{1} & =0.208 \\
\tau_{B} & =\frac{T_{A}}{c_{1} a b^{2}}=\frac{T_{B}}{c_{1} b^{3}} \quad \therefore \quad T_{B}=c_{1} b^{3} \tau_{B} \\
\frac{T_{A}}{T_{B}} & =\frac{\frac{\pi}{2} c^{3} \tau_{B}}{c_{1} b^{3} \tau_{B}}=\frac{\frac{\pi}{2} \cdot \frac{b^{3}}{\pi^{3 / 2}} \tau_{B}}{c_{1} b^{3} \tau_{B}}=\frac{1}{2 c_{1} \sqrt{\pi}} \frac{\tau_{A}}{\tau_{B}}
\end{aligned}
\]
For the same stresses,
\[
\tau_{B}=\tau_{A} \quad \therefore \frac{T_{A}}{T_{B}}=\frac{1}{(2)(0.208) \sqrt{\pi}} \quad \frac{T_{A}}{T_{B}}=1.356
\]

PROBLEM 3.130
Shafts A and B are made of the same material and have the same crosssectional area, but A has a circular cross section and B has a square cross section. Determine the ratio of the maximum torques TA and TB when the two shafts are subjected to the same maximum shearing stress (τA=τB). Assume both deformations to be elastic.
SOLUTION
Let c= radius of circular section A and b= side of square section B.
For equal areas π c^2=b^2

c=(b)/(√(π))

Circle:

τA=(TA c)/(J)=(2 TA)/(π c^3) ∴ TA=(π)/(2) c^3τA

Square:
From Table 3.1,
Ratio:

c1 =0.208
τB =(TA)/(c1 a b^2)=(TB)/(c1 b^3) ∴ TB=c1 b^3τB
(TA)/(TB) =((π)/(2) c^3τB)/(c1 b^3τB)=((π)/(2)·(b^3)/(π^3 / 2)τB)/(c1 b^3τB)=(1)/(2 c1√(π))(τA)/(τB)

For the same stresses,

τB=τA ∴(TA)/(TB)=(1)/((2)(0.208) √(π)) (TA)/(TB)=1.356

Added by Ibrahim A.

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