====== Tensile Stresses in a boiler tube ======
These formulas are for a tube or cylinder where **wall thickness is more than 1/20 of diameter**. For a thick wall tube, scroll to the bottom of this page.

{{ boiler:metallurgy:tube_stresses.png}}

===== Stress in Axial Direction =====
Axial direction is one that goes along the tube longitudinally.

The stress in axial direction at a point in the tube or cylinder wall can be expressed as:

σ<sub>a</sub> = (p<sub>i</sub> r<sub>i</sub><sup>2</sup> - p<sub>o</sub> r<sub>o</sub><sup>2</sup> )/(r<sub>o</sub><sup>2</sup> - r<sub>i</sub><sup>2</sup>) 

where\\

σ<sub>a</sub> = stress in axial direction (MPa, psi)\\
p<sub>i</sub> = internal pressure in the tube or cylinder (MPa, psi)\\
p<sub>o</sub> = external pressure in the tube or cylinder (MPa, psi)\\
r<sub>i</sub> = internal radius of tube or cylinder (mm, in)\\
r<sub>o</sub> = external radius of tube or cylinder (mm, in)\\

===== Stress in Circumferential Direction - Hoop Stress =====
The stress in circumferential direction - hoop stress - at a point in the tube or cylinder wall can be expressed as:

σ<sub>c</sub> = [(p<sub>i</sub> r<sub>i</sub><sup>2</sup> - p<sub>o</sub> r<sub>o</sub><sup>2</sup>) / (r<sub>o</sub><sup>2</sup> - r<sub>i</sub><sup>2</sup>)] - [r<sub>i</sub><sup>2</sup> r<sub>o</sub><sup>2</sup> (p<sub>o</sub> - p<sub>i</sub>) / (r<sup>2</sup> (r<sub>o</sub><sup>2</sup> - r<sub>i</sub><sup>2</sup>))]                                    

where\\

σ<sub>c</sub> = stress in circumferential direction (MPa, psi)\\
r = radius to point in tube or cylinder wall (mm, in) (r<sub>i</sub> < r < r<sub>o</sub>)\\

maximum stress when  r = r<sub>i</sub> (inside pipe or cylinder)\\

===== Stress in Radial Direction =====
Radial direction is one going through the wall thickness, such as from outside surface to the inside surface.\\

The stress in radial direction at a point in the tube or cylinder wall can be expressed as:

σ<sub>r</sub> = [(p<sub>i</sub> r<sub>i</sub><sup>2</sup> - p<sub>o</sub> r<sub>o</sub><sup>2</sup>) / (r<sub>o</sub><sup>2</sup> - r<sub>i</sub><sup>2</sup>)] + [r<sub>i</sub><sup>2</sup> r<sub>o</sub><sup>2</sup> (p<sub>o</sub> - p<sub>i</sub>) / (r<sup>2</sup> (r<sub>o</sub><sup>2</sup> - r<sub>i</sub><sup>2</sup>))]

maximum stress when  r = ro (outside pipe or cylinder)\\


===== Resultant Stress =====
Combined stress in a single point in the cylinder wall cannot be described by a single vector using vector addition. Instead stress tensors (matrixes) describing the linear connection between two physical vectors quantities can be used.

Reference: [[https://www.engineeringtoolbox.com/stress-thick-walled-tube-d_949.html|Link]]
====== Hoop Stresses in a thin walled tube or Cylinder ======


When a thin-walled tube or cylinder is subjected to internal pressure a hoop and longitudinal stress are produced in the wall.

For the thin walled equations below the **wall thickness is less than 1/20 of tube or cylinder diameter**.

The hoop stress is acting circumferential and perpendicular to the axis and the radius of the cylinder wall. The hoop stress can be calculated as

σ<sub>h</sub> = p d / (2 t)                       

where

σ<sub>h</sub> = hoop stress (MPa, psi)\\
p = internal pressure in the tube or cylinder (MPa, psi)\\
d = internal diameter of tube or cylinder (mm, in)\\
t = tube or cylinder wall thickness (mm, in)\\