# 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.