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1 Titre : SSNP161 Essais biaxiaux de Kupfer Date : 10/10/2012 Page : 1/8 SSNP161 Biaxial tests of Summarized Kupfer: Kupfer [1] was interested to characterize the performances of the concrete under biaxial loadings. Two of these tests are modelled in this case test in order to the model compare the experimental data with the results got with Mazars. Behaviors in bi--compression and shears are studied in this case test.

2 Titre : SSNP161 Essais biaxiaux de Kupfer Date : 10/10/2012 Page : 2/8 1 Problem of reference 1.1 Geometry and boundary conditions In these tests, a plate of concrete ( mm ) is subjected to a loading where the ratio of the principal stresses 2 / 1 is fixed ( 3 =0 ). These tests are modelled in two dimensions under the constraint plane ( zz =0 ) using an element quadrangle with 4 nodes (QUAD4). Appear 1.1-a : Modelization 2D and boundary conditions of the biaxial tests Only one quarter of the concrete plate is modelled. Thus, conditions of symmetry are imposed: Following displacements y are blocked on lower edge. Following displacements x are blocked on left edge. Two pressures are imposed on free edges: P 1 and P 2. The ratio between the loads, and implicitly on the principal stresses, is noted : 2 = 1 These pressures follow a linear law of evolution. Time 0 1 P 1 ( MPa ) 0 30 P 2 ( MPa ) 0 30 ω 1.2 Properties of the material Table 1.1-1: Evolution of the pressures For the model of MAZARS, the following parameters were used: Behavior elastic: E= MPa, =0.19 Behavior damaging: d0 = ; Ac=1. 25 ; At=1.0 ; Bc=1965 ; Bt=9 000 ; k =0. 7 These materials parameters induce a limit in compression f c about 33 MPa. (Eq.1)

3 Titre : SSNP161 Essais biaxiaux de Kupfer Date : 10/10/2012 Page : 3/8 1.3 Initial conditions Nothing

4 Titre : SSNP161 Essais biaxiaux de Kupfer Date : 10/10/2012 Page : 4/8 2 Reference solution 2.1 Comparison of the tests and simulations We focused ourselves on two values of ratio = 2 / 1 corresponding to a test of bi--compression ( =0.52 ) and to a test of traction and compression ( = ). The following figure presents the various got results: Appear 2.1-a : Comparison of the curves experimental and numerical during the biaxial test ( =0.52 ) Appears 2.1-b : Comparison of the curves experimental and numerical during the biaxial test =0.52 Let us specify that simulations are controlled in force. So it is not possible to model the phase of rise in temperature. For the ratio =0.52, not taken it into account of the model of the evolution of the Poisson's ratio in bi--compression does not allow to find the long-term strains. However, the numerical results are close to the experimental points at the beginning of simulation. Moreover, the model of

5 Titre : SSNP161 Essais biaxiaux de Kupfer Date : 10/10/2012 Page : 5/8 Mazars allows to find forced it with experimental fracture (the last experimental point corresponds to the fracture of the sample). Then, for ω = , the model provides results very close to the test. Note: To the stabilized version 11.2 of Code_Aster, the model Mazars followed the equations defined in the thesis of Mazars of 1984 [2]. Recently, a reformulation was proposed in order to fill certain gaps of the model of 1984 with knowing the description of the behavior of the concrete in bi--compression and pure shears. This version of 2012 was implemented from the STA It is possible to compare the response of the different version of the Mazars model in order to highlight the improvements of the version It appears well that the model of Mazars of 1984 strength in bi--compression underestimates. Then, for = 0.052, the differences between the results coming from the two models are weak. Let us recall that the improvements of the model of 2012 relate to in particular behavior in bi--compression and pure shears. Consequently, this result is logical. The curve obtained starting from the model of Mazars of 2012 remains closer to the experimental points all the same. 2.2 Bibliographical references [1] H. Kupfer, H.K. Hilsdorf, H. Rüsch, Biaxial Behavior of Concrete under Stress, ACI Newspaper, vol. 66, No 66-62, 1969, pp [2] J. Mazars, A micro description of and macroscale ramming of concrete structure, Engineering Fractures Mechanics, Vol25, 1986, p [3] J. Mazars, F. Hamon, A new strategy to formulate has 3D ramming model for concrete under monotonic, cyclic and severe loadings, Engineering Structures, 2012, Under review

6 Titre : SSNP161 Essais biaxiaux de Kupfer Date : 10/10/2012 Page : 6/8 3 Modelization A 3.1 Characteristic of the modelization One uses a modelization C_PLAN. The ratio of the principal stresses 2 / 1 is fixed at Characteristics of the mesh The mesh contains 1 element of type QUAD Quantities tested and results We test the strains and stresses with the node N1 of the Figure 1.1-a. 3.4 Remarks Standard identification Increment of reference Value of reference Forced Tolerance σ yy 7 ANALYTIQUE Pa 5% Strain ε yy 7 ANALYTIQUE 3,00E-004 5% Stress σ yy 69 ANALYTIQUE Pa 5% Table 3.3-1: Summary of the TEST_RESU the first two points of measurement make it possible to compare the slope at the origin of the stressstrain curve. The last point corresponds to the stress with fracture.

7 Titre : SSNP161 Essais biaxiaux de Kupfer Date : 10/10/2012 Page : 7/8 4 Modelization B 4.1 Characteristic of the modelization One uses a modelization C_PLAN. The ratio of the principal stresses 2 / 1 is fixed at Characteristics of the mesh The mesh contains 1 element of type QUAD Quantities tested and results We test the strains and stresses with the node N1 of the Figure 1.1-a. 4.4 Remarks Standard identification Increment of reference Value of reference Forced Tolerance yy 6 ANALYTIQUE Pa 5% Strain yy 6 ANALYTIQUE 1,05E-004 5% Stress yy 46 ANALYTIQUE Pa 5% Table 4.3-1: Summary of the TEST_RESU the first two points of measurement make it possible to compare the slope at the origin of the stressstrain curve. The last point corresponds to the stress with fracture.

8 Titre : SSNP161 Essais biaxiaux de Kupfer Date : 10/10/2012 Page : 8/8 5 Summary of the results the version of the model of MAZARS of 2012 makes it possible to find the stress with experimental fracture with an error lower than 5%. Note: the gain compared to the model of 1984 is especially visible in bi--compression i.e. for the ratio =0.52. On the other hand, the strain at failure is far away from the test because this model does not take into account the evolution of the Poisson's ratio in bi--compression.

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