pyflange.fatigue

Fatigue analysis pipeline for a flange bolt.

Evaluates the cyclic stress history, total cumulative fatigue damage, and expected fatigue life for a bolt in a flanged connection under external bending moment histories.

The analysis transforms the flange shell moment Markov matrix into a bolt stress Markov matrix using the segment's non-linear transfer functions, accounting for axial force, bolt bending, bolt diameter effects, and dynamic stress multiplication.

Bases: DoubleSlopeFatigueCurve

Bolt fatigue S-N curve according to IEC 61400-6 AMD1.

Constructs a design bilinear S-N curve for threaded bolts with logarithmic slopes $m_1 = 3$ (for $N \le 2 \times 10^6$) and $m_2 = 5$ (for $N > 2 \times 10^6$), with a knee point located at $N_{12} = 2 \times 10^6$ cycles.

The characteristic reference stress range $\Delta \sigma_c$ at 2 million cycles is adjusted for bolt size effects according to IEC 61400-6 AMD1:

• For $d \le 30\text{ mm}$: $$\Delta \sigma_c = \Delta S_{\text{ref}}$$
• For $30\text{ mm} < d \le 72\text{ mm}$: $$\Delta \sigma_c = \Delta S_{\text{ref}} \cdot \left(\frac{0.030}{d}\right)^{0.1}$$
• For $d > 72\text{ mm}$: $$\Delta \sigma_c = \Delta S_{\text{ref}} \cdot \left(\frac{0.030}{d}\right)^{0.1} \cdot \left(\frac{0.072}{d}\right)^{0.25}$$

The knee point stress range for the design curve is obtained by dividing the characteristic strength by the material partial safety factor:

$$\Delta S_{12} = \frac{\Delta \sigma_c}{\gamma_M}$$

import numpy as np
from pyflange.fatigue import BoltFatigueCurve, MarkovMatrix

curve = BoltFatigueCurve(0.080)       # M80 bolt, default fatigue class
N = curve.N(40e6)                     # cycles to failure at a 40 MPa range

mkvm = MarkovMatrix(range=np.array([20e6, 40e6]),
                    mean=np.array([300e6, 300e6]),
                    cycles=np.array([1e6, 1e4]),
                    duration=20)      # e.g. 20 years
D = curve.cumulated_damage(mkvm)      # Miner's sum

Bases: MultiSlopeFatigueCurve

Bilinear (double-slope) Wöhler (S-N) fatigue curve.

Implements a two-slope S-N curve commonly used in structural design standards (e.g., Eurocode 3, DNV, IEC standards). The curve transitions from slope $m_1$ in the lower-cycle / higher-stress regime ($N \le N_{12}$) to slope $m_2$ in the higher-cycle / lower-stress regime ($N > N_{12}$) at an intersection knee point $(\Delta S_{12}, N_{12})$.

Abstract base class for Wöhler (S-N) fatigue strength curves.

This class provides the interface for evaluating fatigue endurance (number of cycles to failure $N$ as a function of stress range $\Delta S$, and vice versa), as well as calculating fatigue damage and cumulative damage using the Palmgren-Miner linear cumulative damage hypothesis:

$$D = \sum_{i} \frac{n_i}{N(\Delta S_i)}$$

This class should not be instantiated directly; use one of its concrete subclasses such as SingleSlopeFatigueCurve, DoubleSlopeFatigueCurve, or BoltFatigueCurve.

A Markov matrix representation of a cyclic load history.

Encapsulates a discrete load or stress spectrum resulting from cycle counting (e.g., rainflow counting). The load history is characterized by arrays of cycle ranges, mean values, and associated cycle counts, along with an overall duration.

The physical quantity represented can be stress (Pa), force (N), bending moment (N·m), etc., provided that consistent units are maintained throughout downstream calculations.

Bases: FatigueCurve

Multi-slope composite Wöhler (S-N) fatigue curve.

Represents an S-N curve formed by combining multiple SingleSlopeFatigueCurve segments. For any given stress range or cycle count, the effective capacity is evaluated as the upper envelope (maximum endurance / maximum allowable stress range) among all constituent curve segments.

Bases: FatigueCurve

Wöhler (S-N) fatigue curve with a single logarithmic slope.

Represents a linear S-N relationship in log-log space governed by the power-law equation:

$$N \cdot \Delta S^m = a = \Delta S_{\text{ref}}^m \cdot N_{\text{ref}}$$

where $m$ is the inverse slope exponent, $\Delta S_{\text{ref}}$ is a reference stress range, and $N_{\text{ref}}$ is the corresponding cycle count at failure.