A machine component is subjected to two-dimensional stresses. The tensile stress in the X direction varies from 40 to 100 N/mm^2 while the tensile stress in the Y direction varies from 10 to 80 N/mm^2. The frequency of variation of these stresses is equal. The corrected endurance limit of the

Question

A machine component is subjected to two-dimensional stresses. The tensile stress in the X direction varies from 40 to 100 N/mm² while the tensile stress in the Y direction varies from 10 to 80 N/mm². The frequency of variation of these stresses is equal. The corrected endurance limit of the component is 270 N/mm². The ultimate tensile strength of the material of the component is 660 N/mm². Determine the factor of safety used by the designer.

Accepted Answer

[latex] ext { Given }\left(\sigma_{x} ight)_{\max }=100 N / mm ^{2} [/latex].

[latex] \left(\sigma_{x} ight)_{\min .}=40 N / mm ^{2} \quad\left(\sigma_{y} ight)_{\max }=80 N / mm ^{2} \quad\left(\sigma_{y} ight)_{\min } [/latex].

[latex] =10 N / mm ^{2} \quad S_{u t}=600 N / mm ^{2} \quad S_{e}=270 N / mm ^{2} [/latex].

Step I Mean and amplitude stresses

[latex] \sigma_{x m}=\frac{1}{2}(100+40)=70 N / mm ^{2} [/latex].

[latex] \sigma_{x a}=\frac{1}{2}(100-40)=30 N / mm ^{2} [/latex].

[latex] \sigma_{y m}=\frac{1}{2}(80+10)=45 N / mm ^{2} [/latex].

[latex] \sigma_{y a}=\frac{1}{2}(80-10)=35 N / mm ^{2} [/latex].

[latex] \sigma_{m}=\sqrt{\left(\sigma_{x m}^{2}-\sigma_{x m} \sigma_{y m}+\sigma_{y m}^{2} ight)} [/latex].

[latex] =\sqrt{\left[(70)^{2}-(70)(45)+(45)^{2} ight]} [/latex].

[latex] =61.44 N / mm ^{2} [/latex].

[latex] \sigma_{a}=\sqrt{\left(\sigma_{x a}^{2}-\sigma_{x a} \sigma_{y a}+\sigma_{y a}^{2} ight)} [/latex].

[latex] =\sqrt{\left[(30)^{2}-(30)(35)+(35)^{2} ight]} [/latex].

[latex] =32.79 N / mm ^{2} [/latex].

Step II Construction of modified Goodman diagram

[latex] an heta=\frac{\sigma_{a}}{\sigma_{m}}=\frac{32.79}{61.44}=0.534 ext { or } heta=28.09^{\circ} [/latex].

The modified Goodman diagram for this example is shown in Fig. 5.55.

Step III Permissible stress amplitude
Refer to Fig. 5.55. The co-ordinates of the point X are obtained by solving the following two equations simultaneously.

[latex] \frac{S_{a}}{270}+\frac{S_{m}}{660}=1 [/latex] (a).

[latex] \frac{S_{a}}{S_{m}}= an heta=0.534 [/latex] (b).

∴ [latex] S_{a}=152.88 N / mm ^{2} [/latex].

[latex] S_{m}=286.29 N / mm ^{2} [/latex].

Step IV Factor of safety

[latex] (f s)=\frac{S_{a}}{\sigma_{a}}=\frac{152.88}{32.79}=4.66 [/latex].

Question 5.19: A machine component is subjected to two-dimensional stresses...

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