Code_Aster Titre : Réalisation d'une étude de génie civil sous charge[...] Responsable : M'JAHAD Sofia

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1 Titre Réalisation d'une étude de génie civil sous charge[...] Responsable M'JAHAD Sofia Date 12/11/2014 Page 1/50 Clé U Realization of a study of civil engineer under seismic loading Summary The purpose of this document is to give advices to carry out studies of civil engineer under seismic loading. One presents to it the use of the numerical models of the reinforced concrete structures under earthquake available in. One points out the methods of calculating in nonlinear transitory dynamics first of all. One describes then modelings finite elements and the models of steel and concrete behavior available. One presents finally the various stages to be carried out within the framework of a seismic nonlinear dynamic study (prior studies, resolution of calculation and postprocessing).

2 Titre Réalisation d'une étude de génie civil sous charge[...] Responsable M'JAHAD Sofia Date 12/11/2014 Page 2/50 Clé U Contents 1 Introduction Methods of calculating transitory nonlinear dynamics Operator DYNA_NON_LINE Choice of the temporal diagram of integration Choice of the step of time of calculation Implicit schemes Explicit diagrams Choice of modeling finite elements Introduction Description of local modeling (solid elements) Modeling of the concrete Modeling of the reinforcements Modeling of the connection between the concrete and the reinforcements Description of semi-total modeling multi-layer hull Modeling of the concrete Modeling of the reinforcements Modeling of the connection between the concrete and the reinforcements Description of semi-total modeling multifibre beam Description of total modeling General information Modeling of reinforced concrete plate Modeling of the connections Connection hull hull Connection 3D beam and hull beam Connection post floor Connection 3D hull Modeling of nonlinear connections (nodes of framework; damper ) using discrete elements Other structural elements modélisables Modeling of a liner Modeling of prestressing Which type of modeling adopt? Introduction Local modeling (solid elements) Semi-total modeling multi-layer hull Semi-total modeling multifibre beam Modeling of total hull Choice of the model of behavior... 16

3 Titre Réalisation d'une étude de génie civil sous charge[...] Responsable M'JAHAD Sofia Date 12/11/2014 Page 3/50 Clé U Introduction Experimental observations Specificities of the seismic studies Algorithm of Borst Description of the models of behavior of steel Description of the models of concrete behavior Model ENDO_ISOT_BETON Model ENDO_ORTH_BETON Localization related to the nonlocal damage and models Model GLRC_DM Model GLRC_DM coupled with elastoplasticity (VMIS_CINE_LINE) Model MAZARS Which type of model behavior to adopt? Introduction Model ENDO_ISOT_BETON Model ENDO_ORTH_BETON Model MAZARS_GC Model GLRC_DM Setting in facts of the case Grid General advices Massive local modeling and multi-layer hull Initial conditions Limiting conditions Loading seismic Mono-support Multi-support General precautions on the temporal signal Damping Definition Nonlinear case Other advices into nonlinear Analyses preliminary to nonlinear transitory dynamic calculation Analysis of the mass Modal analysis Elastic linear transitory calculation dynamic Other analyses Analysis push-over (quasi-static nonlinear monotonous) Cyclic static analysis... 39

4 Titre Réalisation d'une étude de génie civil sous charge[...] Responsable M'JAHAD Sofia Date 12/11/2014 Page 4/50 Clé U Resolution of nonlinear transitory dynamic calculation Introduction Assistance with the convergence of calculation Temporal evolution of the loading Convergence criteria Algorithm of Newton Not linearities materials Diagram of temporal integration Sequence of calculations Postprocessing Analysis of the total behavior of the structure Maximum displacements and efforts Determination of the spectrum of oscillator Eigen frequencies Assessment of dissipated energies Analysis of the local behavior of the structure Introduction Local modeling (solid elements) Multi-layer hulls Multifibre beams Total hull Discrete elements carrying laws of linear behavior or not Case existing tests and studies CAS-tests Temporal diagrams Modelings and associated models of behavior Loading Damping of Rayleigh Postprocessing Examples of studies carried out Floor of the P4 type Veils in shearing SAFE T PUG model Conclusions Bibliography... 49

5 Titre Réalisation d'une étude de génie civil sous charge[...] Responsable M'JAHAD Sofia 1 Date 12/11/2014 Page 5/50 Clé U Introduction Within the framework of the seismic revaluation of the buildings of civil engineer, the digital simulations aim to better apprehend their nonlinear behavior. One presents in this document the tools available in in order to model reinforced concrete structures under seismic loading. The purpose of this document is to provide the user of the advices of methodology and the rules of good practice to be followed at the time of the various stages of its calculation. One approaches successively the following points methods of calculating in nonlinear transitory dynamics, the choice of modeling finite elements, the choice of the model of behavior, the setting in facts of the case, analyses preliminary to nonlinear transitory dynamic calculation, the resolution of nonlinear transitory dynamic calculation, postprocessing, existing studies. However one does not approach in this documentation the lawful calculation of dimensioning by spectral method [U ]. 2 Methods of calculating transitory nonlinear dynamics One does not detail in this document the general advices for the realization of a nonlinear transitory dynamic calculation. It is however important to insist on the specificity of such calculations. The use of the dynamic operator DYNA_NON_LINE ([U ] and [R ]) requires additional precautions compared to a nonlinear static calculation STAT_NON_LINE ([U ] and [R ]). It is in particular important to note that, the temporal evolution to represent being of big size, one has an often expensive problem in time CPU. One is thus brought to set up modelings who are not too bulky. In addition, certain options available in nonlinear statics are not licit in dynamics. It is the case in particular piloting of the loading which makes it possible to help the convergence of a static calculation. In dynamics, the history of loading being real, one cannot use this kind of method more such as it is. 2.1 Operator DYNA_NON_LINE The operator DYNA_NON_LINE ([U ] and [R ]) allows to calculate the dynamic evolution of a structure whose material and/or geometry have a nonlinear behavior. One solves the equations of dynamics with a discretization by finite elements on the grid in physical base, methods of temporal integration implicit or explicit for the resolution of the temporal problem, methods of integration of Newton-Raphson for the resolution of the nonlinear incremental problem associated with mechanical balance. This method consists in reiterating on the resolution with a tangent operator. In dynamics, contrary to statics, it cannot exist of operator of resolution strictly noninvertible. This is due to the presence of the matrix of mass in this operator. However, this does not guarantee of anything convergence calculation. The most frequent problems relate to the resolution of the behavior. One gives to the paragraph 7.2 advices of use of the operator DYNA_NON_LINE allowing to improve convergence of the algorithms. 2.2 Choice of the temporal diagram of integration The user can currently choose in DYNA_NON_LINE between four temporal diagrams Implicit schemes

6 Titre Réalisation d'une étude de génie civil sous charge[...] Responsable M'JAHAD Sofia Date 12/11/2014 Page 6/50 Clé U a nondissipative implicit scheme diagram of average acceleration (or trapezoid regulates), a dissipative implicit scheme diagram HHT (which introduces a digital dissipation high frequency), Explicit diagrams a diagram clarifies nondissipative diagram of the centered differences, a dissipative explicit diagram the diagram of Tchamwa-Wielgosz (which introduces a digital dissipation high frequency). One will refer to [10] and [R ] to know in detail the rules of use of these various methods of temporal integrations. Within the framework of the studies civil engineer under seismic loading, one recommends to use the classical implicit scheme of average acceleration which does not bring digital dissipation. One will see in the paragraph 7.2 that one can be brought, in the event of digital instability of calculations (oscillations high frequencies), to use a diagram of the type HHT. Moreover, in the event of nonsevere convergence, one will be able to try to continue calculation by using an explicit diagram. However the operator DYNA_NON_LINE is directed towards the implicit approaches. It is not optimized (vectorized) for the explicit resolutions. It is thus currently recommended to use an explicit diagram with more the greatest caution. 2.3 Choice of the step of time of calculation Implicit schemes The step of time to choose must observe a certain number of conditions [R ] the step of time must be sufficiently small to correctly represent the temporal sampling of the loading; one advises to choose for reasons of precision (criterion of the type Shannon on the cut-off frequency), a step of time such as t 10 1f max, with f max the high frequency to be represented; in addition, it can be interesting to determine an approximation of condition CFL (CurrentFriedrichs-Lewy, [4]) of the problem in order to have a lower limit of the step of time to use. The step of time of stability of condition CFL is given by t c =L/ c with L the length characteristic of the smallest element of the grid and c the celerity of the elastic waves of compression unidimensional given by c= E. The use of a step of time definitely lower (more than one order of magnitude) than condition CFL does not have a physical direction and can cause digital oscillation high frequencies. It will be in particular taken care, during recuttings of the step of time, so that the step of time used remains close to condition CFL (it should be noted that the step of time of stability is generally very weak). In practice, it is necessary in nonlinear mode to make sure of the low sensitivity of the answer obtained for calculations with various steps of time. When one has a result converged for a step of time t 1, one will make sure of the stability of this result by taking a step of time lower or equal to t 1 is satisfactory ,1 t 1. If the answer is identical, the step of time Explicit diagrams These diagrams, contrary to the implicit schemes, are conditionally stable. One must imperatively use a step of time lower than the step of time of stability (condition CFL). In the contrary case, calculation is likely to diverge (one observes for example accelerations abnormally raised). The step of time of

7 Titre Réalisation d'une étude de génie civil sous charge[...] Responsable M'JAHAD Sofia Date 12/11/2014 Page 7/50 Clé U stability is given at the beginning of the operator DYNA_NON_LINE. The option STOP_CFL =OUI allows to make sure that condition CFL is never exceeded. Notice The calculation of the CFL is not programmed for all the elements (in particular the discrete elements are ignored); the CFL estimated by can thus be larger (less penalizing) that the real CFL, with the risks of brutal divergence which results from this. 3 Choice of modeling finite elements 3.1 Introduction One seeks to represent two large families of structural reinforced concrete present elements in the classical buildings of civil engineer beams and posts, floors and veils. One distinguishes three main categories of representations finite elements Local representation (solid elements) In this classical approach finite elements, the constitutive materials (steel and concrete) are modelled separately. One uses massive finite elements for the concrete and the models of associated behaviors are written in 3D. This kind of approach makes it possible to obtain a detailed description of the nonlinear phenomena concerned but its application to the totality of structure of an industrial type can prove to be difficult (prohibitory times CPU and size memory). Moreover, the use of local models of behavior involves important problems of nonconvergence. Semi-total representation Modelings used are standard beams and multi-layer hulls and an integration is carried out in the thickness of the element. The constitutive materials are always modelled separately. One can distinguish multi-layer hulls supported by finite elements of standard hull. Simplifying assumptions associated with the theory of the hulls are adopted (the fields of displacements vary linearly in the thickness of the hull, the transverse constraint zz is worthless). The models of associated behaviors are written in 2D forced plane; multifibre beams supported by finite elements of standard beam. Simplifying assumptions associated with the theory of the beams of Euler are adopted (the sections remain right and perpendicular to average fibre). The models of associated behaviors are written in 1D. The principal interest of this kind of modeling is to be much less expensive in time CPU and the face memory that a local classical representation. It makes it possible of more than represent in a relatively realistic way the structure to be studied. Total representation In this approach, the total behavior of the reinforced concrete is modelled (the components are not considered any more separately). The finite elements supports are elements of structures (beams, hulls) to only one sleep. The associated specific models of behavior are written in aggregate variables (generalized efforts, generalized deformations). Total modelings are generally developed for specific applications (for a kind of component of a structure, for a kind of request, ). This kind of representation is far from expensive in time CPU and the face memory, but the data of modeling require an identification which must be realized carefully. Moreover, this identification is generally valid only for one class of loadings. Figure 3.1-a following illustrates the types of modelings available in.

8 Titre Réalisation d'une étude de génie civil sous charge[...] Responsable M'JAHAD Sofia Date 12/11/2014 Page 8/50 Clé U Figure 3.1-a summary of the types of modelings available.

9 Titre Réalisation d'une étude de génie civil sous charge[...] Responsable M'JAHAD Sofia Date 12/11/2014 Page 9/50 Clé U Description of local modeling (solid elements) Modeling of the concrete Modeling using finite elements massive classics is not approached in this document being given which it in the case of does not present of clean specificities the concrete. One will refer to the general documentation of for any problem of modeling Modeling of the reinforcements One can use the two following types of modeling to represent the reinforcement mainstays elements of bar (BAR, [U ]). These finite elements 3D linear transmit only efforts and axial deformations. On the other hand, in dynamics, the efforts of inertia appear in all the directions. The use of the elements BAR imply to model all the reinforcements with one. This modeling is to be used when reinforcement is complicated and when one seeks to represent in a very fine way the geometrical position of the reinforcements; elements of grid out of membrane (GRILLE_MEMBRANE, [U ]). This modeling GRILLE_MEMBRANE allows to only represent the tablecloths of reinforcement to only one direction of reinforcement working out of membrane. The finite elements supports are elements of surface (TRIA3, QUA4, ). The concept of offsetting does not exist for this modeling. It is thus necessary to position the reinforcements at the good place during the grid of the structure. To represent a complete bed of reinforcement (reinforcement in two orthogonal directions), one duplicates the first bed of GRILLE_MEMBRANE and one defines a second orthogonal bed in the first. Notice It was shown that the mixture of solid elements (concrete) and bars (steel) can introduce singularities source of nonconvergence (application of a specific loading) [ 14]. It is thus preferable, if the problem allows it, to use the elements of grid to represent the reinforcements Modeling of the connection between the concrete and the reinforcements One recommends to make coincide the nodes of the meshs steels and concrete. This makes it possible not to weigh down the size of the problem because one thus avoids introducing relations of connections between the meshs steels and concrete to ensure adherence. It is necessary to make well correspond all the nodes concrete located along the reinforcement with a node steel. The reinforced concrete structure is then represented by the superposition of the elements BAR or GRILLE_MEMBRANE used for steels and of the elements 3D solid masses used for the concrete. It should be noted that this strategy of modeling implies that the steel-concrete connection is perfect. A more realistic modeling of the connection between steels and the concrete is available in 2D in ( JOINT_BA, [R ]). This one is not adapted to our seismic applications. It can be used to study a steel-concrete connection locally (test of wrenching, for example). 3.3 Description of semi-total modeling multi-layer hull Modeling of the concrete One models the concrete using elements of plates or hulls of (DKT, DST, Q4G, COQUE_3D). One does not reconsider in this document the formulation of these elements. One will refer to [U ] for their use. It is pointed out simply that since calculations carried out are nonlinear, one uses a method of integration per layer for these elements. For each layer, one uses a method of Simpson at three points of integration, in the middle of the layer and in skins higher and lower of layer. For N layers the

10 Titre Réalisation d'une étude de génie civil sous charge[...] Responsable M'JAHAD Sofia Date 12/11/2014 Page 10/50 Clé U number of points of integration in the thickness is of 2N 1. For tangent rigidity, one calculates for each layer, in plane constraints, the contribution to the matrices of rigidity of membrane, inflection and coupling membrane-inflection as well as the contribution to the internal generalized efforts. These contributions are added and assembled to obtain the matrix of total tangent rigidity. To treat non-linearities material, one recommends to use from 3 to 5 layers in the thickness for a number of points of integration being worth 7.9 and 11 respectively Modeling of the reinforcements One can use the two following types of modeling to represent the reinforcement mainstays elements of bar ( BAR, cf ); elements of grid (GRILLE_EXCENTRE, [U ]). This modeling GRILLE_EXCENTRE allows to represent the tablecloths of reinforcement to only one direction of reinforcement with offsetting. One thus defines the position of the bed of reinforcements compared to neutral fibre of the reinforced concrete hull. To represent a complete bed of reinforcement (reinforcement in two orthogonal directions), one duplicates the first bed of GRILLE_EXCENTRE and one defines a second orthogonal bed in the first. One nets only one hull which one duplicates in in order to create the groups of meshs corresponding to the reinforcements (operator CREA_MAILLAGE ). All the groups of meshs are based on the same nodes. Notice In this semi-total representation of multi-layer hull, it is not possible to model the transverse reinforcements Modeling of the connection between the concrete and the reinforcements As with the 3.2.3, one recommends to make coincide the nodes of the meshs steels and concrete. This makes it possible not to weigh down the size of the problem because one thus avoids introducing relations of connections between the meshs steels and concrete to ensure adherence. If elements are used GRILLE_EXCENTRE for the reinforcements, the nodes steels and concrete coincide naturally because the meshs are duplicated and are based in fact on the same nodes. The reinforced concrete structure is represented by the superposition of modelings GRILLE_EXCENTRE used for the reinforcements and hull used for the concrete. 3.4 Description of semi-total modeling multifibre beam Multifibre modeling beam (element POU_D_EM, [U ] and [R ]) is based on the resolution of a problem of beam whose heterogeneous section is divided into several fibres. Each fibre is equipped with a uniaxial behavior corresponding to material constituting it, while kinematics is defined by the extension coming from the clean extension (axial deformation) and from the variation of curve of the beam itself. The section can be of an unspecified form and is described using a grid 2D.

11 Titre Réalisation d'une étude de génie civil sous charge[...] Responsable M'JAHAD Sofia Date 12/11/2014 Page 11/50 Clé U y x aciers y z Maillage 2 section(qua4,tri3) Maillage 1 poutre (SEG 2) position des aciers Figure 3.4-a multifibre modeling beam of a reinforced concrete beam. Remarks In this semi-total representation of multifibre beam, it is not possible to model the transverse reinforcements. The behavior of a reinforced concrete beam is defined by the data material of concrete fibres and the data material of fibres of steel reinforcements. 3.5 Description of total modeling General information In this kind of representation, the elements of structures implemented have only one fibre or layer in the section. The associated homogenized models of behavior are written in aggregate variables (generalized efforts, generalized deformations) without passage by local laws. Most developments existing in the literature the beams concern. In, this kind of total elements of reinforced concrete beam is however not developed. It should be noted that these total modelings are very specific (with a kind of component of a structure, with a kind of request, ) and allow with difficulty to represent in a sufficiently fine way the behavior of complex industrial structures. In addition, the data of modeling require an identification which must be realized carefully Modeling of reinforced concrete plate The modeling of reinforced concrete plate available in is similar to that which was initially developed in the computer code E UROPLEXUS [6] in order to deal with the problems of fast dynamics for the structures of civil engineer. The elements supports are DKT degenerated at only one point of integration in the thickness (elements DKTG). This modeling (Figure a ) use aggregate variables ( N, membrane efforts; M, bending moments;, generalized deformations and, curves) resulting of a model of total behavior (for the

12 Titre Réalisation d'une étude de génie civil sous charge[...] Responsable M'JAHAD Sofia Date 12/11/2014 Page 12/50 Clé U reinforced concrete, model GLRC_DM). This model makes it possible to simulate the behavior of reinforced concrete plates under cyclic loading. The model of behavior is described GLRC_DM more in detail in the paragraph 4.4. P la q u e e n b é to n a r m é A r m a t u r e s lo n g it u d in a le s N G L R C N N x,n y,n xy,m ε ε xx, ε yy, ε xy, κ x xx,m,κ y yy,κ,m xy xy C b e e o m p o rte m e n t d u é to n a rm é n m e m b r a n a ir e t e n fle x io n M e m b ra n e M G L R C F le x io n Figure a model of total reinforced concrete plate. 3.6 Modeling of the connections It is possible to connect various modelings between them according to the zones of the structure. One does not describe here in detail the various possible approachs of modeling. However some points of modeling are clarified Connection hull hull The connection between two perpendicular hulls can be carried out directly even if, in any rigour, the transmission of the torsional stresses is not exact (the degree of freedom of rotation generating torsion is not transmitted). If one wishes to represent the connection between two perpendicular hulls in a more precise way, it is possible to do it by means of linear relations (in order not to twice count volume with the intersection of the two hulls). For that, one must use the operator AFFE_CHAR_MECA, LIAISON_COQUE, [U ] and [U ]. The multi-layer connection hull total hull is natural because the finite elements concerned have the same degrees of freedom Connection 3D beam and hull beam

13 Titre Réalisation d'une étude de génie civil sous charge[...] Responsable M'JAHAD Sofia Date 12/11/2014 Page 13/50 Clé U There exists in an operator allowing to connect a massive part to a beam ( AFFE_CHAR_MECA, LIAISON_ELEM, OPTION= 3D_POU', [R ] and [U ]), or a hull with a beam (AFFE_CHAR_MECA, LIAISON_ELEM, OPTION= COQ_POU', [R ], [U ] and [U ]). These connections (3D_POU and COQ_POU) allow to connect two parts of grids being prolonged. It is not envisaged to model for example a connection between a hull and a beam coming to be connected perpendicularly. It is thus seen that the connections post floor are not modélisables in this manner. In the case of a connection post - floor, if one seeks to transmit the torsional stresses correctly, one recommends to only carry out the connection using hulls (cf 3.6.1, above) and if required to prolong the post with beams using a connection hull-beam (Figure a). Another possibility, used in statics, consists in adding fictitious beams in the plan of the hull to transmit the efforts coming from the post. This solution is to be used with precaution in dynamics because one must impose on the fictitious beams very low masses. So nodes of these beams are not attached to the hull, one is likely to have a disturbance of the solution (matrix of badly conditioned mass). Plancher (coque multicouche ou globale) Raccord en coque multicouche Raccord coque-poutre Poteau (poutre multifibre) Figure a connection post floor. If the torsional stresses are not dominating, one will be able to directly connect the beam and the hull (common node). However with a nonlinear behavior of the floor, that can cause a strong dependence with the grid even difficulties of convergence because of local concentrations of efforts Connection post floor One recommends to use the option PLAQ_POUT_ORTH keyword LIAISON_ELEM DE the operator AFFE_CHAR_MECA, [U ]. One must have envisaged the definition as a preliminary trace of the section of beam in the grid of the floor with a group of meshs named. One recommends to place the node end of the post in the geometrical centre of gravity of the trace of the section of beam in the floor. One will be able to consult CAS-test SDLX301 [V ] building with floor-columns. Indeed, to proceed thus makes it possible to respect as well as possible the assumptions of validity as well of the theory of the beams as that of the plates. The theory of the plates does not authorize the application of a specific effort (that creates a singularity certainly manageable in elasticity but causing difficulties into nonlinear) Connection 3D hull One generally recommends fictitiously to prolong the hull in the solid mass 3D in order to ensure the conditions kinematics enter the two parts.

14 Titre Réalisation d'une étude de génie civil sous charge[...] Responsable M'JAHAD Sofia Date 12/11/2014 Page 14/50 Clé U Massif 3D Coque Figure a connection 3D hull Modeling of nonlinear using discrete elements connections (nodes of framework; damper ) One can represent the behavior of a connection post-beam, post-floor or other by discrete elements DIS_T (translation) and DIS_TR (translation and rotation) [U ] and of the models of linear behavior (characteristic linear data in the operator AFFE_CARA_ELEM [U ]) or of nonlinear behavior (parameters materials defined with the operator DEFI_MATERIAU [U ], and called via the operand RELATION in the operator BEHAVIOR [U ]). To ensure an antiseismic protection, it is proposed to have the dampers within the structure of Génie Civil or between two buildings. These devices are designed to exert one-way efforts on connections which one can regard as specific. However as their anchoring is in general distributed, one will be able to benefit from the various structural connections presented to the paragraphs above. One will choose discrete elements with two nodes in translation DIS_T. For example, one will be able to employ DIS_ECRO_CINE [R ] allows to model in a total way (yield stress, work hardening nonlinear, ultimate load) the nonlinear behavior of nodes of framework in the steel or reinforced concrete gantries (girder-pole connection, beam-beam, beam-floor, or even veil-floor on the whole of the nodes of intersection enter the grids plates of the veil and the floor ), but by supposing a total decoupling between points along the arris line. One will be able to consult CAS-test SSND102 [V ]. DIS_VISC [R ] allows to model shock absorbers with one-way efforts with behavior of the type nonlinear generalized Zener, with law power. One will be able to consult CAS-test SSND101 [V ]. These modelings are usable within a static or dynamic analysis transitory on physical basis. It is possible also to refer directly to a special law of behavior assigned to a pair of dependent nodes (non-linearities located), within a transitory dynamic analysis on modal basis (operators DYNA_TRAN_MODAL [U ] and DYNA_VIBRA [U ]). For example, one will be able to employ 3.7 DIS_VISC [R ] shock absorbers with one-way efforts with behavior of the type nonlinear generalized Zener, with law power. One will be able to consult CAS-test SDND107 [V ]. ANTI_SISM [U ] shock absorbers with one-way efforts with behavior coupled nonlinear displacement-speed. This law results from a formulation resulting from the reference [bib22]; one will be able to consult CAS-test SDND120 [V ]. Other structural elements modélisables

15 Titre Réalisation d'une étude de génie civil sous charge[...] Responsable M'JAHAD Sofia Date 12/11/2014 Page 15/50 Clé U Modeling of a liner The liner is a metal hull placed in internal skin of the enclosure guaranteeing the sealing in the event of accidental escape. In order to model it, there exist two possibilities when the concrete is modelled by solid elements (local representation 3D), the liner is modelled directly by a hull with the real position, when the concrete is modelled by hulls (multi-layer or total), the liner is modelled by a hull offset compared to the average layer of the concrete hull Modeling of prestressing The cables of prestressed out of steel are put in tension in order to compress the structural concrete of civil engineer. One does not present here the methodology of implementation of prestressing in. One will refer to [U ] who describes in detail the realization of a study with cables of prestressing. It will be noted simply that one can use DEFI_CABLE_BP [U ] with either a massive modeling 3D or a modeling hull of the type DKT or DKTG (GLRC). Moreover, it is not necessary to make coincide the nodes of the cables and the concrete nodes. The order DEFI_CABLE_BP indeed allows to also create connections kinematics which will bind the nodes of the cable with the nodes of the concrete of the surrounding mesh. On the other hand, that generates a large number of multipliers of Lagrange which will weigh down calculation. There is thus a compromise to find between facility to carry out the grid and cost of calculation. The introduction of too a large number of relations kinematics can become problematic for a transitory seismic study already expensive in time. 3.8 Which type of modeling adopt? Introduction The choice of modeling by finite elements is closely related to the models of behavior which are associated for him. It is thus necessary to study the models usable (concrete in particular) for the seismic studies before answering entirely this question. In the chapter 4 according to, one describes the concrete and steel models available in and one takes stock (Table 4) various possible strategies of modeling. In order to choose a modeling adapted to a given problem, it is necessary to answer the following questions which type of structure does one seek to model? beam, post/flagstone, veil, simple/complex geometry, which is the size of the problem? which type of loading does one impose? membrane (traction/compression), inflection, shearing, which types of results does one seek to analyze? total sizes (displacements, efforts with the supports, spectra of floor, ), local sizes (constraints in the concrete, deformations in steels, ). One has below the advantages and the disadvantages related to each type of modeling. One answers these questions for each type of modeling. Notice It is possible to connect various modelings between them according to the zones of the structure (multi-layer hulls, total hulls, multifibre beams, ) Local modeling (solid elements)

16 Titre Réalisation d'une étude de génie civil sous charge[...] Responsable M'JAHAD Sofia Date 12/11/2014 Page 16/50 Clé U Advantages it makes it possible to finely represent complex geometries of the nodes type of frameworks or zones which one seeks to model with precision (including the whole of longitudinal and transverse reinforcement), it makes it possible to represent all the types of loading, it gives access the total and local sizes. Disadvantages it is more expensive in time CPU and the face memory that semi-total and total modelings, the solid elements mixture (concrete) and bars (steel) can introduce singularities source of nonconvergence, the delocalization is not available with DYNA_NON_LINE (cf ) Semi-total modeling multi-layer hull Advantages it is adapted to the modeling of the mean structures of standard hull (flagstone and veil), it makes it possible to represent all the types of loading, it makes it possible to reduce the size of the problem compared to a modeling in solid elements. Disadvantages it does not make it possible to represent transverse reinforcement finely, certain features are not available for this kind of modeling (great transformations, delocalization (cf ), ) Semi-total modeling multifibre beam Advantages it is adapted to the modeling of the mean structures of standard beam, it makes it possible to reduce the size of the problem compared to a modeling in solid elements, it is inexpensive in time CPU and the face memory, it can be combined with the use of other elements of structure (multi-layer hulls, ). Disadvantages it does not make it possible to represent transverse reinforcement, it is not adapted to the loadings of type shearing, it is rather adapted to the relatively simple structures. However one can plan to simulate the total behavior of relatively complex structures out of reinforced concrete (see the benchmark PUG 2000, 9.2.3) Modeling of total hull Advantages it is adapted to the modeling of the mean structures of standard hull (flagstone and veil), it is inexpensive in time CPU and the face memory, it can be combined with the use of other elements of structure (multi-layer hulls, ). Disadvantages it does not make it possible to finely represent the mechanical phenomena and the local answers (forced, ). The behavior of the structure is homogenized. the implementation of total models of behavior requires an identification of the parameters being able to be difficult. 4 Choice of the model of behavior

17 Titre Réalisation d'une étude de génie civil sous charge[...] Responsable M'JAHAD Sofia 4.1 Date 12/11/2014 Page 17/50 Clé U Introduction The concrete is a complex material made up by grains of very different scales centimetres for the aggregates, millimetres for sands, tens of microns for cements Each one of these components has different mechanical properties and the interfaces between components involve important heterogeneities in material. Moreover, the placement of the concrete during construction is likely to generate spatial of the nonuniform components. In addition, although these phenomena are not taken into account here, it is important to note that the concrete is a material multiphasic (presence of water and vapor in the interstices) and growing old. It undergoes in the course of the time of the phenomena of thermohydration, drying and creep for example Experimental observations Traction Behaviour in traction is of fragile type. One observes an abrupt reduction in the constraint when one reaches the tensile strength ( Figure a ). The order of magnitude of resistance in traction is of approximately 10 times lower than that of resistance in compression. The behavior is quasi linear and reversible until the rupture. Cracking develops in the orthogonal direction with the loading. For cyclic behaviour in traction, one observes a loss of rigidity during cycles (reduction in the elastic module in the event of refill), an appearance of unrecoverable deformations when one discharges starting from a nonlinear state. Compression The behavior of the concrete in compression is of ductile type. Observation of the curve of stress-strain in compression (Figure b ) allows to distinguish 3 phases up to levels of constraints reaching approximately 40% of the maximum constraint to the peak ( c ), the behavior is quasi elastic; from 40 to 100% of c, the behavior becomes gradually nonlinear. Near the peak the behavior is strongly unelastic. Cracking develops in the direction parallel with the loading. A voluminal phenomenon of dilation is observed (increase in the Poisson's ratio). In the event of discharge, unrecoverable deformations appear; beyond the peak, the behavior observed becomes lenitive the slope post-peak becomes negative. For cyclic behaviour in compression, one observes a loss of rigidity during cycles (reduction in the elastic module in the event of refill); an appearance of unrecoverable deformations when one discharges starting from a nonlinear state; hystereses of the cycles of load - discharge. Cyclic Observation of the curve of stress-strain in cyclic traction and compression (Figure c ) highlight two important aspects the dissymmetry of the thresholds in traction and compression ( c =10 t approximately); refermeture of the cracks. Rigidity in compression is taken again when the cracks are closed (unilateral effect).

18 Titre Réalisation d'une étude de génie civil sous charge[...] Responsable M'JAHAD Sofia Figure a experimental answer of the concrete in cyclic traction (resulting from [ 8]). Date 12/11/2014 Page 18/50 Clé U Figure b experimental answer of the concrete in cyclic compression (resulting from [12]). Figure c experimental answer of the concrete in traction and compression (resulting from [21]) Specificities of the seismic studies In the seismic studies, it is first of all imperative to correctly represent the cracking of the concrete in traction abrupt reduction in the constraint post-peak, reduction in the module of discharge, appearance of unrecoverable deformations. Being given the cyclic aspect of the seismic loadings, it is then paramount to take into account the unilateral aspect of the concrete dissymmetry of the thresholds, refermeture of the cracks (taken again rigidity). Lastly, according to the level of constraint reached in compression in the study, it is necessary to correctly represent the ductile nonlinear behavior of the concrete in compression nonlinear increase in the constraint to the peak then softening, reduction in the module of discharge, appearance of unrecoverable deformations. The models able to represent (more or less precisely) these phenomena are the following ENDO_ISOT_BETON [R ], ENDO_ORTH_BETON [R ], MAZARS_GC [R ], GLRC_DM [R ].

19 Titre Réalisation d'une étude de génie civil sous charge[...] Responsable M'JAHAD Sofia Date 12/11/2014 Page 19/50 Clé U Other concrete models exist in but their use is not recommended. It is the case for example models of Mazars [R ], Double Drücker-Prager [R ] and BETON_REGLE_PR [U ] (elastic nonlinear and thus nondissipative). These models do not make it possible to simulate refermeture of the cracks and are not of this fact not adapted to the cyclic case of loading. 4.2 Algorithm of Borst Before detailing the various models available, one recalls that there exists in a general method of integration of the nonlinear models of behaviors 3D in plane constraints (method of Deborst, [R ]). This one consists in making worthless the constraint zz in the resolution of the problem. The method of Borst moreover was generalized with the models 1D. One imposes an assumption of uniaxial constraint ( yy = zz =0 ). This fact the models of behaviors 3D can be used for modelings of multifibre beams. The keyword RESI_CPLAN_RELA allows to slacken the convergence criteria of the method (checking of the condition of plane constraints). One advises for the seismic studies to preserve RESI_CPLAN_RELA with 10 6 (value by ). Moreover, it is recommended to take a value of RESI_CPLAN_RELA identical to that of the relative residue on balance RESI_GLOB_RELA. If the convergence criteria are not respected, calculation stops (or there is recutting of the step of time). One can use the keyword ITER_CPLAN_MAXI in order to better satisfy the condition with plane constraint. By this parameter is fixed at 1 (value sufficient for the nonlenitive models). If one observes problems of convergence during iterations due to the algorithm of Borst, one recommends to increase the value of this parameter. One will be able to take for example ITER_CPLAN_MAXI equal 5 or 10. If convergence criteria N are not respected after the defined iteration count, calculation stops (or there is recutting of the step of time). The choice of the value of ITER_CPLAN_MAXI depends on the operator of resolution used in the method of Newton (cf 7.2.3) if the tangent matrix is used, one can take the value by initially ITER_CPLAN_MAXI=1. In the event of nonconvergence, one will be able to test a higher value; if the elastic matrix or the matrix of discharge is used it is recommended to take a value directly of ITER_CPLAN_MAXI higher (one advises to make between 5 and 10 iterations) if not the condition of plane constraints is likely not to be respected. 4.3 Description of the models of behavior of steel One briefly presents in this paragraph the models of behavior usable to represent the material steel within the framework of the seismic studies. One summarizes in Table 1 various models available according to modeling used. Modeling BAR Modeling multifibre POU_D_EM VMIS_ISOT_LINE X X VMIS_CINE_LINE X X PINTO_MENEGOTTO X X GRILLE_ISOT_LINE Modeling GRILLE_MEMBRANE GRILLE_EXCENTRE X

20 Titre Réalisation d'une étude de génie civil sous charge[...] Responsable M'JAHAD Sofia Date 12/11/2014 Page 20/50 Clé U GRILLE_CINE_LINE X GRILLE_PINTO_MEN X Table 1 models of behavior of steel. One does not detail in this document the models of behavior of steels. These models are described in [R ] and [U ]. Let us note simply that the model of Von Mises with symmetrical linear isotropic work hardening (VMIS_ISOT_LINE) and the model of Von Mises with symmetrical linear kinematic work hardening ( VMIS_CINE_LINE) are classical elastoplastic models; the model PINTO_MENEGOTTO allows to represent the uniaxial elastoplastic behavior of the reinforcements of the reinforced concrete. This model translates for it not linearity of the work hardening of the bars under cyclic loading and takes into account the Bauschinger effect. It makes it possible of more than simulate the buckling of the reinforcements in compression. In the seismic studies, the levels of deformation in steels remain most of the time relatively low. The use of an elastoplastic model of behavior of Von Mises with kinematic work hardening is sufficient. When one wishes to represent in a finer way the plastic behavior of steels, one can use the model of Pinto - Menegotto. 4.4 Description of the models of concrete behavior One presents in this paragraph the models of behavior usable to represent the material concrete within the framework of the seismic studies. One summarizes in Table 2 various models available according to modeling used. massive 3D 2D plane Contraintes ENDO_ISOT_BETON X X (De Borst) X (De Borst) ENDO_ORTH_BETON X X (De Borst) X (De Borst) MAZARS X X GLRC_DM 2D Global X 1D Fibre X X Table 2 models of concrete behavior Model ENDO_ISOT_BETON This three-dimensional model ([R ] and [1 ]) is based on the theory of the damage (Figure 4.4.1a and Figure b ), the variable of damage being scalar (isotropic damage). The principal interests of this model are its simplicity (two or three parameters of entry) and its thermodynamically founded theoretical writing. Determination of the parameters of the model There are 3 parameters for the model ENDO_ISOT_BETON [U ] besides the classical elastic parameters E (Young modulus) and (Poisson's ratio) SYT is the maximum constraint in simple traction. The value of this parameter is a classical experimental data of the concrete. SYT in general about 1 to 4 MPa is (Figure a ); D_SIGM_EPSI is the slope of the curved post-peak in traction (Figure a ). This parameter plays a paramount role in the answer of the structure. It is necessary to use a slope post-peak sufficient stiff so that the propagation of the damage is correctly represented by calculation. However the use of a realistic slope post-peak involves important problems of convergence.

21 Titre Réalisation d'une étude de génie civil sous charge[...] Responsable M'JAHAD Sofia Date 12/11/2014 Page 21/50 Clé U Indeed, more the behavior of material is lenitive plus convergence becomes difficult. The value of this parameter is roughly from -0.2 to -0.6 times the Young modulus. In practice, one must often decrease the value of this parameter in order not to have nona convergence of calculation; SYC is the yield stress in simple compression. It is pointed out that the model ENDO_ISOT_BETON is quasilinear in compression (not nonlinearity in compression). The use of this optional parameter makes it possible to return the threshold of damage depend on containment [R ]. For constraints lower than SYC, there is a behavior of the realistic linear concrete. Beyond SYC (yield stress), the model is not representative any more of the behavior of the concrete. SYC approximately 10 times the constraint with the peak of traction is worth SYT, in absolute value. If this parameter is not used, the yield stress in compression is taken about SYT, which is too weak. The use of this parameter is thus advised more especially as it does not worsen a priori the problems of convergence. Internal variables V1 value of the damage, V2 indicator of damage (0 for the elastic mode (null damage), 1 if damaged, 2 if broken (damage equal to 1)). ENDO_ISOT_BETON allows to model the fragile behavior of the concrete in traction; the unilateral behavior of the concrete (refermeture of the cracks by a resumption of rigidity). ENDO_ISOT_BETON does not allow to model nonlinear behaviour in compression; anisotropy of material; unrecoverable deformations in compression and traction; loops of hysteresis in load - discharge. Figure a relation in cyclic traction ENDO_ISOT_BETON Figure b relation in cyclic traction and compression ENDO_ISOT_BETON. Model ENDO_ORTH_BETON This three-dimensional model [R ] is based on the theory of the damage ( Figure a and Figure b ). One chooses to introduce two variables of damage a tensor D of a nature 2 relating to the damage created in traction. This makes it possible orthogonally to describe the privileged orientation of the cracks to the direction of greater traction (anisotropic damage); a scalar D relating to the damage created in compression (isotropic damage).

22 Titre Réalisation d'une étude de génie civil sous charge[...] Responsable M'JAHAD Sofia Date 12/11/2014 Page 22/50 Clé U There are 6 parameters for the model ENDO_ORTH_BETON [U ] (TABLE 3), besides the classical elastic parameters E (Young modulus) and (Poisson's ratio). Definition Dimension ALPHA Parameter of coupling without k0 K0 Constant part of the threshold MPa k1 K1 Parameter of the threshold MPa k2 K2 Parameter of the threshold without B ECROB Blocked voluminal energy relating to traction MPa d ECROD Blocked voluminal energy relating to compression MPa Table 3 parameters of the model ENDO_ORTH_BETON. Determination of the parameters of the model for the parameter, a value between 0.85 and 0.9 are recommended; parameters k 0 and B must be identified simultaneously on a simple tensile test; parameters d, k 1 and k 2 must be identified simultaneously on an unconfined compression test. One will imperatively refer to documentation [R ] and the CAS-tests [V ] in which one discusses in detail the choice the parameters the model. Internal variables V1 with V6 tensor of damage of traction; V7 damage of compression. ENDO_ORTH_BETON allows to model the fragile behavior of the concrete in traction; nonlinear behaviour in compression; the unilateral behavior of the concrete (refermeture of the cracks by a resumption of rigidity); anisotropy of material. ENDO_ORTH_BETON does not allow to model unrecoverable deformations in compression and traction; loops of hysteresis in load - discharge.