Primary Stresses in Pressure Vessels

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Introduction to the primary stress of a pressure vessel

Hoop Stress (aka Circumferential Stress):

This stress is like a belt squeezing around the middle of the vessel.
For those cylindrical types, it’s the main stress to watch out for since it runs circles around the vessel, perpendicular to its long axis.
The math for this, for the simpler, thinner walls, goes like this: Hoop Stress = (Pressure × Radius) / Wall Thickness. It’s all about how much pressure you have, how big your vessel is, and how thick the walls are.

When discussing pressure vessels, two main types of stresses come into play due to internal or external pressure: Hoop (Circumferential) Stress and Longitudinal (Axial) Stress. The shape of the pressure vessel—whether it’s cylindrical or spherical—affects how these stresses are distributed. Let’s break down each type of stress and how it impacts the structure of pressure vessels.

Hoop (Circumferential) Stress

Hoop stress, also known as circumferential stress, acts around the circumference of a cylindrical pressure vessel. It is the stress that runs in a circle around the vessel, perpendicular to its axis. In cylindrical vessels, hoop stress is usually the highest type of stress, as it wraps around the vessel’s outer wall.

Direction of Stress: Circumferential, wrapping around the vessel’s walls.
Cause: Internal or external pressure acting around the cylindrical body.
Significance: Hoop stress is often the highest in cylindrical vessels, as it must resist the force attempting to expand the vessel outward.

Mathematical Formula for Hoop Stress

For thin-walled cylindrical pressure vessels, hoop stress can be calculated using the formula:

Longitudinal (Axial) Stress

Longitudinal stress, also referred to as axial stress, acts along the length or axis of the cylindrical vessel. This type of stress is caused by internal or external pressure acting on the ends (or caps) of the vessel. Longitudinal stress is typically lower than hoop stress in cylindrical vessels because the vessel’s structure is better equipped to handle forces along its length than around its circumference.

Direction of Stress: Axial, acting along the length of the vessel.
Cause: Pressure exerted on the end caps of the vessel.
Significance: Although usually lower than hoop stress, longitudinal stress is crucial in assessing the overall structural integrity of the vessel.

Mathematical Formula for Longitudinal Stress

For thin-walled cylindrical pressure vessels, longitudinal stress can be calculated with the formula:

$σlongitudinal=P×r2t\sigma_{\text{longitudinal}} = \frac{P \times r}{2t}$

where:

$σlongitudinal\sigma_{\text{longitudinal}}$ = Longitudinal stress (measured in Pascals or PSI)
$P$ , $r$ , and $t$ are as defined above.

This formula helps evaluate the wall thickness needed to handle the pressure along the vessel’s axis without causing failure.

Stress in Spherical Pressure Vessels

In spherical pressure vessels, the stress distribution differs from that in cylindrical vessels. Because of the uniform shape, stress is evenly distributed in all directions, making it identical along both the circumferential and longitudinal planes. Therefore, a spherical vessel experiences a single type of stress.

Stress Distribution: Uniform in every direction, without any distinction between hoop and longitudinal stresses.
Application: Spherical vessels are ideal for storing gases under high pressure, as they can handle higher stresses uniformly due to their geometry.

Mathematical Formula for Stress in Spherical Pressure Vessels

For thin-walled spherical pressure vessels, the formula to calculate stress is:

$σ=P×r2t\sigma = \frac{P \times r}{2t}$

where:

$σ\sigma$ = Stress in the vessel (measured in Pascals or PSI)
$P$ , $r$ , and $t$ are as previously defined.

This formula reflects the uniform stress distribution in spherical vessels, which allows them to withstand high internal pressures with relatively thin walls compared to cylindrical vessels.

Longitudinal (Axial) Stress:

This one runs top to bottom or head to toe along the vessel.
It kicks in because the pressure ends up pushing on the heads or caps of the cylinder.
The math here is a bit like the hoop stress but slices the pressure effect in half: Longitudinal Stress = (Pressure × Radius) / (2 × Wall Thickness).

For the sphere gang, the stress is all chill and equal no matter where you look. It’s like a perfect zen balance with the formula looking pretty similar to the longitudinal stress for cylinders.

Now, these formulas are keeping it simple and straight for the thinner walls. If you’re dealing with a chunkier vessel or some wild pressure scenarios, you’re gonna need to dive into some heavier math or computer magic.

But wait, there’s more – Secondary and Tertiary Stresses:

Secondary stresses are like the drama caused by things not moving together right, like when parts of the vessel get heated differently or it’s sitting wonky.
Tertiary stresses are the leftovers from making the vessel, like welding marks, which can add their twist to the stress mix.

Special mentions:

Thermal Stresses: When things heat up or cool down unevenly, it can make the vessel throw a fit, especially at weak spots like where pipes or other bits are attached.
Factor of Safety: It’s like adding a buffer or extra padding in your design to make sure you’re very clear of any trouble, considering all the what-ifs.
Fatigue and Cyclic Loading: This is about how the vessel deals with the pressure party starting and stopping over time. Too much of this can lead to cracks even if it seems like everything should be cool.
Corrosion and Material Degradation: Rust or chemical attack can seriously cramp your vessel’s style, thinning walls and setting up for a bad day.
External Loads and Attachments: Other stuff hanging on or pushing on the vessel can bring its own set of stresses, like wind or an earthquake doing a number on it.

Keeping these vessels safe and sound is a mix of smart design, picking the right materials, keeping an eye on them, and staying on top of maintenance. As tech gets better and we get smarter with our tools and sims, we’re always upping our game to keep things safe and efficient.

Need a reliable partner?

Red River specializes in the design and manufacturing of pressure vessels. We also fabricate related items such as prefabricated spools and skid packages.

Reach Out to us today and experience the Red River difference. Where American Made and American Values come together, we care more.

FAQ: Understanding the Primary Stress of Pressure Vessels

What is the primary stress in a pressure vessel?

Primary stress in a pressure vessel refers to the stress caused by internal pressure. This stress is distributed evenly across the vessel’s walls and is the main factor in determining the vessel’s thickness and material selection. It’s crucial in ensuring the vessel can withstand the internal pressure without failure.

How do material choices affect the handling of primary stress in pressure vessels?

Material selection is critical in managing primary stress. Materials must have adequate strength and flexibility to handle the internal pressure. Factors like corrosion resistance, temperature tolerance, and the vessel’s contents (e.g., chemicals, gases) play a significant role in material choice. Common materials include carbon steel, stainless steel, and alloys designed for specific pressure and temperature conditions.

Are there different types of primary stress in pressure vessels?

Yes, primary stress in pressure vessels can be categorized into two types: membrane stress and bending stress. Membrane stress is uniform across the thickness of the vessel wall, while bending stress varies across the wall thickness. The design of the vessel aims to minimize bending stress as it can lead to fatigue and eventual failure.

How does vessel shape impact the management of primary stress?

The shape of a pressure vessel significantly influences how it handles primary stress. Cylindrical and spherical shapes are most common as they distribute stress more evenly and can handle higher pressures more effectively. The design must consider areas like joints and nozzles, as they can be points of stress concentration.

What role does pressure vessel testing play in ensuring safety against primary stress?

Pressure vessel testing is essential for safety. Tests like hydrostatic testing, where the vessel is filled with water and pressurized, help to ensure that the vessel can handle its intended pressure. Non-destructive testing methods, such as ultrasonic or radiographic testing, are also used to detect any flaws or weaknesses in the vessel that could affect its ability to handle primary stress.

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