A cantilever beam made of cold drawn steel 4OC8 (Sut = 600 N/mm^2 and Syt = 380 N/mm^2) is shown in Fig. 5.42. The force P acting at the free end varies from –50 N to +150 N. The expected reliability is 90% and the factor of safety is 2. The notch sensitivity factor at the fillet is 0.9. Determine

Question

A cantilever beam made of cold drawn steel 4OC8 [latex] \left(S_{u t}=600 N / mm ^{2} ext { and } S_{y t}=380 ight. \left.N / m m^{2} ight) [/latex] is shown in Fig. 5.42. The force P acting at the free end varies from –50 N to +150 N. The expected reliability is 90% and the factor of safety is 2. The notch sensitivity factor at the fillet is 0.9. Determine the diameter ‘d’ of the beam at the fillet cross-section.

Accepted Answer

[latex] ext { Given } P=-50 N ext { to }+150 N \quad S_{u t}=600 N / mm ^{2} [/latex].

[latex] S_{y t}=380 N / mm ^{2} \quad R=90 \% \quad(f s)=2 \quad q=0.9 [/latex].

Step I Endurance limit stress for cantilever beam

[latex] S_{e}^{\prime}=0.5 S_{u t}=0.5(600)=300 N / mm ^{2} [/latex].

From Fig. 5.24 (cold drawn steel and [latex]S_{u t}=600 \left. N / mm ^{2} ight)[/latex].

[latex] K_{a}=0.77 [/latex].

Assuming 7.5 < d < 50 mm,

[latex] K_{b}=0.85 [/latex].

[latex] ext { For } 90 \% ext { reliability, } K_{c}=0.897 [/latex].

[latex] ext { From Fig. } 5.5, K_{t}=1.44 [/latex].

From Eq. (5.12)

[latex] K_{f}=1+q\left(K_{t}-1 ight) [/latex] (5.12).

[latex] K_{f}=1+q\left(K_{t}-1 ight)=1+0.9(1.44-1)=1.396 [/latex].

[latex] K_{d}=\frac{1}{K_{f}}=\frac{1}{1.396}=0.716 [/latex].

[latex] S_{e}=K_{a} K_{b} K_{c} K_{d} S_{e}^{\prime} [/latex].

= 0.77 (0.85) (0.897) (0.716) (300)
= 126.11 N/mm².

Step II Construction of modified Goodman diagram
At the fillet cross-section,

[latex] \left(M_{b} ight)_{\max .}=150 imes 100=15000 N - mm [/latex].

[latex] \left(M_{b} ight)_{\min .}=-50 imes 100=-5000 N - mm [/latex].

[latex] \left(M_{b} ight)_{m}=\frac{1}{2}\left[\left(M_{b} ight)_{\max .}+\left(M_{b} ight)_{\min } ight] [/latex].

[latex] =\frac{1}{2}[15000-5000]=5000 N - mm [/latex].

[latex] \left(M_{b} ight)_{a}=\frac{1}{2}\left[\left(M_{b} ight)_{\max }-\left(M_{b} ight)_{\min } ight] [/latex].

[latex] an heta=\frac{\left(M_{b} ight)_{a}}{\left(M_{b} ight)_{m}}=\frac{10000}{5000}=2 [/latex].

[latex] heta=63.435^{\circ} [/latex].

The modified Goodman diagram for this example is shown in Fig. 5.43.
Step III Permissible stress amplitude
Refer to Fig. 5.43. The coordinates of the point X are determined by solving the following two equations simultaneously.

(i) Equation of line AB

[latex] \frac{S_{a}}{126.11}+\frac{S_{m}}{600}=1 [/latex] (a).

(ii) Equation of line OX

[latex] \frac{S_{a}}{S_{m}}= an heta=2 [/latex].

Solving the two equations,

[latex] S_{a}=114.12 N / mm ^{2} ext { and } S_{m}=57.06 N / mm ^{2} [/latex].

Step IV Diameter of beam

[latex] ext { Since } \quad \sigma_{a}=\frac{S_{a}}{(f s)} \quad herefore \quad \frac{32\left(M_{b} ight)_{a}}{\pi d^{3}}=\frac{S_{a}}{(f s)} [/latex].

[latex] \frac{32(10000)}{\pi d^{3}}=\frac{114.12}{2} [/latex].

d = 12.13 mm.

Question 5.12: A cantilever beam made of cold drawn steel 4OC8 (Sut = 600 N...

Related Answered Questions