5. THIN CYLINDRICAL SHELLS

For 4th Semester Polytechnic ME Students
Written by Garima Kanwar | Blog: Rajasthan Polytechnic

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Course Code ME 4002 (Same in MA/MP/MT 4002)
Course Title STRENGTH OF MATERIALS

5. Thin Cylindrical Shells

This chapter deals with the analysis of thin-walled cylindrical shells, which are commonly used in pressure vessels, pipes, and other engineering structures. The primary stresses encountered in such shells are longitudinal stresses and hoop stresses, which need to be understood for the safe design of these structures.

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5.1 Explanation of Longitudinal and Hoop Stresses in the Light of Circumferential and Longitudinal Failure of Shell

1. Longitudinal Stresses:

   • These stresses occur along the length of the cylindrical shell, i.e., in the direction parallel to the axis of the cylinder.
   • Cause: Longitudinal stresses are primarily due to the internal pressure exerted on the cylinder, which tends to expand or contract the length of the shell.
   • Significance: The longitudinal stress is typically lower than the hoop stress in a thin-walled cylinder and is generally less critical for failure analysis.

2. Hoop Stresses:

   • Also known as circumferential stresses, hoop stresses act in the direction perpendicular to the longitudinal axis, around the circumference of the cylinder.
   • Cause: Hoop stresses are caused by the internal pressure pushing against the walls of the cylinder, trying to expand it circumferentially.
   • Significance: Hoop stress is the most critical stress in the design of thin-walled cylinders, as it tends to be the largest stress and is the primary cause of failure in pressure vessels.

3. Circumferential Failure:

   • The failure of a cylindrical shell due to hoop stress occurs when the material’s strength is exceeded by the hoop stress. This is the most common mode of failure for thin-walled pressure vessels.

4. Longitudinal Failure:

   • Longitudinal failure occurs when the longitudinal stress exceeds the material’s tensile strength. While this is possible, it usually happens at higher pressures or when the material is weak in the longitudinal direction.

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5.2 Derivation of Expressions for the Longitudinal and Hoop Stress for Seamless and Seam Shells

The derivation of stress expressions for thin-walled cylindrical shells under internal pressure is based on assumptions of small wall thickness and a uniform distribution of pressure over the cross-section.

Hoop Stress (σₕ)

For a thin-walled cylindrical shell subjected to internal pressure $p$, the hoop stress (also called the circumferential stress) is the most significant stress and is given by the formula:

$$\sigma_h = \frac{p \cdot r}{t}$$

Where:

• $\sigma_h$ = Hoop stress (circumferential stress)
• $p$ = Internal pressure
• $r$ = Radius of the cylinder
• $t$ = Wall thickness of the cylinder

This expression assumes that the wall thickness $t$ is much smaller than the radius $r$ (i.e., the cylinder is thin-walled).

Longitudinal Stress (σₗ)

The longitudinal stress, which is the stress along the length of the cylindrical shell, is given by:

$$\sigma_l = \frac{p \cdot r}{2t}$$

Where:

• $\sigma_l$ = Longitudinal stress
• $p$ = Internal pressure
• $r$ = Radius of the cylinder
• $t$ = Wall thickness of the cylinder

Difference Between Seamless and Seam Shells:

• Seamless Shells: These shells have no welded seam and are considered to be stronger because of the uniformity of the material.
• Seam Shells: These shells have a welded seam that can potentially be a weak point. The stresses around the seam need to be considered in the design, and often these seams are designed to withstand higher stresses.

However, in both cases, the formulas for hoop and longitudinal stresses are derived using the same principles and are generally applicable to both seamless and seam shells.

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5.3 Related Numerical Problems for Safe Thickness and Safe Working Pressure

Example 1: Calculation of Safe Thickness of a Cylindrical Shell

Given:

• Internal pressure $p = 1.5 \, \text{MPa}$
  
• Cylinder radius $r=0.5\text{m}$
• Allowable tensile strength of material $\sigma_{\text{allow}} = 150 \, \text{MPa}$
  

Find: The minimum required wall thickness $t$ to ensure the shell does not fail under pressure.

Solution: For hoop stress, using the formula:

$$\sigma_h = \frac{p \cdot r}{t}$$

We set $\sigma_h = \sigma_{\text{allow}}$ to ensure the material strength is not exceeded.

$$150 = \frac{1.5 \times 10^6 \cdot 0.5}{t}$$

Solving for $t$:

$$t = \frac{1.5 \times 10^6 \times 0.5}{150} = 5 \, \text{mm}$$

Thus, the required wall thickness is 5 mm.

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Example 2: Calculation of Safe Working Pressure for a Cylindrical Shell

Given:

• Cylinder radius $r = 0.75 \, \text{m}$
  
• Wall thickness $t=10\text{mm}$
• Allowable stress $\sigma_{\text{allow}} = 120 \, \text{MPa}$
  

Find: The safe working pressure $p$ for the cylindrical shell.

Solution: For hoop stress, using the formula:

$$\sigma_h = \frac{p \cdot r}{t}$$

We set $\sigma_h = \sigma_{\text{allow}}$.

$$120 = \frac{p \cdot 0.75}{0.01}$$

Solving for $p$:

$$p = \frac{120 \times 0.01}{0.75} = 1.6 \, \text{MPa}$$

Thus, the safe working pressure is 1.6 MPa.

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Key Concepts to Remember:

• Hoop stress is the most critical stress and is typically higher than the longitudinal stress.
• Longitudinal stress is generally half of the hoop stress in thin-walled cylindrical shells.
• The formulas derived are based on the assumptions that the cylinder has a small wall thickness compared to its radius (thin-walled assumption).
• Safe design of cylindrical shells involves ensuring that both hoop stress and longitudinal stress do not exceed the material's allowable stress, and calculations are done for safe thickness and working pressure.

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These calculations are crucial for the safe design and operation of pressure vessels, pipelines, and other cylindrical structures exposed to internal pressures.

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