Multiscale Approach of the Mechanical Behaviour of Reinforced Concrete Structures

The Real Model

M. David 1,2 J.-J. Marigo 2

1 LaMSID-UMR EDF/CNRS 28322 École Polytechnique, LMS

IntroductionSteel reinforcements and prestressing tendons have a great influence on the mechanical behaviour of reinforced concrete structures. The method generally used in the literature to model these reinforcements consists in using bar elements. This simplified model has several drawbacks. In particular, these bar elements introduce a slight mesh dependency, and create local stress concentrations. We propose to build a different model, which reproduces the mechanical non-linear behaviour of steel reinforcements and prestressing tendons, by using the method of asymptotic analysis.

The Industrial Context

� Modelling of the containment efficiency of a concrete vessel (50 m high, 1.2 m thick)

� The steel reinforcements (green) have a diameter of 2 cm, and a spacing of 20 cm

� The prestressing tendons (yellow) have a diameter of 8.4 cm, and a spacing of 40 cm

� Three dimensional computations are unaffordable (~ 109 dof)

� Bar elements are not really satisfactory(stress concentrations, mesh dependency)

� Need for relevant simplified models

The Limit Model

� Identification of a limit model, when

� Using the method of separation of scales, we obtain the following limit behaviour. The fibers form a membrane, which is connected with concrete through a non-linear interface law:

Validation of the Limit Model

� Modelling of the pull-out of a set of reinforcing bars

� Comparison between a 3D reference simulation and the limit model

� Excellent agreement between the twomodels: the relative error on the load-displacement curve is less than 5 %

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Conclusion and Future Works

Work Completed:

• Development of a relevant limit model, accounting for a possible decohesion of the steel-concrete interface

• Implementation of the model in a finite elementcode (Code_Aster)

• Validation of the model and comparisons with detailed 3D computations

Future Work:

• Application to the modelling of large concrete structures

• Use of this model in combination with concrete damage models

References• J.-J. Marigo, C. Pideri : The Effective Behavior of Elastic Bodies Containing Microcracks or Microholes Localized on a Surface.International Journal of Damage Mechanics (Special issue of ESMC 2009)

• M. David, J.-J. Marigo, C. Pideri : Homogenized Interface Model Describing Inhomogeneities Located on a SurfaceJournal of Elasticity (submitted)

� Typical interface law:

� Parameters of the model:

- spacing of the bars

- diameter of the bars

- rigidity of steel

� Study of a set of unidirectional cylindrical shaped steel reinforcements, with a diameter d, a spacing e, and the following material parametersfor steel and concrete:

� Classical mechanicsequations in the continuum:

� Non-linear interface law between steel and concrete, characterized by a potential (for monotonic loadings):

� The mechanical parameters characterizing the behaviour of the membrane and the cohesive law are computed analytically.

� The model is implemented in the finite element code Code_Aster. The elastic behaviour is modelled by membrane elements, and the relative sliding of the membrane by specific cohesive elements

Industrial Application

� Modelling of the opening of a crack in a prestressed reinforced concrete structure

� Pre-stressing tendons modelled in 3D

� Reinforcing bars modelled with the limit model

� Study of the influence of the tendon-concrete interface law

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