# Hoop Stress Calculator

The hoop stress calculator determines the **stresses acting on a thin-walled pressure vessel**. Pressure vessels are specially designed containers used to **hold fluids at a different pressure** than ambient ones. Due to high internal pressure, the parameters like **hoop stress** and **longitudinal stress** become crucial when designing these containers.

Knowledge of these stresses is helpful to design the riveted or welded joints on the body. Various pressure vessels include **boilers, water tanks, petrol tanks, gas cylinders, spray cans, fire extinguishers, pipes,** etc. Read on to understand what is hoop stress, longitudinal stress in a cylinder, and more.

## Stress on a pressure vessel

A pressure vessel is manufactured using rolled-up sheets welded or riveted together. In some cases, **it is also forged.** The manufacturing process depends on various factors like application and required strength. A pressure vessel design includes an **estimation of the stresses** that can cause failure. The shapes for the pressure vessel calculations are simplified as a cylinder or spherical in most cases. The shells are classified as either thick or thin based on their dimensions. If a shell's **wall thickness is not greater than one-tenth of the radius**, it is regarded as a thin shell.

## What are circumferential stress and longitudinal stress?

Consider a thin-walled pressure vessel. When a shell is subjected to a **large amount of internal pressure,** tensile stresses act along both directions. The stress acting **along the tangential direction to the circumference of a sphere** or cylindrical shell is known as **circumferential stress or hoop stress**. In a cylindrical shell, the **stress acting along the direction of the length of the cylinder** is known as **longitudinal stress**.

These stresses are vital parameters when it comes to pressure vessel design. Therefore, the **maximum permissible stress in the material** must not exceed either the circumferential or hoop stress. The failure from hoop stress results in rupturing of a cylindrical shell in two cylinders, whereas the excess longitudinal stress in the cylinder splits the cylinder into two troughs.

For a cylindrical shell having diameter `d`

and thickness `t`

, the circumferential or hoop stress `σ`

is given by the hoop stress equation:_{h}

`σ`_{h} = p * d / (2 * t)

where `p`

is internal pressure. The large cylindrical shells are manufactured with joints, and when the efficiency of the joints is taken into consideration, the circumferential stress equation becomes:

`σ`_{h} = p * d / (2 * t * η_{t})

where `η`

is the efficiency of longitudinal joints because the forces are acting along the longitudinal section. Similarly, the longitudinal stress, considering circumferential joint efficiency, _{t}`η`

is:_{c}

`σ`_{l} = p * d / (4 * t * η_{c})

Now that we know the hoop stress, one can also estimate the ratio of longitudinal stress to hoop stress, which is `0.5`

. The hoop stress acting on a cylindrical shell is double the longitudinal stress, considering ideal efficiency.

For a sphere, the hoop stress of a thin walled pressure vessel is also calculated using similar principle; however, the stress acting on the shell is only of one type, i.e., the hoop stress. The hoop stress formula for a spherical shell is:

`σ`_{h} = p * d / (4 * t * η)

where `η`

is the efficiency of joints.

## Change in shell dimensions

Due to the extreme operating conditions and internal pressure, the shell tends to expand or contract, i.e., the dimensions change due to the stresses. The **change in dimensions is a function of material properties** as well as the stresses. Consider a shell of made a material whose Young's modulus is `E`

and Poisson's ratio, `μ`

. A cylinder has two main dimensions – length and diameter, which would change due to internal pressure. The change in diameter `𝛿d`

is:

`𝛿d = p * d`^{2} * (1 - μ/2) / (2 * t * E)

The change in length `𝛿l`

is written as:

`𝛿l = p * d * l * (0.5 - μ) / (2 * t * E)`

Interestingly, upon rearranging the above equations, the strain `ε`

is a function of stress (either hoop or longitudinal) and material constants. For instance:

`ε = 𝛿d / d = p * d / (2 * t) * ( (1 - μ/2) / E)`

`ε = 𝛿d / d = σ`_{h} * ((1 - μ/2) / E)

As the dimensions of the shell increases, the volume is also affected, it is given by the equation:

`𝛿V = 0.25 * π * (d`^{2} * 𝛿l + 2 * d * l * 𝛿d)

Similarly, the change in dimensions for the spherical shell can be estimated using the equations:

`𝛿d = p * d`^{2} * (1 - μ) / (4 * t * E)

`𝛿V = π * d`^{4} * (1 - μ) / (8 * t * E)

## How to calculate hoop stress?

Now that you know what hoop stress is and its equation. Let's go through the steps to calculate the stresses using this hoop stress calculator.

- Select the
**shape of the shell**, either`Sphere`

or`Cylinder`

. - Enter the
**radius**`r`

or**diameter**`d`

of the shell. - Fill in the
**thickness**of the shell,`t`

. - The calculator returns the
**thickness to diameter ratio**. - Enter the
**internal pressure**on the walls of the shell,`p`

. - Insert
**Young's modulus**`E`

and**Poisson's ratio**`μ`

for the shell material. - The hoop stress calculator will return the respective
**stresses, including shear stress**in pressure vessels and**changes in dimensions**.

## Using the hoop stress calculator

Estimate the hoop stress in a water tank built using riveted joints of efficiency `0.75`

and having an internal pressure of `1.5 MPa`

. Take diameter and thickness of the shell as `3 m`

and `16.667 mm`

respectively.

To find the hoop stress in the spherical tank:

- Select the shape of the shell as
`Sphere`

. - Enter the diameter of the shell,
`d = 3 m`

. - Input the thickness of the shell,
`t = 16.667 mm`

. - Enter the internal pressure on the walls of the shell,
`p = 1.5 MPa`

. - Activate the
`advanced`

mode and set the joint efficiency as`0.75`

. - The hoop stress calculator then uses the circumference stress equation:

`σ`_{h} = p * d / (4 * t * η) = 1.5 * 3000 / (4 * 16.667 * 0.75) = 90 MPa

You can follow similar steps if you wonder how to calculate hoop stress in a pipe by setting the shape to `Cylinder`

, or for any other pressure vessel calculations.

## FAQ

### What is hoop stress?

The stress acting along the tangents of the cross-section of the sphere is known as hoop stress.

### What is hoop stress formula?

The hoop stress formula for a spherical shell with diameter `d`

and thickness `t`

under pressure `p`

is:

`σ`_{h} = p * d / (4 * t * η)

where `η`

is joint efficiency.

### How do I calculate hoop stress of a sphere?

To calculate hoop stress of a sphere:

**Multiply**internal pressure and diameter of the shell.**Divide**the resultant with four times the thickness.**Divide**the resultant with joint efficiency.

### What is longitudinnal stress?

The stress acting along the axial direction in a cylindrical shell due to the internal pressure is known as longitudinal stress.