2. Benchmark solution#

2.1. Calculation method used for the reference solution#

Comparison to other numerical results obtained with Code_Aster (version 14.2 [bib1]) with a \(3D\) mesh of the elbow and straight parts, connected at the ends to straight beams (C modeling). This \(\mathrm{3D}\) mesh contains 1500 HEXA20 meshes. A modeling of the elbow in elements COQUE_3D gave results comparable to calculation \(\mathrm{3D}\) (see [§2.2]).

B modeling uses Plasticity behavior MFront instead of code_aster VMIS_ISOT_LINE behavior (3D calculation reference).

Quantities useful for seismic resistance are also calculated using the following formulas:

(2.1)#\[ \ mathit {EN} = {\ mathrm {\ epsilon}} _ {\ mathit {xx}}\]

(2.2)#\[ \ mathit {ET} =\ frac {{r} _ {\ mathit {moy}}\ times {\ mathrm {\ gamma}}} _ {\ mathit {twist}}}} {2}\]

(2.3)#\[ \ mathit {EFY} = {r} _ {\ mathit {moy}}\ times {\ mathrm {\ kappa}}} _ {y}\]

\[\]

: label: eq-4

mathit {ESTAR} =sqrt {{mathit {EN}}} ^ {2}} + {mathit {ET}} ^ {2} + {left (frac {mathrm {pi}\ timesmathit {pi}pi}timespi}timesmathit {EFY}}} {4}right)} ^ {2}} + {left (frac {mathrm {pi}timespi}timesmathit {EFZ}} {4}right)} ^ {2}}

Calculation of the second type of quantity for the earthquake:

\[\]

: label: eq-5

mathrm {lambda} =frac {etimes {R} _ {c}} {{r} _ {mathit {moy}}} ^ {2}} ^ {2}}

and

(2.4)#\[ k2=\ mathit {max}\ left (\ mathrm {1,}\ frac {\ mathrm {1.65}} {\ mathrm {\ lambda}}\ right)\]

With:

(2.5)#\[ {\ mathit {EFY}} _ {2} =\ frac {{r} _ {\ mathit {moy}}\ times {\ mathrm {\ kappa}}} _ {y}} {y}} {k2}\]

And:

(2.6)#\[ {\ mathit {ESTAR}} _ {2} =\ sqrt {{\ mathit {EN}}} ^ {2} + {\ mathit {ET}} ^ {2} + {\ left (\ frac {\ mathrm {\ pi}\ times\ mathrm {\ pi}\ pi}\ times\ mathrm {\ gamma}}\ times\ mathit {EFY}}} {4}\ right)} ^ {2}} + {\ mathrm {\ pi}\ pi}\ pi}\ pi}\ times\ mathrm {\ gamma}\ times\ mathrm {\ gamma}\ times\ mathit {}}} {4}\ right)} ^ {2}} + {\ left (\ frac {\ mathrm {\ pi}\ times\ mathrm {\ gamma}\ times\ mathit {EFZ}} {4}\ right)}} ^ {2}}}\]

2.2. Benchmark results#

For a moment applied \(\mathrm{Mz}\) in \(D\), moving \(\mathrm{DY}\) from the same point \(D\) is equal to [bib1]:

Moment	\(\mathrm{Dy}\) point \(D\) \((m)\) ** (3D)**	\(\mathrm{Dy}\) point \(D\) \((m)\) ( COQUE_3D )

3.08670D+06	1.09257D—02	1.08875D—02	1.09257D—02
3.48715D+06	1.23431D—02
3.88759D+06	1.37775D—02	1.37381D—02
4.28804D+06	1.52557D—02
4.68848D+06	1.67908D—02
5.08892D+06	1.83836D—02
5.48937D+06	2.00903D—02
5.88981D+06	2.20209D—02
6.29026D+06	2.42545D—02
6.69070D+06	2.68829D—02
7.09115D+06	3.01030D—02

2.3. Precision on the reference results#

Because the reference solution is numerical, the precision can be evaluated from [§2.2] to \(\text{2\%}\) by comparing the 3D and COQUE_3D solutions.

2.4. Bibliographical references#

[1] J.M. PROIX, A. BEN HAJ YEDDER: « Project CACIP: study of a pipe bent under bending ». Rating EDF/DER HI-75/98/001/0

[2] C. BARATTE (SEPTEN), MN. BERTON, N. BLAY (CEA), F. LE BRETON (FRAMATOME - ANP): « Project for the new codification of seismic pipe design criteria ». Note EDF/SEPTEN E-N-ES-MS/01-01004-A.