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): (Assume ISA conditions)
Additionally, compute the longitudinal stress in the fuselage skin (in MPa or MegaPascal):
the circumferential stress in the fuselage skin 46.9 Megapascals
the longitudinal stress in the fuselage skin is 23.45 Megapascals
To calculate the circumferential stress in the fuselage skin, we can use the formula for hoop stress:
σ_circ = pd/2t
where:
σ_circ = circumferential stress
p = internal pressure
d = internal diameter
t = skin thickness
First, we need to find the internal pressure. The cabin pressure is maintained at 80% of the sea level value, so we need to calculate the sea level pressure (p_sl).
Using the International Standard Atmosphere (ISA) conditions, the sea level pressure is approximately 101.325 kPa.
p_sl = 101.325 kPa = 101325 Pa
To calculate the internal pressure (p), we multiply the sea level pressure by the cabin pressure ratio (80% or 0.8):
p = 0.8 * p_sl
p = 0.8 * 101325 Pa
p = 81060 Pa
Next, we can calculate the internal diameter (d) using the fuselage radius (r):
d = 2 * r
d = 2 * 3.2 m
d = 6.4 m
Now, we can substitute the values into the hoop stress formula:
σ_circ = (81060 Pa * 6.4 m) / (2 * 2.4 mm)
σ_circ = (521184 Pa * m) / (4.8 * 10^-3 m)
σ_circ = 108400 Pa
Therefore, the circumferential stress in the fuselage skin is approximately 108.4 MPa.
To calculate the longitudinal stress in the fuselage skin, we can use the formula:
σ_long = pd/4t
Let's substitute the values into the formula:
σ_long = (81060 Pa * 6.4 m) / (4 * 2.4 mm)
σ_long = (521184 Pa * m) / (9.6 * 10^-3 m)
σ_long = 54200 Pa
Therefore, the longitudinal stress in the fuselage skin is approximately 54.2 MPa.
To calculate the circumferential stress in the fuselage skin of the jet, we need to use the formula for hoop stress in a thin-walled cylindrical shell:
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, and
t is the thickness of the fuselage skin.
To find the pressure difference, we need to calculate the difference in pressure between the inside and outside of the cabin. We are given that the cabin pressure is maintained at 80% of the sea level value. The standard atmospheric pressure at sea level is approximately 101.3 kPa.
P = (80 / 100) * 101.3 kPa.
Now, we need to convert the given radius of the fuselage from meters to millimeters and the thickness of the fuselage skin from millimeters to meters to maintain consistency.
r = 3.2 meters * 1000 mm/meter,
t = 2.4 mm * 0.001 meters/mm.
Substituting the values into the formula, we get:
σ_circumferential = (P * r) / t.
Now, let's calculate the values and the final result:
P = (80 / 100) * 101.3 = 80.8 kPa.
r = 3.2 * 1000 = 3200 mm.
t = 2.4 * 0.001 = 0.0024 meters.
σ_circumferential = (80.8 kPa * 3200 mm) / 0.0024 meters.
First, convert kPa to MPa:
80.8 kPa = 0.0808 MPa.
Substituting the values:
σ_circumferential = (0.0808 MPa * 3200 mm) / 0.0024 meters.
Now, convert mm to meters:
3200 mm = 3.2 meters.
Substituting the values again:
σ_circumferential = (0.0808 MPa * 3.2 meters) / 0.0024 meters.
Finally, we can calculate the circumferential stress, σ_circumferential:
σ_circumferential = 0.0808 MPa * (3.2 meters / 0.0024 meters).
Now, substitute the values and compute the result.
To calculate the circumferential stress, we can use the formula:
σ = (p * r) / t
where σ is the circumferential stress, 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 skin.
First, we need to calculate the pressure difference. At an altitude of 6,200 metres, the atmospheric pressure is approximately 43 kPa. To maintain the cabin pressure at 80% of sea level, we need to add an additional pressure of:
0.8 * (101.3 kPa - 43 kPa) = 45.04 kPa
Therefore, the total pressure difference is:
45.04 kPa + 43 kPa = 88.04 kPa
Next, we can substitute the values into the formula:
σ = (88.04 kPa * 3.2 m) / (2.4 mm * 10^-3) = 117.39 MPa
Therefore, the circumferential stress in the fuselage skin is 117.39 MPa.
To calculate the longitudinal stress, we can use the formula:
σ = (p * r^2) / (2t)
where σ is the longitudinal stress, p is the pressure difference, r is the radius of the fuselage and t is the thickness of the skin.
Substituting the values we get:
σ = (88.04 kPa * (3.2 m)^2) / (2*2.4 mm*10^-3) = 309.27 MPa
Therefore, the longitudinal stress in the fuselage skin is 309.27 MPa.