Pipeline Structural Integrity

Pipeline Stress (Longitudinal) Analysis

Evaluate combined longitudinal and hoop stresses in gas transmission and liquid pipelines per ASME B31.8 and B31.4 using von Mises and Tresca failure criteria.

Launch Calculator Combined Stress Theory

B31.8 Combined Limit

0.90 × SMYS × T

Maximum combined equivalent stress

Von Mises

√(SH²+SL²−SHSL)

Octahedral shear stress criterion

Poisson Effect

ν = 0.3

Steel Poisson's ratio for axial stress

Contents

Use this guide when:

  • Evaluating combined stress in buried pipelines
  • Analyzing above-ground pipeline spans
  • Performing ASME B31.8 code compliance checks
  • Assessing thermal stress in restrained pipe

1. Overview

Pipeline stress analysis evaluates the structural adequacy of a pipe by comparing the combined effect of all stress components against code-allowable limits. Unlike simple hoop stress checks, longitudinal stress analysis accounts for the biaxial stress state that exists in every pressurized pipe, plus contributions from temperature change, bending, and external loads.

Hoop Stress

SH = PD/2t

Circumferential stress from internal pressure, the primary design-governing stress in most pipelines.

Longitudinal Stress

SL = SLP + SLT + SB

Axial stress from Poisson effect, thermal change, and bending loads.

Combined Stress

Von Mises / Tresca

Biaxial stress combination using appropriate failure theory.

Code Compliance

Seq ≤ 0.9 SMYS

ASME B31.8 limits combined stress to 90% of SMYS times temperature derating.

Why longitudinal stress matters: While hoop stress alone governs pipe wall thickness selection, longitudinal stress can become significant in restrained buried pipelines where thermal and Poisson effects act together, or in above-ground spans where bending stress adds to the axial load. The combined biaxial stress state may exceed allowable limits even when hoop stress alone passes.

2. Hoop Stress

Hoop stress (circumferential stress) is the primary stress component in a pressurized pipe. It acts perpendicular to the pipe axis and is the governing stress for wall thickness design.

Barlow's Formula (Hoop Stress): S_H = P × D / (2 × t) Where: S_H = Hoop stress (psi) P = Internal pressure (psig) D = Outside diameter (inches) t = Nominal wall thickness (inches)

ASME B31.8 Hoop Stress Limit

The maximum allowable hoop stress depends on the class location, which reflects population density near the pipeline. Higher population areas require greater safety margins.

Maximum Allowable Hoop Stress: S_H ≤ F × E × T × SMYS Where: F = Design factor (class location) E = Longitudinal joint factor (weld type) T = Temperature derating factor SMYS = Specified Minimum Yield Strength
Class Location Design Factor (F) Max %SMYS Description
Class 1 0.72 72% Rural areas with ≤ 10 buildings per mile
Class 2 0.60 60% Semi-rural, 11–46 buildings per mile
Class 3 0.50 50% Suburban, ≥ 46 buildings per mile
Class 4 0.40 40% High-consequence areas, multi-story buildings

Example: Hoop Stress Calculation

Given: 20" OD, 0.500" wall, X52 grade (SMYS = 52,000 psi) Design pressure: 1,000 psig, Class 1 Hoop stress: S_H = 1,000 × 20.000 / (2 × 0.500) = 20,000 psi Allowable: S_allow = 0.72 × 1.0 × 1.0 × 52,000 = 37,440 psi %SMYS = 20,000 / 52,000 = 38.5% Result: PASS (20,000 < 37,440 psi)

3. Longitudinal Stress Components

The total longitudinal stress in a restrained pipeline is the algebraic sum of three components: Poisson stress from internal pressure, thermal stress from temperature change, and bending stress from weight or external loads.

3.1 Poisson Longitudinal Stress (SLP)

When a pressurized pipe is restrained axially (by soil friction in a buried pipe), internal pressure creates not only hoop stress but also an axial tensile stress through the Poisson effect. As the pipe expands radially under pressure, it tries to contract axially, but soil restraint prevents this contraction, generating tensile longitudinal stress.

Poisson Longitudinal Stress: S_LP = ν × S_H Where: ν = Poisson's ratio = 0.3 for steel S_H = Hoop stress (psi) Example (from above): S_LP = 0.3 × 20,000 = 6,000 psi (tensile)

3.2 Thermal Longitudinal Stress (SLT)

When a restrained pipeline operates at a temperature different from installation temperature, the pipe attempts to expand or contract but is prevented by soil friction. This generates longitudinal stress proportional to the temperature difference.

Thermal Longitudinal Stress: S_LT = −E × α × ΔT Where: E = Modulus of elasticity (psi) α = Coefficient of thermal expansion (in/in/°F) ΔT = T_operating − T_installation (°F) For carbon steel: E = 29 × 10&sup6; psi (at ambient) α = 6.33 × 10²&sup6; in/in/°F S_LT per °F = 29e6 × 6.33e-6 = 183.6 psi/°F Example (ΔT = 50°F): S_LT = −29e6 × 6.33e-6 × 50 = −9,179 psi (compressive)
Sign convention: When operating temperature exceeds installation temperature, thermal stress is compressive (negative) because the restrained pipe cannot expand. When operating temperature is below installation temperature, thermal stress is tensile (positive) because the pipe cannot contract.

3.3 Bending Stress (SB)

Bending stress occurs when the pipeline spans between supports without continuous soil support. Common scenarios include above-ground pipe racks, river crossings, wash-out areas, and sections with insufficient soil support due to erosion or settlement.

Bending Stress (simply-supported beam): S_B = M / Z Where: M = Maximum bending moment (in-lbs) Z = Section modulus = I / (D/2) (in³) I = Moment of inertia = π/64 × (D&sup4; − d&sup4;) (in&sup4;) For uniformly loaded beam: M = w × L² / 8 Where: w = Distributed load (lb/in) = total weight per foot / 12 L = Unsupported span length (inches)

Total Longitudinal Stress

Total Longitudinal Stress (algebraic sum): S_L = S_LP + S_LT + S_B Where: S_LP = Poisson stress (always tensile for positive pressure) S_LT = Thermal stress (compressive when T_oper > T_install) S_B = Bending stress (tensile on bottom fiber) Example: S_LP = +6,000 psi S_LT = −9,179 psi S_B = +3,500 psi S_L = 6,000 − 9,179 + 3,500 = +321 psi

4. Combined Stress Criteria

The pipe wall experiences a biaxial stress state with hoop stress (circumferential) and longitudinal stress (axial) acting simultaneously. The appropriate failure theory must be applied to determine whether this combined stress state is within acceptable limits.

4.1 Von Mises Criterion (Octahedral Shear Stress)

ASME B31.8 specifies the von Mises criterion for evaluating combined stress in gas transmission pipelines. This theory is based on the distortion energy concept and predicts that yielding occurs when the octahedral shear stress reaches a critical value.

Von Mises Combined Stress (biaxial, no shear): S_eq = √(S_H² + S_L² − S_H × S_L) Where: S_eq = Equivalent (von Mises) stress (psi) S_H = Hoop stress (psi) S_L = Total longitudinal stress (psi) This is derived from the general von Mises equation: σ_eq = √(σ_1² + σ_2² − σ_1σ_2 + 3τ²) For pipe with no torsion (τ = 0), σ_1 = S_H, σ_2 = S_L.

4.2 Tresca Criterion (Maximum Shear Stress)

The Tresca criterion is more conservative than von Mises and is sometimes used as an alternative check. It predicts failure when the maximum shear stress reaches half the yield strength.

Tresca Combined Stress: S_eq = max(|S_H − S_L|, |S_H|, |S_L|) The Tresca criterion is approximately 15% more conservative than von Mises for typical biaxial pipe stress states. For a pipe where S_H >> S_L: Tresca ≈ S_H Von Mises ≈ S_H × √(1 − S_L/S_H + (S_L/S_H)²)

4.3 Comparison for Typical Pipeline

Scenario SH (psi) SL (psi) Von Mises (psi) Tresca (psi)
Pressure only 20,000 6,000 17,776 20,000
Pressure + thermal (ΔT=50°F) 20,000 −3,179 21,765 23,179
Pressure + thermal + bending 20,000 321 19,839 20,000
Large thermal (ΔT=150°F) 20,000 −21,540 35,956 41,540
When S_L is compressive: Note that when longitudinal stress is compressive (negative, as with large thermal loads), the von Mises combined stress can exceed hoop stress alone. This is the critical scenario where combined stress analysis reveals risks that a simple hoop stress check would miss.

5. Code Compliance

ASME B31.8 provides specific limits for combined stress that are separate from the hoop stress design factor limits.

B31.8 Section 833 Requirements

ASME B31.8 §833.4 Combined Stress Limit: S_eq ≤ k × SMYS × T Where: k = 0.90 for combined stress evaluation SMYS = Specified Minimum Yield Strength (psi) T = Temperature derating factor For X52 pipe at ≤ 250°F: Allowable = 0.90 × 52,000 × 1.0 = 46,800 psi Note: The combined stress factor k = 0.90 applies regardless of class location. The class-location design factor (F = 0.72, 0.60, 0.50, or 0.40) applies only to hoop stress design.

Temperature Derating Factors

Temperature (°F) Derating Factor (T)
≤ 2501.000
3000.967
3500.933
4000.900
4500.867

Utilization Ratio

Utilization Ratio: U = S_eq / (k × SMYS × T) Interpretation: U < 0.50 Very conservative design U = 0.50–0.80 Typical design range U = 0.80–1.00 Near limit, verify assumptions U > 1.00 EXCEEDS ALLOWABLE — redesign required

6. Restrained vs. Unrestrained Pipe

The longitudinal stress state depends fundamentally on whether the pipe is axially restrained. This distinction changes which stress components are present and how they interact.

Restrained Pipe (Buried Pipeline)

In a buried pipeline, soil friction prevents the pipe from moving axially. This restraint means that all thermal expansion and Poisson contraction must be absorbed as stress rather than displacement. The longitudinal stress equation for fully restrained pipe includes both Poisson and thermal components.

Restrained Pipe Longitudinal Stress: S_L = ν × S_H − E × α × ΔT + S_B All three components are present: ν×S_H: Poisson effect (always present with pressure) E×α×ΔT: Thermal restraint stress S_B: Bending from settlement or buoyancy Soil friction develops over the "virtual anchor length": L_a = F_thermal / (f_s × π × D) Where f_s = soil friction per unit area Typically 30–100 ft from a free end before full restraint

Unrestrained Pipe (Above-Ground)

An unrestrained pipe on slide supports or rollers is free to expand axially. Thermal stress does not develop because the pipe can accommodate expansion through displacement. However, the pressure end-cap force creates longitudinal tension.

Unrestrained Pipe Longitudinal Stress: S_L = S_H / 2 + S_B Where: S_H/2 = End-cap pressure stress = P×D/(4t) S_B = Bending stress from weight on spans Note: S_H/2 = (P × A_bore) / A_metal for thin wall = P × D / (4 × t) ≈ S_H / 2 No thermal stress term — pipe expands freely.

Comparison Summary

Parameter Restrained (Buried) Unrestrained (Above-Ground)
Pressure longitudinal ν × S_H (Poisson) S_H / 2 (end cap)
Thermal stress −EαΔT (significant) Zero (free to expand)
Bending stress Settlement/buoyancy only Span weight bending
Critical scenario Large ΔT + high pressure Long spans + high pressure

7. Practical Applications

Pipeline Crossings

Road and river crossings often involve unsupported spans or directional changes that create significant bending stress. The combination of high hoop stress from maximum operating pressure with bending stress from the unsupported weight can push combined stress near code limits.

Thermal Cycling Scenarios

Gas pipelines that undergo frequent pressure cycling also experience temperature changes as gas heats during compression and cools during expansion. A pipeline installed at 70 degrees F that operates at 140 degrees F during compression and drops to 30 degrees F during decompression experiences a total temperature range of 110 degrees F, creating alternating tensile and compressive thermal stresses.

Design Strategies for High Combined Stress

Strategy Effect on Combined Stress Practical Limitation
Increase wall thickness Reduces hoop stress directly Increased material cost and weight
Higher grade pipe (e.g., X65 vs X52) Increases allowable stress Material cost, weldability considerations
Reduce span length Reduces bending stress (L² effect) Requires additional pipe supports
Install at mid-range temperature Reduces peak thermal stress May require pre-heating during construction
Add expansion loops (above-ground) Converts restrained to unrestrained Space requirements, additional fittings

Common Pipe Grades and SMYS Values

API 5L Grade SMYS (psi) SMTS (psi) Typical Application
B (Grade B)35,00060,000Low-pressure gathering
X4242,00060,000Distribution systems
X5252,00066,000Transmission pipelines
X6060,00075,000High-pressure transmission
X6565,00077,000Offshore and high-pressure
X7070,00082,000Large-diameter transmission
X8080,00090,000Ultra-high-pressure projects
Critical takeaway: A pipeline that passes hoop stress checks can still fail combined stress limits when thermal and bending stresses are considered. Always perform combined stress analysis per B31.8 section 833 for restrained buried pipelines and for above-ground sections with significant unsupported spans.

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