Thick Cylinders free study notes

1. Introduction to Thick Cylinders

A thick cylinder is a pressure vessel in which the wall thickness is not negligible compared to its diameter. These cylinders are commonly used where high internal pressures exist, such as:

• Gun barrels
• Hydraulic presses
• High-pressure pipes
• Cylinders in power plants

Unlike thin cylinders, stresses in thick cylinders vary across the thickness, so simple formulas cannot be used.

2. Definition of Thick Cylinder

A cylinder is considered thick if:$$\frac{t}{d} > \frac{1}{10}$$

Where:

• $t$t = thickness
• $d$d = internal diameter

3. Types of Stresses in Thick Cylinder

When subjected to internal and/or external pressure, three stresses develop:

(a) Radial Stress (σr\sigma_rσr​)

• Acts perpendicular to the surface
• Varies from maximum at inner surface to minimum at outer surface

(b) Hoop Stress (σθ\sigma_\thetaσθ​)

• Acts tangentially
• Maximum at inner surface
• Decreases towards outer surface

(c) Longitudinal Stress (σl\sigma_lσl​)

• Acts along the axis
• Usually constant across thickness (for closed ends)

4. Lame’s Theory of Thick Cylinders

To determine stress distribution, Lame’s equations are used.

General Equations

$\sigma_r = A – \frac{B}{r^2}$σr​=A−r2B​

$\sigma_\theta = A + \frac{B}{r^2}$σθ​=A+r2B​

Where:

• $\sigma_r$σr​ = radial stress
• $\sigma_\theta$σθ​ = hoop stress
• $r$r = radius at any point
• $A, B$A,B = constants determined from boundary conditions

5. Boundary Conditions

To find constants $A$A and $B$B:

Case 1: Internal Pressure Only

• At inner radius $r_i$ri​: $$\sigma_r = -p_i$$σr​=−pi​
• At outer radius $r_o$ro​: $$\sigma_r = 0$$σr​=0

Case 2: Internal and External Pressure

• At $r_i$ri​: $\sigma_r = -p_i$σr​=−pi​
• At $r_o$ro​: $\sigma_r = -p_o$σr​=−po​

(Negative sign indicates compressive stress)

6. Maximum Stresses

Maximum Hoop Stress

• Occurs at inner surface

$$\sigma_{\theta,\text{max}} = A + \frac{B}{r_i^2}$$σθ,max​=A+ri2​B​

Minimum Hoop Stress

• Occurs at outer surface

$$\sigma_{\theta,\text{min}} = A + \frac{B}{r_o^2}$$σθ,min​=A+ro2​B​

7. Stress Distribution Characteristics

• Radial stress is compressive and decreases outward
• Hoop stress is tensile and maximum at inner surface
• Stress variation is non-linear (parabolic)
• Maximum failure risk is at inner surface

8. Longitudinal Stress (Closed Cylinder)

For cylinders with closed ends:$$\sigma_l = \frac{p_i r_i^2 – p_o r_o^2}{r_o^2 – r_i^2}$$σl​=ro2​−ri2​pi​ri2​−po​ro2​​

9. Compound Cylinders

To withstand very high pressures, cylinders are sometimes made of multiple layers (compound cylinders).

Purpose:

• Reduce maximum hoop stress
• Increase strength
• Improve safety

Method:

• Shrink fitting outer cylinder over inner cylinder
• Induces compressive stress at inner surface

10. Failure Theories Applied

Design of thick cylinders uses:

• Maximum principal stress theory
• Maximum shear stress theory
• Distortion energy theory

11. Applications

• Hydraulic cylinders
• Gun barrels
• Pressure vessels
• High-pressure pumps
• Nuclear reactors

12. Comparison: Thin vs Thick Cylinder

Feature	Thin Cylinder	Thick Cylinder
Stress distribution	Uniform	Varies across thickness
Radial stress	Negligible	Significant
Formula used	Simple	Lame’s equations
Thickness ratio	$t/d \leq 1/10$t/d≤1/10	$t/d > 1/10$t/d>1/10
Accuracy	Approximate	More accurate

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