Fatigue Curve Fitting for Steel Material Specimens

Running head: MECH2450 ENGINEERING COMPUTATIONS 2
MECH2450 ENGINEERING COMPUTATIONS 2
Name of the Student
Name of the University
Author Note

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Introduction:
The fatigue characteristics of a material is commonly described by its fatigue limit or
through the failure curves like the S-N curve, Wöhler curve or fatigue curve. The relation
between the amplitude of cyclic stress and the number of cycles needed for the failure. In the
horizontal axis of the S-N curve the number of cycles Nf (either in logarithmic or in linear
scale) is taken and in the vertical axis the maximum stress amplitude is displayed in either
logarithmic or in linear scale. The different S-N curves for different materials are generally
derived from different fatigue tests of the materials. In the different tests a constant amplitude
of cyclic stress is applied to a series of the specimens of the material until breakdown or
failure occurs. In a few cases the test is intentionally stopped after reaching a specific number
of cycles usually considered very large (typically Nf>10^6). Often times the fatigue curves
are given by Kt=1 which is un-notched specimens that describes the fatigue characteristics of
materials. Now, in this assignment a sample S-N data is collected for steel material specimens
from the web and then that data is fitted by two curve fitting methods. One method is a non-
linear regression and the other method is invented by Kim and Zhang for fatigue curve fitting
and two methods are compared by both numerical testing and graphical overview. Also, the
best fitted model for the collected S-N data is concluded based on the results of graphical and
numerical testing which is obtained through MATLAB coding.
Methodology:
Now, the sample collected S-N curve data as total of 14 data points in two columns.
In the first column the number of cycles to failure is given and in the second column the
stress amplitude in kPa is given.
Now, the data is loaded in MATLAB and then the number of cycles Nf is converted to 10
base log scale and then the stress amplitude is expressed as a function of log(Nf) by least

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square method (Gan et al., 2017). Now, in the Kim and Zhang method the formula for S-N
fatigue curve model is given below.
σ max=σuT ¿ ¿
N0=¿ initial cycle number corresponding to ultimate tensile strength σ uT ~ 0.5.
α , β = model parameters.
σ uTfor steel = 7000000 kPa.
σ max = maximum stress amplitude in kPa
Now, the model parameters are estimated by an approximate model of the above which is
log Δ DfT
Δ N f
≈ log α + β log σmax (1)
where, DfT = 1− σ max
σuT
= fatigue damage at tensile fatigue failure.
Δ operators are difference between ith and i+1th entries.
Now, equation (1) is fitted with least square model from which log α and β are estimated and
the parameters are computed (Burhan & Kim, 2018). Then the two models are fitted with the
given data and the accuracy of fitting is displayed by the outputs of the models.
Additionally, the confidence intervals of the slope coefficients and prediction intervals of the
mean maximum stress amplitude is compared.
Results and Discussion:
The output of the MATLAB script modelfit in which the two models are fitted are shown
below.
Output:

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logmod =
Linear regression model:
y ~ 1 + x1
Estimated Coefficients:
Estimate SE tStat pValue
___________ __________ _______ __________
(Intercept) 2.5103e+06 4.3667e+05 5.7488 9.1812e-05
x1 -3.7431e+05 88174 -4.2451 0.0011371
Number of observations: 14, Error degrees of freedom: 12
Root Mean Squared Error: 8.1e+05
R-squared: 0.6, Adjusted R-Squared 0.567
F-statistic vs. constant model: 18, p-value = 0.00114