To calculate the circumferential stress in the fuselage skin of the jet, we need to follow these steps:
-
Determine the pressure at altitude and the equivalent cabin pressure:
- Standard atmospheric pressure at sea level: \( P_0 \approx 101325 , \text{Pa} \)
- Cabin pressure maintained at 80% of sea level: \[ P_c = 0.8 \times P_0 = 0.8 \times 101325 , \text{Pa} = 81060 , \text{Pa} \]
- The atmospheric pressure at an altitude of 6200 m can be found using the barometric formula or estimated values:
- According to the ISA (International Standard Atmosphere), the pressure at 6200 m is approximately \( 54130 , \text{Pa} \).
-
Determine the pressure difference acting on the fuselage:
- The pressure difference \( \Delta P \) is calculated as: \[ \Delta P = P_c - P_a = 81060 , \text{Pa} - 54130 , \text{Pa} = 26930 , \text{Pa} \]
-
Calculate the effective radius of the fuselage:
- The outer radius \( R_o \) of the fuselage is calculated from the radius and skin thickness: \[ R_o = R + \text{skin thickness} = 3.2 , \text{m} + 0.0024 , \text{m} = 3.2024 , \text{m} \]
- The inner radius \( R_i \) (if needed) can be considered for thin-walled pressure vessel approximations, but since we will use the outer radius for calculations of stress, we can use \( R_o \).
-
Calculate the thickness of the fuselage skin:
- The thickness \( t \) is given as: \[ t = 0.0024 , \text{m} \]
-
Calculate the circumferential stress:
- Using the formula for circumferential stress (\( \sigma_\theta \)) in a thin-walled cylindrical tube: \[ \sigma_\theta = \frac{R_o \Delta P}{t} \]
- Plugging in the values: \[ \sigma_\theta = \frac{3.2024 , \text{m} \times 26930 , \text{Pa}}{0.0024 , \text{m}} \] \[ \sigma_\theta = \frac{86141.312 , \text{Pa m}}{0.0024} \] \[ \sigma_\theta \approx 35883880.08 , \text{Pa} \]
-
Convert to MPa: \[ \sigma_\theta \approx 35.88 , \text{MPa} \]
Thus, the circumferential stress in the fuselage skin is approximately 35.88 MPa.