Pipeline Mechanical Design - Mathematical Methods Reference

Complete mathematical reference for pipeline mechanical design calculations in NeqSim.

Table of Contents

Constants and Parameters

Physical Constants

Symbol Value Unit Description
$g$ 9.81 m/s² Gravitational acceleration
$\rho_{sw}$ 1025 kg/m³ Seawater density
$\rho_{steel}$ 7850 kg/m³ Carbon steel density
$\rho_{conc}$ 3040 kg/m³ Concrete coating density
$E$ 207,000 MPa Young’s modulus (steel)
$\nu$ 0.3 - Poisson’s ratio
$\alpha$ 11.7×10⁻⁶ 1/K Thermal expansion coefficient

API 5L Material Grades

Grade $S_y$ (MPa) $S_u$ (MPa) Grade $S_y$ (MPa) $S_u$ (MPa)
A25 172 310 X60 414 517
B 241 414 X65 448 531
X42 290 414 X70 483 565
X52 359 455 X80 552 621

Design Factors

Code Factor Value Condition
ASME B31.8 $F$ 0.72 Class 1 (rural)
ASME B31.8 $F$ 0.60 Class 2 (semi-developed)
ASME B31.8 $F$ 0.50 Class 3 (developed)
ASME B31.8 $F$ 0.40 Class 4 (high-density)
DNV-OS-F101 $\gamma_m$ 1.15 Material factor
DNV-OS-F101 $\gamma_{SC}$ 0.96 Low safety class
DNV-OS-F101 $\gamma_{SC}$ 1.04 Medium safety class
DNV-OS-F101 $\gamma_{SC}$ 1.14 High safety class

Wall Thickness Formulas

ASME B31.8 - Gas Transmission Pipelines

Minimum wall thickness (Barlow formula):

\[t_{min} = \frac{P_d \cdot D}{2 \cdot S_y \cdot F \cdot E \cdot T}\]
Symbol Description Typical Values
$P_d$ Design pressure MPa
$D$ Outside diameter m
$S_y$ SMYS 290-827 MPa
$F$ Design factor 0.40-0.72
$E$ Joint factor 1.0 (seamless)
$T$ Temperature derating 1.0 (<120°C)

Maximum Allowable Operating Pressure:

\[MAOP = \frac{2 \cdot S_y \cdot t_{nom} \cdot F \cdot E \cdot T}{D}\]

Test Pressure:

\(P_{test} = 1.25 \times MAOP\) (Class 1)

\(P_{test} = 1.40 \times MAOP\) (Class 2-4)

ASME B31.3 - Process Piping

Minimum wall thickness:

\[t_{min} = \frac{P_d \cdot D}{2 \cdot (S_a \cdot E + P_d \cdot Y)}\]
Symbol Description Value
$S_a$ Allowable stress $S_y / 3$
$Y$ Coefficient 0.4 (T ≤ 482°C)

ASME B31.4 - Liquid Pipelines

Minimum wall thickness:

\[t_{min} = \frac{P_d \cdot D}{2 \cdot S_y \cdot F \cdot E \cdot T}\]

Default design factor $F = 0.72$.

DNV-OS-F101 - Submarine Pipelines

Pressure containment wall thickness:

\[t_1 = \frac{(P_{li} - P_e) \cdot (D - t_1)}{2 \cdot f_y \cdot \alpha_U / (\gamma_m \cdot \gamma_{SC})}\]

Iterative solution required.

Design pressure:

\[P_d = P_{inc} + \Delta P_{cont}\] \[P_{inc} = \gamma_{inc} \cdot P_{mop}\]
Symbol Description
$P_{li}$ Local incidental pressure
$P_e$ External pressure
$f_y$ Yield strength
$\alpha_U$ Material strength factor (0.96)
$\gamma_{inc}$ Incidental factor (1.1)

Nominal Wall Thickness

After calculating $t_{min}$:

\[t_{nom} = \frac{t_{min} + t_{corr}}{f_{fab}}\]
Symbol Description Typical Value
$t_{corr}$ Corrosion allowance 3 mm
$f_{fab}$ Fabrication tolerance 0.875 (12.5%)

Stress Analysis Formulas

Hoop Stress (Barlow’s Equation)

\[\sigma_h = \frac{P \cdot D}{2 \cdot t}\]

or for internal diameter:

\[\sigma_h = \frac{P \cdot (D - 2t)}{2 \cdot t}\]

Longitudinal Stress - Restrained Pipe

\[\sigma_L = \nu \cdot \sigma_h - E \cdot \alpha \cdot \Delta T + \sigma_{press}\]

\(\sigma_{press} = \frac{P \cdot D}{4 \cdot t}\) (end-cap effect)

Longitudinal Stress - Unrestrained Pipe

\[\sigma_L = \frac{P \cdot D}{4 \cdot t}\]

Von Mises Equivalent Stress

General form:

\[\sigma_{vm} = \sqrt{\sigma_1^2 + \sigma_2^2 + \sigma_3^2 - \sigma_1\sigma_2 - \sigma_2\sigma_3 - \sigma_1\sigma_3 + 3(\tau_{12}^2 + \tau_{23}^2 + \tau_{13}^2)}\]

For biaxial stress state (pipeline):

\[\sigma_{vm} = \sqrt{\sigma_h^2 + \sigma_L^2 - \sigma_h \cdot \sigma_L}\]

Allowable Stress

\[\sigma_{allow} = \eta \cdot S_y\]
Code $\eta$
ASME B31.8 0.72-0.90
DNV-OS-F101 0.87

Stress Utilization

\[U = \frac{\sigma_{vm}}{\sigma_{allow}}\]

Design is safe when $U < 1.0$.

External Pressure and Buckling

External Pressure at Depth

\[P_e = \rho_{sw} \cdot g \cdot h\]

Convert to MPa: $P_e = \frac{\rho_{sw} \cdot g \cdot h}{10^6}$

Elastic Collapse Pressure (Timoshenko)

\[P_{el} = \frac{2 \cdot E}{1 - \nu^2} \cdot \left(\frac{t}{D}\right)^3\]

Plastic Collapse Pressure

\[P_p = 2 \cdot f_y \cdot \frac{t}{D}\]

Combined Collapse Pressure (DNV)

Solving the quartic equation:

\[(P_c^2 - P_{el}^2)(P_c - P_p) = P_c \cdot P_{el} \cdot P_p \cdot f_o\]

where $f_o$ is the ovality factor.

Simplified approximation:

\[P_c = \frac{P_{el} \cdot P_p}{\sqrt{P_{el}^2 + P_p^2}}\]

Propagation Buckling Pressure

\[P_{pr} = 35 \cdot f_y \cdot \left(\frac{t}{D}\right)^{2.5}\]

External Pressure Check

\[P_e \leq \frac{P_c}{\gamma_m \cdot \gamma_{SC}}\]

If $P_e > P_{pr}$, buckle arrestors required.

Weight and Buoyancy Formulas

Cross-Sectional Areas

Steel cross-section:

\[A_{steel} = \frac{\pi}{4} \left[ D^2 - (D - 2t)^2 \right] = \pi \cdot t \cdot (D - t)\]

Internal cross-section:

\[A_{int} = \frac{\pi}{4} (D - 2t)^2\]

Coating cross-section:

\[A_{coat} = \frac{\pi}{4} \left[ (D + 2t_{coat})^2 - D^2 \right]\]

Concrete cross-section:

\[A_{conc} = \frac{\pi}{4} \left[ (D + 2t_{coat} + 2t_{conc})^2 - (D + 2t_{coat})^2 \right]\]

Weights per Unit Length

Steel weight:

\[w_{steel} = \rho_{steel} \cdot A_{steel}\]

Coating weight:

\[w_{coat} = \rho_{coat} \cdot A_{coat}\]

Concrete weight:

\[w_{conc} = \rho_{conc} \cdot A_{conc}\]

Contents weight:

\[w_{cont} = \rho_{fluid} \cdot A_{int}\]

Total dry weight:

\[w_{dry} = w_{steel} + w_{coat} + w_{conc}\]

Displaced Volume per Unit Length

\[V_{disp} = \frac{\pi}{4} \cdot D_{total}^2\]

where $D_{total} = D + 2t_{coat} + 2t_{conc}$

Submerged Weight

\[w_{sub} = w_{dry} + w_{cont} - \rho_{sw} \cdot g \cdot V_{disp}\]

Required Concrete Thickness for Stability

Solve for $t_{conc}$:

\[w_{target} = w_{dry} + w_{cont} - \rho_{sw} \cdot g \cdot V_{disp}(t_{conc})\]

Thermal Design Formulas

Thermal Expansion

Free expansion:

\[\Delta L = \alpha \cdot L \cdot \Delta T\]

Restrained thermal stress:

\[\sigma_{thermal} = E \cdot \alpha \cdot \Delta T\]

Overall Heat Transfer Coefficient

\[\frac{1}{U \cdot D_o} = \frac{1}{h_i \cdot D_i} + \sum_j \frac{\ln(D_{j+1}/D_j)}{2\pi k_j} + \frac{1}{h_o \cdot D_o}\]
Layer Thermal conductivity $k$ (W/m·K)
Steel 50
3LPE 0.4
PUF 0.025
Concrete 1.5

Temperature Profile

\[T(x) = T_{ambient} + (T_{inlet} - T_{ambient}) \cdot e^{-\frac{U \cdot \pi \cdot D \cdot x}{\dot{m} \cdot c_p}}\]

Required Insulation Thickness

Solve for $t_{ins}$:

\[T_{arrival} = T_{ambient} + (T_{inlet} - T_{ambient}) \cdot e^{-\frac{U(t_{ins}) \cdot \pi \cdot D \cdot L}{\dot{m} \cdot c_p}}\]

Structural Design Formulas

Moment of Inertia

\[I = \frac{\pi}{64} \left[ D^4 - (D - 2t)^4 \right]\]

Support Spacing (Deflection-Based)

Simply supported span:

\[L = \left( \frac{384 \cdot E \cdot I \cdot \delta_{max}}{5 \cdot w} \right)^{0.25}\]

Fixed ends:

\[L = \left( \frac{384 \cdot E \cdot I \cdot \delta_{max}}{w} \right)^{0.25}\]
Symbol Description
$\delta_{max}$ Maximum allowable deflection
$w$ Weight per unit length (N/m)

Expansion Loop Length

U-loop:

\[L_{loop} = \sqrt{\frac{3 \cdot E \cdot D \cdot \Delta L}{\sigma_{allow}}}\]

where $\Delta L = \alpha \cdot \Delta T \cdot L_{anchor}$

Z-loop: $L_{loop} = 1.2 \times$ U-loop result

Omega loop: $L_{loop} = 0.9 \times$ U-loop result

Minimum Bend Radius

Cold bend (API 5L):

\[R_{min} = 18 \cdot D\]

Hot bend:

\[R_{min} = 5 \cdot D\]

Induction bend:

\[R_{min} = 3 \cdot D\]

Natural Frequency (Simply Supported)

\[f_n = \frac{\pi}{2L^2} \sqrt{\frac{E \cdot I}{m_e}}\]

where $m_e$ = effective mass including added mass for subsea.

Vortex Shedding Frequency

\[f_s = \frac{St \cdot V}{D_{total}}\]

Strouhal number $St \approx 0.2$ for cylinders.

VIV Avoidance Criterion

\[f_n > 1.3 \cdot f_s\]

Fatigue Analysis

S-N Curve (DNV-RP-C203)

\[N = \frac{a}{S^m}\]
Curve $a$ $m$ Application
B1 $4.0 \times 10^{15}$ 4.0 Parent metal, good conditions
D $10^{11.764}$ 3.0 Welded joints
E $10^{11.610}$ 3.0 Butt welds
F $10^{11.455}$ 3.0 Fillet welds
W3 $10^{10.970}$ 3.0 Poor quality welds

Fatigue Life

\[\text{Life} = \frac{N}{\text{cycles per year}}\]

Miner’s Rule (Cumulative Damage)

\[D = \sum_i \frac{n_i}{N_i} \leq 1.0\]

where:

Cost Estimation Formulas

Material Cost

\[C_{steel} = w_{steel} \cdot L \cdot P_{steel}\] \[C_{coating} = A_{surface} \cdot L \cdot P_{coating}\]

where $A_{surface} = \pi \cdot D$ (external surface area per meter)

Fabrication Cost

\[C_{welds} = N_{welds} \cdot P_{weld}\] \[N_{joints} = \frac{L}{L_{joint}} + 1\] \[N_{field welds} = N_{joints} - \frac{L}{L_{stalk}}\]

Typical pipe joint length: $L_{joint} = 12.2$ m (40 ft)

Installation Cost

\[C_{install} = L \cdot R_{base} \cdot (1 + f_{depth})\]
Method $R_{base}$ ($/m) $f_{depth}$
Onshore 300 $50 \times$ burial depth
S-lay 800 $2 \times$ water depth / 1000
J-lay 1200 $3 \times$ water depth / 1000
Reel-lay 600 $1.5 \times$ water depth / 1000

Accessories Cost

\[C_{flanges} = N_{flanges} \cdot P_{flange}(class, size)\] \[C_{valves} = N_{valves} \cdot P_{valve}(type, size)\]

Total Project Cost

\[C_{direct} = C_{steel} + C_{coating} + C_{welds} + C_{install} + C_{accessories}\] \[C_{indirect} = C_{direct} \cdot (f_{eng} + f_{test} + f_{conting})\] \[C_{total} = C_{direct} + C_{indirect}\]
Factor Typical Value
Engineering ($f_{eng}$) 10%
Testing ($f_{test}$) 5%
Contingency ($f_{conting}$) 15%

Labor Hours

\[H_{total} = H_{welding} + H_{coating} + H_{install} + H_{testing}\] \[H_{welding} = N_{welds} \cdot h_{weld}\]

where $h_{weld}$ = hours per weld (typically 4-8 hours depending on diameter).

Unit Conversions

From To Multiply by
bar MPa 0.1
psi MPa 0.006895
inch m 0.0254
ft m 0.3048
lb/ft kg/m 1.488
$/ft $/m 3.281

References

  1. ASME B31.3 - Process Piping (2022)
  2. ASME B31.4 - Pipeline Transportation Systems for Liquids and Slurries (2022)
  3. ASME B31.8 - Gas Transmission and Distribution Piping Systems (2022)
  4. DNV-OS-F101 - Submarine Pipeline Systems (2021)
  5. DNV-RP-C203 - Fatigue Design of Offshore Steel Structures (2021)
  6. API 5L - Specification for Line Pipe (2018)
  7. ISO 13623 - Petroleum and Natural Gas Industries — Pipeline Transportation Systems (2017)
  8. Timoshenko, S.P. - Theory of Elastic Stability (1961)
  9. Palmer, A.C. & King, R.A. - Subsea Pipeline Engineering (2008)