1. Presentation of the BETON_REGLE_PR law#

The law of behavior is of the « twice 1D » type in the natural deformation coordinate system (confused with the natural stress coordinate system). We only describe a 1D behavior, which is of the non-linear elastic type.

In compression, it is the law rectangular parabola defined by regulations [bib1]; the constraint is given by the following relationships:

(1.1)#\[\begin{split} \ {\ begin {array} {c}\ mathrm {\ sigma} = {\ mathrm {\ sigma}} _ {y} ^ {c}\ left [1- {\ left (1-\ frac {\ frac {\ mathrm {\ epsilon}}} {{\ mathrm {\ epsilon}}} _ {0}}\ left (1-\ frac {\ mathrm {\ epsilon}}} {\ right)} ^ {n}\ right]\ right]\ text if} 0 {\<\ mathrm {\ epsilon}\ le {\ mathrm {\ epsilon}} _ {0}\ text {(partie parabole)}\\\ mathrm {\ sigma} = {\ mathrm {\ sigma}} _ {y} ^ {c}\ text {si}\ mathrm {\ epsilon} > mathrm {\ epsilon}}} _ {0}\ text {(rectangle part)}\ end {array}\end{split}\]

In traction the law is of the triangle type [bib2]:

(1.2)#\[ \ mathrm {\ sigma} =E\ mathrm {\ epsilon}\ text {\ epsilon}\ text {si}\ mathrm {\ epsilon}\ le\ frac {{\ mathrm {\ sigma}}} _ {y} ^ {\ sigma}} _ {y} ^ {t}} _ {y} ^ {t}} {e}} {E}} {E}} {E}\ text {with}\ mathrm {\ sigma}}} _ {y} ^ {t} + {E} _ {T}\ left [\ mathrm {\ epsilon} -\ frac {{\ mathrm {\ sigma}} _ {y} ^ {t}}} {E}\ right]\]

The associated material parameters are as follows:

\(E\): Young’s modulus;

\({\sigma }_{y}^{t}\): the peak tensile stress (\({f}_{t}\));

\({E}_{T}\): the tangent module (generally between \(\frac{\mathrm{-}E}{3}\) and \(\frac{\mathrm{-}E}{10}\));

\({\sigma }_{y}^{c}\): the maximum compression stress (\({f}_{c}\));

\(n\): the exponent of the law of work hardening in compression;

\({\varepsilon }_{c}\): the deformation at which \({\sigma }_{y}^{c}\) is reached.

The stress-strain response is given in the following figure:

_images/10000200000001E20000016A9A9D54562284D4EF.png

Figure 1-a : 1D stress-strain curve

In order for the original slopes in tension and in compression to be identical, the parameters can be chosen in such a way that:

(1.3)#\[ \ frac {\ mathrm {\ partial}\ sigma} {\ mathrm {\ partial}\ epsilon} (\ epsilon\ mathrm {=} 0)\ mathrm {=}} E\ mathrm {=}\ sigma}} {=} {f} _ {f} _ {f} _ {f} _ {c}}\]