PARC_CL 2.0 crack model for NLFEA of reinforced concrete structures

PARC_CL 2.0crack model for NLFEA of reinforced concrete structures under cyclicloadingsBeatrice Belletti ⇑,Matteo Scolari,Francesca VecchiUniversity of Parma,Department of Engineering and Architecture,Parco area delle Scienze 181/A,43124Parma,Italya r t i c l e i n f o Article history:Received 3March 2017Accepted 8June 2017Available online 4July 2017Keywords:PARC_CL 2.0fixed crack model Reinforced concrete Constitutive model Cyclic loadsNumerical models Shear wallsa b s t r a c tThis paper presents the development of an updated version of PARC_CL 2.0crack model implemented in Abaqus Code using subroutine UMAT.for.The PARC_CL 2.0model is a fixed cracked model for the response prediction of reinforced concrete members.More refined material constitutive relationships were incorporated in the formulation of the model to take into account plastic deformations.Finally,to validate the model,reinforced concrete (RC)panels subjected to cyclic loads,available in literature,are modeled and compared with experimental observations.The PARC_CL 2.0model was able to capture,with reasonable accuracy,the overall behaviour including cyclic shear stress vs.shear strain behaviour,shear stiffness,cyclic stiffness degradation,energy dissipation during the cycles and pinching effect.Ó2017Elsevier Ltd.All rights reserved.1.IntroductionOne of the main characteristics of RC structures is the highly non-linear response to cyclic loading,in particular seismic one.For this reason,realistic constitutive models are required to obtain reasonably accurate simulations of RC members.Numerous are the constitutive models applied for monotonic loading case,as sum-marized by Bazˇant [1]and de Borst [2],while the cyclic ones are less common in literature.Although many models can be used to simulate non-linear response of RC elements,the smeared crack models are more popu-lar than discrete ones due to their capability to treat the cracked solid as a continuum by reducing stiffness properties.On the con-trary,discrete crack models represent cracks as a geometrical dis-continuity using,for example,interface elements.A further distinction can be done between smeared rotating crack model [3]and smeared fixed crack model.The first one assumes that,during loading,the crack pattern should change direction [4–6];the sec-ond one hypothesizes the starting crack pattern as fixed [7,8].In this last case,the prediction of shear stresses generated along the cracks becomes very important (while in the rotating model it is not considered)most of all when the structural behaviour is dom-inated by aggregate interlock phenomena.More complexities,as stiffness degradation in concrete and the Bauschinger effect of steel bars,are introduced by cyclic loads.Areliable numerical model must be able to capture these types of nonlinearity.Existing commercial finite-element codes often have limitations in representing cyclic behaviour,due to idealizations in material models.For example,to solve convergence problems,the tensile behaviour of concrete is commonly assumed to be secant in the unloading/reloading phases even if the experimental evidence demonstrates that irrecoverable tensile strains remain in concrete [9,10].Furthermore the crack closing process implies that the con-crete path does not pass through the origin [11].However,the most desirable model should be characterized by sufficiently accurate response predictions and simplicity in formulation.Therefore,for all these reasons,a Physical Approach for Reinforced Concrete under Cyclic Loading condition (PARC_CL 2.0)is proposed.The PARC_CL 2.0fixed crack model is a develop-ment of the previous PARC versions,described in [12–14]which were successfully applied to the analysis of RC structures subjected to monotonic loading,as reported in [15,16].The present model allows to take into account hysteretic cycles and plastic strains in the unloading phase and it is implemented in subroutine UMAT.for for the analyses of RC members by means of ABAQUS code.In particular,it is possible to assess the static and dynamic behaviour of slabs,structural walls buildings and floors by means of multi layered shell or membrane elements.Moreover,it can be useful for the seismic assessment of structural elements where deforma-tions and crack distributions are fundamental damage indicators (e.g.nuclear power plant facilities),[17].In this paper,the material formulations are firstly presented;after that,the model is validated by comparing the PARC_CL 2.0/10.1016/pstruc.2017.06.0080045-7949/Ó2017Elsevier Ltd.All rights reserved.⇑Corresponding author.E-mail address:beatrice.belletti@unipr.it (B.Belletti).Computers and Structures 191(2017)165–179Contents lists available at ScienceDirectComputers and Structuresjournal homepage:www.elsevi/locate/compstrucresults with the measured data from experimental tests on simplereinforced concrete panels,with different steel bar orientationsand reinforcement ratios,subjected to reversed cyclic loadingavailable in literature,Mansour and Hsu[18].2.Material constitutive models implemented in PARC_CL2.0 2.1.Basic hypothesesThe proposed PARC_CL2.0model is based on a total strainfixedcrack approach,in which at each integration point two referencesystems are defined:the local x,y-coordinate system and the1,2-coordinate system along the principal stress directions.The anglebetween the1-direction and the x-direction is denoted as w, whereas h i is the angle between the direction of the i th order ofbars and the x-direction;a i=h iÀw is the direction of the i th bars with respect to direction1.When the value of the principal tensilestrain in concrete exceeds the concrete tensile limit strain e t,cr for thefirst time,thefirst crack is formed and the1,2-coordinate sys-tem remainsfixed(Fig.1a).2.2.Strainfields2.2.1.Strainfield for concreteThe concrete behaviour is assumed orthotropic both before and after cracking and the total strains at each integration point are cal-culated in the orthotropic1,2-system,Eq.(1):f e1;2g¼½T w Áf e x;y gð1Þwhere½T w is the transformation matrix given by Eq.(2):½T w ¼cos2w sin2w cos wÁsin wsin2w cos2wÀcos wÁsin wÀ2Ácos wÁsin w2Ácos wÁsin w cos2wÀsin2w264375ð2Þ{e1,2}and{e x,y}represent respectively the biaxial strainsfields in 1,2-system and x,y-system,as shown in Eqs.(3)and(4),Fig.1c.f e1;2g¼f e1e2c12g tð3Þf e x;y g¼f e x e y c xy g tð4ÞThe stress-strain behaviour presented herein is calculated on the base of the uniaxial strains in the1,2-coordinate system, according to Eqs.(5)–(7),as shown in Fig.1d.e1¼11Àm2e1þm1Àm2e2ð5Þe2¼m1Àm2e1þ11Àm2e2ð6Þ c12¼c12ð7ÞWhen,after thefirst crack the Poisson’s ratio is assumed to be zero,in Eqs.(5)and(6),the biaxial strains are the same as the uni-axial strains.This condition is illustrated in Fig.1c and d,where the stresses r1and r2are related to the uniaxial strains and represent both the biaxial stresses and the uniaxial stresses.NomenclatureRoman lower case lettersa m average crack spacingb strain hardening ratio of steel reinforcementd max maximum aggregate sizef c cylinder compressive strength of concretef t axial tensile strength of concretef yi yield strength of i th order of reinforcing steell x,l yfinite element dimensions in the x,y-directionss i bar spacingtfinite element thicknessv crack slidingw crack widthRoman upper case lettersE c modulus of elasticity of concreteE hi hardening modulus of the i th order of barsE si modulus of elasticity of the i th order of barsG C fracture energy of concrete in compressionG F fracture energy of plain concrete in tensionR parameter which influences the shape of the hysteretic behaviour of steel reinforcingGreek lower case lettersa i angle between the x i-direction and the1-directionc12biaxial shear strain of concrete in the1,2-coordinate systemc12uniaxial shear strain of concrete in the1,2-coordinate systemc xy shear strain in the x,y-coordinate systeme1biaxial strain of concrete along1-directione1uniaxial strain of concrete along1-directione2biaxial strain of concrete along2-directione2uniaxial strain of concrete along2-directione x strain along x-directione xi strain of the i th order of bars along x i-direction e y strain along y-directionf softening coefficient of concreteh i angle between the direction of the i th order of bars andthe x-directiont Poisson’s ratioq i reinforcement ratio related to the i th order of barsr1concrete stress along1-directionr2concrete stress along2-directionr x i stress of the i th order of bars along x i-directions12shear stress of concrete in the1,2-coordinate system W angle between the1-direction and the x-directionMatrices and vectorsf e1;2g biaxial strainfield for concrete in1,2-coordinate sys-temf e1;2g uniaxial strainfield for concrete in1,2-coordinate sys-temf e x;yg strainfield in x,y-coordinate systemf e x i;y ig strainfield in x i,y i-coordinate systemf r1;2g stressfield of concrete in1,2-coordinate systemf r x;yg overall stressfield,considering the contribution of bothconcrete and bars,in x,y-coordinate systemf r x i;y ig stressfield for the i th order of bars in x i,y i-coordinatesystemf r x;yg c stressfield for concrete in x,y-coordinate systemf r x;yg s;i stressfield for the i th order of bars in x,y-coordinatesystem½D1;2 Jacobian matrix of concrete in1,2-coordinate system ½D x;y overall Jacobian matrix,considering the contribution of both concrete and bars,in x,y-coordinate system½D xi;y i Jacobian matrix of the i th order of bars in x i,y i-coordinate system½T w ;½T#itransformation matrices½T w t;½T#it transpose of the transformation matrices166 B.Belletti et al./Computers and Structures191(2017)165–1792.2.2.Strain field for steelThe reinforcement is assumed smeared in concrete.The steel strain field along the reference system of each bar is obtained rotating the strains in the x,y -system,as shown in Eq.(8):f e x i ;y ig ¼½T #i Áf e x ;y gð8Þwhere ½T h i is the transformation matrix given by Eq.(9):½T h i ¼cos 2h isin 2h i cos h i Ásin h isin 2h i cos 2h iÀcos h i Ásin h i À2Ácos h i Ásin h i 2Ácos h i Ásin h i cos 2h i Àsin 2h i264375ð9Þ2.3.Stress fields2.3.1.Stress field for concreteThe concrete stress field in the 1,2-coordinate system is given by Eq.(10):f r 1;2g ¼r 1r 2s 128><>:9>=>;ð10Þwhere r 1and r 2represent the normal stresses in concrete along 1and 2directions calculated following the relation presented in Sec-tion 2.5.1,while s 12is the shear stress in concrete calculated according to the aggregate interlock model,Section 2.5.2.2.3.2.Stress field for steelThe steel stress field,defined for each i th order bars in the x i ,y i -system,is given by Eq.(11):f r x i ;y ig ¼r x i r y i s x i y i 8><>:9>=>;¼r x i 008><>:9>=>;ð11Þwhere r xi represents the stress along the axis of the i th order of barand it can be calculated following the Menegotto-Pinto’s procedure explained in Section 2.5.3.The dowel action phenomenon is not considered yet in the model;for this reason in Eq.(11)there are not stresses in the direction perpendicular to the axis of the bar.2.3.3.Total stress fieldBoth the concrete and the steel stress fields can be transformed from their local coordinate system to the overall global x ,y coordi-nate system using respectively Eqs.(12)and (13):f r x ;yg c ¼½T w t Áf r 1;2g ð12Þf r x ;y g s ;i ¼½Thi t Áf r x i ;y i gð13ÞFinally the total stress field in the x ,y -system is obtained by assum-ing that concrete and reinforcement behave like two springs placed in parallel,Eq.(14):f r x ;yg ¼f r x ;y g c þX n i ¼1q i f r x ;y g s ;ið14Þ2.4.Stiffness matrix and numerical solution procedureThe proposed PARC_CL 2.0model is based on a tangent approach,in which the Jacobian matrix in the local coordinate sys-tem for each material is composed by derivatives as shown in Eq.(15)for concrete and in Eq.(16)for each i th order of bars.½D 1;2 ¼@r 1@e 11ð1Àm 2Þ@r 1@e 2m ð1Àm 2Þ0@r 2@e 1m ð1Àm 2Þ@r 2@e 21ð1Àm 2Þ00@s 12@c 1226643775ð15Þ½D x i ;y i ¼@rx i@e x i000000026643775ð16ÞTo avoid numerical problems,the stiffness contributes of con-crete became secant in softening branches.When,after the firstcrack the Poisson’s ratio is assumed to be zero,the terms out of the diagonal become zero.The stresses presented herein are calcu-lated according to Eqs.(18),(19)and (29)for concrete and Eq.(30)for steel.Finally,the global stiffness matrix is obtained by Eq.(17).½D x ;y ¼½T w tÁ½D 1;2 Á½T w þX n i ¼1q i ½T h i t Á½D x i ;y i Á½T h ið17ÞThe stiffness matrix is updated until a solution is closely approximated.The main steps of the implemented algorithm are given in the flowchart reported in Fig.2.An iterative procedure was performed until all the equilibrium,compatibility and consti-tutive equations were satisfied.2.5.Adopted constitutive laws2.5.1.Cyclic uniaxial constitutive law for concreteThe tensile envelope curve,presented in Eq.(18)and shown in Fig.3,is characterized by a bilinear stress-strain relation [19]prior to cracking;after cracking,to represent the softened trend,an exponential law is assumed [20].Fig.1.(a)RC membrane element subjected to plane stress state,(b)crack parameters,(c)biaxial strain condition in 1,2-coordinate system,and (d)uniaxial strain condition in 1,2-coordinate system.B.Belletti et al./Computers and Structures 191(2017)165–179167where e t ;el ¼0:9Áf t =E c ;e t ;cr ¼0:00015;e t ;u ¼e t ;cr þ5:136ÁG F =ða m Áf t Þ;c 1¼3;c 2¼6:93.The compressive envelope curve,shown in Fig.3,is character-ized by a first elastic part followed by a parabolic formulation [21]as presented in Eq.(19):r ¼E c Áe e c ;el <e 60f c Á1þ4e Àe c ;el e c ;cr Àe c ;el À2e Àe c ;el e c ;cr Àe c ;el2 !e c ;cr <e <e c ;el f c Á1Àe Àe c ;cr e c ;u Àe c ;cr 2 !e c ;u <e <e c ;cr 0e 6e c ;u8>>>>>>><>>>>>>>:ð19Þwhere e c ;el ¼f c =ð3ÁE c Þ;e c ;cr ¼5Áe c ;el ;e c ;u ¼e c ;cr À1:5ÁG C =ða m Áf c ÞThe value of the fracture energy of concrete in tension G F (Fig.3)is evaluated according to [22].The value of the fracture energy of concrete in compression G C (Fig.3)is assumed equal to 250G F according to [23].The crack spacing a m is assumed constant.Its value is assumed equal to the square root of the average element area,according to [24],while for reinforced concrete it is evaluated by ‘‘a priori”methods based on the average spacing of the cracks l s,max according to Model Code 2010[19].As stated in the introduction,the PARC_CL 2.0crack model allows to consider plastic and irreversible deformations in the unloading phase.As shown in Fig.3,both in tension and compres-sion the unloading paths,for simplicity,are represented by a straight line with slope E c from the experienced maximum tensile strain on the monotonic curve (e t,re )to the plastic tensile strain (e t,pl )in the tensile domain and from the experienced minimum com-pressive strain on the monotonic curve (e c,re )to the plastic com-pressive strain (e c,pl )in the compressive domain.The passage from compression to tension is with zero stress,while to pass from tension to compression it is necessary to determinate the stress with no deformation r cl,cr (Fig.3),called the residual bond stress by Okamura and Maekawa [25],as reported in Eq.(20).r cl ;cr¼Àf t 0:05þ0:03e t ;ree t ;plð20ÞThe biaxial state of concrete in compression [26],due to trans-verse cracking,is also taken into account by reducing the compres-sive stress f c and the compressive peak strain e c,cr with the f coefficient,Eq.(21):f ¼1ð0:85À0:27Áe ?=e Þ;0:46f 61ð21ÞThe cyclic behaviour considering the biaxial state of concrete is shown in Fig.4.The effect is applied only on the envelope curve.During the unloading-reloading steps the value of f is maintained fixed until the compressive deformation of concrete e reaches the minimum compressive strain of the previous cycle e c,re .In Fig.4is shown a typical example of a cycle in which the concrete pass from a biaxial envelope curve with f 1coefficient to a biaxial curve with f 2coefficient,where f 1>f 2.2.5.2.The aggregate interlock effectThe shear stress due to the effect of aggregate interlock is eval-uated on the basis of the crack width,w ,and the crack sliding,v,Fig.1d,according to Gambarova [27]and shown in Fig.5c:r ¼E c Áe 06e <e t ;el 0:9Áf t þ0:1Áf t e Àe t ;el e t ;cr Àe t ;el e t ;el 6e <e t ;cr f t 1þc 1Áe Àe t ;cr e t ;u Àe t ;cr 3 !Áexp Àc 2Áe Àe t ;cr e t ;u Àe t ;cr Àe Àe t ;cr e t ;u Àe t ;cr Áð1þc 31ÞÁexp ðÀc 2Þ&'e t ;cr 6e <e t ;u 0e P e t ;u8>>>>><>>>>>:ð18Þ168 B.Belletti et al./Computers and Structures 191(2017)165–179s12¼ s1Àffiffiffiffiffiffiffiffiffiffi2wmaxs!a3þa4vw31þa4vÀÁvð22Þwhere s¼0:27Áf c;a3¼2:45= s;a4¼2:44Áð1À4= sÞ.Gambarova’s relation can be schematized with a bilinear curve [13]in which the endpoint of the elastic part P(c⁄cr,s⁄)has coordi-nates equal to Eqs.(23)and(24),as shown in Fig.5a.cÃcr¼vÃ=a mð23Þsü s1Àffiffiffiffiffiffiffiffiffiffi2wd maxs!a3þa4vÃw31þa4vÃÀÁ4vÃwð24Þwhere vüf cÁw=a5þa6;a5¼0:366Áf cþ3:333;a6¼f c=110.The bilinear curve is used to define the stress-strain relationship between the shear stress s12and the shear strain c cr,12in the cracked phase of the concrete,Fig.5a.In this phase,the shear mod-ulus,G cr,is derived by Eq.(25):G cr¼s12c12;cr¼sÃvÃÁa m¼GÃcrÁa m if c12;cr6cÃcrs12c12;cr¼sÃc12;crif c12;cr>cÃcr8<:ð25Þwhere G⁄cr represents the secant shear modulus associated to thefirst linear branch in the cracked phase.The s12-c12curve,includingthe elastic part,is shown in Fig.5b,where c f corresponds to theshear strain at the onset of concrete cracking and c⁄defines thepoint after which the shear strain remains constant,Eq.(26).cücÃcrþc fð26ÞAccording to the total strain concept,the PARC_CL2.0crackmodel assumes that the un-cracked concrete,characterized bythe elastic deformation c12,el,and the cracked concrete,character-ized by the cracking deformation c12,cr,behave like two springs inseries,Fig.6.Therefore,the equivalent overall shear modulus,G eq,in the cracked phase can be calculated according to Eq.(27),asschematically reported in Fig.6b.G eq¼GÁG crGþG crð27Þwhere G is the elastic shear modulus,Eq.(28),and G cr is calculatedaccording to Eq.(25).G¼E ctð28ÞB.Belletti et al./Computers and Structures191(2017)165–179169Combining the cracked shear modulus,G cr ,with the elastic shear modulus,G ,using Eq.(27),the overall s 12-c 12behaviour,reported in Fig.6b,can be derived as reported in Eq.(29).s 12¼G Ác 12if c 126c fG Ãeq Ác 12if c f <c 126c ÃsÃif c 12>c Ã8><>:ð29ÞIn Eq.(29),G ⁄eq is calculated using Eq.(27)by substituting G crwith G ⁄cr .In this paper a new formulation of aggregate interlock for cyclic loading is presented.In the unloading phase,a branch with slope G ⁄eq ,as illustrated in Fig.7-a,has been defined.Changing the crack width,w ,the value of G ⁄eq change as shown in Eq.(27)and,as a con-sequence of this,different cyclic curves can be obtained as shown in Fig.7b.2.5.3.Cyclic uniaxial constitutive law for embedded mild steel barsThe constitutive relation for steel,employed in this study,is based on Menegotto-Pinto model [28]and allows to representthe hysteretic stress-strain behaviour of reinforcing steel bar also including yielding,strain hardening and Bauschinger effect,Fig.8.The Menegotto-Pinto formulation,applied axially to the bar,can be expressed following Eq.(30).r x i ¼r ÃÁðr n 0Àr n À1r Þþr n À1rð30Þwherer üb e Ãþð1Àb Þe Ã1þðe ÃÞR n ½1=Rn;e üe x i Àe n À1r e n 0Àe n À1r;b ¼E hiE si;R n ¼R 0Àa 1Ánn À1a 2þn n À1;n n ¼e n r Àe n0:Eq.(30)represents a curved transition from a straight line asymptote with slope E si to another asymptote with slope E hi .The stress r 0and the strain e 0define the intersection point of the two asymptotes of the branch considered (e.g.point A in Fig.8);similarly,r r and e r are the stress and the strain in the point where the last strain reversal occurs (e.g.point B in Fig.8).As shown in Fig.8(r 0,e 0)and (r r ,e r )are updated after each strain reversal.b is the strain-hardening ratio and can be calculated as the ratio between E hi and E si .R n is the parameter thatinfluences170 B.Belletti et al./Computers and Structures 191(2017)165–179the shape of the transient curve and it allows to consider the Bauschinger effect.n n is updated following the strain reversal and its value does not change when reloading occurs after partial unloading,Fig.8;R 0is the value of the parameter R n during the first loading cycle.According to [29],in this paper it is assumed that R 0=20,a 1=18.45and a 2=0.001.3.Validation of PARC_CL 2.0crack modelPARC_CL 2.0crack model has been adopted in Scolari et al.[30]for the prediction of in-plane and out-of-plane behaviour of RC walls.In this paper,in order to assess the efficiency of the proposed PARC_CL 2.0model under cyclic static loading,some experimental tests on RC panels found in literature were investigated.The exper-imental program,referred to Mansour and Hsu [18],was per-formed using the ‘‘Universal Element Tester”facility at the University of Houston and it consists in 12panel specimens tested under reversed cyclic shear stresses.8out of the 12specimens are investigated in this paper by means of NLFEA and PARC_CL 2.0crack model.All panels were 1398Â1398Â178mm in size,except panels CE4,CA4and CB4,which were 1398Â1398Â203mm in size.The specimens are reinforced with two parallel steel grids placed at angles of 45°(CA and CB-series Fig.9a)and 0°(CE-series Fig.9b)to the x -direction.The properties of the panels are summarized in Table 1.The ele-ments were subjected to reversed cyclic principal stresses in the horizontal and vertical directions:the two stresses were main-tained equal in magnitude and opposite in direction to obtain a state of pure shear stress (Fig.9).NLFEA analysis was carried out using a single,4-node mem-brane element with reduced integration (defined M3D4R in [31]).To simulate the same loading condition,an external frame was modeled using truss elements in [31](Fig.9c).At the frame’s end,the cyclic displacement time history was imposed.In Fig.10the experimentally-measured shear stress vs.shear strain responses,in the i ,j system (Fig.9),for CA,CB and CE series are compared with the analytical model predictions.An acceptable level of agreement is observed between model and test results in terms of shear stress capacity,stiffness,ductility,shape of the unloading/reloading loops and pinching characteristics of the response.One of the main purposes of the experimental tests carried out by [32,33]was to investigate the effect of the steel bar orientation on the cyclic behaviour.Indeed,the steel bar orientation produces different cyclic response of the shear members:this becomes evi-dent comparing the hysteretic loops of two RC panels,CA3(Fig.10b)and CE3(Fig.10g).According to Hsu [33],a detailed study of the different behaviors of panels with different steel bars orientation is conducted to check the PARC_CL 2.0validity.Each tested panel can represent an element taken from the web of squat wall subjected to horizontal load V .In Fig.11a the CA3panel,in which the angle a i is 45°,is shown;while in Fig.11b the CE3panel,in which the a i angle is 0°,is presented.The cyclic curves in terms of stress-strain obtained by using the PARC_CL 2.0are shown in Fig.12for concrete and Fig.13for steel.Fig.12shows a reduction in the maximum attained compressive strengths of concrete:panel CA3reaches a value of f c of almost 15MPa (33%of its maximum compressive strength).This is due to the presence of cracks in the orthogonal direction that causes the biaxial state of concrete in compression,Fig.4and Eq.(21).Instead,CE3element remains mainly in tension,reaching a f c value of 3MPa.The cyclic behaviour of steel bars in CA3-panel is shown in Fig.13a:each bar is always in the tensile domain becausetheB.Belletti et al./Computers and Structures 191(2017)165–179171compressive stresses are resisted by concrete.Instead,steel bars in CE3panel are subjected to tension and compression stresses (Fig.13b):it is due to the steel bars orientation that are set parallel to the external applied stresses.The global response in terms of shear strain and stresses are shown in Fig.10b and g:the steel bar orientation produces differ-ences in terms of shear ductility and energy dissipation capacity.When the steel bars are oriented in the 1,2-system (CE3-panel Fig.15b),the hysteretic loops are fully rounded and the behaviour is ductile;while when they are oriented at 45°(CA3-panel Fig.14b),the behaviour is much less ductile.The PARC_CL 2.0model is able to predict the pinched shape as well as the fully rounded of the hysteretic loops:in fact,the obtained curve of the CA3-panel (and of the CA series in general)is severely ‘pinched’near the origin,while the CE-series is fully rounded.3.1.Presence of pinching mechanismTo better understand the pinching effect,the first hysteretic cycle after yielding,obtained by means PARC_CL2.0,is presented in Fig.14.Four points A,B,C,D are chosen to show the correlation between the shear stresses and strain and the corresponding stres-ses and strain in steel and concrete.Point A represents the last point of the shear loading;point B is the minimum shear stress in the positive strain domain while C is the corresponding point in the negative strain domain and they delimit the pinching zone.Point D is the minimum stress point.When the element is unloaded from point A to point B,also the steel stress is reduced (Fig.14e and f):the behaviour of the steel is equal in the x i -direction and y i -direction because of the same reinforcement ratio.Consequently,the concrete in the 1-direction reduces its strain and vertical cracks start to close.At point B,shear stress,steel stresses and concrete stresses are close to zero.Proceeding from point B to point C the vertical cracks in one direction are fully closed and hor-izontal cracks start to open:this region,with a very small shear resistance is called pinching zone.Finally,from point C to point D the vertical strain increases and produces reloading of the steel bars;consequently,the shear stiffness increases.3.2.Absence of pinching mechanismThe PARC_CL 2.0results for CE3panel are illustrated in Fig.15:steel bars are oriented in the direction of the applied stresses so the pinching mechanism is absent.As in the previous panel in the first cycle after yielding is presented:with the help of 4points it is remarked the correlation between materials.Point A represents the maximum shear strain of the cycle;point B is the minimum shear stress in the positive domain,point C is in the negative strain region and it represents the moment in which the steel in the y i -direction reaches yielding (Fig.15f).Point D is the minimum shear strain reached by the panel.When the element is unloaded from point A to point D the panel is subjected to compressive stress in the horizontal direction (1-axis)and to tensile stress in the perpen-dicular direction (2-axis)as demonstrated to the concrete beha-viour.Consequently,also the bars are subjected to the same type of load:the bars along the compressive direction (x i )reduce the stress while the steel in the perpendicular direction (y i )is sub-jected to tensile load:the reinforcement is offering a high shear stiffness from point B to point C.The pinching effect in the two panels can be explained by exam-ining cracked elements,in which the cracks are both vertical and horizontal because of previous cycles of positive and negative shear stresses.For panel CA3,since both the vertical and horizontal cracks are open,the applied stress (r H and r V )must be resisted by the two 45°steel bars.The effect of stresses are separated,as shown in Fig.16a:while the horizontal stress induces a compres-sive stress in the bars,the vertical stress induces an equal tensile stress,so the stresses in the two 45°steel bars cancel eachotherFig.9.Steel bar orientations in test panels:(a)Panels of CE-series (h i =0°),(b)CA-and CB-series (h i =45°)and (c)modeling of the test setup.Table 1Material properties and steel bar arrangement of test panels.SeriesPanelConcrete Steel in x-i directionSteel in y-i directionh i (°)f c (MPa)G F (N/mm)G C (N/mm)q x-i (%)f y (MPa)q y-i (%)f y (MPa)CECE2490.14736.780.54424.10.54424.10CE3500.14836.90 1.20425.4 1.20425.40CE4470.14636.50 1.90453.4 1.90453.40CACA2450.14536.200.77424.10.77424.145CA344.50.14536.20 1.70425.4 1.70425.445CA4450.14536.20 2.70453.4 2.70453.445CBCB3480.14736.63 1.70425.40.77424.145CB4470.14636.502.70453.40.67424.145172 B.Belletti et al./Computers and Structures 191(2017)165–179。