A modern jet flies at an altitude of 6,200 metres, during its climb to cruise altitude. For additional passenger comfort, the cabin pressure is maintained at 80% of the sea level value.
Given that the fuselage radius is 3.2 metres and that the skin thickness is 2.4 mm, compute the circumferential stress in the fuselage skin (in MPa or MegaPascal)
Additionally, compute the longitudinal stress in the fuselage skin (in MPa or MegaPascal):
To find the circumferential stress and longitudinal stress in the fuselage skin, we can make use of the formulas for stress in a thin-walled cylindrical pressure vessel.
Circumferential stress (σθ) is the stress that occurs in the direction perpendicular to the length of the cylinder or fuselage. It can be calculated using the formula:
σθ = (P * r) / t
where P is the pressure difference between the inside and outside of the fuselage, r is the radius of the fuselage, and t is the thickness of the fuselage skin.
Longitudinal stress (σL) is the stress that occurs along the length of the cylinder or fuselage. It can be calculated using the formula:
σL = (P * r) / (2t)
Given values:
Fuselage radius (r) = 3.2 metres
Fuselage skin thickness (t) = 2.4 mm = 0.0024 metres
Pressure difference (P) = 80% of the sea level value (Assuming sea-level pressure is 101325 Pascals)
First, let's calculate the pressure difference:
Sea-level pressure = 101325 Pascals
Pressure difference = 80% of 101325 = 0.8 * 101325 = 81060 Pascals
Now we can calculate the circumferential stress:
σθ = (P * r) / t = (81060 * 3.2) / 0.0024 = 108480000 / 0.0024 = 45100000 Pascals = 45.1 MPa
Therefore, the circumferential stress in the fuselage skin is 45.1 MPa.
Next, let's calculate the longitudinal stress:
σL = (P * r) / (2t) = (81060 * 3.2) / (2 * 0.0024) = 108480000 / 0.0048 = 22560000 Pascals = 22.56 MPa
Therefore, the longitudinal stress in the fuselage skin is 22.56 MPa.
So, the circumferential stress is 45.1 MPa and the longitudinal stress is 22.56 MPa in the fuselage skin.
To start, we need to determine the external pressure acting on the fuselage skin at the given altitude.
The atmospheric pressure decreases as we climb in altitude. At sea level, the standard atmospheric pressure is approximately 101.325 kPa.
To determine the pressure at 6,200 meters, we can use the barometric formula:
P = P0 * (1 - (L * h) / T0) ^ ((g * M) / (R * L))
Where:
P0 = sea level pressure (101.325 kPa)
L = temperature lapse rate (0.0065 K/m)
h = altitude (6,200 m)
T0 = sea level temperature (288.15 K)
g = acceleration due to gravity (9.80665 m/s^2)
M = molar mass of Earth's air (0.0289644 kg/mol)
R = ideal gas constant (8.31446261815324 J/(mol·K))
Plugging in the values:
P = 101.325 * (1 - (0.0065 * 6,200) / 288.15) ^ ((9.80665 * 0.0289644) / (8.31446261815324 * 0.0065))
Simplifying:
P = 101.325 * (1 - (0.0403 / 288.15)) ^ 3.48552
P = 101.325 * (1 - 0.000139827) ^ 3.48552
P = 101.325 * (0.999860173) ^ 3.48552
P ≈ 101.325 * 0.999343460
P ≈ 101.160 kPa
The external pressure acting on the fuselage skin at 6,200 meters altitude is approximately 101.160 kPa.
Now we can calculate the circumferential stress in the fuselage skin.
The circumference of the fuselage can be calculated using the formula:
C = 2 * π * r
Where:
r = radius of the fuselage (3.2 m)
C = 2 * π * 3.2
C ≈ 20.1 m
The circumference of the fuselage is approximately 20.1 meters.
The circumferential stress can be calculated using the formula:
σ_circ = (P * r) / t
Where:
P = pressure (101.160 kPa)
r = radius of the fuselage (3.2 m)
t = skin thickness (2.4 mm = 0.0024 m)
σ_circ = (101.160 * 10^3 * 3.2) / 0.0024
σ_circ ≈ 134,880,000 / 0.0024
σ_circ ≈ 56,200,000 N/m^2 or 56.2 MPa
The circumferential stress in the fuselage skin is approximately 56.2 MPa.
To calculate the longitudinal stress in the fuselage skin, we use the formula:
σ_long = P * r / (2 * t)
Using the same values as before:
σ_long = (101.160 * 10^3 * 3.2) / (2 * 0.0024)
σ_long ≈ 134,880,000 / 0.0048
σ_long ≈ 28,100,000 N/m^2 or 28.1 MPa
The longitudinal stress in the fuselage skin is approximately 28.1 MPa.
To calculate the circumferential stress in the fuselage skin, we can use the formula for hoop stress:
𝜎_circumferential = 𝑃 × 𝑟 / 𝑡
Where:
𝜎_circumferential is the circumferential stress
𝑃 is the pressure difference between the inside and outside of the fuselage
𝑟 is the radius of the fuselage
𝑡 is the thickness of the skin
In this case, the pressure inside the fuselage is 80% of the sea level pressure. We need to convert the altitude into pressure using the international standard atmosphere model. At an altitude of 6,200 meters, the pressure is approximately 48.096 kPa.
Now, we can calculate the circumferential stress:
𝜎_circumferential = (48.096 × 1000) × 3.2 / (2.4 × 10^-3)
𝜎_circumferential ≈ 64.128 MPa
Therefore, the circumferential stress in the fuselage skin is approximately 64.128 MPa.
To calculate the longitudinal stress in the fuselage skin, we can use the formula for axial stress:
𝜎_longitudinal = 𝑃 × 𝑟 / (2 × 𝑡)
Using the same values as before, we can calculate the longitudinal stress:
𝜎_longitudinal = (48.096 × 1000) × 3.2 / (2 × 2.4 × 10^-3)
𝜎_longitudinal ≈ 106.88 MPa
Therefore, the longitudinal stress in the fuselage skin is approximately 106.88 MPa.