The bearings at A and B exert only x and z components of force on the steel shaft. Determine the shaft's diameter to the nearest millimeter so that it can resist the loadings of the gears. Use the maximumdistortion-energy theory of failure with

Question

The bearings at A and B exert only x and z components of force on the steel shaft. Determine the shaft's diameter to the nearest millimeter so that it can resist the loadings of the gears. Use the maximumdistortion-energy theory of failure with [latex]\sigma_{ ext {allow }}=200 \mathrm{MPa}[/latex].

Accepted Answer

Maximum resultant moment

[latex]M=\sqrt{1250^{2}+250^{2}}=1274.75 \mathrm{~N} \cdot \mathrm{m} \\sigma_{1,2}=\frac{\sigma_{x}}{2} \pm \sqrt{\frac{\sigma_{x}^{2}}{4}+ au_{x y}^{2}}[/latex]

Let [latex]a=\frac{\sigma_{x}}{2}, b=\sqrt{\frac{\sigma_{x}^{2}}{4}+ au_{x y}^{2}}\\sigma_{1}=a+b, \quad \sigma_{2}=a-b[/latex]

Require,
[latex]\sigma_{1}^{2}-\sigma_{1} \sigma_{2}+\sigma_{2}^{2}=\sigma_{ ext {allow }}^{2}\a^{2}+2 a b+b^{2}-\left[a^{2}-b^{2} ight]+a^{2}-2 a b+b^{2}=\sigma_{ ext {allow }}^{2}\a^{2}+3 b^{2}=\sigma_{ ext {allow }}^{2}\\frac{\sigma_{x}^{2}}{4}+3\left(\frac{\sigma_{x}^{2}}{4}+ au_{x y}^{2} ight)=\sigma_{ ext {allow }}^{2}\\sigma_{x}^{2}+3 au_{x y}^{2}=\sigma_{ ext {allowu }}^{2}[/latex]

[latex]\left(\frac{M c}{\frac{\pi}{4} c^{4}} ight)^{2}+3\left(\frac{T c}{\frac{\pi}{2} c^{4}} ight)^{2}=\sigma_{ ext {allow }}^{2}[/latex]

[latex]\frac{1}{c^{6}}\left[\left(\frac{4 M}{\pi} ight)^{2}+3\left(\frac{2 T}{\pi} ight)^{2} ight]=\sigma_{ ext {allow }}^{2}\c^{6}=\frac{16}{\sigma_{ ext {allow }}^{2} \pi^{2}} M^{2}+\frac{12 T^{2}}{\sigma_{ ext {allow }}^{2} \pi^{2}}\c=\left[\frac{4}{\sigma_{ ext {altow }}^{2} \pi^{2}}\left(4 M^{2}+3 T^{2} ight) ight]^{\frac{1}{6}}\=\left[\frac{4}{\left(200\left(10^{\circ} ight) ight)^{2}(\pi)^{2}}\left(4(1274.75)^{2}+3(375)^{2} ight) ight]^{\frac{1}{6}}[/latex]

= 0.0203 m = 20.3 mm

d = 40.6 mm

Use

d = 41 mm

Question 11.50: The bearings at A and B exert only x and z components of for...

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