Eurocode1993 S-N curves

This document shows how to obtain the Eurocode1993 S-N curves (see EN 1993-1-9 (2005) ) from the category as given in the  Preferences  in Ashes.


Note:  the category as defined in the Eurocode is a number that corresponds to the stress that gives 100% damage at two million cycles. This value is called reference stress  and is noted 
$$\Delta\sigma_C$$

The S-N curves are defined in the Eurocode as three straight lines corresponding to three regions in a log-log plot, as shown in the figure below:


1 First region


In the first region, corresponding to 
$$N<5\cdot10^6$$
, the equation for the curve is
$$S=N^{m_1}\cdot a_1$$

In this region, 
$$m_1$$
is given as being equal to 
$$-1/3$$
.
$$a$$
can be found following the definition of the category  of the curve. In this example, we are defining the category 100 curve. This means that for a stress 
$$S_1 = 100\text{ MPa}$$
, we have 
$$N_1 = 2\cdot10^6$$
cycles. This yield
 
$$a_1 = \frac{100}{(2\cdot10^6)^{-1/3}}$$

Thus the first region of the curve corresponds to the following graph:



2 Second region

In the second region, corresponding to 
$$5\cdot 10^6
, the equation for the curve is
$$S=N^{m_2}\cdot a_2$$

In this region, 
$$m_2$$
 is given as 
$$-1/5$$
.
The lower limit of region 2 
$$N = 5\cdot10^6$$
corresponds to the upper limit of region 1, which yields
$$N_{12}^{m_2}\cdot a_2=N_{12}^{m_1}\cdot a_1$$

Solving for 
$$a_2$$
 gives
$$a_2=a_1\cdot N_{12}^{(m_1-m_2)}$$


The first and second region correspond to the following graph:




Note:  the stress range corresponding to 
$$N=5\cdot10^6$$
 cycles is called constant amplitude fatigue limit  and is noted 
$$\Delta\sigma_D$$
. By definition, 
$$\Delta\sigma_D=0.737\Delta\sigma_C$$
.
The stress range corresponding to 
$$N=10^8$$
cycles is called cut off limit  and is noted 
$$\Delta\Sigma_L$$
. By definition, 
$$\Delta\Sigma_L=0.549\Delta\sigma_D=0.405\Delta\sigma_C$$




3 Third region

This regions corresponds to 
$$N>10^8$$
 and is defined by a horizontal line such that 
$$S=\Delta\sigma_L$$
. This implies that any stress range lower than 
$$\Delta\sigma_L$$
 will not produce any damage. This region is commonly known as the endurance limit .

The figure below sums up the S-N curves as defined in the Eurocode:





Note that in practice region 3 is only considered if no stress range had an amplitude above the cut-off limit . This means that once the simulation is finished, the stress ranges should be checked and if none of them is above the cut-off limit, the fatigue life is considered infinite.
If this is not the case, if one or more of the stress ranges are above the cut-off limit, the fatigue life estimation are computed assuming that there is no region 3 and that region 2 extends to infinity .

In Ashes, only the latter case is considered. Therefore, the S-N curves will be as shown in the figure below.