SSNS106 Damage of a reinforced concrete plate under requests varied with model GLRC_DM

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1 Titre : SSNS106 - Endommagement d une plaque plane sous so[...] Date : 01/03/2013 Page : 1/67 SSNS106 Damage of a reinforced concrete plate under requests varied with model GLRC_DM Summarized: This test the model validates damage of reinforced concrete GLRC_DM plate (see [R ]) for varied cyclic loadings: tension/compression, alternating bending, shears in the plane and their combinations. The analyses are made in static (STAT_NON_LINE). The results are compared with those of a modelization multi-layer, in which one represents steels of the three-dimensions functions of reinforcement by the elasticity and the concrete by the model of behavior ENDO_ISOT_BETON (see [R ]). This test can be used usefully basic as departure to fix the parameter setting of this model in the various situations of loading likely to occur in practice. To supplement, one treats two modelizations with kit between behavior GLRC_DM and elastoplastic behavior with isotropic hardening, in order to represent the appearance of residual strains as expected in reality. The modelizations L and M taking into account test thermomechanics in thermal strains equivalent in term of forces to the mechanical loadings of the modelizations A and B.

2 Titre : SSNS106 - Endommagement d une plaque plane sous so[...] Date : 01/03/2013 Page : 2/67 1 Problem of reference 1.1 Geometry l e l Appears 1.1-a: geometry of the square reinforced concrete Length plate : l= 1.0 m ; Thickness: e=0.1m. Coating of the three-dimensions functions of lower and higher reinforcement: 0.01 m. 1.2 Properties of the material All the parameters of model GLRC_DM, elastics and nonlinear, are identified from corresponding tests in the modelizations A and B, except for the Young's modulus modified in the test D (shears) in order to reduce the error in the linear field and to thus better validate the damage part. I.e. one identifies: m Effective Young's modulus of membrane E éq effective Poisson's ratio of bending m f effective Young's modulus of bending E éq effective Poisson's ratio of membrane f membrane Force of the threshold of cracking in tension N D (noted NYT GLRC ) Coefficient of damage in tension mt Bending moment of the threshold of cracking in bending ( MYF GLRC ) M D noted Coefficient of damage in bending f These parameters are calculated starting from the characteristics of the materials steel (models acier acier elastoplastic E a e, Eecr ) and concrete (via the model ENDO_ISOT_BETON E b b EIB SYT EIB, to see [ R ] ), and are checked by chock from the modelization A and thanks to the modelization B.

3 Titre : SSNS106 - Endommagement d une plaque plane sous so[...] Date : 01/03/2013 Page : 3/67 With the assessment, here the values of the characteristics of the materials and parameters GLRC_DM : modelization A and B C D and E F G H, I and J K E a, MPa e acier, MPa acier E ecr, MPa E b, MPa b D_SIGM_EPSI 0,2 E b 0,2 E b 0,2 E b 0,2 E b 0,2 E b 0,2 E b 0,2 E b EIB SYT EIB, MPa m E éq, MPa f E éq, MPa m f mt c ,02 f NYT GLRC, N / m MYF GLRC, N c Note:: it is noted that the value of EIB, cf [R ], is comparable contrary to that used in model GLRC_DM, cf [R ]. 1.3 Boundary conditions and loadings One considers various modelizations A, B, C, D and E for various types of loadings characteristic and various behaviors of the plate. In all the cases, the loadings are the displacements (rotations) imposed on edges of the plate differently for each modelization. The modelizations considered are: modelization A ( 3): traction and compression - pure tension; modelization B ( 4): alternate pure bending; modelization C ( 5): coupling of traction and compression and bending; modelization D ( 6): pure shears and distortion in the plane; modelization E ( 7): coupling of bending and shears in the plane; modelization F ( 8): pure traction and compression with kit_dll of elastoplastic behavior endommageable (GLRC_DM + Von Mises);

4 Titre : SSNS106 - Endommagement d une plaque plane sous so[...] Date : 01/03/2013 Page : 4/67 modelization G ( 9): pure traction and compression with kit_ddl of elastoplastic behavior endommageable (GLRC_DM + Von Mises); modelization H ( 1010): traction and compression pure tension, high requests; modelization I ( 1111): alternated pure bending, high requests; modelization J ( 1212): coupling traction and compression and bending, requests high; modelization K ( 13): compression - tension with ALPHA_C=100 ; 1.4 Initial conditions At the beginning displacements and the forced are worth zero everywhere.

5 Titre : SSNS106 - Endommagement d une plaque plane sous so[...] Date : 01/03/2013 Page : 5/67 2 Reference solution the reference solution is obtained by multi-layer a semi-globale modelization out of plate, where the mesh and the loading are the same ones as for the modelizations with model GLRC_DM corresponding. One models the concrete and the reinforcements separately. For each three-dimensions function of reinforcements, one considers a layer which behaves only in the longitudinal meaning of reinforcements. Thus there will be 4 layers for reinforcements. Moreover, several analytical results with the model GLRC_DM could be established. 2.1 Models On the same mesh one defines 5 models representing the reinforced concrete plate: 1 models DKT for the concrete and 4 models GRILL for reinforcements (2 following direction X, 2 following direction Y for the lower parts and higher). The rate of reinforcement for each three-dimensions function of reinforcements is of m 2 / m. The position of reinforcements (lower or higher) is defined by key word EXCENTREMENT under the key word factor GRILL in the operator AFFE_CARA_ELEM, who is worth ±0.04 m. The cracking of the concrete is modelled by constitutive law ENDO_ISOT_BETON, while it is supposed that steel always remains in the elastic domain. 2.2 Properties of the materials Concrete (ENDO_ISOT_BETON models) : Young's modulus: E b = MPa Steel : Poisson's ratio: b =0.20 Threshold of damage in simple tension SYT EIB : 3.4 MPa Lenitive slope: 0.2 E b ( γ EIB =5 ). Young's modulus: E a = MPa acier Limit of linearity e : MPa acier Slope post-elastic E écrouis : = E a =300 MPa. Loi de comportement du béton Loi de comportement de l'acier 6.0E E+08 Contrainte (N/m 2 ) 4.0E E E E E E E E E E E E E E E E E+07 Contrainte (N/m 2 ) 6.0E E E E E E E E E+07 Déformation -8.0E+08 Déformation Appear 2.2-a: rational curves of the materials (for multi-layer model).

6 Titre : SSNS106 - Endommagement d une plaque plane sous so[...] Date : 01/03/2013 Page : 6/67 3 Modelization A 3.1 Characteristic of the modelization Traction and compression - pure tension. A6 Modelization: DKTG Boundary conditions : Fixed support in A 1 ; Appear 3.1-a: mesh and boundary conditions. DX =0.0 on the edge A 1 A 3 ; DX =U 0 f t on the edge A 2 A 4 ; where U 0 = m and f t represent the amplitude of the cyclic loading according to the parameter (of pseudo-temps) t. For checking well the model, one considers two functions of loading as follows: 1.0 Fonction de chargement f1 1.0 Fonction de chargement f f(t) f(t) t(sec) -1.0 t(sec) Appear 3.1-b: Functions of loading f1 (left) and f2 (right). Note: the extreme strain is: , that is to say well in-on this side transition in plasticity of steels. Time step of integration: Characteristics of the mesh Many nodes: 9. Number of meshes: 8 TRIA3 ; 8 SEG2.

7 Titre : SSNS106 - Endommagement d une plaque plane sous so[...] Date : 01/03/2013 Page : 7/67

8 Titre : SSNS106 - Endommagement d une plaque plane sous so[...] Date : 01/03/2013 Page : 8/ Simple analytical solution Besides the computation of reference, one does a very simple analytical calculation to check the model as well as the software. With this intention, one considers a homogeneous beam EM traction and compression whose properties of the materials are the same ones as those of the homogeneous plate. One imposes the same boundary conditions on the beam. Then by considering the constitutive law considered (elastic endommageable in tension and linear elastic in compression), one calculates the force FX corresponding to DX imposed. 3.4 Values tested and results for the function of loading f1 One and the compares the average reaction forces according to Ox the axis average displacements according to the axis Oy obtained by the modelization multi-layer (reference) and by that resting on the model GLRC_DM, in term of relative differences; the tolerance is taken in absolute value on these relative differences: Standard identification of reference Value of reference Tolerance TENSION - ELASTIC PHASE t=0,25 relative Difference of the forces FX in A2 A4 AUTRE_ASTER relative Difference of displacements DY in AUTRE_ASTER A2 A4 TENSION - PHASE DAMAGE t=1,0 relative Difference of the forces FX in A2 A4 AUTRE_ASTER 0.1, relative Difference of displacements DY in AUTRE_ASTER 0.1, A2 A4 TENSION - PHASE relative t=1,5 Difference UNLOADING of the forces FX in A2 A4 relative Difference of displacements DY in A2 A4 COMPRESSION - PHASE CHARGEMENT (always elastic) t=3,0 AUTRE_ASTER 0.1, AUTRE_ASTER 0.1, relative Difference of the forces FX in A2 A4 AUTRE_ASTER relative Difference of displacements DY in A2 A4 COMPRESSION - PHASE relative t=3,5 Difference UNLOADING of the forces FX in A2 A4 relative Difference of displacements DY in A2 A4 AUTRE_ASTER 0.1, AUTRE_ASTER AUTRE_ASTER 0.1,7 10-1

9 Titre : SSNS106 - Endommagement d une plaque plane sous so[...] Date : 01/03/2013 Page : 9/67 compared Diagrams force FX (forces N xx ) displacement DX in tension/compression for the loading f1 : 6.0E+05 FX - DX 4.0E+05 charge 2.0E+05 discharge FX(N) 0.0E E+05 reloads -4.0E E+05 discharge FX_multicouche FX_GLRC FX_ANALYT -8.0E E E E E E E E E E-04 DX(m) Diagrams compared displacement DY (due to the effect Fish) according to time: 7.0E E E-05 DY - Temps DY_multicouche DY_GLRC 4.0E-05 DY(m) 3.0E E E-05 discharge refill 0.0E E E E t(sec)

10 Titre : SSNS106 - Endommagement d une plaque plane sous so[...] Date : 01/03/2013 Page : 10/67 Diagrams from the evolution of the damage of model GLRC_DM ( d 1 for the upper face and d 2 the lower face) according to time: 1.2 Variable d'endommagement d1 & d2 - Temps 1.0 d1=d charge t(sec) From the variables with damage, one also tests the dissipated energy which is written: [R ] E=k 0 d 1 d 2 with here k 0 = J /m 2 dissipated energy thus the same profile has as the curve above. 3.5 Values tested and results for the function of loading f2 One and the compares the average forces of reaction according to Ox the axis average displacements according to the axis Oy obtained by the modelization multi-layer (reference) and by that resting on the model GLRC_DM, in term of relative differences; the tolerance is taken in absolute value on these relative differences: Standard identification of reference Value of reference tolerance COMPRESSION - ELASTIC PHASE t=0,25 relative Difference of the forces N xx in A2 A4 AUTRE_ASTER relative Difference of displacements DY in A2 A4 AUTRE_ASTER COMPRESSION - PHASE DAMAGE t=1,0 relative Difference of the forces N xx in A2 A4 AUTRE_ASTER relative Difference of displacements DY in A2 A4 AUTRE_ASTER COMPRESSION - PHASE relative t=1,5 Difference UNLOADING of the forces N xx in AUTRE_ASTER A2 A4 relative Difference of displacements DY in A2 A4 AUTRE_ASTER TENSION - PHASE CHARGEMENT (always elastic) t=3,0 relative Difference of the forces N xx in A2 A4 AUTRE_ASTER 0.1, relative Difference of displacements DY in A2 A4 AUTRE_ASTER 0.1, TENSION - PHASE relative t=3,5

11 Titre : SSNS106 - Endommagement d une plaque plane sous so[...] Date : 01/03/2013 Page : 11/67 Difference UNLOADING of the forces N xx in AUTRE_ASTER 0.1, A2 A4 relative Difference of displacements DY in A2 A4 AUTRE_ASTER 0.1, compared Diagrams FX (forces N xx ) displacement DX in tension/compression for the loading f2 :: 6.0E E+05 FX - DX Then load-discharge in tension 2.0E+05 FX(N) 0.0E E E+05 Load-discharge in compression -6.0E+05 FX_multicouche FX_GLRC FX_ANALYT -8.0E E E E E E E E E E-04 DX(m) Diagrams compared displacement DY (due to the effect Fish) according to time: 4.0E E-05 DY - Temps DY_multicouche DY_GLRC DY(m) 2.0E E E E-05 Load-discharge in compression Then load-discharge in tension -2.0E E t(sec)

12 Titre : SSNS106 - Endommagement d une plaque plane sous so[...] Date : 01/03/2013 Page : 12/67 Diagrams of the evolution of the damage of model GLRC_DM ( d 1 for the upper face and d 2 the lower face) according to time: 1.2 variable d'endommagement d1 & d2 - Temps d1=d load-discharge in tension t(sec) 3.6 Remarks According to the preceding curves, one notes that the model GLRC_DM represents the total behavior of the reinforced concrete in pure traction and compression in a satisfactory way. The relative error of model GLRC_DM compared to the reference solution is acceptable. It should be noted that the difference between the model GLRC_DM and ENDO_ISOT_BETON is most important during the phase of damage: the behavior of the concrete in tension is then lenitive and one finds a slope negative in the model of multi-layer reference, in spite of reinforcement (layers of steel and layers ENDO_ISOT_BETON) whereas one of the assumptions of model GLRC_DM is not to model the softening of the reinforced concrete. Being based on the assumption of the isotropic equivalent material (see [R ]), the model GLRC_DM over-estimates the effect Fish slightly. One checks also the symmetry of the response according to the selected meaning of load in compression-tension or the reverse, according to the loading f1 or f2.

13 Titre : SSNS106 - Endommagement d une plaque plane sous so[...] Date : 01/03/2013 Page : 13/67 4 Modelization B 4.1 Characteristic of the modelization alternate Pure bending. Appear 4.1-a: mesh and boundary conditions Modelization: DKTG Boundary conditions: DRY =0.0 on the edge A 1 A 3 DRY = R 0 f t on the edge A 2 A 4, where R 0 = and f t is the amplitude of the cyclic loading according to the parameter (of pseudo-temps) t, For checking well the model, one regards three functions of loadings as: Appear 4.1-b: Positive bending, then negative bending Appears 4.1-c: Negative bending, then positive bending Appears 4.1-d: Two cycles of alternating bending

14 Titre : SSNS106 - Endommagement d une plaque plane sous so[...] Date : 01/03/2013 Page : 14/67 Notes: the extreme strain of steels is: , that is to say in-on this side transition in plasticity of steels. Increment of integration: 0.05 s. 4.2 Characteristics of the mesh Many nodes: 9. Number of meshes: 8 TRIA Simple analytical solution Besides the computation of reference, one does a very simple analytical calculation to check the model as well as the code. With this intention, a homogeneous beam is considered whose properties of the materials are the same ones as those of the homogeneous plate. One imposes the same boundary conditions on the beam. Then by considering the constitutive law considered (elastic endommageable in tension and linear elastic in compression), one calculates the force MY corresponding to DRY imposed. 4.4 Quantities tested and results for the function of loading f1 One and the compares the average moments according to Oy the axis average rotations according to the axis Ox obtained by the modelization multi-layer (reference) and by that resting on the model GLRC_DM, in term of relative differences; certain tolerances are taken in absolute value, others in relative values (from a value of NON-regression, they are then noted R ), on these relative differences: Standard POSITIVE BENDING - PHASE elastic t=0,25 relative Difference of the moments M yy in identification of reference Value of reference A2 A4 relative Difference of rotations DRX in A2 A4 NON_REGRESSION POSITIVE BENDING - PHASE DAMAGE t=1,0 relative Difference of the moments M yy in Tolerance AUTRE_ASTER AUTRE_ASTER A2 A4 relative Difference of rotations DRX in A2 A4 NON_REGRESSION POSITIVE BENDING - PHASE relative t=1,5 Difference UNLOADING of the moments M yy in AUTRE_ASTER A2 A4 relative Difference of rotations DRX in A2 A4 NON_REGRESSION NEGATIVE BENDING - PHASE elastic t =2,25 relative Difference of the moments M yy in AUTRE_ASTER A2 A4 relative Difference of rotations DRX in A2 A4 NON_REGRESSION NEGATIVE BENDING - PHASE DAMAGE t=3,0 relative Difference of the moments M yy in AUTRE_ASTER A2 A4 relative Difference of rotations DRX in A2 A4 NON_REGRESSION NEGATIVE BENDING - PHASE relative t=3,5

15 Titre : SSNS106 - Endommagement d une plaque plane sous so[...] Date : 01/03/2013 Page : 15/67 Difference UNLOADING of the moments M yy in AUTRE_ASTER A2 A4 relative Difference of rotations DRX in A2 A4 NON_REGRESSION compared Diagrams moment/rotation in cyclic bending for the loading f1 : 1.5E+04 MY - RY 1.0E+04 load MY(N.m) 5.0E E+00 discharge s -5.0E+03 discharge -1.0E+04 refill (opposite bending) MY_multicouche MY_GLRC MY_ANALYT -1.5E E E E E E E E E E E E E E-03 RY Diagrams compared rotation DRX (due to the effect Fish) according to time for the loading f1 : 8.0E E E E-04 charge RX - Temps discharge RX_multicouche RX_GLRC -5.4E-20 RX -2.0E E E E E-03 refill (opposite bending) discharge s -1.2E t(sec)

16 Titre : SSNS106 - Endommagement d une plaque plane sous so[...] Date : 01/03/2013 Page : 16/67 Diagrams from the evolution with the damage from model GLRC_DM ( d 1 for the upper face and d 2 the lower face) according to time: 4.5 Quantities tested and results for the function of loading f2 One and the compares the average moments according to Oy the axis average rotations according to the axis Ox obtained by the modelization multi-layer (reference) and by that resting on the model GLRC_DM, in term of relative differences; certain tolerances are taken in absolute value, others in relative values (from a value of NON-regression, they are then noted R ), on these relative differences: Standard POSITIVE BENDING - PHASE elastic t=0,25 relative Difference of the moments M yy in identification of reference Value of reference Tolerance AUTRE_ASTER A2 A4 relative Difference of rotations DRX in A2 A4 AUTRE_ASTER POSITIVE BENDING - PHASE DAMAGE t=1,0 relative Difference of the moments M yy in AUTRE_ASTER A2 A4 relative Difference of rotations DRX in A2 A4 AUTRE_ASTER POSITIVE BENDING - PHASE relative t=1,5 Difference UNLOADING of the moments M yy in AUTRE_ASTER A2 A4 relative Difference of rotations DRX in A2 A4 AUTRE_ASTER NEGATIVE BENDING - PHASE elastic t=2,25 relative Difference of the moments M yy in AUTRE_ASTER A2 A4 relative Difference of rotations DRX in A2 A4 NON_REGRESSION NEGATIVE BENDING - PHASE DAMAGE t=3,0 relative Difference of the moments M yy in AUTRE_ASTER A2 A4 relative Difference of rotations DRX in A2 A4 NON_REGRESSION NEGATIVE BENDING - PHASE relative t=3,

17 Titre : SSNS106 - Endommagement d une plaque plane sous so[...] Date : 01/03/2013 Page : 17/67 Difference UNLOADING of the moments M yy in AUTRE_ASTER A2 A4 relative Difference of rotations DRX in A2 A4 NON_REGRESSION One checks well that these results are identical to those obtained with the loading f1 (in opposite meaning). 4.6 Quantities tested and results for the function of loading f3 One and the compares the average moments according to Oy the axis average rotations according to the axis Ox obtained by the modelization multi-layer (reference) and by that resting on the model GLRC_DM, in term of relative differences: Standard POSITIVE BENDING - PHASE elastic t=4,25 relative Difference of the moments M yy in identification of reference Value of reference Toleranc e AUTRE_ASTER A2 A4 relative Difference of rotations DRX in A2 A4 AUTRE_ASTER POSITIVE BENDING - PHASE DAMAGE t=5,0 relative Difference of the moments M yy in AUTRE_ASTER A2 A4 relative Difference of rotations DRX in A2 A4 AUTRE_ASTER POSITIVE BENDING - PHASE relative t=1,5 Difference UNLOADING of the moments M yy in AUTRE_ASTER A2 A4 relative Difference of rotations DRX in A2 A4 AUTRE_ASTER NEGATIVE BENDING - PHASE elastic t=2,25 relative Difference of the moments M yy in AUTRE_ASTER A2 A4 relative Difference of rotations DRX in A2 A4 NON_REGRESSION NEGATIVE BENDING - PHASE DAMAGE t=3,0 relative Difference of the moments M yy in AUTRE_ASTER A2 A4 relative Difference of rotations DRX in A2 A4 NON_REGRESSION NEGATIVE BENDING - PHASE relative t=3,5 Difference UNLOADING of the moments M yy in AUTRE_ASTER A2 A4 relative Difference of rotations DRX in A2 A4 NON_REGRESSION compared Diagrams moment/rotation in cyclic bending for the loading f3 :

18 Titre : SSNS106 - Endommagement d une plaque plane sous so[...] Date : 01/03/2013 Page : 18/67 1.5E+04 MY - RY 1.0E+04 MY(N.m) 5.0E E E E+04 MY_multicouche MY_GLRC MY_ANALYT -1.5E E E E E E E E E E E E E E-03 RY Diagrams compared rotation DRX (due to the effect Fish) according to time for the loading f3 : 1.5E E-03 RX - Temps RX_multicouche RX_GLRC 5.0E-04 RX 0.0E E E E t(sec) Diagrams of the evolution of the damage of model GLRC_DM ( d 1 for the upper face and d 2 the lower face) according to time: One notes that the second cycle of alternating bending does not cause a new damage, like foreseeable.

19 Titre : SSNS106 - Endommagement d une plaque plane sous so[...] Date : 01/03/2013 Page : 19/ Remarks In consideration of the preceding curves, it is found that with a precise retiming of the parameters of model GLRC_DM, the results of model GLRC_DM are very close to those of the computation of reference. That wants to say that the model GLRC_DM can represent the behavior of reinforced concrete slabs well in alternate pure bending. It should be noted that on the level of rotation according to X (due to the effect Fish), after appearance of the damage, the difference between the two models appears clearly to the detriment of the response provided by the model GLRC_DM (based on the assumption of the isotropic equivalent material, to see [R ]).

20 Titre : SSNS106 - Endommagement d une plaque plane sous so[...] Date : 01/03/2013 Page : 20/67 5 Modelization C 5.1 Characteristic of the modelization Coupling of traction and compression and bending. Figure 5.1-a : 5.1-a Mesh and boundary conditions Modelization: DKTG Boundary conditions: coupling of Traction and compression and Bending: DX =0.0 and DRY =0.0 on the edge A 1 A 3 DX =U 0 f t and DRY = R 0 f t the edge A 2 A 4, where U 0 = , R 0 = and f t is the amplitude of the cyclic loading according to the parameter (of pseudo-temps) t. Two types of loading are considered: The same function f1 of loading for the membrane and bending (synchronous case): 1. 0 F o n c t i o n d e c h a r g e m e n t ( m e m b r a n e e t f l e x i o n ) 0. 5 f ( t ) t ( s e c ) Appear 5.1-b: function of loading f1 the function f2 of loading of membrane twice faster than that of bending (in practice the frequencies of membrane of a slab are higher than those of bending): F o n c t i o n d e c h a r g e m e n t p o u r f l e x i o n F o n c t i o n d e c h a r g e m e n t p o u r t r a c t i o n - c o m p r e s s i o n f ( t ) f ( t ) t ( s e c ) Appear 5.1-c: function of loading for the traction and compression t ( s e c ) Appears 5.1-d: function of loading for bending

21 Titre : SSNS106 - Endommagement d une plaque plane sous so[...] Date : 01/03/2013 Page : 21/67 Foot-note: the extreme strain of steels is , that is to say in-on this side transition in plasticity of steels. Increment of integration: 0.05 s.

22 Titre : SSNS106 - Endommagement d une plaque plane sous so[...] Date : 01/03/2013 Page : 22/ Characteristics of the mesh Many nodes: 9. Number of meshes: 8 TRIA3 ; 8 SEG Quantities tested and results : first loading (even function of loading for membrane and bending) One compares the average forces according to the axis Ox, average displacements according to the axis Oy, the average moments according to the axis Oy and the average rotations according to the axis Ox obtained by the modelization multi-layer S (reference) and by that resting on the model GLRC_DM, in term of relative differences; the tolerances are taken in relative value on these relative differences (NON-regression): Standard identification of reference Value of reference ELASTIC Tolerance PHASE t=0,25 relative Difference of the forces N xx in A2 A4 NON_REGRESSION relative Difference of displacements DY in A2 A4 NON_REGRESSION relative Difference of the moments M yy in A2 A4 NON_REGRESSION relative Difference of rotations DRX in A2 A4 NON_REGRESSION PHASE DAMAGE t=1,0 relative Difference of the forces N xx in A2 A4 NON_REGRESSION relative Difference of displacements DY in A2 A4 NON_REGRESSION relative Difference of the moments M yy in A2 A4 NON_REGRESSION relative Difference of rotations DRX in A2 A4 NON_REGRESSION PHASE relative t=1,5 Difference UNLOADING of the forces N xx in A2 A4 NON_REGRESSION relative Difference of displacements DY in A2 A4 NON_REGRESSION relative Difference of the moments M yy in A2 A4 NON_REGRESSION relative Difference of rotations DRX in A2 A4 NON_REGRESSION PHASE elastic t=2,25 relative Difference of the moments M yy in A2 A4 NON_REGRESSION relative Difference of rotations DRX in A2 A4 NON_REGRESSION PHASE RECHARGING t=3,0 relative Difference of the forces N xx in A2 A4 NON_REGRESSION relative Difference of displacements DY in A2 A4 NON_REGRESSION relative Difference of the moments M yy in A2 A4 NON_REGRESSION relative Difference of rotations DRX in A2 A4 NON_REGRESSION PHASE relative t=3,5 Difference UNLOADING of the forces N xx in A2 A4 NON_REGRESSION relative Difference of displacements DY in A2 A4 NON_REGRESSION relative Difference of the moments M yy in A2 A4 NON_REGRESSION relative Difference of rotations DRX in A2 A4 NON_REGRESSION

23 Titre : SSNS106 - Endommagement d une plaque plane sous so[...] Date : 01/03/2013 Page : 23/67 compared Diagrams of the force FX ( forces N xx ) according to the displacement DX imposed for the loading f1 : 4.0E+05 FX - DX 2.0E E+00 FX(N) -2.0E E E+05 FX_multicouche FX_GLRC -8.0E E E E E E E E E E-04 DX(m) Compared diagrams of the moment M yy according to the rotation DRY imposed for the loading f1 : MY - RY 1.5E E E+03 MY(N.m) 0.0E E E E+04 MY_multicouche MY_GLRC -2.0E E E E E E E E-03 RY

24 Titre : SSNS106 - Endommagement d une plaque plane sous so[...] Date : 01/03/2013 Page : 24/67 Compared diagrams displacement DY (due to the effect Fish) for the loading f1 : 8.0E E-05 DY - Temps DY_multicouche DY_GLRC 4.0E-05 DY(m) 2.0E E E E t(sec) Compared diagrams rotation DRX (due to the effect Fish) for the loading f1 : 1.5E E-03 RX - Temps RX_multicouche RX_GLRC 5.0E-04 RX 0.0E E E E t(sec) Diagrams of the evolution of the damage of model GLRC_DM ( d 1 for the upper face and d 2 the lower face) according to time:

25 Titre : SSNS106 - Endommagement d une plaque plane sous so[...] Date : 01/03/2013 Page : 25/ Quantities tested and results : second loading (membrane twice faster than bending) One compares the average forces according to the axis Ox, average displacements according to the axis Oy, the average moments according to the axis Oy and the average rotations according to the axis Ox obtained by the modelization multi-layer (reference) and by that resting on the model GLRC_DM, in term of relative differences; the tolerances are taken in relative value on these relative differences (NON-regression): Standard identification of reference Value of reference ELASTIC Tolerance PHASE t=0,2 relative Difference of the forces N xx in A2 A4 NON_REGRESSION relative Difference of displacements DY in A2 A4 NON_REGRESSION PHASE elastic t=0,25 relative Difference of the moments M yy in A2 A4 NON_REGRESSION relative Difference of rotations DRX in A2 A4 NON_REGRESSION PHASE DAMAGE t=0,5 relative Difference of the forces N xx in A2 A4 NON_REGRESSION relative Difference of displacements DY in A2 A4 NON_REGRESSION PHASE DAMAGE t=1,0 relative Difference of the moments M yy in A2 A4 NON_REGRESSION relative Difference of rotations DRX in A2 A4 NON_REGRESSION PHASE relative t=1,5 Difference UNLOADING of the forces N xx in NON_REGRESSION A2 A4 relative Difference of displacements DY in A2 A4 NON_REGRESSION relative Difference of the moments M yy in A2 A4 NON_REGRESSION relative Difference of rotations DRX in A2 A4 NON_REGRESSION PHASE elastic t=2,25 relative Difference of the moments M yy in A2 A4 NON_REGRESSION relative Difference of rotations DRX in A2 A4 NON_REGRESSION PHASE elastic t=2,5 relative Difference of the forces N xx in A2 A4 NON_REGRESSION relative Difference of displacements DY in A2 A4 NON_REGRESSION PHASE RECHARGING t=3,0 relative Difference of the moments M yy in A2 A4 NON_REGRESSION relative Difference of rotations DRX in A2 A4 NON_REGRESSION PHASE relative t=3,5 Difference UNLOADING of the forces N xx in NON_REGRESSION A2 A4 relative Difference of displacements DY in A2 A4 NON_REGRESSION relative Difference of the moments M yy in A2 A4 NON_REGRESSION relative Difference of rotations DRX in A2 A4 NON_REGRESSION

26 Titre : SSNS106 - Endommagement d une plaque plane sous so[...] Date : 01/03/2013 Page : 26/67 compared Diagrams of the force FX (forces N xx ) according to the displacement DX imposed for the loading f2 : 6.0E+05 FX - DX 4.0E E+05 FX(N) 0.0E E E E+05 FX_multicouche FX_GLRC -8.0E E E E E E E E E E-04 DX(m) Compared diagrams moment M yy according to the rotation DRY imposed for the loading f2 : 1.5E+04 MY - RY 1.0E+04 MY(N.m) 5.0E E E E+04 MY_multicouche MY_GLRC -1.5E E E E E E E E-03 RY

27 Titre : SSNS106 - Endommagement d une plaque plane sous so[...] Date : 01/03/2013 Page : 27/67 Compared diagrams displacement DY (due to the effect Fish) for the loading f2 : 7.0E E E-05 DY_multicouche DY_GLRC DY - Temps 4.0E-05 DY(m) 3.0E E E E E E E t(sec) Compared diagrams rotation DRX (due to the effect Fish) for the loading f2 : 1.5E E-03 RX - Temps RX_multicouche RX_GLRC 5.0E-04 RX 0.0E E E E t(sec) Diagrams of the evolution of the damage of model GLRC_DM ( d 1 for the upper face and d 2 the lower face) according to time:

28 Titre : SSNS106 - Endommagement d une plaque plane sous so[...] Date : 01/03/2013 Page : 28/67 Remarks These results are obtained by means of the material parameters which were identified starting from the tests A (for the parameters of membrane) and B (for the parameters of bending). Although the results of model GLRC_DM in pure traction and compression and pure bending are very satisfactory compared to the multi-layer computation of reference, the error of model GLRC_DM in coupling of membrane bending in the nonlinear phase is notable. It is noted that the response in elastic phase is right and that the difference is due to the criterion of appearance of the damage. By reference to the curves, one notes that in coupling of membrane bending, the thresholds of (et) damage N D M D already identified starting from the tests pure traction and compression and pure bending, give an overestimation in the event of coupling bending-membrane compared to the reference solution. That generates a notable error in the following phases. One proposes to lower the values from N D and M D 10% in order to decrease this error. (cf [R ] models GLRC_DM, 3.2.1).

29 Titre : SSNS106 - Endommagement d une plaque plane sous so[...] Date : 01/03/2013 Page : 29/67 6 Modelization D 6.1 Characteristic of the modelization Distortion and pure shears in the plane. Appear 6.1-a: Mesh and boundary conditions Modelization: DKTG L=1.0 m. Boundary conditions (see figure above on the right) so that the plate is subjected to a pure distortion: xy must be constant or with pure shears: forces are applied. Consequently, one applies the field of displacement following to edges of the plate for the distortion: { u = D x y 0 = 1 u y = D 0 x 2 u x, y u y, x = D 0 Thus: one imposes a fixed support in A 1, u x = D 0 y, u y =0 on the edge A 1 A 3, u x =0, u y = D 0 x the edge A 1 A 2, u x = D 0 y, u y = D 0 L the edge A 2 A 4, u x = D 0 L, u y =D 0 x on the edge A 3 A 4, where D 0 = and f t represent the amplitude of the cyclic loading according to the parameter (of pseudo-temps) t, definite like: F o n c t i o n d e c h a r g e m e n t 1. 0 E E f ( t ) 0. 0 E E E t ( s ) Appear 6.1-b: function of loading Increment of integration: 0.05 s.

30 Titre : SSNS106 - Endommagement d une plaque plane sous so[...] Date : 01/03/2013 Page : 30/67 For the shears, the following forces are applied: one imposes F y = D 0 on A 2 A 4, one imposes F x = D 0 on A 4 A 3, one imposes F y = D 0 on A 3 A 1, one imposes F x = D 0 on A 1 A 2, 6.2 Characteristic of the mesh Nodes: 121. Meshes : 200 TRIA3 ; 40 SEG Quantities tested and results For the distortion, one compares the shears N xy obtained by the two modelizations ; the tolerances are taken in absolute value on these relative differences : Standard identification of reference Value of reference Toleranc e DISTORTION POSITIF - PHASE elastic t=0,25 relative Difference of the forces N xy in B AUTRE_ASTER DISTORTION POSITIF - PHASE DAMAGE t=1,0 relative Difference of the forces N xy in B AUTRE_ASTER DISTORTION POSITIF - PHASE relative t=1,5 Difference UNLOADING of the forces N xy in B AUTRE_ASTER DISTORTION NEGATIF - PHASE CHARGEMENT t=3,0 relative Difference of the forces N xy in B AUTRE_ASTER DISTORTION NEGATIF - PHASE relative t =3,5 Difference UNLOADING of the forces N xy in B AUTRE_ASTER Diagram shears N xy (in the plane) according to time: , , 0 V _ m u l t i c o u c h e V _ G L R C _ D M , , 0 t 0, 0 0, 0 0, 5 1, 0 1, 5 2, 0 2, 5 3, 0 3, 5 4, , , , , 0

31 Titre : SSNS106 - Endommagement d une plaque plane sous so[...] Date : 01/03/2013 Page : 31/67 Diagram shears N xy (in the plane) according to D 0 imposed: V _ m u l t i c o u c h e V _ G L R C _ D M , , , , 0 3,5 0 E 3,0 0 E 2,5 0 E D 0 0, 0 2,0 0 E 1,5 0 E 0 4 1,0 0 E 0 4 5,0 0 E 0,0 0 E ,0 0 E 0 5 1,0 0 E 0 4 1,5 0 E 0 4 2,0 0 E 0 4 2,5 0 E 0 4 3,0 0 E 0 4 3,5 0 E , , , , 0 Diagram of the evolution of the damage of model GLRC_DM ( d 1 =d 2 ) according to time: 2, 0 1, 8 D 1 1, 6 1, 4 1, 2 1, 0 0, 8 0, 6 0, 4 0, 2 t 0, 0 0, 0 0, 5 1, 0 1, 5 2, 0 2, 5 3, 0 3, 5 4, 0 For the shears, one makes tests of non regression on the shear strains xy : Standard identification of reference Value of reference SHEARS POSITIF - PHASE elastic t=0,1 Strains of cisaillemen xy in B NON_REGRESSION 3,013 DISTORTION POSITIF - PHASE D DAMAGE t=0, Strains of cisaillemen xy in B NON_REGRESSION 2, Toleranc e Remarks In order to have a better agreement between the model GLRC_DM and the reference (models multilayer) in pure distortion, it was necessary to modify the Young's modulus of E=35620 MPa with E=42500 MPa compared to the modelizations A, B, C, knowing that in pure distortion steels are not charged.

32 Titre : SSNS106 - Endommagement d une plaque plane sous so[...] Date : 01/03/2013 Page : 32/67 It is checked that the shears obtained with Code_Aster at time t=0,37427, just with appearance of the first damage produce the elastic theoretical value: = N D 1 m 1 2 m 1 mt 2 m 1 mc 1 m 2 mc mt D N xy = 2 2 m k 0 2 mc mt that is to say: N D xy = N /m.

33 Titre : SSNS106 - Endommagement d une plaque plane sous so[...] Date : 01/03/2013 Page : 33/67 7 Modelization E 7.1 Characteristic of the modelization Coupling bending - shears in the plane. Modelization: DKTG L=1.0 m. Boundary conditions (see figure below): Appear 7.1-a: mesh one imposes a fixed support in A 1, and Appear 7.1-b: Boundary conditions u x = D 0 y, u y =0 on the edge A 1 A 3, u x =0, u y = D 0 x and DRY =0.0 the edge A 1 A 2 u x = D 0 y, u y = D 0 L and DRY = R 0 f t on the edge A 2 A 4, u x = D 0 L, u y =D 0 x on the edge A 3 A 4, where D 0 = f t, R 0 = in oiling of the parameter (of pseudo-temps) t, definite like: F o n c t i o n d e c h a r g e m e n t 7.2 Characteristics of the mesh 1. 0 E and f t represent the amplitude of the cyclic loading 5. 0 E f ( t ) 0. 0 E E E t ( s )

34 Titre : SSNS106 - Endommagement d une plaque plane sous so[...] Date : 01/03/2013 Page : 34/67 Many nodes: 121. Number of meshes: 200 TRIA3 ; 40 SEG Quantities tested and results One evaluates by tests of NON-regression at various times the results got by modelization GLRC_DM : Standard identification of reference Values of reference tolerance To t=1,0 Displacement DX in A2 NON_REGRESSION Displacement DZ in A2 NON_REGRESSION Force N yy in A2 NON_REGRESSION Variable of damage d1 in A1 NON_REGRESSION Variable of damage d2 in A1 NON_REGRESSION To t=2,8 Displacement DX in A4 NON_REGRESSION Displacement DZ in A4 NON_REGRESSION Force N yy in A4 NON_REGRESSION Variable of damage d1 in A1 NON_REGRESSION Variable of damage d2 in A1 NON_REGRESSION To t=3,0 Variable of damage d1 in A1 NON_REGRESSION Variable of damage d2 in A1 NON_REGRESSION compared Diagrams models multi-layer-models GLRC_DM bending moment M yy according to time: , , 0 M Y _ m u l t i c M Y _ G L R C _ D M 5 0 0, 0 0, 0 0, 0 0, 5 1, 0 1, 5 2, 0 2, 5 3, 0 3, 5 4, , , 0 t , 0

35 Titre : SSNS106 - Endommagement d une plaque plane sous so[...] Date : 01/03/2013 Page : 35/67 Compared diagrams models multi-layer-models GLRC_DM shears N xy according to time: , , 0 V _ M u l t i c V _ G L R C _ D M , , , 0 0, 0 0, 0 0, 5 1, 0 1, 5 2, 0 2, 5 3, 0 3, 5 4, , , 0 t , 0 Compared diagrams models multi-layer-models GLRC_DM of the shears bending moment N xy according to the distortion: 4,0E+04 Effort tranchant V - D0 3,0E+04 2,0E+04 1,0E+04 V(N) 0,0E+00-1,0E+04-2,0E+04-3,0E+04 V_multicouche V_GLRC -4,0E+04-1,5E-04-1,0E-04-5,0E-05 0,0E+00 5,0E-05 1,0E-04 1,5E-04 D0(m) Compared diagrams models multi-layer-models GLRC_DM of the bending moment M yy according to rotation: MY - RY 1,5E+03 1,0E+03 5,0E+02 MY(N.m) 0,0E+00-5,0E+02-1,0E+03 MY_multicouche MY_GLRC -1,5E+03-6,0E-03-4,0E-03-2,0E-03 0,0E+00 2,0E-03 4,0E-03 6,0E-03 RY

36 Titre : SSNS106 - Endommagement d une plaque plane sous so[...] Date : 01/03/2013 Page : 36/67 Diagram of the evolution of the damage of model GLRC_DM ( d 1, d 2 ) according to time: 1, 4 1, 2 D 1 D 2 1, 0 0, 8 0, 6 0, 4 0, 2 t 0, 0 0, 0 0, 5 1, 0 1, 5 2, 0 2, 5 3, 0 3, 5 4, 0 Diagram of the evolution of the surface density of dissipated energy (in J /m 2 ) of model GLRC_DM according to time : 2 5, 0 é n e r g ie d is s ip é e 2 0, 0 1 5, 0 1 0, 0 5, 0 0, 0 0, 0 0, 5 1, 0 1, 5 2, 0 2, 5 3, 0 3, 5 4, 0 t It is checked, confer [R ], that with the data of the benchmark, one a: k 0 =9, J /m 2, from where surface densities of dissipated energy: Time d 1 d 2 dissipated energy J /m 2 t=2,0 s t=4,0 s

37 Titre : SSNS106 - Endommagement d une plaque plane sous so[...] Date : 01/03/2013 Page : 37/67 8 Modelization F pure Traction and compression elastoplastic behavior endommageable ( GLRC_DM + Von Mises). In this test, one is interested in the elastoplastic behavior. One can insert a plastic behavior into the response of model GLRC_DM via a kit which makes it possible to put in series the model GLRC_DM with a plastic model of classical Von Mises. This kit consists in imposing the same tensor of the stresses on the two models and cumulating the two tensors of the strains. 8.1 Characteristics of the modelization Modelization: DKTG L=1.0 m. Boundary conditions: Fixed support in A 1 ; DX =0.0 on the edge A 1 A 3 ; Appears 8.1-a: mesh and boundary conditions DX =U 0 f t on the edge A 2 A 4 ; where U 0 = m and f t as follows represent the amplitude of the cyclic loading according to the parameter (of pseudo-temps t ): F o n c t i o n d e c h a r g e m e n t 1. 0 E E f ( t ) 0. 0 E E E t ( s ) Increment of integration: 8, Characteristics of the mesh Many nodes: 9. Number of meshes: 8 TRIA3 ; 8 SEG2.

38 Titre : SSNS106 - Endommagement d une plaque plane sous so[...] Date : 01/03/2013 Page : 38/ Quantities tested and Standard Identification results of reference Value of reference Toleranc e To t=0,017 tension - elastic phase Displacement DY in A4 NON_REGRESSION membrane Force N xx in A4 AUTRE_ASTER With t=0,085 tension - phase damage Displacement DY in A4 NON_REGRESSION membrane Force N xx in A4 NON_REGRESSION To t=0,085 tension - phase plasticity + damage membrane Force N xx in A4 NON_REGRESSION With t=2,04 tension - phase discharges membrane Force N xx in A4 NON_REGRESSION With t=0,017 tension - elastic phase Density of total strain energy in slab in A2 NON_REGRESSION Density of membrane strain energy in slab in A2 NON_REGRESSION Density of total strain energy in slab net M1 NON_REGRESSION Density of strain energy of bending in slab nets M1 AUTRE_ASTER With t=0,085 tension - phase damage Density of total strain energy in slab in A2 NON_REGRESSION Density of total strain energy in slab 1.5 M1 nets NON_REGRESSI ON Density of strain energy of bending in slab nets M1 AUTRE_ASTER At t=1,0 end of load Density of strain energy of bending in slab nets M1 AUTRE_ASTER With t=0,017 tension - elastic phase Strain energy in slab NON_REGRESSION Work external NON_REGRESSION With t=0,085 tension - phase damage Strain energy in slab NON_REGRESSION Work external NON_REGRESSION At t=1,0 end of load Strain energy in slab NON_REGRESSION

39 Titre : SSNS106 - Endommagement d une plaque plane sous so[...] Date : 01/03/2013 Page : 39/67 9 Modelization G 9.1 Characteristic of the modelization Pure shears in the plane elastoplastic behavior endommageable ( GLRC_DM + Von Mises). Modelization: DKTG L=1.0 m. Boundary conditions (see figure below): Appear 9.1-a: mesh one imposes a fixed support in A 1, Appear 9.1-b: boundary conditions u x = D 0 y, u y =0 on the edge A 1 A 3 u x =0, u y = D 0 x and DRY =0.0 the edge A 1 A 2, u x = D 0 y, u y = D 0 L and the edge A 2 A 4, u x = D 0 L, u y =D 0 x on the edge A 3 A 4, where D 0 = f t, and f t represent the amplitude of the cyclic loading according to the parameter (of pseudo-temps) t, definite like (increment of integration: ):

40 Titre : SSNS106 - Endommagement d une plaque plane sous so[...] Date : 01/03/2013 Page : 40/67 F o n c t i o n d e c h a r g e m e n t 1. 0 E E f ( t ) 0. 0 E E E t ( s )

41 Titre : SSNS106 - Endommagement d une plaque plane sous so[...] Date : 01/03/2013 Page : 41/ Characteristics of the mesh Many nodes: 121. Number of meshes: 200 TRIA3 ; 40 SEG Quantities tested and Standard Identification results of reference Values of reference tolerance To t=0,25 Displacement DY in A4 AUTRE_ASTER 0.2, membrane Force N yy in A4 AUTRE_ASTER To t=1,0 membrane Force N yy in A4 NON_REGRESSION , To t=2,0 Displacement DX in A4 NON_REGRESSION , membrane Force N yy in A4 NON_REGRESSION , To t=2,0 membrane Force N yy in A4 NON_REGRESSION , To t=3,0 membrane Force N yy in A4 NON_REGRESSION ,0 10-4

42 Titre : SSNS106 - Endommagement d une plaque plane sous so[...] Date : 01/03/2013 Page : 42/67 10 Modelization H 10.1 Characteristic of the modelization T rac tion Compression pure. Modelization: DKTG Boundary conditions: Fixed support in A 1 ; DX =0.0 on the edge A 1 A 3 ; Appear 10.1-a: mesh and boundary conditions. DX =U 0 f t on the edge A 2 A 4 ; where U 0 = m and f t represent the amplitude of the cyclic loading according to the parameter (of pseudo-temps) t. For checking well the model, one considers a function of loading as follows: 1.0 Fonction de chargement f1 0.5 f(t) t(sec) Appear 10.1-b: function of loading: tension, then compression Notes: the extreme strain is: , that is to say approximately one the third of the strain of transition in plasticity of steels. Time step of integration: Characteristics of the mesh Many nodes: 9. Number of meshes: 8 TRIA3 ; 8 SEG2.

43 Titre : SSNS106 - Endommagement d une plaque plane sous so[...] Date : 01/03/2013 Page : 43/ Multi-layer reference solution LABORD_1D Besides the reference solution presented to the 2, one creates one second reference solution using constitutive law LABORD_1D [R ]. This second reference solution is obtained by a modelization 1D out of beam multifibre, where the loading is the same one as that of the multilayer solutions and GLRC_DM. One models the concrete and the reinforcements separately. This solution rising from a computation 1D, it will not be able to take into account the effects of fish. The length of the beam is taken equal to 1m while its section is 1x 0.1m in order to correspond to dimensions of the plate used for computation multi-layer and total Meshes and models It is necessary to define two meshes: a longitudinal 1D mesh of the beam, including 2 nodes and 1 element (POU_D_EM). A B Fi gu re a mesh 2D of the 10.3-a: mesh beam for LABORD_1D concrete cross section. With the mesh 1D, one associates beam the model multifibre. The mesh 2D is in fact only the representation of fibers used for computation multifibre. On this mesh, one will add two fibers of steel using operator DEFI_GEOM_FIBRE and factor key word the FIBER. The surface of each fiber of reinforcements is of m 2 and their position in the thickness is 0.0,± 0.04 m the cracking of the concrete is modelled by constitutive law LABORD_1D, while it is supposed that steel always remains in the elastic domain Properties of the materials Concrete (LABORD_1D models) : Young's modulus: E b = MPa Steel : Poisson's ratio: b =0.0 Appear 10.3-b: mesh section for LABORD_1D Initial threshold of damage in tension (positive): Y 01 =341.0 J.m 3 Initial threshold of damage in compression (positive): Y 02 = J.m 3 Stress of total reclosing of cracks in compression: f =3.5 MPa Parameters characteristic of the material: A 1 = ; A 2 = B 1 =1.2 ; B 2 = =0.867 MPa ; 2 = 35 MPa Young's modulus: E a = MPa acier Limit of linearity e : MPa acier Slope post-elastic E écrouis : = E a =300 MPa.

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