Step-by-Step Calculation with the Cylinder Stress Formula

Step-by-Step Calculation with the Cylinder Stress Formula: A Comprehensive Guide

Cylindrical structures are ubiquitous in engineering, appearing in everything from pressure vessels and pipelines to structural components and rotating machinery. Understanding the stresses that these cylinders experience is crucial for ensuring their structural integrity and preventing failures. This article provides a comprehensive, step-by-step guide to calculating stresses in cylinders, covering various loading conditions and geometries. We will explore the underlying formulas, discuss real-world applications, and work through detailed examples to solidify your understanding.

Understanding Cylinder Stress: An Introduction

Before diving into the calculations, let's define what we mean by "cylinder stress." In general, we are concerned with the stresses induced within the cylinder wall due to internal or external pressure, axial loads, or thermal gradients. These stresses can be categorized as: Hoop Stress (Circumferential Stress): Stress acting in the circumferential direction (tangential to the cylinder wall). Longitudinal Stress (Axial Stress): Stress acting along the length of the cylinder. Radial Stress:Stress acting in the radial direction (perpendicular to the cylinder wall).

The magnitude of these stresses depends on several factors, including the cylinder's geometry (radius and wall thickness), the applied pressure, and the material properties. We’ll consider both thin-walled and thick-walled cylinder scenarios.

Thin-Walled Cylinder Stress Formulas

Thin-walled cylinders are defined as cylinders where the wall thickness (t) is significantly smaller than the radius (r), typically t/r < 0.1. In these cases, we can make simplifying assumptions that lead to relatively straightforward stress calculations.

Hoop Stress (σ_h)

The hoop stress in a thin-walled cylinder subjected to internal pressure (p) is given by:

σ_h = (p r) / t

Where: σ_h = Hoop stress (Pa or psi)

p = Internal pressure (Pa or psi)

r = Radius of the cylinder (m or in)

t = Wall thickness of the cylinder (m or in)

Longitudinal Stress (σ_l)

The longitudinal stress in a thin-walled cylinder subjected to internal pressure (p) is given by:

σ_l = (p r) / (2 t)

Notice that the longitudinal stress is half the hoop stress. This is because the pressure acts over the circular area at the end of the cylinder.

Example 1: Calculating Stresses in a Thin-Walled Pressure Vessel

A cylindrical pressure vessel has an internal diameter of 1 meter and a wall thickness of 10 mm. It is subjected to an internal pressure of 2 MPa. Calculate the hoop stress and longitudinal stress in the vessel wall.

Solution

1.Convert Units: Ensure all units are consistent.

Radius, r = 0.5 meters

Thickness, t = 0.01 meters

Pressure, p = 2 MPa = 2 x 10⁶ Pa

2.Calculate Hoop Stress:

σ_h = (p r) / t = (2 x 10⁶ Pa 0.5 m) /

0.01 m = 100 x 10⁶ Pa = 100 MPa

3.Calculate Longitudinal Stress:

σ_l = (p r) / (2 t) = (2 x 10⁶ Pa 0.5 m) / (2

0.01 m) = 50 x 10⁶ Pa = 50 MPa

Therefore, the hoop stress is 100 MPa, and the longitudinal stress is 50 MPa.

Thick-Walled Cylinder Stress Formulas (Lamé's Equations)

When the wall thickness is comparable to the radius (t/r > 0.1), the thin-walled cylinder assumptions are no longer valid. We must use Lamé's equations to accurately determine the stress distribution. Lamé's equations account for the radial stress variation across the cylinder wall.

Hoop Stress (σ_h)

σ_h = p_i (r_i² (r_o² + r²)) / (r² (r_o² - r_i²)) - p_o (r_o² (r_i² + r²)) / (r² (r_o² - r_i²))

Radial Stress (σ_r)

σ_r = p_i (r_i² (r_o² - r²)) / (r² (r_o² - r_i²)) - p_o (r_o² (r² - r_i²)) / (r² (r_o² - r_i²))

Where:

p_i = Internal pressure (Pa or psi)

p_o = External pressure (Pa or psi)

r_i = Internal radius (m or in)

r_o = External radius (m or in)

r = Radius at which stress is being calculated (r_i ≤ r ≤ r_o) (m or in)

Note that these equations allow you to calculate the stress at any radial location (r) within the cylinder wall. The maximum hoop stress usually occurs at the inner surface (r = r_i).

Longitudinal Stress (σ_l)

For a thick-walled cylinder with closed ends, the longitudinal stress is constant across the wall thickness and can be calculated as:

σ_l = (p_i r_i² - p_o r_o²) / (r_o² - r_i²)

Example 2: Calculating Stresses in a Thick-Walled Cylinder

A thick-walled cylinder has an internal radius of 50 mm and an external radius of 100 mm. It is subjected to an internal pressure of 50 MPa and an external pressure of 10 MPa. Calculate the hoop stress and radial stress at the inner and outer surfaces of the cylinder. Also, calculate the longitudinal stress.

Solution

1.Convert Units: Ensure all units are consistent. In this case, we'll use MPa and mm.

r_i = 50 mm

r_o = 100 mm

p_i = 50 MPa

p_o = 10 MPa

2.Calculate Hoop Stress at Inner Surface (r = r_i = 50 mm):

σ_h = 50 (50² (100² + 50²)) / (50² (100² - 50²)) - 10 (100² (50² + 50²)) / (50² (100² - 50²))

σ_h = 50 (2500 12500) / (2500 7500) - 10 (10000 5000) / (2500 7500)

σ_h = 50 (12500/7500) - 10 (5000/7500) = 50 (5/3) - 10 (2/3) = 83.33 -

6.67 =

76.66 MPa

3.Calculate Hoop Stress at Outer Surface (r = r_o = 100 mm):

σ_h = 50 (50² (100² + 100²)) / (100² (100² - 50²)) - 10 (100² (50² + 100²)) / (100² (100² - 50²))

σ_h = 50 (2500 20000) / (10000 7500) - 10 (10000 12500) / (10000 7500)

σ_h = 50 (2/3) - 10 (5/3) = 33.33 -

16.67 =

16.66 MPa

4.Calculate Radial Stress at Inner Surface (r = r_i = 50 mm):

σ_r = 50 (50² (100² - 50²)) / (50² (100² - 50²)) - 10 (100² (50² - 50²)) / (50² (100² - 50²))

σ_r = 50 (7500/7500) - 10 (0/7500) = 50 - 0 = 50 MPa

5.Calculate Radial Stress at Outer Surface (r = r_o = 100 mm):

σ_r = 50 (50² (100² - 100²)) / (100² (100² - 50²)) - 10 (100² (100² - 50²)) / (100² (100² - 50²))

σ_r = 50 (0/7500) - 10 (7500/7500) = 0 - 10 = -10 MPa

6.Calculate Longitudinal Stress:

σ_l = (50 50² - 10 100²) / (100² - 50²) = (50 2500 - 10 10000) / (10000 - 2500) = (125000 - 100000) / 7500 = 25000 / 7500 = 3.33 MPa

Therefore, the hoop stress is 76.66 MPa at the inner surface and

16.66 MPa at the outer surface. The radial stress is 50 MPa at the inner surface and -10 MPa at the outer surface. The longitudinal stress is

3.33 MPa. Note the negative sign for radial stress at the outer surface indicating compressive stress.

Beyond Simple Pressure: Additional Considerations

While internal pressure is a common loading scenario, cylinders often experience other loads and conditions that must be considered: Axial Loads:Cylinders can be subjected to tensile or compressive axial loads. The stress due to axial loads is simply σ = F/A, where F is the axial force and A is the cross-sectional area of the cylinder wall.

Thermal Stress: Temperature gradients within the cylinder wall can induce thermal stresses. These stresses depend on the coefficient of thermal expansion of the material, the temperature difference, and the material's elastic modulus. Calculating thermal stress often requires more advanced finite element analysis (FEA) techniques.

Bending Loads: If a cylinder acts as a beam, it will experience bending stresses. The bending stress can be calculated using the bending stress formula: σ = My/I, where M is the bending moment, y is the distance from the neutral axis, and I is the area moment of inertia of the cylinder's cross-section.

Combined Loading: In many real-world scenarios, cylinders are subjected to a combination of loads. In such cases, the principle of superposition can be used to determine the total stress. However, it's essential to consider the interaction between different stress components using stress transformation techniques (e.g., Mohr's circle).

How do you calculate hoop stress in thin-walled cylinders?

As shown above, hoop stress is calculated using the formula σ_h = (p r) / t, where 'p' is the internal pressure, 'r' is the radius, and 't' is the wall thickness. This formula is applicable when the wall thickness is significantly smaller than the radius (t/r < 0.1).

What is the difference between true stress and engineering stress?

Engineering stress is calculated using the original cross-sectional area of the material, while true stress is calculated using the instantaneous cross-sectional area under load. Engineering stress is simpler to calculate but less accurate at larger deformations, whereas true stress provides a more accurate representation of the stress state during deformation.

When should principal stress formulas be applied in design?

Principal stress formulas should be applied when dealing with combined loading scenarios or complex geometries where stresses act in multiple directions. Principal stresses represent the maximum and minimum normal stresses at a point and are crucial for failure analysis using theories like maximum shear stress theory or distortion energy theory (von Mises). Calculating principal stresses helps engineers identify critical locations and prevent yielding or fracture.

Common Pitfalls and Misconceptions

Using Thin-Walled Formulas for Thick-Walled Cylinders: This is a common mistake. Always check the t/r ratio to determine whether the thin-walled or thick-walled formulas are appropriate. Using the thin-walled formula on a thick-walled cylinder will result in significant errors.

Ignoring External Pressure: In many cases, external pressure is negligible compared to internal pressure. However, in situations where external pressure is significant (e.g., submersible vehicles), it must be included in the calculations.

Forgetting Units: Always ensure that all units are consistent before performing calculations. Mixing units can lead to errors in the final result.

Neglecting Stress Concentrations: At points of geometric discontinuity (e.g., holes, welds), stress concentrations can occur. These stress concentrations can significantly increase the local stress levels and should be considered in the design.

Conclusion

Calculating stresses in cylinders is a fundamental skill for mechanical engineers. By understanding the underlying formulas and the assumptions behind them, you can accurately predict the stress distribution in cylindrical structures and ensure their structural integrity. This article has provided a comprehensive guide to calculating stresses in both thin-walled and thick-walled cylinders, covering various loading conditions and geometries. By carefully considering the factors discussed and avoiding common pitfalls, you can confidently design safe and reliable cylindrical components for a wide range of engineering applications. Remember that more complex scenarios involving combined loading or thermal gradients may require more advanced analysis techniques, such as finite element analysis.