A machine part in the form of cantilever beam is subjected to fluctuating load as shown in the figure. The load varies from 800 N to 1600 N. The modified endurance, yield and ultimate strengths of the material are 200 MPa, 500 MPa and 600 MPa, respectively. The factor of safety of the beam using modified Goodman criterion is __________ (round off to one decimal place).

The correct answer is 2.

1. Identify the given parameters from the problem description and the figure:
Let us note down the parameters of the beam geometry and the material strengths:
- Width of the beam cross-section, b = 12 mm = 0.012 m
- Depth of the beam cross-section, d = 20 mm = 0.020 m
- Length of the cantilever beam (from the figure), L = 100 mm = 0.1 m
- Ultimate strength, σ_ut = 600 MPa
- Yield strength, σ_yt = 500 MPa
- Modified endurance strength, σ_e = 200 MPa

2. Determine the fluctuating bending moments:
The cantilever beam is subjected to a fluctuating point load at its free end, varying from P_min = 800 N to P_max = 1600 N.
The maximum bending moment occurs at the fixed support of the cantilever beam:
$$M_{\text{max}}=P_{\text{max}}\times L=1600\text{N}\times 0.1\text{m}=160\text{N-m}$$
Similarly, the minimum bending moment is:
$$M_{\text{min}}=P_{\text{min}}\times L=800\text{N}\times 0.1\text{m}=80\text{N-m}$$

3. Calculate the maximum and minimum bending stresses:
For a rectangular cross-section, the section modulus is:
$$Z=\frac{b\times d^2}{6}$$
Therefore, the bending stress is given by:
$$\sigma =\frac{M}{Z}=\frac{6\times M}{b\times d^2}$$
Now, calculate the maximum bending stress (σ_max) and minimum bending stress (σ_min):
$${\sigma }_{\text{max}}=\frac{6\times 160}{0.012\times {0.020}^2}=200\times {10}^6\text{Pa}=200\text{MPa}$$
$${\sigma }_{\text{min}}=\frac{6\times 80}{0.012\times {0.020}^2}=100\times {10}^6\text{Pa}=100\text{MPa}$$

4. Calculate the mean and amplitude components of stress:
The mean stress (σ_m) is:
$${\sigma }_m=\frac{{\sigma }_{\text{max}}+{\sigma }_{\text{min}}}{2}=\frac{200+100}{2}=150\text{MPa}$$
The stress amplitude (σ_a) is:
$${\sigma }_a=\frac{{\sigma }_{\text{max}}-{\sigma }_{\text{min}}}{2}=\frac{200-100}{2}=50\text{MPa}$$

5. Apply the Modified Goodman Criterion:
The equation for the Modified Goodman line is given by:
$$\frac{{\sigma }_m}{{\sigma }_{\text{ut}}}+\frac{{\sigma }_a}{{\sigma }_e}=\frac{1}{N}$$
Where N is the factor of safety. Substituting the calculated stress values and the given material strengths:
$$\frac{150}{600}+\frac{50}{200}=\frac{1}{N}$$
Simplifying each term:
$$\frac{1}{4}+\frac{1}{4}=\frac{1}{N}$$
$$\frac{2}{4}=\frac{1}{N}\Rightarrow N=2$$
Additionally, checking for static yield failure (since it is a modified Goodman criterion):
$${\sigma }_{\text{max}}=200\text{MPa}<{\sigma }_{\text{yt}}=500\text{MPa}$$
This indicates that yielding does not govern. Therefore, the factor of safety is N = 2.