Thin-Walled Pressure Vessel Equation

Thin-Walled Pressure Vessel Equation (specifically for hoop stress). Because 3D-printed parts have anisotropic properties—meaning they are stronger along the “grain” of the print—we must apply your safety factor to the material’s yield strength.

1. The Hoop Stress Formula

For a cylindrical manifold, the hoop stress (σ) is the stress exerted on the walls by the internal fluid pressure. The formula is:

σ=tP⋅r​

Where:

• P = Internal pressure
• r = Internal radius of the manifold
• t = Wall thickness

2. Solving for Maximum Pressure (Pmax​)

We rearrange the formula to solve for pressure, incorporating your safety factor (Sf​):

Pmax​=r⋅Sf​σyield​⋅t​

Where:

• σyield​ = Tensile yield strength of PA12-CF (approx. 76 MPa as established in your technical data).
• Sf​ = Safety factor (e.g., 3.0 for dynamic/pressurized applications).

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3. Step-by-Step Calculation Example

Let’s assume a manifold design with:

• Internal Radius (r): 10 mm
• Wall Thickness (t): 3 mm
• Yield Strength (σyield​): 76 MPa
• Safety Factor (Sf​): 3.0

Calculation:

1. Effective Yield Strength: 76 MPa/3.0=25.33 MPa
2. Max Pressure: (25.33 MPa⋅3 mm)/10 mm
3. Result: 7.6 MPa (or approx. 76 bar / 1,100 PSI)

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4. Critical Engineering Considerations for 3D Printing

While the math provides the theoretical limit, 3D-printed parts have specific “weak points” that the formula does not account for:

• Layer Adhesion (Z-Axis): The formula assumes a homogenous material. In 3D printing, the interlayer bond is weaker than the plastic itself. If the manifold is printed with the Z-axis perpendicular to the hoop stress, multiply your safety factor by an additional 1.5 to account for potential layer delamination.
• Infill Density: Ensure the manifold is printed with 100% solid infill. Voids or low-infill patterns will lead to premature failure under pressure.
• Permeability: 3D-printed walls are rarely “airtight” at high pressures. You may need to incorporate a thin internal resin coating or increase the “wall line count” in your slicer settings to prevent fluid weeping through the printed layers.

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