# Eurocode1993 S-N curves

This document shows how to obtain the Eurocode193 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 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
$5\cdot 10^6<N<10^8$
, 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.

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