Vibrational Stress Analysis – Liquid

Introduction

This application calculates the stress induced on a pipeline from vibrational construction equipment, such as impact pile drivers, hydraulic breakers, or jackhammers. The vibrations caused by operating this equipment stresses the pipeline, potentially causing integrity issues. This calculator allows you the user to make accurate estimates for the vibrational stress added to the pipeline.

Notably, the calculator does not calculate ANY of the loading stresses (both from the soil, and from the equipment itself). Further analysis using one of our crossing calculators, such as CEPA Grid Load, may be necessary.

Peak Particle Velocity

The peak particle velocity (PPV) caused by the vibrational equipment can induce stress on the pipe. The method to calculate PPV differs depending on the equipment used.

Impact Pile Driver

For an impact pile driver:

\text{PPV}_{\text{Impact Pile Driver}} = \text{PPV}_\text{ref} \left(\frac{25}{D}\right)^n \times \sqrt{\frac{E_\text{equip}} {E_\text{Ref}} }

text{PPV}{text{Impact Pile Driver}} = text{PPV}text{ref} left(frac{25}{D}right)^n times sqrt{frac{E_text{equip}} {E_text{Ref}} }

Where:
PPV_{impact pile driver} − Peak particle velocity at the pipe cause by the pile driver (in/sec)
PPV_ref = 0.65 in/sec for a reference pile driver at 25 feet.
𝐷 − Pipe burial depth (ft)
n – Soil attenuation coefficient
E_ref = 36,0000 ft – lb, rated energy of reference pile driver.
E_equip − Rated energy of impact pile driver (ft – lb)

Hydraulic Breaker

For a hydraulic breaker:

\text{PPV}_{\text{Hydraulic Breaker}} = \text{PPV}_\text{ref} \left(\frac{25}{D}\right)^n \times \sqrt{\frac{E_\text{equip}} {E_\text{Ref}} }

text{PPV}{text{Impact Pile Driver}} = text{PPV}text{ref} left(frac{25}{D}right)^n times sqrt{frac{E_text{equip}} {E_text{Ref}} }

Where:
PPV_{hydraulic breaker} − Peak particle velocity at the pipe cause by the pile driver (in/sec)
PPV_ref = 0.24 in/sec for the reference hydraulic breaker at 25 feet.
𝐷 − Pipe burial depth (ft)
n – Soil attenuation coefficient
E_ref = 5000 (ft – lb), rated energy of the reference hydraulic breaker
E_equip − Rated energy of hydraulic breaker(ft – lb)

General Equipment

For vibratory rollers, large bulldozers, caisson drilling, jackhammers, and crack-and-seat Operations:

\text{PPV}_{\text{Equipment}} = \text{PPV}_\text{ref} \left(\frac{25}{D}\right)^n

text{PPV}{text{Equipment}} = text{PPV}text{ref} left(frac{25}{D}right)^n

Where:
PPV_Equipment − Peak particle velocity at the pipe cause by the pile driver (in/sec)
𝐷 − Pipe burial depth (ft)
n – Soil attenuation coefficient
PPV_ref = Reference PPV at 25 feet for the given equipment, specified in the table below.

Soil Attenuation Coefficient

The soil attenuation coefficient used for the PPV calculations above depends on the soil type near the equipment:

Stress Analysis

Using the PPV, we can calculate the total hoop and longitudinal stress on the pipeline, ignoring stress caused by loading.

Vibrational Stress

The vibrational stress on the pipeline is given by:

\sigma_{\text{vibration}} = 365.6 \cdot \text{PPV} + 446

sigma_{text{vibration}} = 365.6 cdot text{PPV} + 446

Where:
σ_vibration – Vibration-induced stress (psi)
PPV – Peak particle velocity (in/sec)

Hoop Stress

We calculate the individual components of hoop stress, and add them together to get a total hoop stress.

Barlow Stress

The Barlow stress, or the hoop stress caused by internal pressure, is given by:

\sigma_{H,\text{Barlow}} = \frac{PD}{2t}

sigma_{text{Barlow}} = frac{PD}{2t}

Where:
σ_H,Barlow − Hoop stress from internal pressure (psi)
𝑃 − Pipe internal pressure (psig)
𝐷 − Pipe outside diameter (in)
𝑡 − Pipe wall thickness (in)

Total Hoop Stress

The total hoop stress applied to the pipe is given below.

\sigma_{H} = \sigma_\text{vibration} + \sigma_{H,\text{Barlow}}

sigma_{text{hoop}} = sigma_text{vibration} + sigma_text{Barlow}

where:
σ_H − Total hoop stress on the pipeline (psi)

The maximum allowable total hoop stress is given by:

\sigma_{H,\text{max}} = \text{SMYS} \times F \cdot E \cdot T

sigma_{text{hoop,max}} = text{SMYS} times FET

where:
σ_H,max − Maximum allowable hoop stress on the pipe (psi)
SMYS – Specified minimum yield strength of the pipe (psi)
F – Design Factor
E – Longitudinal Joint Factor
T – Temperature Derating Factor

Longitudinal Stress

The total longitudinal stress on the pipe is a combination of stress from the internal pressure, thermal expansion, and vibration.

Internal Pressure

The longitudinal stress caused by the hoop stress is given by:

\sigma_{L, \text{Barlow}}=0.3 \times \sigma_{H,\text{Barlow}}

sigma_{L, text{Barlow}}=0.3times sigma_text{Barlow}

Where:
σ_L,Barlow − Longitudinal stress caused by the internal pressure (psi)

Thermal expansion

The longitudinal stress caused by thermal expansion is given by:

\sigma_{L, \text{Thermal}}=EC(T_1 - T_2)

sigma_{L, text{Thermal}}=EC(T_1 – T_2)

Where:
σ_L,Thermal − Longitudinal stress caused thermal expansion/contraction (psi)
𝐸 − Modulus of Elasticity (psi)
𝐶 − Coefficient of Linear Expansion (in/℉)
𝑇₁ − Installation temperature (℉)
𝑇₂ − Operation temperature (℉)

Total Longitudinal Stress

The total longitudinal stress is given as:

\sigma_L = \sigma_{L, \text{Barlow}} + \sigma_{L, \text{Thermal}} + \sigma_\text{vibration}

sigma_L = sigma_{L, text{Barlow}} + sigma_{L, text{Thermal}} + sigma_text{vibration}

Where:
σ_L− Total longitudinal stress (psi)

The maximum allowable total longitudinal stress is given by:

\sigma_{L,\text{max}}  = \frac{ \text{max allowable stress (\%)}}{100} \times \text{SMYS}

Where:
σ_{L, max} − Maximum allowable longitudinal stress on the pipeline (psi)
max allowable stress (%) – The user specified maximum allowable combined stress (typically 90%)
SMYS – Specified minimum yield strength (psi)

Equivalent Stress

The equivalent stress on the pipeline is estimated via von Mises equation:

\sigma_{eq} = \sqrt{\sigma_H^2 + \sigma_L^2 - \sigma_H \sigma_L}

sigma_{text{eq}} = sqrt{sigma_H^2 + sigma_L^2 – sigma_H sigma_L}

Where:
σ_Eq − Total equivilant stress on the pipeline (psi)

The maximum allowable equivalent stress is given by:

\sigma_{eq,\text{max}} = \frac{ \text{max allowable stress (\%)}}{100}\times \text{SMYS}

Where:
σ_{Eq, max} − Maximum equivalent stress on the pipeline (psi)
max allowable stress (%) – The user specified maximum allowable combined stress (typically 90%)
SMYS – Specified minimum yield strength (psi)

Case Guide

Part 1: Create Case

Input Parameters

Part 2: Outputs/Reports

Results

• Calculated and Allowable Hoop Stress( psi)
• Calculated and Allowable Longitudinal Stress (psi)
• Calculated and Allowable Total Stress (psi)

References

• Robert B. Francini, William Nik Baltz – Blasting and Construction Vibrations Near Existing Pipelines: What Are Appropriate Levels, IPC2008-64325.
• California Department of Transportation – Transportation- and construction-induced vibration guidance manual.
• ASME B31.4 – Pipeline Transportation Systems for Liquids and Slurries
• API 5L, API 5LS and API 5LX – Specification of Pipe Grade
• ASTM – Various – Weld Joint Factor

FAQ