The overlapping crack model for uniaxial and eccentric concrete compression tests

Transcription

1 Magazine of onrete Researh, 9, 61, No. 9, November, doi: /mar The overlapping rak model for uniaxial and eentri onrete ompression tests A. arpinteri, M. orrado, G. Manini and M. aggi olitenio di Torino An analytial/numerial model, referred to as the overlapping rak model, is proposed in the present paper for the analysis of the mehanial behaviour of onrete in ompression. Starting from the experimental evidene of strain loalisation in uniaxial ompression tests, the present model is based on a ouple of onstitutive laws for the desription of the ompression behaviour of onrete: a stress strain law until the ahievement of the ompression strength and a stress displaement relationship desribing the post-peak softening behaviour. The displaement would orrespond to a fititious interpenetration and therefore the onept of overlapping rak in ompression is analogous to the ohesive rak in tension. Aording to this approah, the slenderness and size-sale effets of onrete speimens tested under uniaxial ompression are interpreted from an analytial point of view. Then, implementing the overlapping rak model into the finite element method, eentri ompression tests are numerially simulated and ompared with experimental results. The influene of the size-sale, the speimen slenderness, as well as the degree of load eentriity, is disussed in detail, quantifying the effet of eah parameter on the dutility of onrete speimens. Notation b thikness of the speimen h depth of the ross-setion of the speimen D oeffiient of influene for the applied fore {D w } T vetor of the oeffiients of influene for the nodal displaements e load eentriity E modulus of elastiity E i tangent modulus E 1 seant modulus from the origin to the peak ompression stress ó {F} vetor of nodal fores F ultimate ompression fore F u ultimate tensile fore G rushing energy of onfined onrete G,0 rushing energy of unonfined onrete G F frature energy {K } vetor of the oeffiients of influene for the applied fore [K w ] matrix of the oeffiients of influene for the nodal displaements l length of the speimen Department of Strutural Engineering and Geotehnis, olitenio di Torino, orso Dua degli Abruzzi 24, 10129, Torino, Italy. (MAR ) aper reeived 7 July 8; last revised 20 February 9; aepted 11 Marh 9 applied fore s E brittleness number for plain onrete in ompression {w} vetor of nodal displaements w overlapping displaements w r ritial overlapping displaement w t opening displaements w t r ritial rak opening displaement ä speimen shortening å strain å strain at maximum load of ó å onstitutive law º slenderness of the speimen W total rotation of the speimen ó stress ompression strength ó Introdution The ompression behaviour of onrete, and in partiular the ultimate strength and post-peak branh, have a predominant role in the design of onrete and onrete-based strutures. Strutural design, in fat, is usually onduted by omparing an ation with a resistane evaluated on the basis of the ultimate strength. On the other hand, the post-peak behaviour is fundamental for a orret evaluation of the dutility, as, for X (Online) (rint) # 9 Thomas Telford Ltd

2 arpinteri et al. example, for the evaluation of the ultimate axial deformation of olumns or the rotational apaity of reinfored onrete (R) beams. In this ontext, the size-sale effets play an important role, beause the harateristi parameters of onrete are measured on speimens at a laboratory sale, that are far from the dimensions of a real struture. The problem of redution of the ompression strength by inreasing the element size the so-alled size-effet has been investigated in detail in the literature. 1,2 The orret evaluation of the onstitutive parameters is also ompliated by many other testing aspets. The Round Robin programme arried out by the Reunion Internationale des Laboratoires et Experts des Materiaiux, Systemes de onstrution et Ouvrages (RILEM) Tehnial ommittee 148-SS 3 demonstrated that the strength of onrete depends on the frition between the onrete and loading platen, as well as the slenderness of the speimen. 4 With dereasing slenderness, an inrease of speimen strength is measured when rigid steel loading platens are used. However, when fritionreduing measures are used, for example by inserting a Teflon TM sheet between the steel loading platen and the onrete speimen, the ompression strength measured on prisms or ylinders beomes almost independent of the slenderness ratio, l/h. 4,5 Moreover, the effet of frition disappears when the speimen slenderness is higher than 2.5. All the experiments of the aforementioned Round Robin programme revealed also that, in the softening regime, dutility (in terms of stress and strain) is a dereasing funtion of the slenderness. Furthermore, a lose observation of the stress plotted against post-peak deformation urves showed that a strong loalisation of deformations ours in the softening regime, independently of the loading system, onfirming the earlier results by van Mier. 6 The phenomenon of strain loalisation in ompression, evidened in many other experimental programmes on onrete and rok, 7 11 suggests that, in the softening regime, energy dissipation takes plae over an internal surfae rather than within a volume, in lose analogy with the behaviour in tension. These two interonneted phenomena may explain the size-sale effets on dutility. Owing to strain loalisation, the post-peak branh of the stress strain urve is no longer a true material property, but it beomes dependent on the speimen size and slenderness. Based on these experimental evidenes, Hillerborg 12 and Markeset 13 proposed to model failure of onrete in ompression (rushing) by means of strain loalisation over a length proportional to the depth of the ompressed zone, or defined as a material harateristi length. 14 In this way, the ó å relationship used to desribe the softening regime make it possible to address the issue of size effets, although the length over whih the loalisation ours beomes a free parameter, usually defined by the best fitting of experimental data. Based on the evidene that the post-peak dissipated energy referred to a unitary surfae an be onsidered as a material parameter 7,10 and, onsequently, that the post-peak stress displaement relationship is independent of the speimen size, 8,9 arpinteri et al. 15,16 and orrado 17 have reently proposed modelling the proess of onrete rushing using an approah analogous to the ohesive rak model, whih is routinely adopted for modelling the tensile behaviour of onrete. In tension, the loalised displaement is represented by a rak opening, while in ompression it would be represented by an interpenetration. This new approah, also based on the frature mehanis onepts, is referred to as the overlapping rak model and assumes a stress displaement law as a material parameter for the post-peak behaviour of onrete in ompression. It is remarkable to note that, whereas the lassi problem of uniaxial ompression has reeived great attention from the sientifi ommunity, the state-ofthe-art literature shows that only a few experimental studies have been proposed for the analysis of the mehanial behaviour of onrete in eentri ompression, and the related problem of size-sale effets is largely unsolved. The present paper introdues the mathematial aspets of the overlapping rak model for the desription of onrete rushing in ompression, exploring the analogies between tensile and ompression tests. A new numerial algorithm is then proposed, based on the finite element method whih is able to desribe the non-linear behaviour of eentrially loaded speimens in ompression. This model is validated by means of a omparison between the numerial preditions and the results of the experimental tests arried out by Debernardi and Taliano. 22 Finally, using this model, the important issue of size-sale effets in eentri ompression is addressed, showing the limitations of the existing design formulae. Overlapping rak model for the desription of onrete rushing In strutural design, the most frequently adopted onstitutive laws for onrete in ompression desribe the material behaviour in terms of stress and strain (elasti-perfetly plasti, paraboli-perfetly plasti, Sargin s parabola et.). This approah, whih implies an energy dissipation within a volume, does not permit the mehanial behaviour to be orretly desribed by varying the strutural size. On the ontrary, size-sale effets are attributable to strain loalisation within one or more transversal or inlined bands. 2 6 The present formulation adopts the stress displaement relationship proposed by arpinteri et al. 15,16 between the ompression stress and the interpenetration, in lose analogy with the ohesive model. The main hypotheses are detailed below. 746 Magazine of onrete Researh, 9, 61, No. 9

3 The overlapping rak model for uniaxial and eentri onrete ompression tests The onstitutive law used for the undamaged material is a linear-elasti stress strain relationship haraterised by the values of the elasti modulus, E, the ompression strength, ó, and the ultimate elasti strain, å (see Figure 1). The rushing zone develops when the maximum ompression stress reahes the onrete ompression strength. () The proess zone is perpendiular to the prinipal ompression stress. (d) The damaged material in the proess zone is assumed to be able to transfer a ompression stress between the overlapping surfaes. Suh stresses are assumed to be a dereasing funtion of the interpenetration, w (see Figure 1). The following simple linear softening law an be used (see also the pioneering Hillerborg model 12 ), although more ompliated shapes ould also be adopted ó ¼ ó 1 w w r where w is the interpenetration, w r is the ritial value of the interpenetration orresponding to the ondition of ó ¼ 0 and ó is the ompression strength. The rushing zone is then represented by a fititious overlapping, whih is mathematially analogous to the fititious rak in tension. It is important to note that, (1) from the mathematial point of view, the overlapping displaement is a global quantity, and therefore it permits the strutural behaviour to be haraterised without the need for modelling the atual failure mehanism of the speimen into the details, whih may vary from pure rushing to diagonal shear failure, to splitting, depending on its size-sale and slenderness. 5 The proposed model, based on a fititious interpenetration, permits a true material onstitutive law to be obtained, independent of the strutural size. This approah, in whih the post-peak stress displaement relationship is onsidered as a material onstitutive law, is experimentally onfirmed by van Vliet and van Mier 8 and by Jansen and Shah, 9 who onsidered speimens with different slenderness. Moreover, this assumption is more general, as it an be extended to speimens haraterised by different sizes. To demonstrate this, the uniaxial ompression tests arried out by Ferrara and Gobbi 23 on plain onrete speimens are onsidered. The slenderness varied from 0.5 to 2 and the sale range was 1:2:4. The experimental non-dimensional stress against average strain urves are shown in Figure 2, where S, M and L identify the three onsidered dimensions. As expeted, the elasti / S λ 0 5 λ M λ 0 5 λ L λ 0 5 λ ε: mm/mm ε ε / S λ 0 5 λ M λ 0 5 λ L λ 0 5 λ G w r Figure 1. Double onstitutive law introdued by the overlapping rak model for onrete in ompression w 0 3 w: mm Figure 2. Uniaxial ompression tests on speimens with different dimension and slenderness. 23 S, M and L denote, respetively, small, medium and large speimens. ó å relationship; ó w post-peak relationship Magazine of onrete Researh, 9, 61, No

4 arpinteri et al. behaviour is independent of any geometrial parameter, the slope of the orresponding branh being equal to the tangent elasti modulus of the material (Figure 2). However, the post-peak behaviour is largely influened by the slenderness and the sale of the speimen. As a onsequene, the stress strain relationship annot be assumed as a material property. The stress displaement urve for the softening regime an be obtained by omputing the post-peak loalised interpenetration through subtrating the elasti expansion, aused by the redution of the applied stress in the post-peak regime, to the total shortening of the speimen (see Figure 2). As a result of suh an operation, the experimental stress displaement urves related to speimens with different size or slenderness ollapse onto a very narrow band, demonstrating that the ó w relationship is able to provide a slenderness and sizesale independent onstitutive law of onrete in ompression. It is worth noting that the rushing energy, G, defined as the area below the post-peak softening urve of Figure 1, is now a true material parameter, sine it is not affeted by the strutural size. Dahl and Brinker 7 arried out a series of uniaxial ompression tests with the aim of measuring the dissipated energy per unit ross-setional area. They obtained values of about 50 N/mm and laimed that this dissipated energy beomes independent of the speimen size if the speimen is large enough. An empirial formulation for alulating the rushing energy has reently been proposed by Suzuki et al., 10 based on the results of uniaxial ompression tests arried out on plain and transversally reinfored onrete speimens. In the present study, the rushing energy is omputed aording to the aforementioned empirial equation, whih onsiders the onfined onrete ompression strength by means of the stirrups yield strength and the stirrups volumetri ontent 10 (see Figure 3) G ¼ G,0 þ k2 a p e ó ó ó 2 (2a) G : N/mm G,0 : N/mm kpe / a displaement, w r 1 mm, is one order of magnitude higher than the ritial opening displaement in tension (see also the experimental results by Jansen and Shah 9 ). Finally, it is noted that, in the ase of onrete onfinement, the rushing energy, omputed using Equation 2a, and the orresponding ritial value for rushing interpenetration, inrease onsiderably (Figure 3). 70 : Ma Figure 3. rushing energy against stirrups onfinement; and ompression strength where ó is the average onrete ompression strength, k a is a parameter depending on the stirrups strength and volumetri perentage and p e is the effetive lateral pressure. The rushing energy for unonfined onrete, G,0, an be alulated using the following expression (see Figure 3) G,0 ¼ 80 50k b (2b) where the parameter k b depends on the onrete ompression strength. 10 By varying the onrete ompression strength from 20 to 90 Ma, Equation 2b gives a rushing energy ranging from 30 to 58 N/mm (Figure 3). It is worth noting that G,0 is between two and three orders of magnitude higher than the tensile frature energy, G F, whereas the ritial value for the overlapping Uniaxial ompression tests Aording to the overlapping rak model, the mehanial behaviour of a plain onrete speimen subjeted to uniaxial ompression an be desribed by three simplified stages, analogously to the model proposed by arpinteri 24 for onrete slabs in tension. The speimen behaves elastially without any damage or loalisation zones, Figure 4. The displaement of the upper side is ä ¼ ål ¼ ó E l for å < å (3) After reahing the ultimate ompression strength ó, the deformation starts to loalise in a rushing band. The behaviour of this zone is desribed by 748 Magazine of onrete Researh, 9, 61, No. 9

5 The overlapping rak model for uniaxial and eentri onrete ompression tests 0 0 l ε l ε w w r l w 0 w w r () (d) Figure 4. Subsequent stages in the deformation history of a speimen in ompression the softening law shown in Figure 1, whereas the outside part of the speimen behaves elastially, Figure 4(). The displaement of the upper side an be omputed as the sum of the elasti deformation and the interpenetration displaement w ä ¼ ó E l þ w for w < w r (4) Introduing the softening law of Equation 1 into Equation 4, a one-to-one orrespondene is obtained between ä and ó ä ¼ ó E l þ w r 1 ó for w < w r (5) ó While the rushing zone overlaps, the elasti zone expands at progressively dereasing stresses. At this stage, the loading proess will be stable if it is displaement-ontrolled, that is if the external displaement ä is imposed. However, this is only a neessary and not a suffiient ondition for stability. () When ä > w r, onrete in the rushing zone is ompletely damaged and it is unable to transfer stresses, Figure 4(d). The ompression stresses vanish and the ondition of omplete interpenetration (stage 3) beomes ó ¼ 0 for ä > w r (6) When w r. å l, the softening proess is stable if it is displaement-ontrolled, beause the slope dó/dä at stage 2 is negative (Figure 5). When w r ¼ å l, this elementary model predits an infinite slope and a sudden drop in the load bearing apaity under displaement ontrol (Figure 5). Finally, when w r, å l, the slope dó/dä of the softening branh beomes positive (snap-bak), and a negative jump ours, as shown in Figure 5(). Rearranging Equation 5 ä ¼ w r þ ó l E w r (7) ó The same onditions just obtained from a geometrial point of view (Figure 5), may also be given by the analytial derivation of Equation 7. Normal softening ours for dä/dó, 0, that is for l E w r, 0 (8) ó whereas atastrophial softening (snap-bak) ours for dä/dó > 0 l E w r > 0 (9) ó Equation 9 may be rearranged as follows w r =2b å ðl=bþ < 1 (10) 2 where b is the speimen width. The ratio (w r /2b) is dimensionless and is a funtion of the material properties and of the strutural size s E ¼ w r 2b ¼ G (11) ó b where G ¼ 1 2 ó w r is the rushing energy (Figure 1). The energy brittleness number in ompression, s E, analogous to that proposed by arpinteri in ,25 for ohesive rak propagation in tension, desribes the sale effets typial of frature mehanis, that is the dutile brittle transition when the size-sale inreases. Equation 10 may be rewritten in the following form s E å º < 1 (12) 2 where º ¼ l/b is the speimen slenderness. Therefore, when the size-sale and the speimen slenderness are relatively large and the rushing energy is relatively low, the global strutural behaviour beomes brittle. The single values of parameters s E, å and º are not responsible for the global brittleness or dutility of the struture onsidered, but only their ombination B ¼ s E =å º. When B < 1 2, the onrete Magazine of onrete Researh, 9, 61, No

6 arpinteri et al. U E i E1 G ε ε w r w Figure 6. Improved onstitutive laws for the overlapping rak model O ε l w r δ relationship provided by the model ode is adopted up to the ahievement of the onrete ompression strength (see Figure 6) U ó ð ¼ E i=e 1 Þðå=å Þ ðå=å Þ 2 ó 1 þ ½ðE i =E 1 Þ 2Šðå=å Þ (13) O ε l w r U δ where ó is the ompression strength; ó is the atual value of the ompression stress; å is the ompression strain; å ¼ (å 1 in the original model ode notation); E i is the tangent modulus; E 1 is the seant modulus from the origin to the peak ompression stress, ó. Moreover, the following stress displaement ubi relationship desribing the softening regime is introdued (see Figure 6), whih has been omputed by imposing ó ¼ ó at w ¼ 0, ó ¼ 0 at w ¼ w r, and horizontal tangents in the same points ó ó ¼ 2 w w r þ w w (14) r O Figure 5. Stress displaement response: normal softening; vertial drop; () atastrophi softening (snap bak) () speimen of Figure 4 shows a mehanial behaviour whih an be defined as brittle or atastrophi. In this ase, a bifuration of the global equilibrium ours, sine, when point U in Figure 5() is reahed, the global unloading may our along two alternative paths when the external displaement is dereased: the elasti UO or the virtual softening U. A more realisti simulation of onrete ompression tests an be performed by introduing more sophistiated onstitutive laws. In the following, in order to take into aount the non-linear behaviour of onrete in the inreasing branh, a well-known stress strain w r ε l δ A omparison between the numerial preditions obtained using Equations 13 and 14 and the experimental results of uniaxial ompression tests arried out by Jansen and Shah 9 on speimens haraterised by different slenderness and onrete strength are shown in Figures 7 and 8. Both experimentally and numerially it is possible to apture snap-bak branhes if a monotoni inreasing funtion of time is assumed as the ontrol parameter. Usually, the experimental tests are arried out using a irumferential displaement ontrol (see Hudson et al. 11 and Jansen et al. 27 ). However, this method is not suitable in the ase of very slender speimen, as those shown in Figure 8, where the failure zone does not always develop in the middle of the speimen. For this reason, Jansen and Shah 9 adopted an alternative method, whih is a linear ombination of fore and displaement, originally proposed by Okubo and Nishimatsu. 28 Aording to this method, a part of the elasti deformation is subtrated from the total speimen deformation, leaving the inelasti deformation as a stable feedbak signal (for more details see Jansen and Shah 9 ). Note that a similar approah is adopted in the proposed numerial proedure, where the overlapping displaement is the ontrol parameter used to determine 750 Magazine of onrete Researh, 9, 61, No. 9

7 The overlapping rak model for uniaxial and eentri onrete ompression tests 47 9 Ma λ 2 5 λ 3 5 λ 4 5 λ Ma / / / 0 3 δ: mm Ma λ 2 5 λ 3 5 λ 4 5 λ δ: mm Ma λ 2 5 λ 3 5 λ 4 5 λ δ: mm Figure 7. Analytial and experimental 9 nondimensional stress against total shortening in the ase of normal strength onrete. ó denotes the stress at the peak load ó, aording to Equation 1, and then ä, aording to Equation 4. The proposed numerial model, based on a more sophistiated relationship with respet to the linear one, exhibits a satisfatory predition apability. In good agreement with the experiments, the mehanial behaviour beomes more brittle, with the appearane of snap-bak instability by inreasing the speimen slenderness and the onrete ompression strength. Eentri ompression tests / 0 3 δ: mm λ 2 5 λ 3 5 λ 4 5 λ 5 5 Figure 8. Analytial and experimental 9 nondimensional stress against total shortening in the ase of high strength onrete. ó denotes the stress at the peak load Overlapping Elasti portion Elasti portion Desription of the numerial algorithm In this setion, a simplified version of the numerial algorithm developed by arpinteri et al. 16 desribing the mehanial behaviour of R beams in bending is presented in order to simulate the behaviour of plain onrete speimens subjeted to eentri ompression by means of the overlapping rak model. In lose analogy with the behaviour of onrete speimens subjeted to uniaxial ompression, all of the non-linear ontributions in the post-peak regime are loalised along the middle ross-setion where interpenetration takes plae, while the two half-speimens exhibit an elasti behaviour, as shown in Figure 9. It is assumed that the stress distribution in the middle ross-setion is linear-elasti until the maximum ompression stress reahes the onrete ompression strength. When this threshold is reahed, onrete rushing is assumed to take plae and a fititious overlapping rak propagates towards the opposite vertial side of the speimen. Outside the overlapping zone, the material is assumed to behave linear-elastially. Aording to the overlapping rak model, the stresses in the overlapping zone are assumed to be a funtion of the amount of interpenetration and beome equal to 0 9 S up S low S up S low Figure 9. Idealisation of the speimen and definition of the total rotation Magazine of onrete Researh, 9, 61, No

8 arpinteri et al. zero when the interpenetration is larger than the ritial value w r, as shown in Figure 1. If the external fore is applied outside the entral ore of inertia, the material behaviour on the tensile side is desribed by means of the well-known ohesive rak model. 25 The middle ross-setion of the speimen an be subdivided into finite elements by n nodes (Figure 10). In this sheme, overlapping or ohesive stresses are replaed by equivalent nodal fores by integrating the orresponding pressures or trations over eah element size. Suh nodal fores depend on the nodal losing or opening displaements aording to the overlapping or ohesive softening laws. The vertial fores, F, ating along suh a rosssetion an be omputed as follows ffg ¼ ½K w Šfwgþ fk g (15) where {F} is the vetor of nodal fores, [K w ] is the matrix of the oeffiients of influene for the nodal displaements, {w} is the vetor of nodal displaements, {K } is the vetor of the oeffiients of influene for the applied fore and is the applied axial fore. The oeffiients of influene [K w ] have the physial dimension of a stiffness and are omputed a priori with a finite element analysis by applying a unitary displaement to eah of the nodes shown in Figure 10. In the generi situation shown in Figure 10, the following equations an be onsidered, taking into aount the linear overlapping softening law (Equation 16a), the undamaged zone (Equation 16b) and the linear ohesive softening law (Equation 16) F i ¼ F 1 w i w for i ¼ 1,...,(p 1) (16a) r w i ¼ 0 for i ¼ p,..., m (16b)! F i ¼ F u 1 wt i w t for i ¼ (m þ 1),..., n (16) r l Node 1 h F i F i Node n Node 1 Node p Node m Node n Figure 10. Finite element nodes along the middle rosssetion; fore distribution with ohesive rak in tension and overlapping rak in ompression Equations 15 and 16 onstitute a linear algebrai system of (2n) equations in (2n+1) unknowns, namely {F}, {w} and. A possible additional equation an be hosen: it is possible to set either the fore in the ohesive rak tip, m, equal to the ultimate tensile fore, or the fore in the overlapping rak tip, p, equal to the ultimate ompression fore. In the numerial sheme, the situation whih is loser to one of these two possible ritial onditions is hosen. This riterion will ensure the uniqueness of the solution on the basis of physial arguments. The driving parameter of the proess is the position of the rak tip that the onsidered step has reahed in the limit resistane. Only this tip is moved when passing to the next step. In lose analogy with ontat mehanis, where the area of ontat is unknown a priori and has to be determined using a non-linear numerial ontrol sheme, in the present work the extension of the overlapping zone in eentri bending tests has to be determined iteratively for eah value of the applied load. However, a main differene with ontat mehanis is that the equilibrium solution to be found in the urrent problem is governed by the stress-overlapping displaement in Equation 1 instead of by the Signorini-Fihera boundary onditions (see Wriggers 29 and aggi et al. 30 ). Finally, at eah step of the algorithm, it is possible to alulate the speimen rotation, W, defined in Figure 9 as follows W ¼ fd w g T fwgþ D (17) where {D w } is the vetor of the oeffiients of influene for the nodal displaements, with physial dimensions of [L] 1, and D is the oeffiient of influene for the applied fore with physial dimensions of [F] 1. omparison between model preditions and experimental results In this setion, the omparison between the numerial preditions and the experimental results of the testing programme by Debernardi and Taliano 22 on 15 plain onrete speimens subjeted to eentri ompression is arried out. The dimensions of the speimens, shown in Figure 11, were kept onstant, whereas five different degrees of eentriity were onsidered, varying between 0 and 48 mm. The mean value of the ompression strength, determined on ten ubes with 6 m sides, was equal to 56 N/mm 2, with standard deviation of 3. 5 N/mm 2. In order to redue the influene of the boundary onditions due to frition between the speimen and the loading platens, the deformations were measured in the entral part of the speimen, whih was further subdivided into three parts. The positioning of the measuring instruments, shown in Figure 11, with partiular regard to the devies 3 8, permits the deformations of three portions of the speimen to be evaluated, eah one having a length of 112 mm. The most important aspets of the experi- 752 Magazine of onrete Researh, 9, 61, No. 9

9 The overlapping rak model for uniaxial and eentri onrete ompression tests e Loading axis e Speimen axis 50 ross-setion y x 25 x 500 DD Figure 11. Eentri ompression tests 22 : typial speimen dimensions ; arrangement of the measuring instruments ments were the orret positioning of the speimen and the appliation of the load at a predetermined level of eentriity. Both extremities of the speimen were onfined by means of speial stirrups to prevent the opening of longitudinal raks. The servo-hydrauli testing mahine operated in strain-ontrolled onditions by applying a load suh that the deformation in the most ompressed fibres, measured by means of the DD1 gauge, inreased at a onstant rate up to failure. By this proedure, the softening stage until failure an be followed after the ahievement of the peak load. As a result of the experiments, the applied load and the deformation, l, reorded by the several extensometers related to the length l, were aquired. The rotations of eah part of the speimen an be omputed as ð W i ¼ l left l right Þ (18) h where h is the distane between two measuring bases opposite to eah other. Hene, the total rotation of the analysed speimens, with a length equal to 336 mm, is given by the sum of the rotations of the three portions. It is worth noting that, by subdividing the total length of the speimen into three parts, a loalisation of deformations in the entral part has been put into evidene. The length of the speimens assumed for the simulations is equal to l ¼ 336 mm, as the length of the speimen supplied with the measuring devies in the testing programme (see Figures 10 and 11). In the numerial sheme, the middle ross-setion of the onrete speimen is disretised into 100 finite elements and the oeffiients of influene entering Equation 15 are preliminarily determined using the finite element method. The experimental tests are also simulated by means of the stress strain relationships provided by the model ode for modelling onrete in ompression. In partiular, Equation 13 is adopted to desribe the whole inreasing branh of the stress strain diagram and the first part of the desending branh for values of ó ó > 0. 5, or, equivalently for å < å,lim. For å. å,lim, the desending branh of the ó å diagram has to be desribed by the following equation 2! 1 2 å 2 ó ¼ 4 î å,lim =å 2 å,lim =å å 4 (19) å 1 þ î ó å,lim =å with å 2 4½ å,lim =å ½ðE i =E 1 Þ 2Š þ 2 å,lim =å ð Ei =E 1 Þ î ¼ 2 (20) å,lim =å ½ð Ei =E 1 Þ 2Šþ 1 The values of E i, E 1 and å,lim are given in Table of the model ode for different values of onrete ompression strength. In the appliation of Equations 13, 19 and 20, it was assumed that all the ross-setions have the same behaviour, and that the total rotation is given by multiplying the urvature of one of these setions for the speimen height. The numerial results are ompared with the experimental ones in the W diagrams for different eentriities (see Figure 12). First, it is worth noting that, exept the ase of e ¼ 12 mm, a perfet agreement is obtained between the numerially predited and the experimentally evaluated softening branhes, onfirming the good predition apability of the proposed model, in spite of the simple linear softening relationship adopted. Moreover, the numerial model aptures the experimentally observed derement of the maximum applied load due to the ontemporaneous presene of bending moment and axial fore. The disrepany between the numerial and the experimental urves in the Magazine of onrete Researh, 9, 61, No

10 arpinteri et al. 600 Experimental Numerial 600 Experimental Numerial Applied load: kn 400 Applied load: kn Experimental Numerial 600 Experimental Numerial Applied load: kn 400 Applied load: kn () (d) Figure 12. Numerial and experimental applied load against total rotation diagrams for the speimens tested by Debernardi and Taliano, 22 by varying the eentriity: e = 12 mm; e = 24 mm; () e = 36 mm; (d) e = 48 mm inreasing branh suggests that a more omplex onstitutive law, with a non-linear ontribution, should be onsidered instead of the linear-elasti one, in order to improve the desription of the real behaviour. Finally, a general good agreement is evidened between the urves obtained by the appliation of the onstitutive law provided by model ode 90 (dashed lines in Figure 12) and the experimental results. This agreement is probably owing to a oinidene, beause, as shown in the next setion, the real behaviour of the speimens is sale-dependent, whereas model ode 90 preditions ompletely disregard the size-sale effets. Size-sale and slenderness effets in eentri ompression tests In this setion, a study of the size-sale and slenderness effets on the behaviour of onrete prisms subjeted to eentri ompression is presented. To this aim, three different strutural sizes, haraterised by ross-setion dimensions, b 3 h, equal to , , mm, and three different slendernesses, º ¼ 1. 0, 2. 2, 4. 0, are onsidered. Besides, the following four values of eentriity are explored: e ¼ 0. 08h, e ¼ 0. 16h, e ¼ 0. 24h, e ¼ 0. 32h. Unfortunately, it is impossible to arry out a diret numerial against experimental omparison owing to the lak of experimental data in the literature. The size-sale effets, for different slendernesses and for an eentriity equal to 0.16h, are shown in the non-dimensional applied load versus total rotation urves of Figure 13. It is worth noting that, independently of the prism slenderness, the post-peak mehanial behaviour is size-sale dependent. In partiular, the softening regime exhibits a dutile-to-brittle transition by inreasing the speimen size for a onstant slenderness. Furthermore, the value of the maximum applied load results to be a slightly dereasing funtion of the strutural size. As mentioned before, from Figure 13 it is dedued that the onstitutive law provided by model ode 90, whih assumes an energy dissipation within a volume, does not apture the size-sale effets. It is interesting to note that, in the ase of º ¼ 1. 0, model ode 90 urve is very lose to the behaviour of the smallest speimen (Figure 13). In the ase of º ¼ 2.2, it agrees with the response of the intermediate speimen (Figure 13), whereas for º ¼ 4. 0 it is lose to that of the largest one (Figure 13()). The effet of slenderness is investigated in the nondimensional load against total rotation diagram of Figure 14 for a given speimen size (ross-setion equal to mm) and a given eentriity (e ¼ 0. 08h). 754 Magazine of onrete Researh, 9, 61, No. 9

11 The overlapping rak model for uniaxial and eentri onrete ompression tests λ A: B: : 300 e 8 h h 150 mm / bh A / bh λ 4 0 λ 2 2 λ B λ 2 2 A: B: : 300 Figure 14. Numerially predited non-dimensional applied load against total rotation urves for speimens with different slenderness / bh A h 150 mm λ B / bh e 8 h e 0 16 h e 4 h e 0 32 h / bh λ 4 0 A: B: : 300 A Figure 15. Numerially predited non-dimensional load against total rotation urves for given speimen dimension and slenderness and different load eentriity () As expeted, the stiffness of the elasti branh is a dereasing funtion of the slenderness, sine it is proportional to the speimen height. orrespondingly, little inrement of the softening slope is evidened. The effet of the load eentriity is shown in Figure 15 for º ¼ 2. 2 and ross-setion dimensions equal to mm. The inrement in the eentriity, e, produes a redution in the stiffness of the elasti B 4 Figure 13. Numerially predited size-sale effets by varying the speimen slenderness and for a given load eentriity e ¼ 0. 16h 5 branh, owing to the inrease in the bending moment. At the same time, the mehanial behaviour beomes undoubtedly more dutile. This result puts into evidene the important ontribution on dutility of the post-peak regime of onrete in ompression in the ase of high strain gradient, as, for example, in a reinfored onrete olumn subjeted to an eentri axial fore. Finally, the non-dimensional load against total rotation urves for different values of the brittleness number in ompression, s E, defined by Equation 11, are shown in Figure 16. The values of the slenderness and the eentriity are, respetively, equal to 4 and 0.08h. Speimens haraterised by the same value of s E exhibit the same mehanial behaviour. A dutile-to-brittle transition is observed by dereasing the brittleness number from to This transition an be obtained either by inreasing the speimen dimension, b, or by inreasing the ompression strength, ó,orby Magazine of onrete Researh, 9, 61, No

12 arpinteri et al. / bh 0 D A B E H G F 4 e 8 h λ 4 A: se 006 B: se 012 : se 021 D: se 043 E: se 086 F: se 171 G: se 343 H: s 686 E slope (see Figure 16). From the strutural point of view, is has been shown that the peak load is a dereasing funtion of the load eentriity (see Figure 15). More importantly, the strutural dutility, evaluated as the area below the load against total rotation diagram, turns out to be a dereasing funtion of the speimen size, for given values of slenderness and load eentriity (see Figure 13). This sheds a new light on the sizesale effets in eentri ompression tests, whih are ompletely disregarded by the design formula proposed by model ode 90. Figure 16. Numerially predited non-dimensional applied load against total rotation diagrams by varying the energy brittleness number in ompression, s E dereasing the rushing energy, G. In the ase of large strutural dimensions and very low rushing energy, a atastrophi failure (snap-bak) is obtained, as learly evidened by urve A in Figure 16. onlusion In the present paper, a theoretial model and a numerial algorithm have been proposed for the analysis of the mehanial behaviour of onrete speimens subjeted to uniaxial or eentri ompression tests. The onept of overlapping rak in ompression, whih is analogous to the ohesive rak in tension, makes it possible synthetially to haraterise the mehanial response of quasi-brittle materials in ompression without simulating eah speifi failure mode. In fat, when the slenderness dereases, a transition from splitting to rushing ollapse takes plae in reality. A similar transition an our by varying the sizesale of the element. In spite of this, the use of a global quantity, represented by the overlapping rak displaement, has the advantage that it defines a true size- and slenderness-independent onstitutive law. The good agreement between the analytial preditions and the experimental results in the ase of uniaxial ompression tests demonstrates the reliability of the proposed approah (see Figures 7 and 8). Moreover, from the dimensional analysis point of view, it is remarkable to note that neither the individual values of the rushing energy, the ompression strength nor the speimen size are responsible for the dutile-tobrittle transition in the mehanial response, but rather only their funtion s E, whih defines an energy brittleness number in ompression analogous to that proposed in tension by arpinteri in As far as the numerial simulations of eentri ompression tests are onerned, the use of the extension of the fititious rushing zone and the length of the tensile rak as the driving parameters is highly effetive, as it follows the desending branh of the load-rotation diagram with either negative or positive Referenes 1. Bažant Z.. and Xiang Y. Size effet in ompression frature: splitting rak band propagation. Journal of Engineering Mehanis, ASE, 1997, 123, No. 2, arpinteri A., Ferro G. and Monetto I. Sale effets in uniaxially ompressed onrete speimens. Magazine of onrete Researh, 1999, 51, No. 3, Reunion Internationale des Laboratoires et Experts des Materiaiux, Systemes de onstrution et Ouvrages (RILEM) Tehnial ommittee 148-SS. Strain-softening of onrete in uniaxial ompression. Materials and Strutures, 1997, 30, No. 4, Indeliato F. and aggi M. Speimen shape and the problem of ontat in the assessment of onrete ompressive strength. Materials and Strutures, 8, 41, No. 2, arpinteri A., iola F. and ugno N. Boundary element method for the strain-softening response of quasi-brittle materials in ompression. omputers and Strutures, 1, 79, No. 4, van Mier J. G. M. Strain Softening of onrete under Multiaxial ompression. hd thesis, Eindhoven University of Tehnology, The Netherlands, Dahl H. and Brinker R. Frature energy of high-strength onrete in ompression. Frature of onrete and Rok: Reent Developments. roeeding of the International onferene on Reent Developments in the Frature of onrete and Rok, ardiff, Wales, 1989, pp van Vliet M. and van Mier J. Experimental investigation of onrete frature under uniaxial ompression. Mehanis of ohesive-fritional Materials, 1996, 1, No. 1, Jansen D.. and Shah S.. Effet of length on ompressive strain softening of onrete. Journal of Engineering Mehanis, 1997, 123, No. 1, Suzuki M., Akiyama M., Matsuzaki H. and Dang T. H. onentri loading test of R olumns with normal- and highstrength materials and averaged stress-strain model for onfined onrete onsidering ompressive frature energy. roeedings of the 2 nd fib ongress, Naples, 6, ID 3 13 (on D-ROM). 11. Hudson J. A., Brown E. T. and Fairhurst. Shape of the omplete stress strain urve for rok. roeedings of the 13th Symposium on Rok Mehanis, University of Illinois, Urbana, Illinois, 1971, Hillerborg A. Frature mehanis onepts applied to moment apaity and rotational apaity of reinfored onrete beams. Engineering Frature Mehanis, 1990, 35, No. 1 3, Markeset G. and Hillerborg A. Softening of onrete in ompression: loalization and size effets. ement and onrete Researh, 1995, 25, No. 4, Bažant Z.. Identifiation of strain-softening onstitutive relation from uniaxial tests by series oupling model for loalization. ement and onrete Researh, 1989, 19, No. 6, arpinteri A., orrado M., aggi M. and Manini G. ohesive versus overlapping rak model for a size effet analysis of 756 Magazine of onrete Researh, 9, 61, No. 9

13 The overlapping rak model for uniaxial and eentri onrete ompression tests R elements in bending. Frature Mehanis of onrete Strutures. roeedings of the 6 th International FraMoS onferene, atania, 7, 2, arpinteri A., orrado M., aggi M. and Manini G. A numerial approah to modelling size effets on the flexural dutility of R beams. RILEM Materials and Strutures. doi: /s y. 17. orrado M. Effetti di sala sulla apaità di rotazione plastia di travi in alestruzzo armato. hd thesis, olitenio di Torino, Torino, 7, in Italian. 18. Hillerborg A., Modeer M. and etersson. E. Analysis of rak formation and rak growth in onrete by means of frature mehanis and finite elements. ement and onrete Researh, 1976, 6, No. 6, etersson. E. rak growth and development of frature zones in plain onrete and similar materials. Lund Institute of Tehnology, Tehnial report, LUTVDG/TVBM arpinteri A. Interpretation of the Griffith instability as a bifuration of the global equilibrium. Appliation of Frature Mehanis to ementitious omposites, roeedings of a NATO Advaned Researh Workshop, Evanston, USA, 1984, pp arpinteri A. usp atastrophe interpretation of frature instability. Journal of the Mehanis and hysis of Solids, 1989, 37, No. 5, Debernardi. G. and Taliano M. Softening behaviour of onrete prisms under eentri ompressive fores. Magazine of onrete Researh, 1, 53, No. 4, Ferrara G. and Gobbi M. E. Strain Softening of onrete under ompression. ENEL-RIS Laboratory, Milano, Italy, Report to RILEM ommittee 148-SS. 24. arpinteri A. Snap-bak and hyperstrength in lightly reinfored onrete beams. Magazine of onrete Researh, 1988, 40, No. 145, arpinteri A. Size effets on strength, toughness, and dutility. Journal of Engineering Mehanis, ASE, 1989, 115, No. 7, omité Euro-International du Béton. EB FI Model ode Thomas Telford, Lausanne. EB Bulletin No. 213/ 214, Jansen D.., Shah S.. and Rossow E.. Stress strain results of onrete from irumferential strain feedbak ontrol testing. AI Materials Journal, 1995, 92, No. 4, Okubo S. and Nishimatsu Y. Uniaxial ompression testing using a linear ombination of stress and strain as the ontrol variable. International Journal of Rok Mehanis and Mineral Siene and Geomehanis, 1985, 22, No. 5, Wriggers. omputational ontat Mehanis. Wiley, hihester, aggi M., arpinteri A. and Zavarise G. A unified interfae ostitutive law for the study of frature and ontat problems in heterogeneous materials. Analysis and Simulation of ontat roblems. Leture Notes in Applied and omputational Mehanis (Wriggers. and Nakenhorst U. (eds)). Springer Verlag, Berlin, 6, Vol. 27, pp Disussion ontributions on this paper should reah the editor by 1 May 2010 Magazine of onrete Researh, 9, 61, No