# Longitudinal Stress

## Definition - What does Longitudinal Stress mean?

Longitudinal stress is defined as the stress produced when a pipe is subjected to internal pressure. The direction of the longitudinal stress in a pipe is parallel to the longitudinal axis of its centerline axis, which means that the stress acts in the direction of the pipe's length. Longitudinal stresses are classed as normal stresses and are tensile.

Closure of the ends of thin-walled pipes and the resulting build-up of internal fluid pressure induces the development of three types of mutually perpendicular stresses. In addition to longitudinal stress, circumferential or hoop stress and radial stress also occur, although the latter is minor compared to the others.

Longitudinal stress is also known as axial stress.

L | Longitudinal Stress |

Hoop Stress | |

r | Radial Stress |

L | Cylinder Length |

d | Cylinder Diameter |

t | Cylinder Thickness |

P | Internal Pressure |

Figure 1: Three Types of Stresses Induced in a Pipe Subjected to Internal Pressure (source)

## Trenchlesspedia explains Longitudinal Stress

It is essential to understand and evaluate longitudinal stresses when designing a pipe. The build-up of stresses in a pipe when internal pressure is applied can result in its failure. In these cases, the least invasive and most efficient methods are used during repair or replacement. Trenchless operations such as sliplining, pipe bursting, fold-and-formed pipe, and cured-in-place pipe (CIPP) are useful in rehabilitating and replacing damaged pipes.

### Calculation of Longitudinal Stress

When evaluating longitudinal stresses, there are two main forces:

Bursting Force – This is the force created due to the liquid's internal pressure in a pipe, which damages the pipe through bursting.

Resisting Force – When the pipe is subjected to internal pressure, forces counteract the failure. This force is known as the resisting force.

If the bursting force exceeds the resisting force, then the pipe will be prone to failure. In calculating the longitudinal stress of a pipe, it is considered to be in equilibrium. Under equilibrium, the bursting force equates to the resisting force.

L | Longitudinal Stress |

p | Pressure |

d | Cylinder Diameter |

t | Cylinder Thickness |

Figure 2: Longitudinal Stress in a Cylinder (source)

As per Figure 2, the longitudinal stress is found as follows:

Bursting Force, FB=Pressure x Area=p×4×d2

Resisting Force, FR=Resisting Area×Longitudinal Stress= πdt ×L

FB=FR

p×4d2=πdt× L

∴Longitudinal Stress, L=pd4t

The Relationship between Longitudinal Stress and Circumferential (Hoop) Stress

Determination of stresses in a thin-walled pipe focuses on the two principal stresses that a pipe of this nature would be exposed to, longitudinal and circumferential. Circumferential stress acts along the pipe's circumference, with failure tending to split the pipe into two halves. The longitudinal stress in a pipe is smaller than the circumferential stress. The formula for circumferential stress demonstrates this.

Circumferential Stress | |

p | Pressure |

d | Cylinder Diameter |

l | Cylinder Length |

t | Cylinder Thickness |

Figure 3: Parameters for Determining Circumferential Stress in a Cylinder (source)

Bursting Force, FB=Pressure x Area=p×d×l

Resisting Force, FR=Resisting Area×Circumferential Stress= 2tl ×

FB=FR

pdL=2tl×

∴Circumferential Stress, =pd2t

Comparing both formulas shows that the circumferential stress in a pipe under equilibrium is twice that of the longitudinal stress.