Generate an S-N Curve – Fatec Engineering

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This article presents two methods to determine average (50% probability of failure) constant amplitude (CA) S-N curves.

List of Symbols

Symbol	Unit	Description
N	Cycles	Life to failure
P_ü		Probability of survival
R		Stress ratio
R_m	MPa	Material ultimate strength
R_max	MPa	Strength of structural item (e.g. static strength of a riveted joint)
S_a	MPa	Stress amplitude in cycle
S_a,max,N=1	MPa	Allowable maximum stress amplitude in cycle at N=1, i.e. upper asymptote of S-N curve corrected for the mean stress
S_f	MPa	Fatigue limit
S_m	MPa	Mean stress of cycle
S_max	MPa	Maximum stress in cycle
S_min	MPa	Minimum stress in cycle
c, k		Coefficients in Basquin and Strohmeyer equation
C₁, C₂	MPa	Coefficients in Weibull equation C₁ = S_f C₂ = S_max,N=1
C₃, C₄		Coefficients in Weibull equation

General

An SN curve is a method to represent fatigue life data.
On the horizontal axis number of cycles to failure N is given on logarithmic scale. On the vertical axis the stress amplitude S_a (or sometimes the maximum stress level S_max) is given, either on logarithmic or linear scale. These stresses are net section stresses.
The S-N curve has two asymptotes, one for which S_a equals the fatigue limit S_f and one for which S_max equals the static strength R_max of the structural item (in case of SN curves for materials this will be the ultimate strength R_m).
$S_{a,max,N=1}=R_{max}-S_{m}=R_{max}\frac{1-R}{2}$ Between the asymptotes, the SN curve is often more or less linear on log-log scale. A schematic representation of an SN curve is shown in figure 1.
S-N curves are given for a specific value of S_m or R (=S_min/S_max).
There exist various mathematical expressions for SN curves. The most commonly used expression is the Basquin equation:
$S_{a}^{k}\cdot N=c\; \; for\;\; S_{a}>S_{f}\;\;and\;\;S_{a}<R_{max}-S_{m}$ Another well-known expression is the Weibull equation:
$S_{a}=C_{1}+\left ( C_{2}-C_{1} \right )\cdot e^{-\left [ \frac{logN}{C_{3}} \right ]^{C_{4}}}$ Or:
$log\left ( N \right )=C_{3}\left [ln\left ( \frac{C_{2}-C_{1}}{S_{a}-C_{1}} \right ) \right ]^{1/C_{4}}\;\;for\;\;C_{1}<S_{a}<C_{2}$ In these equations, C1 represents the lower asymptote (fatigue limit) and C2 represents the upper asymptote.
Further, the Strohmeyer equation is often used:
$\left ( S_{a}-S_{f} \right )^{k}\cdot N=c\;\;for\;\;S_{a}>S_{f}\;\;and\;\;S_{a}<R_{max}-S_{m}$ There are different approaches to perform the tests. In this article, the EN 6072 [1] and the JIS Z 2273 [2] test methods are discussed.

Figure 1: Schematic (i.e. not corresponding with a specific equation) representation of a material S-N curve for R=-1.

Test Methods

EN 6072

This method is described and specified in [1] and is commonly used. In short, the method implies that at least 10 specimens are tested, all at different constant amplitude stress levels, such that the life (number of cycles to failure) will be in the range of 10⁴ to 3∙10⁶ cycles. It is common practice to use 1 specimen for a static test to estimate the upper asymptote of the curve. This is however not necessary; the upper asymptote can be estimated from the material strength.
In general, this test method is used to generate test data for determination of the coefficients for the Weibull curve. The fitting method is presented in [3] and can be best described as a least square method applied on both S and N. The statistical method is mostly applied such that it gives all 4 coefficients of the Weibull equation. An alternative to this application is to fix the values for the upper and lower asymptote (determine them on beforehand) and to use the method to determine only to determine the shape coefficients of the curve. There are some reasons why such an alternative is preferred:
– EN 6072 makes no distinction between failures and run-outs. This makes the outcome of the analysis less accurate.
– The actual value of S_f (lower asymptote) determined using [3] depends on the chosen stress levels, especially if run-outs are used in the analysis.
Using alternatives such as [2] avoids those problems.

JIS Z 2273

This method is described and specified in [2]. In short (see also figure 2), the method implies that at least 14 specimens are tested. 8 specimens are used to determine the finite life part of the SN curve, all at different constant amplitude stress levels; and 6 specimens are used to determine the fatigue limit using the staircase method [4].
The coefficients for the finite life part of the SN curve are often determined using the least square method applied on N.
Since the fatigue limit is separately determined, it should give a more accurate estimation than the EN 6072 method.

Figure 2: Example of S-N curve according to [2].

References

[1]     EN 6072, Test Methods – Constant Amplitude Testing, 2010.
[2]     JIS Z 2273, General Rules for Fatigue testing of Metals, 1978.
[3]     Gecks, M. & Och, F., Ermittlung dynamischer Festigkeitskennlinien durch nichtlineare Regressionsanalyse, DLR, 1977.
[4] Staircase Test Method.

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