Lesson 1.6: Thin-Walled Pressure Vessels

Learn pressure vessel analysis through pneumatic actuator casings, covering hoop and longitudinal stress calculations, wall thickness design, and safety considerations for pressurized components.

🎯 Learning Objectives

By the end of this lesson, you will be able to:

1. Calculate hoop and longitudinal stresses in cylindrical pressure vessels
2. Design wall thickness for pneumatic actuator casings
3. Apply safety factors and failure criteria for pressurized systems
4. Analyze stress concentrations around openings and joints

🔧 Real-World System Problem: Pneumatic Actuator Casing

Pneumatic actuators are essential components in industrial automation, robotics, and mechatronic systems. The cylindrical pressure vessel casing must safely contain high-pressure air while providing mounting points, ports, and sealing surfaces.

System Description

Pneumatic Actuator Components:

• Cylindrical Casing (contains pressurized air)
• Piston Assembly (converts pressure to linear force)
• End Caps (seal the cylinder ends)
• Ports and Fittings (air inlet/outlet connections)
• Mounting Brackets (attachment points)

The Pressure Vessel Challenge

A pneumatic actuator casing experiences complex stress states:

Engineering Question: How do we determine the minimum wall thickness for a pneumatic actuator casing operating at 6 bar (600 kPa) pressure while maintaining adequate safety margins?

Why Pressure Vessel Analysis Matters

Consequences of Poor Design:

• Catastrophic failure with potential for injury
• Leakage leading to performance degradation
• Excessive deformation affecting actuator precision
• Premature fatigue failure in cyclic applications

Benefits of Proper Design:

• Safe operation within design pressure limits
• Reliable performance over service life
• Cost optimization through appropriate wall thickness
• Regulatory compliance with pressure vessel codes

📚 Fundamental Theory: Pressure Vessel Mechanics

Basic Pressure Vessel Geometry

For thin-walled pressure vessels, we assume:

• Wall thickness t << radius r (typically t/r < 0.1)
• Stress is uniform across wall thickness
• Internal pressure p acts normal to all surfaces

Hoop Stress (Circumferential Stress)

Internal pressure creates hoop stress that tries to burst the cylinder:

🔄 Hoop Stress Formula

Where:

• = Hoop stress (Pa)
• = Internal pressure (Pa)
• = Internal radius (m)
• = Wall thickness (m)

Physical Meaning: Hoop stress acts circumferentially, trying to expand the cylinder radially. This is typically the maximum stress in cylindrical pressure vessels.

Longitudinal Stress (Axial Stress)

Pressure acting on the end caps creates longitudinal stress in the cylinder walls:

↕️ Longitudinal Stress Formula

Where:

• = Longitudinal stress (Pa)
• = Internal pressure (Pa)
• = Internal radius (m)
• = Wall thickness (m)

Physical Meaning: Longitudinal stress acts along the cylinder axis due to pressure force on the end caps. Note that it’s exactly half the hoop stress.

• Stress Comparison
• Spherical Vessels
• Design Criteria

📊 Pressure Vessel Stress Comparison

Hoop Stress:

Longitudinal Stress:

Stress Ratio:

Physical Meaning: Hoop stress is always twice the longitudinal stress in cylindrical pressure vessels, making it the critical design parameter.

🔧 Application: Pneumatic Actuator Casing Design

Let’s design a pneumatic actuator casing for an industrial automation system.

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System Parameters:

• Industrial pneumatic actuator casing (cylindrical pressure vessel)
• Working pressure: p = 6 bar = 0.6 MPa
• Internal diameter: 100 mm (radius r = 50 mm)
• Material: Aluminum 6061-T6 (σ_yield = 270 MPa)
• Safety factor: 4.0
• Pressure cycles: 10⁶ cycles (fatigue consideration)
• Design standards: ASME Boiler & Pressure Vessel Code, ISO 4393

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Step 1: Calculate Required Wall Thickness

Click to reveal wall thickness calculations

1. Determine allowable stress:

2. Calculate minimum thickness from hoop stress:

   From , solving for thickness:

3. Check longitudinal stress:

   Since σ_l < σ_allowable ✅, hoop stress governs the design

4. Select standard thickness:

   Design thickness: 3.0 mm (standard aluminum sheet with safety margin)

Step 2: Verify Design with Selected Thickness

Click to reveal design verification calculations

1. Actual hoop stress with t = 3.0 mm:

2. Actual longitudinal stress:

3. Actual safety factors:

   • Hoop stress: SF = 270/10.0 = 27.0
   • Longitudinal stress: SF = 270/5.0 = 54.0
   • Both exceed required SF = 4.0 by large margins

4. Design assessment:

   Very conservative design appropriate for personnel safety, stress concentrations, and manufacturing variations

Step 3: Stress Concentration and Fatigue Assessment

Click to reveal stress concentration and fatigue analysis

1. Stress concentration factors:

   • Ports and fittings: Kt = 2-4
   • Mounting holes: Kt = 2.5-3.5
   • Welds and joints: Kt = 1.5-2.5
   • Local stresses can be 2-4× nominal values

2. Fatigue life assessment:

   • Pressure cycles: 0 to 6 bar (fully reversed)
   • Stress amplitude: σa = (10.0 - 0)/2 = 5.0 MPa
   • Aluminum endurance limit ≈ 110 MPa
   • Fatigue safety factor: 110/5.0 = 22 ✅ Excellent

3. Design recommendations:

   • Round all corners to minimize stress concentrations
   • Add reinforcement around openings
   • Use higher local safety factors near discontinuities
   • Consider dynamic pressure effects (up to 150% static)

📊 Pressure Vessel Design Summary

🎯 Advanced Analysis: Fatigue and Dynamic Loading

Fatigue Considerations

Pneumatic actuators experience cyclic pressure loading:

Pressure Cycles:

• Minimum pressure: 0 bar (atmospheric)
• Maximum pressure: 6 bar (working pressure)
• Stress amplitude:

Fatigue Life Assessment: For aluminum, the endurance limit is typically 35-40% of ultimate strength.

With ultimate strength ≈ 310 MPa:

• Endurance limit ≈ 110 MPa
• Applied stress amplitude: 5.0 MPa
• Fatigue safety factor: 110/5.0 = 22 ✅ Excellent fatigue resistance

Dynamic Pressure Effects

Rapid pressurization can create higher stresses than static analysis predicts:

🛠️ Design Guidelines and Best Practices

Wall Thickness Selection Rules

• Minimum Requirements
• Practical Considerations
• Safety Factors

Code Requirements:

• ASME: Minimum 1.5 mm for steel
• ISO: Minimum 2.0 mm for aluminum
• Structural stability (prevent buckling)
• Corrosion allowance (0.5-1.0 mm)

Common Design Mistakes

📋 Summary and Next Steps

In this unit, you learned to:

1. Calculate hoop stress (σ = pr/t) and longitudinal stress (σ = pr/2t)
2. Design wall thickness using appropriate safety factors
3. Consider stress concentrations and fatigue in real systems
4. Apply pressure vessel design principles to mechatronic components

Key Design Principle: Hoop stress governs cylindrical pressure vessel design

Critical Design Formula: Minimum thickness = pr/(σ_allowable)

Coming Next: We begin Chapter 2 with Lesson 2.1, analyzing shear force and bending moment distributions in robotic arm segments, transitioning from axial loading to bending-dominated structural behavior.

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🎓 Chapter 1 Summary: Foundations in Mechatronic Structures

You’ve completed the foundational units covering axial loading, thermal effects, torsion, and pressure vessels. These principles form the basis for understanding how mechatronic components respond to forces, temperature changes, and internal pressures.

Lessons Completed:

1. Crank-slider connecting rod analysis (stress, strain, material properties)
2. CNC actuator shaft design (axial loading, material selection)
3. Linear actuator compound rod (multi-material load sharing)
4. Heated piston-cylinder system (thermal stress and expansion)
5. Geneva mechanism crankshaft (torsional stress and angular deformation)
6. Pneumatic actuator casing (pressure vessel hoop and longitudinal stress)

Next: Chapter 2 focuses on bending-dominated structural behavior in beams and complex loading scenarios.

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