Question a) determine the principal stresses at point Bb)determine the magnitude of the maximum in-plane shear stress at point B The internal loadings at a cross section through the 190-mm-diameter drive shaft of a turbine consist of an axial force of 12.5 kN, a bending moment of 1.2 kNm, and a torsional moment of 2.25 kNm.. 12.5 kN 1.2 kN.m 2.25 kN.m
a) determine the principal stresses at point B
b)determine the magnitude of the maximum in-plane shear stress at point B
Solved 1 Answer
See More Answers for FREE
Enhance your learning with StudyX
Receive support from our dedicated community users and experts
See up to 20 answers per week for free
Experience reliable customer service
\begin{array}{ll}d=190 \mathrm{~mm} & A=\frac{\pi}{4}(190)^{2} \\F_{a}=12.5 \mathrm{kN} & I=\frac{\pi}{64}(190)^{4} \\M=1.2 k \mathrm{~N}-\mathrm{m} & J=\frac{\pi}{32}(190)^{4}\end{array}Shaft Cross-sectionBending moment M=1.2 \mathrm{kN}-\mathrm{m}, acts at A and does not act at 'B', as can be seen from the obove diagramAt B\text { Axial stress }=\sigma_{a}=\frac{F_{a}}{A}=\frac{12.5 \times 10^{3} \mathrm{~N}}{\frac{\pi}{a} \times 190^{2}}=-0.441 \mathrm{MPa}Due to torsional moment, twisting Shear stress geverated at B\tau=\frac{T(d / 2)}{J}=\frac{2.25 \times 10^{3} \mathrm{~N}(0.19 \mathrm{~m}) \times 10^{6} \mathrm{MPa}}{\frac{\pi}{32}(190)^{4} \mathrm{~mm}^{4}}\tau=3.34 \mathrm{MPa}Principal stress formula\begin{aligned}\sigma_{1,2} & =\frac{1}{2}\left[\sigma_{x}+\sigma_{y} \pm \sqrt{\left(\sigma_{x}-\sigma_{y}\right)^{2}+4 \tau^{2}}\right] \\& =\frac{1}{2}\left[-0.441+0 \pm \sqrt{(-0.441-0)^{2}+4(3.34)^{2}}\right]\end{aligned}(a)\begin{array}{l}\sigma_{1}=3.128 \mathrm{mPa} \text { and } \\\sigma_{2}=-3.57 \mathrm{MPa}\end{array}(B) Maximum in-plane shear stress at B\begin{aligned}\tau_{\text {max }} & =\left|\frac{\sigma_{1}-\sigma_{2}}{2}\right| \\& =\left|\frac{3.128-(-3.57)}{2}\right| \\\tau_{\text {max }} & =3.348 \mathrm{MPa}\end{aligned}Mons circle ...