Static & Thermal Loads

Before the engine evaluates extreme environmental events, it must establish the baseline operating state of the pipeline. This involves calculating the continuous static loads (weight and fluid) and the displacement-driven loads (thermal expansion).

2.1 Dead Weight & Fluid Mass

The engine automatically calculates the distributed mass along every element. Unlike manual structural solvers where the engineer must calculate and apply uniform distributed loads (UDLs), this engine derives the forces directly from the pipe section properties.

The total mass per unit length ($m_{total}$) is the sum of the steel, coating, and internal fluid:

$m_{total} = (A_{pipe} \rho_{steel}) + (A_{coat} \rho_{coat}) + (A_{int} \rho_{fluid}) + w_{added}$

Where:

• $A_{pipe}$, $A_{coat}$, $A_{int}$: Cross-sectional areas of the steel, coating, and internal bore.
• $\rho_{steel}$: Density of carbon steel (defaulted to $7850 \text{ kg/m}^3$).
• $\rho_{fluid}$: Density of the operating fluid (e.g., $1000 \text{ kg/m}^3$ for water, $0$ for dry gas).
• $w_{added}$: Any user-defined additional UDL.

Valve & Flange Masses

Point masses for flanges are applied directly to the nodes as downward forces. Inline rigid valves have their mass distributed evenly across their specified length.

2.2 Fixed-End Forces (FEF)

A standard 12-DOF stiffness matrix only accepts loads applied directly at the nodes. To apply our continuous gravitational UDL ($w = m_{total} \times g$) to the beam, the engine calculates the equivalent Fixed-End Forces and injects them into the global force vector.

The continuous load is rotated into the element's local coordinate system ($w_x, w_y, w_z$), and the resulting nodal shears and moments are applied to each end of the element of length $L$:

Axial / Shear Forces: $F_{x} = \frac{w_x L}{2} \quad , \quad F_{y} = \frac{w_y L}{2} \quad , \quad F_{z} = \frac{w_z L}{2}$

Bending Moments: $M_{z} = \frac{w_y L^2}{12} \quad , \quad M_{y} = \frac{w_z L^2}{12}$

(Note: The signs are inverted for the far node to maintain equilibrium).

2.3 Thermal Expansion

Pipelines expand when subjected to hot process fluids, creating immense thermal strain. The engine evaluates the maximum temperature differential ($\Delta T$) between the installation temperature and the design extremes (Hot or Cold).

The thermal force ($F_{th}$) generated by a completely restrained pipe is calculated using the material's coefficient of thermal expansion ($\alpha$) and Elastic Modulus ($E$):

$F_{th} = E A_{pipe} \alpha \Delta T$

Matrix Injection

Because thermal expansion is an internal element force, it is applied as equal and opposite axial loads to the local nodes of the element:

• Node A (Start): $-F_{th}$
• Node B (End): $+F_{th}$

When assembled into the global matrix, these forces cancel out at continuous joints, leaving only the net expansion forces pushing outward against the boundary restraints (anchors, elbows, and soil friction).

2.4 Internal Pressure

Internal pressure acts radially to create hoop stress and axially to create longitudinal stress.

While pressure elongation (the Bourdon effect) is negligible in standard rigidly-supported plant piping, the longitudinal stress generated by the pressure ($S_p$) is heavily governed by ASME B31. The engine calculates this stress and superimposes it directly during the final code-compliance pass:

$S_p = \frac{P D_o}{4 t_{eff}}$

Where $t_{eff}$ accounts for mill tolerances and corrosion allowances.