A machine member is subjected to fluctuating stress σ = σ₀cos(8πt). The endurance limit of the material is 350 MPa. If the factor of safety used in the design is 3.5 then the maximum allowable value of σ₀ is __________ MPa (round off to 2 decimal places).

The correct answer is 100.

Step 1: Analyze the given stress function
The stress subjected to the machine member is a fluctuating stress given by:
$$\sigma ={\sigma }_0\cos \left(8\pi t\right)$$

Step 2: Determine maximum and minimum stresses
Since the cosine function varies between -1 and 1:
- The maximum stress (${\sigma }_{\text{max}}$) occurs when $\cos \left(8\pi t\right)=1$:
$${\sigma }_{\text{max}}={\sigma }_0$$
- The minimum stress (${\sigma }_{\text{min}}$) occurs when $\cos \left(8\pi t\right)=-1$:
$${\sigma }_{\text{min}}=-{\sigma }_0$$

Step 3: Calculate mean stress and stress amplitude
The mean stress (${\sigma }_m$) is calculated as:
$${\sigma }_m=\frac{{\sigma }_{\text{max}}+{\sigma }_{\text{min}}}{2}=\frac{{\sigma }_0+(-{\sigma }_0)}{2}=0$$
The stress amplitude (${\sigma }_a$) is calculated as:
$${\sigma }_a=\frac{{\sigma }_{\text{max}}-{\sigma }_{\text{min}}}{2}=\frac{{\sigma }_0-(-{\sigma }_0)}{2}={\sigma }_0$$

Step 4: Apply fatigue design criteria
We are given:
- Endurance limit of the material, ${\sigma }_e=350\text{MPa}$
- Factor of safety, $\text{FOS}=3.5$
Since the mean stress (${\sigma }_m$) is zero, any fatigue design criteria (Soderberg, Goodman, or Gerber) simplifies to the same relation because the mean stress term vanishes. Using the Soderberg line equation:
$$\frac{{\sigma }_m}{{\sigma }_y}+\frac{{\sigma }_a}{{\sigma }_e}=\frac{1}{\text{FOS}}$$
Substituting ${\sigma }_m=0$, ${\sigma }_a={\sigma }_0$, ${\sigma }_e=350$, and $\text{FOS}=3.5$:
$$0+\frac{{\sigma }_0}{350}=\frac{1}{3.5}$$
Solving for ${\sigma }_0$:
$${\sigma }_0=\frac{350}{3.5}=100\text{MPa}$$