1. Reference problem#
1.1. Description#
The nonlinear rheological junction model JONC_ENDO_PLAS is evaluated by the response it provides in terms of the rotation—moment curve, in the local coordinate system: “MZ” obtained as a function of the rotations” DRZ “, regardless of the orientation of the local coordinate system and the other directions that remain linear elastic.
1.2. Modelizations#
The models tested are all with DIS_TR elements on SEG2 meshes. The stiffness characteristics of the discrete elements, which are used for the prediction of the nonlinear algorithm, are therefore of the type: K_ TR_D_L.
Note: The units of the parameters should agree with the unit of efforts and moments, the unit of lengths, and the unit of time of the problem [R5.03.17]. For all models the units are homogeneous to [N], [m], [s].
1.2.1. Modeling A#
This modeling makes it possible to test the non-linear static cyclic behavior of the behavior relationship JONC_ENDO_PLAS.
The elements 1 to 8 are each identified by a group of elements (L1 to L8) and the nodes 1 to 16 by groups of nodes (P1 to P16).
Figure 1.2.1-a : Mesh
The orientation of the local coordinates of the elements is aligned with the local coordinate system for the elements 1, 2, 5, and 6. This coordinate system is rotated by an angle of 30° around the X axis for elements 3, 4, 7 and 8.
1.3. Properties of the material and characteristics of the elements#
1.3.1. Modeling A#
The parameters of the off-plane flexural behavior model JONC_ENDO_PLAS are obtained from the analysis of the reinforced concrete section: geometry and characteristics of the materials, cf. [R5.03.17], and they are entered via the command DEFI_MATERIAU [U4.43.01].
Table 1.3.1-a : parameters of the model JONC_ENDO_PLAS , modeling A.
parameter |
KE
|
KP |
KDP |
KDM |
RDP |
RDM |
MYP |
MYM |
Elastic stiffness |
Pente plastique
|
Tangent stiffness |
Tangent stiffness |
Seuil d’endommagement
|
Damage threshold |
Plastic threshold |
Plastic threshold |
|
worthiness |
1.E6 |
5.E4 |
2.E5 |
1.E5 |
1.E-3 |
-1.5E-3 |
2.E3 |
-2.5E3 |
These parameter values were chosen to facilitate the analytical calculation and the verification of the model, and do not correspond to a real reinforced concrete section.
The elastic stiffness characteristics “K_ TR_D_L “, of the element DIS_TR are in the local coordinate system of each element:
1.E8, 2.E8, 5.E8, 10.E8, 10.E8, 20.E8, 1.E6
1.4. Boundary conditions and loads#
Since the discrete element is a SEG2, one of the nodes is blocked, on the other a generalized displacement condition is imposed. Nodes 1, 3, 5, 7, 9, 9, 9, 9, 9, 11, 11, 13 and 15 are embedded (zero displacement and rotation are required). Rotating or rotating and moving loads are imposed on nodes 2, 4, 6, 6, 8, 10, 12, 14, and 16
These loads are defined as a function of time, over the time interval \(t\in [:ref:`0s,24s <0s,24s>\)] `. Four load functions are defined, functions A, B, C, and D.
Table 1.4-a : **** load functions as a function of time: function A.**
jiffy |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
worthiness |
0.001 |
-0.001 |
0.003 |
-0.003 |
0.005 |
-0.005 |
|||||||
jiffy |
13 |
14 |
15 |
16 |
17 |
18 |
19 |
20 |
21 |
22 |
23 |
24 |
|
worthiness |
0.011 |
-0.011 |
0.015 |
-0.015 |
0.020 |
-0.020 |
|||||||
Table 1.4-b **: ** loading functions as a function of time: function B.
jiffy |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
worthiness |
-0.001 |
0.001 |
-0.003 |
0.003 |
-0.005 |
0.005 |
|||||||
jiffy |
13 |
14 |
15 |
16 |
17 |
18 |
19 |
20 |
21 |
22 |
23 |
24 |
|
worthiness |
-0.011 |
0.011 |
-0.015 |
0.015 |
-0.020 |
0.020 |
|||||||
Table 1.4-c **: ** load functions as a function of time: function C.**
instant |
0 |
12 |
24 |
value |
0.02 |
Table 1.4-d **: ** load functions as a function of time: function D.**
instant |
0 |
12 |
24 |
value |
0.01 |
These loads are taxed as follows:
A load corresponding to function A is imposed in rotation around the global Z axis (DRZ) for elements 1 and 3.
A load corresponding to the B function imposed in rotation around the global Z axis (DRZ) for elements 2 and 4.
A load corresponding to the C function is imposed in rotation around the global Z axis (DRZ) for elements 5 and 7.
A load corresponding to the C function is imposed in rotation around the global Z axis (DRZ), accompanied by a load corresponding to the D function in translation (X, Y, and Z directions) and in rotation around the global X and Y axes (DRX and DRY) and in rotation around the global X and Y axes (and) for elements 6 and 8.
Figure 1.4-a : **Rotational loading imposed around the axis:math:`Z`**global. **