Constitutive law (in 2D) for steel-concrete connection: JOINT_BA

Transcription

1 deault itre : Loi de comportement (en 2D) pour la liaison acier [] Date : 26/03/203 Page : /24 Constitutive law (in 2D) or steel-concrete connection: JOIN_BA Summarized: Constitutive law JOIN_BA describes the phenomenon o degradation and racture o connection between the steel bars and the concrete, in reinorced concrete structures his documentation presents the theoretical writing in the thermodynamic rame and the numerical integration o the model, as well as the parameters which manage the model For his use, one will lean on the inite elements o the type joined (see the document [R30609]) already existing in the code Licensed under the terms o the GNU FDL (

2 deault itre : Loi de comportement (en 2D) pour la liaison acier [] Date : 26/03/203 Page : 2/24 Contents Introduction3 2 Short description o the bond steelbéton3 3 Writing théorique5 3 Presentation o the modèle5 32 Analysis o the damage in the direction tangentielle7 33 Analysis o the damage in the direction normale9 34 Analysis o the contribution o the crack riction by glissement0 35 Abstract o the équations 36 Form o the matrix tangente2 4 Integration numérique3 4 Computation o the part riction o cracks with an integration method implicite3 42 the algorithm o résolution4 43 Local variables o the modèle5 5 Parameters o the loi5 the 5 parameters initiaux5 5 the parameter hpen 5 52 the parameter G or modulate stiness o the liaison6 the 52 parameters o endommagement7 52 the limit o elastic strain or threshold o perect dependancy 7522 the parameter o damage AD or the transition o the small strains to great slidings 7523 the parameter o damage BD 8524 limit o strain or threshold o great slidings 9525 the parameter o damage A 2D 9526 the parameter o damage B 2D 2053 parameters o damage on the direction normale20 53 limit o strain N or threshold o large displacements the parameter o damage A DN the parameter o damage B DN 2054 parameters o riction 254 the material parameter o riction o cracks 2542 the material parameter o kinematic hardening 2543 the parameter o inluence o containment 2255 Abstract o the paramètres22 6 Bibliographie24 7 Vériication24 8 Description o the versions o the document24 Licensed under the terms o the GNU FDL (

3 deault itre : Loi de comportement (en 2D) pour la liaison acier [] Date : 26/03/203 Page : 3/24 Introduction constitutive law JOIN_BA describes the phenomenon o degradation and racture o existing connection between the steel bars (smooth or ribbed) and the concrete surrounding it Key point or the structural design out o reinorced concrete, the purpose o the modelization o steel-concrete connection is the representation as well as simpliication o this complex phenomenon o interaction between the two materials which develops in the interace and which undergoes an increasing degradation when certain thresholds o strength are exceeded, speciic or each material he structural models which do not take into account the linkage eects, are generally unable to predict the localization o cracks as well as the networks created In addition, the degradation o the stiness o connection increases the period o vibration, reduced the capacity o dissipation o energy and conduit to a signiicant redistribution o the internal orces (according to Bertero, 979, c [bib2]) Constitutive law JOIN_BA is described in the rame o the thermodynamics o the irreversible processes: the writing and L `use o a classical model o material coupling cracking and riction make it possible to integrate in a robust way o the ine mechanisms nonlinear concomitant into the particular description o the kinematics o sliding his last point enables us not to resort to the classical modelizations o type contact very oten used in this context in spite o the many sources o numerical instabilities hus, in monotonic loading the taking into account o the coupling normal orce shears make it possible to treat cases o strong multiaxial pressures; in cyclic, the hysteretic behavior and the corresponding dissipations are expressed thanks to the coupling between the state o damage and kinematic hardening he use o an implicit scheme makes it possible to obtain a robust implementation he paragraph [ 2] described in short orm the phenomenon o the bond steel concrete he paragraph [ 3] presents the thermodynamic writing o the constitutive law, while the paragraph [ 4] speciies the stage o numerical integration o the model he parameters which manage the model and which could be obtained starting rom the properties o the implied materials, are described in the paragraph [ 5] 2 Short description o steel-concrete connection Conceptually, the phenomenon o connection corresponds to the physical interaction o two dierent materials, which occurs on a zone o interace by allowing the transer and the continuity o the orces and the stresses between the two bodies in contact In the case o the reinorced concrete structures, this phenomenon is also known as the stiness o tension which develops around an element o reinorcement, partially or completely drowned in a volume o concrete he tensile orces which appear inside the reinorcement are transormed into shearing stresses on surace, and are transmitted directly to the concrete in contact which will balance them inally, and vice versa he response o the group will depend on the capacity o the concrete to become deormed as much as steel, since steel will tend to slip inside the concrete surrounding it he phenomenon o connection corresponds to this capacity o the concrete to become deormed and to be degraded locally by creating a species o layer, or wraps, around the reinorcement, whose kinematical and material properties dier rom those moreover concrete or used reinorcement he phenomenon can be broken up into three well deined mechanisms: a chemical dependancy o origin, a mechanism o riction between two rough suraces (steel-concrete or concrete-concrete), a mechanical action created by the presence o the veins o the steel bar on the neighbouring concrete According to this decomposition, one can deduce clearly that or a bar smoothes, the dominating mechanism is riction between the two materials, while or a bar ribbed (in French usually called reinorcement ha: High Dependancy ), the mechanism dominating is the mechanical interaction between suraces When reinorcement is consisted the strands with steel wire ropes, it is possible to Licensed under the terms o the GNU FDL (

4 deault itre : Loi de comportement (en 2D) pour la liaison acier [] Date : 26/03/203 Page : 4/24 control or combine the various mechanisms since they are unction directly o the surace o the cables Licensed under the terms o the GNU FDL (

5 deault itre : Loi de comportement (en 2D) pour la liaison acier [] Date : 26/03/203 Page : 5/24 Connection will undergo a dierent degradation according to the type o loading applied, either monotonous, or cyclic In addition, among the most important parameters which inluence the behavior o connection, one can quote: ) characteristics o the loading, 2) geometrical characteristics o the steel bar, 3) spacing between active bars, 4) characteristics o the concrete, 5) containment by passive reinorcement, 6) side pressure At the time o the study o a cylindrical bar drowned in an ininite medium, one can identiy the surace o discontinuity where one will place the linkage eects, which develops in a certain zone o concrete issured and crushed around the steel bar A a given moment, this surace will correspond to cylindrical crack created during the coalescence o cracks o shears By looking at the network o cracks, one can suppose that, in ideal conditions, the plane o cracking is always perpendicular (normal direction) to the surace o the bar and parallel (tangential direction) to its longitudinal axis (see [Figure 2-a]) hat enables us to project the components o displacement on the normal and tangential direction o the plane o cracking, and consequently to obtain the corresponding strains and stresses Sd béton écrasé glissé F τ 0 C D E S CORPS A A A2 n B t B2 CORPS B CORPS A A n A2 B B2 CORPS B t Appear 2-a: real description o the phenomenon o connection and simpliication inite elements: coordinated in the local coordinate system o the element o interace used as support o theoretical model 3 JOIN_BA Writing the ormulation presented here was developed in the rame o the thermodynamics o the irreversible processes; it gives the constitutive relation between the normal orce, the shearing stress and the sliding by considering the inluence o the cracking o the concrete, riction and the various couplings in the phenomenon For that, the constitutive relations which connect the tensor o the stresses and the tensor o the strains must include: the cracking o the material o interace by shears the inelastic strains because o sliding the hysteretic behavior due to riction the coupling between the tangential response and the normal stresses 3 Presentation o the model One is placed in the rame o a ormulation planes in 2D, in the deinite local coordinate system [Figure 2-a] he tensors o the stresses σ and the strains ε are written: Licensed under the terms o the GNU FDL (

6 deault itre : Loi de comportement (en 2D) pour la liaison acier [] Date : 26/03/203 Page : 6/24 ε= ε N ε τ ε τ 0 and = N éq 3-0 where σ N is the normal stress and σ τ is the shear stress o the element o interace; ε N corresponds to the normal strain and ε τ the tangential strain he normal strain in the tangential direction with the interace is regarded as null his mode o strain or an element o dependancy is with strain energy null he normal and tangential behaviors being regarded as uncoupled on the level rom the state, the thermodynamic potential obtained rom the ree energy o Helmholtz is expressed in the ollowing way: = 2 [ N E N N E D N N G D G D 2 ] H z éq 3-2 where ρ is the density, E is the Young modulus, D N is the local variable o normal damage and D the local variable o tangential damage, both being related to the cracking and ranging between 0 and G is the modulus o stiness or o shears, ε is the unrecoverable deormation induced by sliding with riction o cracks, α is the local variable o kinematic hardening, γ is a material parameter and z, the variable o pseudonym isotropic hardening by damage, with its unction o consolidation Η z - and + deine respectively the positive and negative parts tensor considered One can notice in the equation [éq 3-2] that in the normal direction, the damage will be activated during the appearance o the positive strains produced by tensile orces, while i the strains are negative because o eects o compression, the behavior will remain elastic With regard to the tangential part o the behavior, one can recognize a classical coupling elasticity-damage as well as a new term allowing to associate with the state elasticity-endommageable, a state o sliding with riction he coupling between sliding and cracking is possible thanks to the presence o the variable o damage like multiplier in the second element o the right part o the equation [éq 3-2] he state models are obtained classically by derivative o the thermodynamic potential, and thus make it possible to deine the associated thermodynamic variables he normal stress is expressed like: ={ N = E N si N 0 éq 3-3 N D N E N si N 0 and the total shear stress like: = =G D G D éq 3-4 One can also deine the shear stress due to the sliding with riction (strain ε τ s ): Note:: = =G D éq 3-5 Such a ormulation moves away amply rom a classical ormulation o coupling plasticity damage he assumption bringing to the introduction o the damage into the stress by sliding is based on an Licensed under the terms o the GNU FDL (

7 deault itre : Loi de comportement (en 2D) pour la liaison acier [] Date : 26/03/203 Page : 7/24 experimental observation which is that all the inelastic phenomena in a brittle material come rom the growth o cracks he rate o energy restored by damage-riction can be written like: Υ = ρ ψ D = 2 ε G ε 2 ε ε G ε ε = Υ D Υ éq 3-6 In this last statement, Υ D corresponds to the rate o energy restored by damage and Υ to the rate o energy restored by riction o cracks he state model o kinematic hardening brings to the deinition o the stress o recall: X = = éq 3-7 Concerning the model o hardening o the isotropic damage, it is expressed by: Ζ = ρ ψ =Η' z éq 3-8 z It is necessary or us now in the connection to clariy in a more detailed way the evolution o the damage mechanism, in other words to speciy the statement o Η z For a low value o damage, the mechanism which prevails is the interaction o the concrete with the veins o the steel bar, while or a value much larger, it is the riction between the concrete and the steel which prevails During the evolution o the damage, 2 principal phases could be identiied: ) the irst phase corresponds to transverse a stable crack growth related to the presence o veins on steel (positive apparent hardening o the law o evolution), 2) the second does not utilize more but the coalescence o these transverse cracks bringing not to consider but the mechanisms o riction more (negative hardening towards a residual stress o riction) 32 Damage in the tangential direction the law o evolution o the damage analyzes is divided into three stages: region o perect dependancy, region o transition o small strains to the great slidings, region o maximum strength o connection and degradation until ultimate residual strength o identiy these regions, two thresholds are established: ) the threshold o perect dependancy ε, 2) the threshold o continuity beore coalescence o cracks ε 2 hus, by taking again the statements related to the damage with knowing that o rate o energy restitution [éq 3-6] and that o the local variable associated with isotropic hardening [éq 3-8], one can note: ) a true separation between L damage and the riction o the cracks (what makes it possible to amend only the law D evolution o the damage without aecting the part riction ), 2) the partition in two parts o isotropic hardening since one has two dierent stages in the damage Licensed under the terms o the GNU FDL (

8 deault itre : Loi de comportement (en 2D) pour la liaison acier [] Date : 26/03/203 Page : 8/24 From now on we will write or hardening related to the variable o damage: ={ Z = Z, si 2 =H z éq the 32- z Z Z 2, si 2 components Z and Z 2 are expressed in the ollowing way: Z =[ Y A D Z 2 =[ y 2 A 2D G 2 ln z ]2 z ] z éq 32-2 éq 32-3 One also deines the unction threshold D which depends on D and which S `writes like: D = D Z 0 éq the 32-4 thresholds which manage the law o evolution o the damage are also expressed in terms Υ D (see [Figure 32-a]) he irst statement corresponds to the threshold o perect dependancy and S `written: Y =Y elas = 2 G éq 32-5 Where Y is the initial threshold o damage deined according to the limiting strain o perect dependancy, which will correspond to the limiting strain o shears or tension concrete beore the initialization o the damage In addition, Y 2 is the threshold o initiation o coalescence o microphone - cracks which is deined according to the initial tangential strain o the great slidings ε 2 : Y 2 = G éq 32-6 Licensed under the terms o the GNU FDL (

9 deault itre : Loi de comportement (en 2D) pour la liaison acier [] Date : 26/03/203 Page : 9/24 Figure 32-a: construction o the unctions thresholds in terms o energy the laws o evolution o the local variables in the rame o the standard associated models make it possible to obtain derivative o the multiplier o damage D : Ḋ= D D = D Y et ż= D D D Z = D éq 32-7 By means o in addition the condition o consistency, one obtains the statement o the damage: D = Y Y D exp{ A D [ *{ A 2D Y D Y B 2D } 2 B D Y G D Y ] } éq 32-8 In this statement, one can identiy the part which corresponds to the region o the transition o A D the small strains to the great slidings with two parameters: and B D, as well as the inal part o damage in mode 2, with the parameters A 2D and B 2D It should be noted that the relation Y D Y is managed by a unction o Macaulay, ie this dierence in energy must be always positive or null he unctions which manage isotropic hardening in the tangential direction are expressed like: Z =Y D Y ; éq 32-9 Z 2 ={ 0, si Y Y D Y 2 éq 32-0 Y D Y 2, si Y 2 Y D According to these statements, one can notice that Ζ 2 is not taken into account in the region o transition rom the small strains to great slidings 33 Damage in the normal direction analyzes the two most important mechanisms which can appear on the normal direction are the detachment between the concrete and the steel bars, and the penetration o reinorcement in the body o the concrete hese two conditions can be interpreted respectively like an opening or a closing o crack, and can be described by a particular constitutive law in the normal direction uncoupled rom the tangential behavior In order to simpliy the resolution or compression between suraces, one decided to allow a small penetration between those, which implies that N 0, and by adopting an elastic constitutive law, one will have: N =E N si N 0 éq the 33- cases o the decoherence o the interace can be described by a behavior endommageable in the normal direction, that is to say: Licensed under the terms o the GNU FDL (

10 deault itre : Loi de comportement (en 2D) pour la liaison acier [] Date : 26/03/203 Page : 0/24 N = D N E N si N 0 éq 33-2 with D N scalar variable o the damage in the normal direction, calculated with the ollowing statement: ={ 0 si N N D N éq 33-3 si A DN Y DN Y N N N B DN In this statement, two material parameters, A DN and B DN, control decoherence by the damage in tension o the concrete In addition, Y N is the threshold o damage deined in term o energy, equivalent to the elastic threshold in the normal direction Y elas N and which is expressed like: Y N =Y elas N = 2 N E N éq 33-4 N being limiting strain o perect dependancy, which corresponds to the limiting strain o the concrete in tension beore the initialization o the damage It should be mentioned that when the detachment or the crack opening reached the maximum value o strength to tension, no shear orce must be transmitted between the two materials: it east is the single condition in which the scalar variable o damage in the tangential direction becomes because o the damage in the normal direction 34 Analyzes contribution o the crack riction by sliding With regard to the sliding part o the ormulation, one supposes that it has a behavior pseudo-plastic, with nonlinear kinematic hardening Initially introduced by Armstrong & Frederick, 966, c [bib], nonlinear kinematic hardening makes it possible the ormulation to overcome the principal disadvantage o the model o kinematic hardening o Prager, namely, the linearity o the state model which connects the orces associated with kinematic hardening Here, the nonlinear terms are added in the potential o dissipation he criterion o sliding takes the classical shape o the unction threshold o Drucker-Prager which takes into account the eect o radial containment on the sliding: = X c I 0 éq 34- Here X is the stress o recall, c is a parameter related to the material, translating the inluence o containment, while I corresponds to the irst invariant o the tensor o the stresses, which or our case is expressed like: I = 3 r [ ]= 3 N éq 34-2 Licensed under the terms o the GNU FDL (

11 deault itre : Loi de comportement (en 2D) pour la liaison acier [] Date : 26/03/203 Page : /24 In addition, the initial threshold or the sliding is 0 Moreover, by considering the principle o maximum plastic dissipation, the laws o evolution can be derived rom the statement o the plastic potential which is: φ p = σ Χ c I 3 4 a Χ 2 éq 34-3 Where a is a material parameter It should be mentioned that the quadratic term Χ makes it possible some to introduce the non-linearity o kinematic hardening he laws o evolution or the strain o sliding as or kinematic hardening take the ollowing shapes: the multiplier o sliding = p et = p X éq 34-4 is calculated numerically by imposition o the condition o consistency 35 Summarized equations We show here, abstract o the equations which constitute the constitutive law o the bond steel - concrete: Free energy o Helmholtz Function threshold State models Dissipation Laws o evolution = 2 [ N E N N E D N N G D G D 2 ] H z D =Y D Y Z 0 ; = X c I 0 ={ E N si N 0 N D N E N si N 0 ; =G D G D ; =G D Y = D = Y D Y ; X = = ; Z = z = H ' z Ḋ= D D Y D = D = p ; = p X ; ż= D D Z = D ; Licensed under the terms o the GNU FDL (

12 deault itre : Loi de comportement (en 2D) pour la liaison acier [] Date : 26/03/203 Page : 2/24 36 Form o the tangent matrix In order to ensure the robustness and the eectiveness o the model in the numerical establishment and or the total analysis o massive structures, it is necessary to calculate the tangent matrix, which can be given rom the ollowing statement: éq 36- = H Ater some analytical computations, one can by means o deduce the statement rom the tangent modulus the condition o consistency and the respective laws o evolution: With G g / H = / p / G D g / X 2 / 2 p / X = D Y D Y D =[ g h ' ' g h g ' h h ] G 2 Where, g and h are the ollowing unctions, obtained thanks to [éq 32-8]: éq 36-2 éq 36-3 g =exp{ A D [ = Y Y D éq B G D Y D Y ] } éq 36-5 h= A 2 D Y D Y B 2D éq 36-6 Note:: In practice in Aster, the tangent matrix was not established, only the secant matrix is used either H = E D 0 N 0 G D Licensed under the terms o the GNU FDL (

13 deault itre : Loi de comportement (en 2D) pour la liaison acier [] Date : 26/03/203 Page : 3/24 4 Numerical integration the separation in two parts in the ormulation: damage sliding, enables us to treat each one o it separately hus, the integration o the damage part is carried out explicitly by the deinition o two suraces threshold On the other hand, the sliding part is solved in an implicit way by a classical method with knowing the algorithm o the type return-mapping suggested by Ortiz & Simo, c [bib4], which will ensure convergence in an eective way 4 Computation o the part riction o cracks with an implicit integration method the eects on connection associated with the phenomenon o riction with cracks can be calculated in the rame o a behavior pseudo-plastic with a nonlinear kinematic hardening For the establishment with the integration method suggested, we will carry out a linearization o the unction threshold around the current values o the associated local variables With the iteration i, surace threshold is written: i i = : i i i X : X i X i 0 éq 4- According to the equations [éq 3-7], [éq 3-8], and [éq 36-5], one a: X = = p X = G D Which one can discretize in the ollowing way: = G D p X = X i X i = = X p éq 4-2 éq 4-3 éq 4-4 = i i = G D = G D éq 4-5 By combining these statements with the statement o surace threshold and by writing that is equal to zero, one can deduce the increment rom multiplier to each iteration i : p i = i G D p i i X p X i éq 4-6 Ater obtaining the value o Δλ, one can substitute it in the equations [éq 4-4] and [éq 4-5] in order to bring up to date the thermodynamic orces and X he iterations will have to continue until the moment when the condition o consistency is checked Licensed under the terms o the GNU FDL (

14 deault itre : Loi de comportement (en 2D) pour la liaison acier [] Date : 26/03/203 Page : 4/24 42 he algorithm o resolution In a general way, one seeks to check the equilibrium o structure at every moment, in an incremental orm As clariied previously, or the damage a simple scalar equation makes it possible to obtain the corresponding value, which makes it possible to avoid a recourse to the iterative methods On the other hand, an iterative method is applied or integration o the riction part o cracks hen, the algorithm is the ollowing: ) Geometrical reactualization: ε = ε + s ( ) ( ) u n+ 2) Elastic prediction: ( 0) ( ε ) = ( ε ) n+ n ; e ( 0) ( ε ) = ( ε ) ( ε ) n+ n+ n+ 0 n = n n ; 3) Evaluating o the threshold: ( 0) 0 ( )? φ n+ YES: ( ε ) ( ) i SO, end o the cycle; so NON, beginning o the iterations (0) n n+ = ε + NON: i = 0 = 4) Plastic correction: i / n ( e ) ( e ) (0) ε ; = ε n+ n+ i n ; 0 n = n σ ; + = σ (0) n n+ G D p i i / n / X n p i / X n i i n = n G D p / i p / X i i i n = n p / X i 5) Checking o convergence: i 0 n OL n? i SO, end o the cycle; so NON, to continue the iterations in (iv) YES: ( i + ) σ n+ = σ n+ ; i n = n ; e e ( ε ) = ε ( σ n n ) n+ +, α + ; e ( ε ) = ( ε ) ( ε ) n+ n+ n+ NON: i = i + Licensed under the terms o the GNU FDL (

15 deault itre : Loi de comportement (en 2D) pour la liaison acier [] Date : 26/03/203 Page : 5/24 43 Local variables o the model We show here the local variables stored in each Gauss point in the implementation o the model: Number o local variable Meaning physical D N : Scalar variable o the damage in the normal direction 2 D : Scalar variable o the damage in the tangential direction 3 z : Scalar variable o isotropic hardening or the damage in mode 4 z : Scalar variable o isotropic hardening or the damage in mode 2 5 : Strain o sliding cumulated by riction o cracks 6 : Value o kinematic hardening by riction o the cracks 5 Parameters o the model the constitutive law presented here is controlled by 4 parameters, o which 3 manage and the response in the normal direction others aect the response in the tangential direction In addition, the Young modulus is recovered starting rom the elastic data provided by the operator ELAS, which must always appear in the command ile hese parameters, or the analytical statements which make it possible to obtain them, were obtained or determined rom the computational simulation o the experimental tests carried out by Eligehausen and al, 983, c [bib3] he realization o multiple simulations made it possible question to determine a relation between the geometrical and material characteristics o the materials in (steel and concrete) and the parameters which manage interace the model 5 he initial parameters 5 the parameter hpen the joined element unctioning on the notion o jump o displacement, it is necessary to introduce a dimension characteristic o the zone o degraded interace making it possible to deine the notion o strain in the interace With this intention it was introduced the principle o penetration between suraces: the parameter hpen makes it possible to deine this zone surrounding the bar o steel his parameter corresponds to the possible maximum penetration which depends on the thickness o the compressed concrete - crushed At the same time, hpen dissipation o energy in the element as well as the kinematics o the sliding manages In order to give a reerence to the user or the choice o this parameter, one proposes to calculate it starting rom the diameter o the bar d b and the relative area o the veins α sr deined by: α sr = k F R sin β π d b c éq 5- where k is the number o veins on the perimeter; F R the transverse area o a vein; β is the angle between the vein and the axis longitudinal o the steel bar; and c between veins center in center is the measured distance Finally, hpen will be calculated with the statement: h pen =d b sr éq 5-2 Licensed under the terms o the GNU FDL (

16 deault itre : Loi de comportement (en 2D) pour la liaison acier [] Date : 26/03/203 Page : 6/24 According to Eligehausen and al, the reinorcements usually used in the United States have values o α sr enters 005 and 008 For the smooth bars, since one needs a small value o hpen, one proposes values o sr enters 0005 and 002 he ollowing table gives the values o hpen according to the diameter o the bar: Diameter ( mm ) relative Area Hpen ( mm ) Description 8 00 (008) 0 Bar commercial smoothes ribbed commercial Bar Bars commercial ribbed ribbed commercial Bar ribbed commercial Bar the unit o hpen must o course correspond to the unit used or the mesh 52 he parameter G or Generally modulates stiness o connection, because o diiculty in measuring the strains by shears, the modulus o stiness o a material is calculated starting rom the Poisson's ratio and Young modulus, current parameters obtained in experiments However, or our case, the interace is a pseudo-material whose characteristics must depend on the properties corresponding to the materials in contact, steel and concrete Being given that the material which one expects to damage is the concrete, one proposes to initially use or connection the same value o G or the studied concrete but it can be higher up to a value similar to the value o the Young modulus E, when one increases the value o hpen In the case o reinorcements with stiness higher than those o the current commercial bars (because o a provision or special geometry o the veins), one can make a correction o the value chosen, by multiplying the modulus o stiness by a coeicient o correction calculated starting rom the relative areas o the commercial bars, with the statement: C arm = hen, the modulus o stiness o connection G will be: α SR barre éq 52- α SR barre comm G liai =C arm G beton éq 52-2 In the last statements, G lia is the modulus o stiness o connection; G beton is the modulus o stiness o the concrete; C arm is the coeicient o correction per reinorcement; SR barre, relative area o the veins o the bar concerned; and α SR barre com, relative area o the veins o the commercial bar o the same diameter (preerably, 008 ) Licensed under the terms o the GNU FDL (

17 deault itre : Loi de comportement (en 2D) pour la liaison acier [] Date : 26/03/203 Page : 7/24 52 he parameters o damage 52 the limit o elastic strain or threshold o perect dependancy to deine the threshold o perect dependancy, one considers that the damage by shears must during initiate the going beyond a certain threshold o strain So one proposes to adopt the limiting strains o the concrete in tension, ie, enters 0 4 and , which correspond to the shearing stresses enters 05 and 4 MPa in perect dependancy 522 he parameter o damage AD or the transition o the small strains to the great slidings In this region, the law o evolution o the damage is expressed in term o strains and its construction depends on the deinite elastic slope or the linear behavior (shearing stress versus strain) in the region o perect dependancy: this parameter controls the value o the stress compared to the sliding in the transition o small strains to the great slidings he determination o the value o this parameter is a key and delicate point model, since the evolution o the damage must be carried out with certain conditions noticed by several researchers; or example: the strength o connection is directly proportional to the compressive strength o the concrete However, as one increases the strength o the concrete, the behavior becomes more rigid, bringing to the brittle racture o connection, the particular stiness o the reinorcement, which is related on the diameter and the quantity o the veins on surace, must increase the strength o connection, the relation between the elasticity moduli o the two materials concerned must manage the kinematics o connection directly From computational simulations which one carried out, one observed that this value is located between a minimum o and a maximum o 5, and that it will have to be adjusted according to the selected test o reerence Optionally, one proposes a statement which makes it possible to adopt an initial value and which depends on the particular characteristics o the materials: A D = SR ' c 30 E a E b éq 522- In the last statement, E b will be calculated with the statement provided in the section A2, 2 o BAEL' 9: In the two last statements, one a: E b =000 ' c / 3 éq ) ' c, compressive strength o the concrete in MPa ; 2) E a, elasticity modulus o steel, in MPa ; 3) E b, elasticity modulus o the concrete, in MPa ; 4) α sr, relative area o the veins o the bar concerned Licensed under the terms o the GNU FDL (

18 deault itre : Loi de comportement (en 2D) pour la liaison acier [] Date : 26/03/203 Page : 8/24 [Figure 522-a] gives some a graphic comparison COMPARAISON DE AD - Loi locale de la liaison - ad=0 ad=29 ad=38 20 SXY (MPa) -contrainte de cisaillement AO SXY (MPa) deo EXY deo EXY Appear 522-a: Comparison o A D : the purpose o growth o the strength o connection 523 the parameter o damage B D his parameter is sotening the shape o the curve o behavior, like acilitating the transition rom the elastic slope towards the nonlinear region It can have a value understood enters 0 and 05 (never higher than 05 since it is the equivalent o the square root o the ormula) One can advise to adopt the value o 03 or ordinary computations (See [Figure 523-a]) 25 VARIAION DE Bd - Loi locale de la liaison - 20 SXY (MPa) -contrainte de cisaillement AO Bd=0 Bd=02 Bd=025 Bd=03 Bd=05 SXY (MPa) deo EXY deo EXY Appear 523-a: Comparison o B D : Modiication o the curvature Licensed under the terms o the GNU FDL (

19 deault itre : Loi de comportement (en 2D) pour la liaison acier [] Date : 26/03/203 Page : 9/ the limit o strain 2 or threshold o the great slidings According to several authors, the great slidings are overall higher than mm displacement, but that is an indicator which depends on the orm and dimensions o the specimens tested; thereore, it is proposed that this strain never exceeds 00 (adimensional value) In a more precise way, one proposes to apply the ollowing statement: 2 = h pen 2 A n D C A D = n h pen 2 A 4 D 0 9 A D 4 éq 524- In this statement, one applied a sigmoid unction whose coeicients C and n make it possible to adjust the kinematical eect o A D on the sliding, ie, when connection becomes more resistant because o an increase in stiness, the sliding is reduced gradually One adopted the values 90 and 40 respectively, but they are always optional 2 he choice o the value o the limit o strain is very important because it introduces a more or less great brittleness o the response by translation o the threshold o transition o the small strains to the great slidings his brittleness is related to the stiness o the concrete via the parameter A D It should be noted that the ollowing parameters which manage the damage must be also adjusted at the local level to ensure the correct continuity o behavior in shears o connection and to thus be able to obtain the desired or expected response o a real system steel connection concrete 525 he parameter o damage A 2D the damage, such as it was conceived in the model, obeys two laws o evolution which are expressed using one only variable classical scalar which will ensure the coherence o the damage he parameters o each o the 2 models are independent and numerically stable, but they are likely to generate serious errors in the continuity o the behavior i one does not pay attention to the shape o the local curve stress-strain: to see the case o the curve shown in graphics o [Figure 525-a], with a value A 2D = 0 3 MPa We are not able to propose an analytical relation or the choice o this parameter, but the gained experience enables us to airm that the value o this parameter must be understood enters 0 3 and MPa roughly 40 COMPARAISON DE A2D - Loi locale de la liaison Ad2 = e-3 Ad2 = 3e-3 Ad2 = 6e-3 Ad2 = 9e-3 Ad2 = 2e SXY (MPa) -contrainte de cisaillement AOdeo EXY Licensed under the terms o the GNU FDL (

20 deault itre : Loi de comportement (en 2D) pour la liaison acier [] Date : 26/03/203 Page : 20/24 Appear 525-a: Comparison o A 2D : damage and racture o connection 526 the parameter o damage B 2D his parameter, which supplements the law o evolution o damage in great slidings, controls not only the growth o the strength o connection or the shape o the curve o behavior with the peak and in the region post-peak, but also the kinematics o the response, which implies the determination o the sliding or the maximum shearing stress as well as the amplitude o the curve to the peak o the behavior hen, although the values o the parameters o damage A 2D and B 2D will have to be adjusted at the same time when one builds the curve o behavior o connection in order to respect the continuity o the pace, one can say that the value o B 2D is inversely proportional to the amplitude o sliding at the top, ie, a value o 08 allows great broader slidings in the top than a value o 2, or example For practical cases, one recommends to use a value understood enters 08 and to reproduce a coherent curve o behavior (See [Figure 526-a]) 25 VARIAION DE B2D - Loi locale de la liaison - SXY (MPa) -contrainte de cisaillement AO Bd2 = 07 Bd2 = 08 Bd2 = 09 Bd2 = 0 Bd2 = deo EXY Appear 526-a: Comparison A 2D o : damage and racture o connection 53 the parameters o damage on the normal direction 53 the limit o strain N or threshold o large displacements In a way similar to the elastic behavior in the tangential direction, one considers that decoherence must during initiate the going beyond a certain threshold o strain We propose to adopt a value enters 0 4 and he parameter o damage A DN his parameter compared to the controls primarily the slope o degradation o the normal stress strain due to the opening o the interace We propose to use a minimal value o 0 MPa, which corresponds to a degradation similar to that o the concrete Nevertheless, i one wishes to have a behavior o connection even more brittle, it is enough to increase this value Licensed under the terms o the GNU FDL (

21 deault itre : Loi de comportement (en 2D) pour la liaison acier [] Date : 26/03/203 Page : 2/ he parameter o damage B DN In combination with the preceding parameter, this parameter controls the damage o connection, in particular the shape o the curve o behavior in phase post-peak We propose to use a value equal to, or,2 or more marked curves ADN = e-3 ADN = e-2 ADN = e- SYY (MPa) -Contrainte Normale d'adherence Déormation Normale Appear 533-a: behavior o connection on the normal direction at the time o the opening o the interace (normal tension on connection) 54 he parameters o riction 54 the material parameter o riction o cracks One o the assets o the model suggested here is that it is able to take into account the eects o riction o the cracks, which, in the case o monotonic loading, appears by a positive contribution to the shear strength o connection; in addition, in the cyclic cases o loadings, it is obvious that the pace o the loops o hysteresis depends directly on the choice o the value o this material parameter However, the corresponding values were not gauged, since we did not simulate tests with cyclic loadings yet to validate them emporarily, one proposes to use values lower than 0 MPa, with a maximum value o equal to 0 MPa 542 he material parameter o kinematic hardening On [Figure 542-a], one can appreciate that the reduction in the value o increases hysteretic dissipation, but also the strength o shears and the strain residual pseudo-plastic hat is very important or the cyclic modelization o connection since in reality, when one exceeds the peak o maximum strength, one notices that at the time o the discharge there is no more elastic contribution o the sliding, ie the strain residual pseudo-plastic corresponds exactly to the total sliding reached In other keys, once connected all the cracks in the potential o racture, longitudinal and tangential layer with the steel bar, the single strength which will prevent the displacement o reinorcement is the riction resistance o connection, produced by the contact and the tangle o the asperities between suraces concrete concrete Licensed under the terms o the GNU FDL (

22 deault itre : Loi de comportement (en 2D) pour la liaison acier [] Date : 26/03/203 Page : 22/24 As previously, our experiment is restricted: one proposes to use a maximum value o 0 MPa which gives correct results or applications in monotonic loading, and which seems suitable or cyclic loadings EFFE DE a SUR LES BOUCLES DE HYSERESIS - Frottement des issures SXY (MPa) -contrainte de cisaillement AO γ 00 a 4 γ 00 a γ 00 a 5 γ 00 a deo EXY Appear 542-a: Comparison o a : eects on the loops o hysteresis in cyclic 543 the parameter o inluence o containment c In our model, the inluence o containment was taken into account thanks to the application o this parameter which controls these eects on the connection, and which appears by an increase in the maximum shearing stress as by the increase in maximum displacement to the peak when containment increases For the calibration, we carried out simulations with containments o 0 5, 0 and 5 MPa, by means o always a value o 0 or this parameter It was noticed that i one wants to produce a kinematical translation o the sliding caused by containment, it is enough D `to adopt a value o 2 or 5 (adimensional) Optionally, it is advised to maintain the value o 0 or ordinary computations 55 Summarized parameters to acilitate the use o the model, the ollowing table presents a synthesis o all the parameters o the model o behavior It is pointed out that the values or the statements suggested have only one indicative value, and that the arbitrary combination can give inaccurate and unexpected results compared to the hoped behavior o connection; in other words, a bad choice o the parameters can produce a strong stiness or a weak response o the steel-concrete interace Licensed under the terms o the GNU FDL (

23 deault itre : Loi de comportement (en 2D) pour la liaison acier [] Date : 26/03/203 Page : 23/24 Parameter Unit Value suggested Analytical statement h pen mm - h pen =d b sr G liai MPa - G liai =C arm G beton ε - A D - B D - min 0x0-4 max 5x0-3 min 0 max 50 A D = SR ' c 30 E a E b min 0 max 05 2 ε = h pen 2 A 4 D 0 9 A D 4 A 2D MPa- min 0x0-4 max 9x0-2 min 08 B 2 D - max 5 MPa max 00 a MPa - min 00 max 0 c - 0 (value recommended) α - min 0-4 max MPa - min 0x0 - (value recommended, not gauged) A D N B D N - (value recommended, not gauged) Variables concerned d b sr C arm G beton c E has E B Diameter o the bar relative Area o the veins Coeicient o correction by reinorcemen ts Modulates stiness o the concrete Compressive strength o the concrete (MPa) Elasticity modulus o steel Elasticity modulus o the concrete Licensed under the terms o the GNU FDL (

24 deault itre : Loi de comportement (en 2D) pour la liaison acier [] Date : 26/03/203 Page : 24/24 6 Bibliography ARMSRONG, PJ & FREDERICK, CO : A Mathematical Representation o the Multiaxial Bauschinger Eect GEGB ; Carryorward RD/B/N, 73,966 BERERO 2VV: Concrete Seismic behavior o structural linear elements (beams and columns) and to their connections Euro-International committee o Concrete (CEB); News bulletin No 3; Paris, France, 979 ELIGEHAUSEN 3R, POPOV EP & BERERO VV: Local jump stress-slipway relationships o deormed bars under generalised excitations University o Caliornia; Carryorward No UCB/EERC - 83/23 o the National Science Foundation, 983 ORIZ 4MR & SIMO JC : Year analysis o has new class o constitutive integration algorithms or elastoplastic relations International Newspaper or Numerical Methods in Engineering; Vol 23, pp ,986 Checking 7 constitutive law JOIN_BA is checked by the cases ollowing tests: SNA 2 est o axisymmetric wrenching (Borderie & Pijaudier- Pooch) or the study o steel-concrete connection: model JOIN_BA [V60 26 Validation o constitutive law JOIN_BA (steel-concrete connection) in 2D plane [V603 2] SSNP 26] Description 8 o the versions o the document Aster Author (S) Organization (S) Description 74 - PONNELLE, N DOMINGUEZ EDF R & D /AMA initial o modiications 7,4 ext Licensed under the terms o the GNU FDL (