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 

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 
, the equation for the curve is
$$S=N^{m_1}\cdot a_1$$

In this region, 
is given as being equal to 
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, 
 is given as 
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 
$$a_2=a_1\cdot N_{12}^{(m_1-m_2)}$$

The first and second region correspond to the following graph:

Note: the stress range corresponding to 
 cycles is called constant amplitude fatigue limit and is noted 
. By definition, 
The stress range corresponding to 
cycles is called cut off limit and is noted 
. By definition, 

3 Third region

This regions corresponds to 
 and is defined by a horizontal line such that 
. This implies that any stress range lower than 
 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.