Boundary Conditions & Non-Linearity

Piping systems rarely behave linearly. Pipes lift off resting supports, close gaps on limit stops, and drag across steel beams, generating immense friction.

To solve these non-linearities, the Pipe Stress FEA Engine utilizes an iterative Penalty Method coupled with a Coulomb friction convergence algorithm. This allows the solver to handle dynamic changing states without modifying the size of the global stiffness matrix.

2.1 The Penalty Method

When a node is assigned a Directional Support (e.g., a resting Y-support or an axial limit stop), the engine does not simply delete that Degree of Freedom (DOF) from the matrix. Instead, it injects an artificial "Penalty Stiffness" ($k_p = 10^{12} \text{ N/m}$) into that specific DOF.

For a support defined in a local coordinate system, the local $6 \times 6$ penalty matrix $[K_p]$ is rotated and added to the global stiffness matrix $[K]$:

$[K_{mod}] = [K] + [T]^T [K_p] [T]$

This method ensures the matrix remains numerically stable while effectively locking the pipe in that specific direction.

2.2 Gap Closures & Limit Stops

Many piping supports allow free movement until a physical gap is closed (e.g., a $+X$ limit stop with a 25mm gap). The engine solves this using an iterative status-checking loop.

1. Initial Assumption: All gapped supports are assumed to be open (inactive). The system is solved.
2. Displacement Check: The solver checks the local displacement ($\Delta_{local}$) of the node.
3. Activation: If the pipe has moved further than the allowable gap ($\Delta_{local} > \text{gap}$), the support is flagged as Active.
4. Penalty Force Injection: On the next iteration, the penalty stiffness is applied, and a corresponding penalty force ($F_p$) is injected to "push" the pipe back to the gap boundary:

$F_p = k_p \times \text{gap}$

The solver repeats this process, releasing supports if the reaction forces reverse, until the system finds a stable equilibrium where no gaps are being violated.

2.3 Coulomb Friction — Elastic Stick-Slip Model

Friction opposes thermal expansion and drastically alters the stress profile of a pipeline. The engine models classical Coulomb friction ($\mu$) on all resting and directional supports using the elastic-Coulomb stick-slip model — the same approach used by CAESAR II and by penalty friction in commercial FEA.

Once the solver determines which supports are active (bearing weight or hitting a stop), it extracts the reaction forces on the engaged axes to calculate the Normal Force ($N$):

$N = \sqrt{R_{active\_1}^2 + R_{active\_2}^2 + ...}$

The friction capacity is then $F_{max} = \mu N$.

Stick: a real tangent stiffness

While a support sticks, it contributes a lateral spring of stiffness $k_f$ (10⁸ N/m) to the system matrix, anchored at a slider datum $\vec{a}$ in the sliding plane. Because the spring is inside the matrix, the solve itself finds the friction equilibrium implicitly — this eliminates the force-chasing oscillation that plagues explicit "drag force" friction schemes. The friction force is simply the spring force:

$\vec{F}_{fric} = -k_f (\vec{u} - \vec{a}), \qquad |\vec{F}_{fric}| \le \mu N$

Slip: radial return mapping

When the spring's trial force exceeds the capacity, the support slips: the force is capped at $\mu N$ opposing the slide, and the datum is moved so the spring carries exactly $\mu N$ at the current position (the same radial return mapping used in plasticity):

$\vec{a} \leftarrow \vec{u} - \frac{\mu N}{k_f}\,\hat{F}_{trial}$

A support that lifts off ($N \to 0$) carries no friction and its datum tracks the pipe, so a later re-contact engages force-free.

Convergence criteria

Following CAESAR II practice, convergence is judged on restraint-status stability, not displacement alone. The solve is accepted when, between successive iterations:

• the engaged/disengaged state of every gap and one-way support is unchanged;
• every friction support's stick/slip mode is unchanged, its normal force varies less than 1%, and (sticking) its force is stable or (slipping) its slide direction rotates less than 15°;
• the displacement change — and the estimated remaining creep, extrapolated from the geometric convergence ratio (Richardson) — are below 0.1%.

Steadily sliding systems are accelerated by a synchronized Aitken extrapolation: when the whole system creeps geometrically with a stable ratio, every slipping datum jumps by the remaining geometric sum simultaneously. If a model still resists convergence at full load, the solver falls back to ramped load stepping (friction is path-dependent, so a gradual ramp is both more physical and more stable). Supports that genuinely alternate between engaged and disengaged beyond 5 reversals are frozen engaged with a named warning.

The whole procedure is deterministic — the same model always produces the same result.

2.4 Dynamic Snubbers

Hydraulic and mechanical snubbers act as non-linear velocity-dependent restraints.

During Sustained (Weight) and Expansion (Thermal) load cases, the pipe moves slowly. The engine treats snubbers as Free nodes, allowing the pipe to pass through them without resistance.

However, during Occasional Load Cases (Wind, Seismic) or Modal Analysis, the velocity of the pipe is assumed to be instantaneous. The engine performs a dedicated "Locked Linear" solver pass, where all snubbers are injected with the penalty stiffness ($k_p$) along their active axis, forcing them to act as rigid anchors against the dynamic event.

The locked pass also freezes the operating contact state: every fixed support direction and every one-way/gapped direction that was engaged in the operating solution stays engaged for the occasional overlay (the overlay is a small increment about the operating position, so gaps do not re-open). Friction is omitted from the overlay, matching CAESAR II's default treatment of occasional and dynamic load cases.