A young engineer has come up with a 'brilliant idea': an aircraft with a square fuselage for more space. Since all aircraft have a circular fuselage cross section you suspect there must be a downside to this plan and you suspect it has to do with stress in the fuselage skin due to the cabin pressure.
To convince the young engineer, you have to determine the 'circular' stress in the fuselage due to the cabin pressure as a function of R,P(in) , P(out) and t . To do so, make use of the figure below.
I'm sorry, but I cannot see any figure attached to your question. Could you please provide more information or describe the figure in detail?
young engineer has come up with a 'brilliant idea': an aircraft with a square fuselage for more space. Since all aircraft have a circular fuselage cross section you suspect there must be a downside to this plan and you suspect it has to do with stress in the fuselage skin due to the cabin pressure.
To convince the young engineer, you have to determine the 'circular' stress in the fuselage due to the cabin pressure as a function of R,P(in) , P(out) and t . Give an equation for the 'circular' skin stress phi
To determine the stress in the fuselage due to cabin pressure, we can consider a small element of the fuselage skin. Let's assume this element is a strip with a width "t" and a length "R" in the circumferential direction of the circular cross-section.
The force acting on this strip due to the pressure difference across it can be calculated using Pascal's Law. Pascal's Law states that the pressure exerted at any point in an enclosed fluid is transmitted equally in all directions.
In this case, the pressure inside the cabin (P(in)) is greater than the pressure outside the cabin (P(out)). The force acting on the strip due to this pressure difference is given by:
Force = (P(in) - P(out)) * Area
The area of the strip can be approximated as the product of the width "t" and the length "R". Therefore, the force becomes:
Force = (P(in) - P(out)) * t * R
To determine the stress in the fuselage, we need to divide the force by the cross-sectional area of the strip. The cross-sectional area of the strip can be approximated as the product of the width "t" and the circumference of the circular cross-section, which is 2πR. Therefore, the stress in the fuselage due to cabin pressure is given by:
Stress = Force / (t * 2πR)
Substituting the expression for force, we get:
Stress = (P(in) - P(out)) * t * R / (t * 2πR)
The width "t" cancels out, and we are left with:
Stress = (P(in) - P(out)) / (2πR)
Therefore, the stress in the fuselage due to cabin pressure is directly proportional to the pressure difference across the fuselage (P(in) - P(out)) and inversely proportional to the circumference of the circular cross-section (2πR).
This calculation assumes that the fuselage is thin-walled, and the stress is uniform across the cross-section. In reality, the stress distribution might be more complex, depending on the specific design and structural considerations of the aircraft.
To determine the stress in the fuselage due to cabin pressure, we need to consider the forces acting on a small element of the fuselage skin.
[image: Figure of a cross-section of an aircraft fuselage with dimensions labelled]
Let's consider a small element of the fuselage skin with width Δw and height Δh. The top and bottom surfaces of this element experience a net force in the outward direction due to the pressure difference between the inside and outside of the fuselage. The pressure difference creates a resultant force that acts perpendicular to the surface.
The area of each surface of the element is given by Δw * t, where t is the thickness of the fuselage skin. The pressure difference across the element is (P(in) - P(out)), where P(in) is the internal pressure and P(out) is the external pressure.
The net force exerted on each surface of the element can be calculated using the formula:
F = Pressure * Area
In this case, the force on the top surface of the element is (P(in) * Δw * t), while the force on the bottom surface is (P(out) * Δw * t). Since the force on the top surface is greater, it will cause tension in the fuselage skin.
Now, let's consider the circular element with radius 'R'. The vertical forces exerted on the fuselage skin are balanced by the internal and external stresses acting circumferentially around the fuselage. This circumferential stress is what we are interested in.
The circumference of the circular element is given by 2πR. So, the net force acting circumferentially on the element is:
Net Force = (P(in) - P(out)) * Δw * t
This net force creates stress in the fuselage skin, and we can calculate this stress by dividing the net force by the cross-sectional area of the element. The cross-sectional area is given by 2πR * t.
Stress = (Net Force) / (Cross-sectional Area)
= [(P(in) - P(out)) * Δw * t] / [2πR * t]
= (P(in) - P(out)) * Δw / (2πR)
So, the circular stress in the fuselage due to the cabin pressure can be expressed as (P(in) - P(out)) * Δw / (2πR).
Now that we have the formula, we can plug in the values of R, P(in), P(out), and Δw to calculate the stress. Remember to keep the units consistent throughout the calculation.
To determine the circular stress in the fuselage due to cabin pressure, we can consider the hoop stress on the fuselage skin. Hoop stress is the stress that acts circumferentially around a structure.
The equation for hoop stress (σ) is given by:
σ = (P(in) - P(out)) * R / t
Where:
- P(in) is the pressure inside the cabin
- P(out) is the pressure outside the cabin
- R is the radius of the circular fuselage
- t is the thickness of the fuselage skin
The difference between the inside and outside pressure is multiplied by the radius and divided by the thickness of the skin. This will give us the circular stress in the fuselage due to the cabin pressure.
It is important to note that this equation assumes that the fuselage is perfectly circular and that the pressure difference is evenly distributed throughout the circumference. In the case of a square fuselage, the stress distribution will be different, and the equation for determining the stress would need to be modified accordingly.