Topics covered
- Introduction
- Circumferential or Hoop Stresses
- Longitudinal Stresses
- Radial Stress
Introduction to Induced Stresses:
When the ends of the pipe ends are closed and pipe is subjected to an internal pressure ‘P’ there are various stresses that develop in the pipe. Each element of pipe are subjected to the below mentioned stresses which act in the direction as shown in the fig.1.
- Circumferential (hoop) stress sH
- Longitudinal Stress sL
- Radial Stress sR
CIRCUMFERENTIAL OR HOOP STRESS: sH
The effect of this may split the pipe into two halves as shown in fig.2. The failure of the pipe in two halves in fact is possible across any plane, which contains diameter and axis of the pipe. Elements resisting this type of failure would be subjected to stress and direction of this stress is along the circumference. Hence the above stress is called Circumferential or Hoop Stress.
If –
D = Diameter of the pipe
L = Length of the pipe
t = thickness of the pipe.
Then
Bursting force, FB | = | Pressure * Area |
= | P * D * L | |
Resisting force, F_{R} | = | Resisting metal area * Stress, sH |
Equating F_{B} & F_{R} | ||
P * D * L | = | 2t * L * sH |
∴ t | = | (P * D)/ 2 * sH |
or sH | = | (P * D) /( 2 * t) ______________________(1) |
This equation is used for calculating the thickness of pipe so as to withstand pressure ‘P’ where s H is allowable circumferential stress.
LONGITUDINAL STRESS:sR
Considering that the pipe ends are closed and pipe is subjected to an internal pressure ‘P’ the pipe may fail as shown in Fig.3. Elements resisting this type of failure would be subjected to stress and direction of this stress is parallel to the longitudinal direction of the pipe. Hence this stress is called longitudinal stress.
Then
Bursting force, FB | = | Pressure × Area |
= | P * (πD * D)/4 | |
Resisting force, FR | = | Resisting metal area x Stress, sL |
= | π D t * sL (when t is significantly small as compared to D) | |
Equating F_{B} & F_{R} | ||
P * (πD * D) /4 | = | π D t * sL |
∴ t | = | (P D)/4 * sL |
or sL | = | (P D)/(4 * t) _________________________ (2) |
NOTES:
1) On comparing equations 1 & 2, it is clear that when a pipe having diameter ‘D’ and thickness ‘t’ is subjected to an internal pressure ‘P’, the induced circumferential tress is double the induced longitudinal stress.
2) Normally, the pipe is considered as a thin wall cylinder i.e. t < D/6
3) Usually D is substituted by Do (outside diameter) in order to have higher safely margin.
RADIAL STRESS: sR
Radial stress is a stress in directions coplanar with but perpendicular to the symmetry axis.
The radial stress for a thick-walled pipe is equal and opposite to the gauge pressure on the inside surface, and zero on the outside surface.
The radial stress is always compressive.
Each element of the pipe is subjected to radial stress which acts in radial direction as shown in Fig.4 and calculated as
sR = P
wonderful explanation!
its so useful insight as it bridges theoritical and practical aspect