# S-N curves

Update (2021): Comparing S-N curves between standards

1. ISO 19902 and DNV-RP-C203 (DOI: 10.13140/RG.2.2.14995.20006)
2. BS 7608 and DNV-RP-C203 (DOI: 10.13140/RG.2.2.28416.97289)

In order to set a suitable design criteria, I am looking to compare two classes of S-N curves for a fatigue design, viz., E and F2, and I cannot find a handy plot to refer to, and it is frustrating when standards fail to include. So, I channel it to write some code to roll my own:

The basic S-N curve equation is as follows, which one may know is from Paris-Erdogan law (fracture mechanics):

$$N = k_1 \cdot S^{-m}$$

The standard does describe it in its logarithmic form, which is as follows:

$$\log N = \log k_1 - m \cdot \log S$$

and then it goes on to furnish its two sets of key components that form parts of the equation — highlighted below.1

For graphing purposes, the above can also be written as:

$$S = \left(\frac{N}{k_1}\right)^{-\frac{1}{m}}$$

For example, and I am writing this for myself since I struggle with logarithms, if

$$log_{10}k_1 = 12.18$$

then,

$$k_1 = (10)^{12.18}$$

Code for plotting hotspot stresses versus number of cycles:

#!/usr/bin/env python3
# encoding: utf-8
"""
sncurves.py -- 2016 ckunte
May 7: Initial commit.
Apr 29, 2020: Code simplified
Dec 27, 2020: basex is now base (since matplotlib V3.3)
Feb 17, 2021: A practical stress range is set for structural steel
"""
import numpy as np
import matplotlib.pyplot as plt

# a = log_10(k1)
a = [
12.18, 16.13,  # TJ
14.61, 17.01,  # B
13.23, 16.47,  # C
11.78, 15.63,  # D
11.62, 15.37,  # E
11.40, 15.00,  # F
11.23, 14.71,  # F2
11.00, 14.33,  # G
10.57, 13.62,  # W1
]
m = [3.0, 3.5, 4.0, 5.0]  # Slope
r = [1.8e6, 1.0e5, 4.68e5, 1.0e6]  # Range limit for curves

def style():
plt.rcParams["grid.linestyle"] = ":"
plt.rcParams["grid.linewidth"] = 0.5
plt.grid(True)

def sncurve(curve, r_start, r_mid, r_end, a1, m1, a2, m2, graphcolor):
# For slope 1 (m1)
n = np.arange(r_start, r_mid, 1.0e3)
s = (n / 10**a1) ** (-1 / m1)
plt.loglog(n, s, base=10, color=graphcolor, linewidth=1.0, label=curve)
# For slope 2 (m2)
n = np.arange(r_mid, r_end, 1.0e3)
s = (n / 10**a2) ** (-1 / m2)
return plt.loglog(n, s, base=10, color=graphcolor, linewidth=1.0)

def main():
# Plot all
style()
sncurve("TJ curve", 1.0e3, r[0], 1.0e9, a[0], m[0], a[1], m[3], "black")
sncurve(" B curve", 1.0e3, r[1], 1.0e9, a[2], m[2], a[3], m[3], "magenta")
sncurve(" C curve", 1.0e3, r[2], 1.0e9, a[4], m[1], a[5], m[3], "blue")
sncurve(" D curve", 1.0e3, r[3], 1.0e9, a[6], m[0], a[7], m[3], "orange")
sncurve(" E curve", 1.0e3, r[3], 1.0e9, a[8], m[0], a[9], m[3], "green")
sncurve(" F curve", 1.0e3, r[3], 1.0e9, a[10], m[0], a[11], m[3], "olive")
sncurve("F2 curve", 1.0e3, r[3], 1.0e9, a[12], m[0], a[13], m[3], "brown")
sncurve(" G curve", 1.0e3, r[3], 1.0e9, a[14], m[0], a[15], m[3], "deeppink")
sncurve("W1 curve", 1.0e3, r[3], 1.0e9, a[16], m[0], a[17], m[3], "olivedrab")
plt.legend(loc=0)
plt.xlabel("Number of cycles, N")
plt.ylabel("Hotspot stress, $\sigma$ (MPa)")
return plt.savefig("sncurves.svg")

if __name__ == "__main__":
main()

1. Playing with logarithms is fraught with error, as they are not the same as plain algebra — Leonhard Euler’s gift to the world, which reminds me I should jog my memory from high-school. There is a very nice podcast recording (mp3) on Euler’s e that also discusses the history of logarithms, which was in ancient days allowed to transform “complicated” multiplication into simple addition, which makes perfect sense in a world that had no computers, no calculators, and no slide rules.