# 23 Facts On Radial Stress: The Complete Beginner’s Guide

Internal pressure and external pressure compress the pressure vessel radially, resulting compression stresses called radial stress, the sign convention in common use regards compressive stresses as negative. Radial stress is represented by $\sigma _{r}$

All the three principal stresses (hoop, axial and radial) act on a pressure vessel are mutually perpendicular to each other. Among all the three stresses $\sigma _{r}$ acts in the direction of radius of the cylinder or sphere.

Pressures act in different directions on a cylindrical or spherical object  which are named as Axial, Radial and Tangential stresses.

The Radial Stresses can be formulated as a function of internal pressure and ambient pressure and the inner and outer radii of a pressure vessel. On the inside surface of the cylinder, the $\sigma _{r}$ is the same as the internal pressure.

On the outside, it is the same as the external pressure ( 14 psi or 0.1 MPa). Through the thickness of the cylinder, it varies almost linearly between those values. If we consider a cylindrical pipe carrying fluid, different types of loads like weight loads(pipe weight, fluid weight, etc.), pressures( internal and external design and operating pressures), temperature change, occasional loads(slug force, surge force) create stresses in a piping system.

These loads try to deform the pipe and due to inertia effect the pipe will create some internal resisting force in the form of stresses.

## What is Radial Stress in Pressure Vessel?

The Radial Stresses act differently on a pressure vessel depending on its wall thickness and the shape of vessel.

If the inner surface of a cylinder experience pressure, then the maximum stresses will develop in the inner surface and if the outer surface undergo pressure force then maximum stresses will be acting on the outer surface.

Pressure Vessels are large containers specially designed to keep liquids and gases, the inside pressure is always different from the outside pressure, the inner pressure of a pressure vessel is usually maintained at a higher side. Cellular organisms and arteries of our body are the natural example of pressure vessels.

Vacuum containing pressure vessels are maintained at a lower inner pressure than the atmosphere.

Generally for a pressure vessel we can assume that the material used is isotropic, the strains from the pressures are small and the wall thickness of the vessel is much smaller than outer and inner radius of the container. Aerosol cans, scuba divining tanks and large industrial containers, Boilers etc are the examples of pressure vessels.

## What is Radial Stress in Pipelines?

Radial stress in pipelines is due to the internal pressure inside the pipe created by the fluid or gas.

Radial Stress is acting in pipelines in the form of a normal stress and acts parallel to the pipe radius. The value remains within the range of internal design pressure and atmospheric pressure act on inner and outer surface respectively. The $\sigma _{r}$ which is developed perpendicular to the surface is given by $\sigma _{r}=-p$.

In comparison to other normal stresses acting in pipelines the value of the $\sigma _{r}$ is significantly lesser, for this reason, the Longitudinal stress and circumferential stress are only considered for pipe designing purposes. $\sigma _{r}$ is generally ignored.

## How to calculate Radial stress in pipe?

Radial Stress is a normal stress present in pipe wall, acts in a direction parallel to pipe radius.

$\sigma _{r}$ is acting in pipelines in the form of a normal stress and acts parallel to the pipe radius. The value remains within the range of internal design pressure and atmospheric pressure act on inner and outer surface respectively.

Let us consider the $\sigma _{r}$s in a pressurized pipe, the cross-section of the pipe wall is characterized by its inside radius and outside radius.

$\sigma _{r}=-P_{int}$

$\sigma ^{r}=-P_{amb}$

Minus sign is due to the compressive nature of the stresses.

At an arbitrary location inside the pipe wall forces cause compression which is counteract by the material of the pipe wall.

Value of the compressive stress through out the pipe wall thickness, the expression for stress distribution inside the pipe wall is given by Lame’s theorem.

The expression for

$(\sigma_{r})_{R}$=

$\frac{p_{i}r_{i}^{2}-p_{o}r_{o}^{2}}{r_{o}^{2}-r_{i}^{2}}$+

$\frac{r_{i}^{2}r_{o}^{2}(p_{o}-p_{i})}{r^{2}(r_{o}^{2}-r_{i}^{2})}$

The expression contains many fixed value like $r_{o}$,$r_{i}$,$p_{i}$,$p_{o}$ only radius(r) is only variable.

$\sigma _{r}=C_{1}+\frac{C_{2}}{r^{2}}$

In other words  $\sigma _{r}\alpha \frac{1}{r^{2}}$

Radial Stress is decreased from inside pressure value to outside pressure value.

Maximum $\sigma _{r}$ is simply the internal pressure value of the pipe

$\sigma _{rmax}=p_{int}$

The normal stress that acts towards or away from the central axis of the cylinder is known as Radial Stress.

A set of equations known as Lames equations are used to calculate the stresses acting on a pressure vessel. In case of a pipe $\sigma _{r}$ varies between Internal pressure and Ambient Pressure.

$\sigma _{r}=A-\frac{B}{r^{2}}$

$\sigma _{\theta }=A+\frac{B}{r^{2}}$

Where,A and B are the constant of integration and can be solved by applying boundary conditions.

## Radial Stress Formula for Thick Cylinder

A pressure vessel is considered as thick when $\frac{D}{t}< 20$ where ‘D’ is the diameter of the vessel and ‘t’ is the wall thickness.

In the case of a thick cylinder, the stresses acting are mainly Hoop’s Stress or circumferential stress and Radial Stress. Due to the internal pressure acting inside the vessel, some stresses are developed in the inner wall of the vessel along the radius of the vessel which is known as the Radial Stresses.

Lame’s equation is used to quantify the stresses acting on a thick cylinder. The $\sigma _{r}$ for thick cylinder at a point r from the axis of the cylinder is given below

$(\sigma_{r})_{R}$=

$\frac{p_{i}r_{i}^{2}-p_{o}r_{o}^{2}}{r_{o}^{2}-r_{i}^{2}}$+

$\frac{r_{i}^{2}r_{o}^{2}(p_{o}-p_{i})}{r^{2}(r_{o}^{2}-r_{i}^{2})}$

Where $r_{i}$=inner radius of the cylinder

$r_{o}$=outer radius of the cylinder

$p_{i}$=inner absolute pressure

$p_{o}$=outer absolute pressure

At the inside surface of the cylinder wall the $\sigma _{r}$ is maximum and is equal to $p_{i} – p_{o}$ i.e. gauge pressure.

## Radial stress formula for conical cylinder

The effect of Radial Stress in case of a thin cylinder is not zero but does not worth to consider its effect for design and analysis.

In case of a thin cylinder the hoop stress and axial stresses are much larger than $\sigma _{r}$, therefore for a thin cylinder the Radial Stress is generally ignored. In case of a thick cylinder $\sigma _{r}$ generated is equivalent to gauge pressure on the inner surface of the cylinder and zero on the outer surface.

## Radial Stress Formula for Sphere

The stresses act normal to the walls of the sphere are Radial Stresses.

The $\sigma _{r}$ acting on the outer wall of a sphere is zero since the outer wall is a free surface.

$\sigma _{r}$ formula for a sphere is $\sigma _{r}=-\frac{p_{i}}{2}$,for mid thickness t/2

$\sigma _{r}=-p$, for inner radius

$\sigma _{r}=0$, for outer radius

Radial Stresses are always compressive in nature.

The radial Stress in a pressure vessel is generated due to the action of internal pressure exerted by the inside fluid and the ambient pressure on the outer surface. At an arbitrary location inside the wall of the pressure vessel forces cause compression which is counteract by the material of the wall.

$p_{i}$ and $p_{e}$ compress the shell radially, generating $\sigma _{r}$, as per the convention of continuum mechanics, these stresses are negative.

The $\sigma _{r}$ at the inner and outer radius are respectively

$\sigma _{ri}=-p_{i}$

$\sigma _{re}=-p_{e}$

The stresses are uniformly distributed through the thickness of the structure, the arithmetic mean of stresses will give the radial stress $\sigma _{r}$,

$\sigma _{r}=\frac{\sigma r_{i}+\sigma r _{e}}{2}$

$\sigma _{r}=-\frac{p_{i}+p_{e}}{2}$       Eq(1)

Where$p_{i}=0$, $p_{e}=0$,

Eq(1) gives

$\sigma _{r}=-\frac{p_{i}}{2}$

$\sigma _{r}=-\frac{p_{e}}{2}$

Radial Stresses act in the radial direction of a pressure vessel and just like the tangential or hoop stress it is also

responsible for diametrical deformation of a vessel.

In general Radial Stress is compressive in nature acting between the inner and outer surface of a cylindrical vessel and as per the convention of continuum mechanics, Radial stresses are negative.

## Is Radial Stress a Principal Stress?

Yes, Radial Stress is a Principal Stress.

Radial Stress is the stress towards or away from the principal axis of a pressure vessel. In the case of a thick cylinder, the stress distribution is across the thickness of the cylinder. The maximum $\sigma _{r}$ is obtained at the inner radius of the cylinder.

## Is Radial Stress Shear Stress?

Shear Stress($\tau$) is the component of stress which is coplanar with the cross section of a material.

Due to the shear expansion of a structure Radial stresses are developed which act on the normal direction of the interface. As a result the shear stress strength of the interface is greatly enhanced which in turn greatly improve the ultimate bearing capacity of anchorage structure.

Shear stress classified as Direct shear stress and torsional shear stress. Starting from 1960s, anchorage structure in the form of temporary and permanent reinforcement has been used frequently in Civil as well as Mining Engineerings

## Is Radial Stress Normal Stress?

A Radial Stress is a normal stress coplanar to symmetry axis but acting perpendicularly to the symmetry axis.

Normal Stresses are always act in a direction normal to the face of the crystal structure of a material, they exist both in compressive and tensile nature. Radial Stresses are a type of normal stress and compressive in nature.

Conclusion:

To wrap up the article we can state that Stresses acting in the radial direction of a pressure vessel ($\sigma _{r}$ have great importance just like the two other principal stresses( Hoop and Axial) especially in designing a thick-walled cylindrical or spherical pressure vessel.

Sangeeta Das

I am Sangeeta Das. I have completed my Masters in Mechanical Engineering with specialization in I.C Engine and Automobiles. I have around ten years of experience encompassing industry and academia. My area of interest includes I.C. Engines, Aerodynamics and Fluid Mechanics. You can reach me at https://www.linkedin.com/in/sangeeta-das-57233a203/