An Analytical Formulation of Stress-Block Parameters for Confined Concrete

Transcription

1 The Open Constrution and Building Tehnology Journal, 8,, Open Aess An Analytial Formulation o Stress-Blok Parameters or Conined Conrete Frano Braga, Rosario Gigliotti, Mihelangelo Laterza* and Mihele D Amato Department o Strutural and Geotehnial Engineering, University o Rome, La Sapienza, Italy; Department o Strutural Engineering, University o Basiliata, Potenza, Italy Abstrat: In order to evaluate the apaity o RC members, the main odes allow the use o stress-strain laws that an reprodue losely the real behaviour o onrete, as opposed to parabola-retangular or equivalent retangular diagrams. Both setional strength and dutility depend on the law o onrete, thereore they are inluened by the oninement o members, as evidened in the literature. In this paper a possible design approah is presented, based on lassi setion analysis methods. The method uses parameters that represent the stress-strain law o onined onrete. The studies arried out show that suh parameters an be hosen through simple relationships depending on the strength o non-onined onrete, on the amount and geometry o longitudinal and transverse reinorement, and on the geometry o the setion. At this aim some numerial analyses have been perormed using an analytial model o onined onrete, apable o taking into aount all the mentioned eets, even in the ase o various soures o oninement, when dierent types o hoops and external elements (FRP wrappings, steel plates, et.) are used. More in detail, the setion interation diagrams or the dierent limit states requires the deinition o an appropriate upper bound or the strain o onrete. Thereore the study ouses on the possibility o using stress-bloks depending on the maximum stain assumed, or on the level o residual stress aepted in onrete aording to a speiied limit state. Further studies will extend the parametri analysis in order to obtain design equations to be implemented in odes. INTRODUCTION In the last years the main design odes have adopted the perormane approah. By onsidering the important limit states, this approah deines the perormane level requested to the overall struture and its members. Reerring to the RC strutures, the main researhes regard the modelling o materials and members, and the development o the inite elements (or non linear analysis) apable to desribe the non-linear phenomena. It is well known that the RC buildings behaviour is essentially due to the lexural response o the members and their axial load level. Neozeland Design Code 3 (995) [] and Euroode 8 (EC8) [] link the transversal reinorement bars to the global and loal dutility behaviour. The transversal reinorement ontributes to better the strutural dutility. This or two reasons: it impedes the bukling o longitudinal bars when the setion reahes high urvatures and modiies the onrete stress-strain relationship inreasing strength and dutility. In this way we have an inreasing o setion urvature and onsequently o global dutility. For these reasons the oninement o the onrete has a very important role in strutural engineering, and in reent years it regards strengthening o the existing buildings, too. *Address orrespondene to this author at the Department o Strutural Engineering, University o Basiliata, Potenza, Italy; mihelangelo.laterza@unibas.it In order to evaluate the apaity o the RC ross-setions, the main odes allow to use simpliied relationships or onrete under ompression (parabola-retangle and equivalent retangular distribution, also alled stress-blok). As an alternative, Euroode (EC) [3], ACI 38 [4] and NZS 43 [5] allow to use stress-strain laws that reprodue losely the real behaviour o the ompressive onrete. Aording to previous researhes about ompressive onrete, this paper has the aim to provide a irst approah to evaluate the apaity o RC setions by the use o stressbloks or onined onrete. An analytial evaluation was arried out using a model that estimates analytially the oninement eets on the ore onrete. By onsidering the interation between onrete and transversal/longitudinal bars and/or transversal strengthening (as FRP), this model provides a global parameter alled equivalent onining pressure that allows us to evaluate the stress-strain relationship or onined onrete. Moreover, analysis have been made supposing plain setion remains plane ater bending, strain in bounded reinorement bar is the same as that in the surrounding onrete, tensile strength o onrete is negleted. The results have allowed to propose a irst ormulation o stress-blok parameters (depending to materials strength, setion geometry and oninement level) to use within the RC setion equilibria. It has been also investigated the inluene o ompressive onrete maximum strain on the apaity o the setion /8 8 Bentham Open

2 38 The Open Constrution and Building Tehnology Journal, 8, Volume Braga et al. FLEXURAL STRENGTH OF RC SECTION In order to evaluate the RC setion apaity, the ollowing lassial assumptions are made: non-linear onstitutive laws or materials. The maximum strains assumed depend by the state limit onsidered. plain setion remains plane ater bending; the strain in bonded reinorement is the same as that in the surrounding onrete; onrete tensile strength is negleted (or all limit states where onrete is raked). Fig. (). Stress and strain diagrams along the setion. For eah deormation proile ixed along ross setion, the M-N ouple in balane with internal ores is desribed by: N C A n = + () si si i= n h h M = C y + Asi si di i= () STRESS-BLOCKS In previous equations instead o real -, it is allowed the use o a retangular equivalent diagram or ompressive onrete alled stress-blok. The EC also urnishes the stress-strain relationship or onined onrete and its ultimate strain, but does not indiate the upper limit or eah limit state.,..9.8 Eetive strength alulated Eetive strength EC Eetive height o ompression alulated Eetive height o ompression EC.7 EC =.8 per 5MPa =. per 5MPa k k =.8 ( 5 ) 4 per5 < 9 MPa =. ( k 5 ) per5 < 9 MPa k k k k (MPa) Fig. (3). Stress-blok parameters o unonined onrete in aording to EC. The Fig. (4) shows stress-bloks adopted by some national odes [-9] ACI 38 CSA A3.3 (994) e CSA S6 () NZS 3 (5) EC (4) NS 3473 (995) CEB-FIB Model Code (99) D.M. 4/9/ (MPa) Fig. (). Stress-strain relationship or onrete and equivalent retangular diagram. By ixing the maximum onrete strain m, the stressblok is desribed through the eetive strength d and the eetive height o ompression zone m. These parameters are given by: m d = d m (3) m d = d m(,5) m (4) Reerring to unonined onrete, EC permits to use either a parabola-retangular or a simpliied diagram as the retangular equivalent one, whose parameters are given as a untion o onrete strength (Fig. 3) (MPa) Fig. (4). Stress-blok parameters urnished by some national odes. As the EC, the ACI 38, NZS 3 (995) and Italian ode D.M. 4/9/5 [6] too, allow to use - diagram whih an reprodue more losely the real behaviour o onrete. ANALYTICAL MODEL USED FOR CONFINED CONCRETE To obtain stress-blok parameters or onined onrete, numerial analyses have been made using a ompletely analytial model o onined onrete, proposed by three o the authors o the present paper (Braga, Gigliotti and Laterza []).

3 An Analytial Formulation o Stress-Blok Parameters The Open Constrution and Building Tehnology Journal, 8, Volume 75 The key assumption o the model is that the inrement o stress in the onrete setion is produed without any out-oplane strain. This means that the oninement exerised by the transverse reinorements should take plae in plane strain onditions. In general terms, reerring to a generially representative ross setion, the stress-strain relationship or onrete an be written as ollows: ( ) ( ) ( ) = + (5) Z Z Z Z Z Z where z ( z ) and z ( z ) are the stress o onined and unonined onrete, respetly; z is the related strain. z ( z ) is the strength inrement related to the triaxial state. This inrement is proportional to onining pressures, aused by the transverse reinorements and their distribution, to the harateristis and to the strutural element dimensions. With reerene to the previous hypothesis, the strength inrement z ( z ) due to oninement is onstant inside the setion, and is given by: = Bl (6) Z where B = l S E l SE E l C E I A 3 8 E Es As S E l + 5 Es Is ( + ) z 4 { s 89 s s s( ) } C = I + + l A + 35 s( ) s( 5) r,max (7) Fig. (6). Distribution o vertial onining pressures. The model takes into aount the distribution o onining pressures oered by longitudinal reinorements (Fig. 6). By assuming that the pressures distributed along the strutural element are uniorm, the onining pressure rm is redued by the ollowing parameter: k sl 3 45 l = l st lon st st where l = ; = ; st = S l lon () when the bending stiness o longitudinal bars beomes negligible, only the arhing ation between two stirrups spreads the onining pressures along the olumn. Thus, it an be assumed that ksl k, where k is given by the relationship (Sheikh and Uzumeri, 98) []: k S = 4 lst () r/r =,5 rm r/r = rm R The eetive onining pressure due to square stirrups beomes: = k () r sl rm Fig. (5). Conining pressure along internal irumerenes. Fig. (5) shows, in polar oordinates, the oninement pressure. On ylinders internal to setion (square or irular), the mean onining pressure rm is given by: By using the analytial ormulation o the onining pressures obtained or square setion and/or irular setions with a single stirrup (S and/or C onigurations), the pressures applied by other highly omplex onigurations o transverse reinorements an also be obtained. In suh ases, in order to alulate the onining pressure r, the priniple o superposition is used and applied to the pressures rs alulated or eah stirrup and/or FRP/Steel wrapping (Figs. 7-8). rm = Bl (8) and substituting into Eq. (6): (, ) ( ) = = = (9) z r nm z rm z rm The rm pressure is a onstant varying the radius o ylinders. Thereore, this parameter is appropriate to desribe in every point the onining state inside the setion. rm is also alled equivalent onining pressure. Fig. (7). Square and irular reinorement onigurations.

4 76 The Open Constrution and Building Tehnology Journal, 8, Volume Braga et al. Fig. (8). External jakets (FRP and Steel). For example, or the setions onined by internal and external stirrup (Fig. 9), the onining pressure is given by: A = + (3) r ri re Ae Reerring S4 setion (Fig. 9), r is obtained by: b b (a) r = + r+ r L L (4) b (b) r = rl + r (5) L Fig. (9). Setion S4. As examples, onseutively stress-strain relationships are plotted and evaluated or two analyzed ases. Figures also show the onining pressures (as a untion o longitudinal strain o onrete) separately applied by internal and external reinorements and the resultant onining pressure rm. Fig. (). S3 setion type with strengthened by FRP Conining pressures and stress-strain relationship. For S3 setion type, it an be observed that the onining pressure remains onstant beyond the point orresponding to the yielding stress o steel. Instead, the result onining pressure due to FRP inreases ater the yielding o internal stirrup until wrapping ailure ours. FLEXURAL STRENGTH OF RC CONFINED SEC- TION For studying the onined setion are requested two dierent laws or the onrete: a onined - or ore and an unonined one or over. Fig. (). S3 setion type. Conining pressures and stress-strain relationship. Fig. (). Conined setion: stress diagrams and internal ores. The M-N interation urve, or eah limit state, is obtained ixing the material longitudinal strains and turning the strain proile around them. By reerring to Ultimate Limit State (ULS), Fig. (3) shows the deormation ields and M-N interation urve in aording to EC. In general, the upper limits or material deormations depend on the perormane requirements.

5 An Analytial Formulation o Stress-Blok Parameters The Open Constrution and Building Tehnology Journal, 8, Volume 77 Strain diagrams at the Ultimate Limit State (ULS) A s G A s sm sy N Corresponding interation diagram 5 y 4 u s 3 M Unonined Conined Conined Conined Flexural load (kn m) Unonined Conined Conined Conined Fig. (3). Strain diagrams and interation urve in aording to Euroode. By reerring to ULS, the EC and D.M. 4/9/5 assume or strain ompressive onrete the limit o 3,5. Instead, this limit is equal to 3 or ACI 38 and NZS 3 (5). As shown in previous experimental tests these limits are reerred to unonined onrete setions []. To analytially evaluate the inluene o maximum deormation assumed on setion strength, it has been utilized the sotware Response [3]. This sotware puts no limit to the maximum material deormations. One ixed the axial load N, it researhes the strain proile to whih the maximum lexural strength orresponds. The analysis have been made reerring to a square setion (L=5 mm) and using our - relationships or onrete, obtained rom unonined onrete strength ( =3 MPa) by dierent oninement levels (Fig. 4). Axial stress (MPa) Unonined Conined- Conined- Conined Axial strain Fig. (4). Stress-strain relationships onsidered. Three dierent amount o longitudinal bars have been onsidered (minimum, medium, maximum aording to EC8) Axial strain (mm/m) Fig. (5). Interation urves and maximum longitudinal strains o onrete reerred to mean amount o longitudinal bars Axial strain (mm/m) Unonined Conined Conined Conined Axial strain (mm/m) Unonined Conined Conined Conined Fig. (6). Maximum longitudinal strains o onrete reerred to minimum and maximum amount o longitudinal bars. The preliminary study shows that: or eah ase analyzed, the strain m orresponding to maximum setion strength is onstant in a wide range o axial ore N; -

6 78 The Open Constrution and Building Tehnology Journal, 8, Volume Braga et al. m inreases with inreasing o level oninement inside the setion and it results nearly independent rom the longitudinal bars perentage; unonined onrete m is lose by the maximum strain values adopted by some odes or the ULS. In partiular, this value is equal to the 3,5 aording to EC and to the 3 or the ACI 38 and NZS 3 (5). As an alternative, the interation urves an be obtained by onventional method, whih requests the deinition o material maximum strains. In this way, m o onrete an be ixed by the ratio: = (6) m where is the peak o onined stress-strain urve, while m orresponds to m onsiderated. Next igures (Figs. 7-) show onvetional interation urves obtained onsidering dierent values. By using the onventional method, the interation urve whih is assoiated the maximum lexural strength o RC setion (Response urves) is obtained, too. However, the appropriate value is variable in the ases onsidered g=. g=.93 Response g=.95 g=.9 g=.85 g=.8 g=.75 g= Flexural load (kn m) Fig. (7). Interation urves obtained varying the maximum onrete strain (unonined onrete and mean amount o longitudinal bars) g=. g=.95 Response g=.95 g=.9 g=.85 g=.8 g=.75 g= Flexural load (knm) Fig. (8). Interation urves obtained varying the maximum onrete strain (Conined onrete and mean amount o longitudinal bars) g=. g=.963 Response g=.95 g=.9 g=.85 g=.8 g=.75 g= Flexural load (knm) Fig. (9). Interation urves obtained varying the maximum onrete strain (Conined onrete and mean amount o longitudinal bars) g=. g=.977 Response g=.95 g=.9 g=.85 g=.8 g=.75 g= Flexural load (knm) Fig. (). Interation urves obtained varying the maximum onrete strain (Conined3 onrete and mean amount o longitudinal bars) Unonined Conined Conined Conined3.75 Unonined Conined Conined Conined3 Fig. (). and parameters relative to the analysis onsidered

7 An Analytial Formulation o Stress-Blok Parameters The Open Constrution and Building Tehnology Journal, 8, Volume 79 It an be also observed that the setion apaity tends to derease i m inreases. In all ases onsidered the maximum strain o onrete is more than 3 or 3,5. Suh values are, on the ontrary, lose to unonined ases, as numerial analysis have shown. Fig. () shows and stress-blok parameters obtained, as a untion o. EVALUATION OF STRESS-BLOCK PARAMETERS As previously reported, main odes allow to use stressstrain relationship that reprodues losely the real behaviour o onined onrete. However, they do not urnish both stress-blok parameters and upper limit or strain o ompressive onrete or eah limit state. Moreover, main odes utilize the volumetri ratio o onining reinorement as a parameter whih indiates the oninement level along the RC member. However, as Fig. () shows, the geometrial volumetri ratio o onining reinorement s is not appropriated to indiate the oninement level along the olumn. Inat, it is not the same or the dierent types o transversal reinorement even i s is a onstant in all ases onsidered. Axial stress (MPa) Fig. (). Stress-strain relationship o onrete or dierent types o reinored setion. (=., L=5 mm, l = mm, tr =8 mm, =3 MPa, y =43 MPa). In order to obtain a irst analytial ormulation o stressblok parameters or onined onrete, analytial analysis were arried out. Three dierent values o unonined onrete strength (, 3, 4 MPa) and three dierent o lateral tie length L (3, 4, 5 mm) have been onsidered. Stress-blok parameters obtained are reported as a untion o transversal reinorement spaing S and o maximum onrete strain m by parameter. parameter Unonined S S3 S4- S4- S Axial strain In Fig. (3) are plotted values o all the analyzed ases. It an be observed that or eah value onsidered the results are less sattered around medium value, unless or less than, g=. g=.95 g=.9 g=.85 g=.8 g=.75 g= Fig. (3). Variation o parameter with spaing hoop or various values. Fig. (4) shows three urves reerred to the smallest, the medium and the biggest values obtained as a untion o. Aording to the results, it ould propose the ollowing expression or : =,5 +,9 (7) The eetive strength is given by: SB = = ks = ksk (8) where is the unonined onrete strength k s is the ratio o onined onrete strength to unonined onrete strength k is the ratio o the olumn ompressive strength ( ) to the ylindrial speimen strength ( ). For the ULS, Sheik et Al. (99) [4] have proposed or k the ollowing values: k =,85 pern = N k = pern Nb (9) with a linear interpolation or intermediate ases. In the previous relations, N is the axial load applied, and N b is the balaned axial load y = -.5x x -.86 R = = m/.8 mean minimum maximum Fig. (4). parameter: minimum, mean and maximum value or eah onsidered. For N=N is equal to, while as shown by obtained results, or = (orresponding to the maximum lexural.75.7

8 8 The Open Constrution and Building Tehnology Journal, 8, Volume Braga et al. strength o the setion) assumes a value next to.9. Thereore it ould propose: * SLU = k =,85 pern = N * SLU = k =,9 pern Nb () Finally, or ULS we assume only one value or a k equal to.85 whih is the same or main international odes. In general, or the dierent limit states, the strength o stress blok ould be expressed as shown: * ' SB = = ks () where * =, 5 +,85 () Parameter In ollowing igures are represented the main results related to parameter. an be plotted as a untion o transversal reinorement spaing and o onrete strength. The numerial analysis show that is little sensitive to the lateral length tie g= g=.95 g=.9 g=.85 g=.8 g=.75 g= Fig. (5). Variation o parameter with spaing hoop or various values ( = MPa) g= g=.95 g=.9 g=.85 g=.8 g=.75 g= Fig. (6). Variation o parameter with spaing hoop or various values ( =3 MPa) g= g=.95 g=.9 g=.85 g=.8 g=.75 g= Fig. (7). Variation o parameter with spaing hoop or various values ( =4 MPa). In Figs. (8-3) are plotted values or three o onsidered, ixing =.95 y = -.7x R =.6675 y = -.84x R =.4468 y = -.343x R =.559 = MPa =3 MPa =4 MPa Fig. (8). Variation o parameter with spaing hoop or various values ( =.95) =.9 y = -.88x R =.7747 y = -.66x R =.584 y = -.344x R = = MPa =3 MPa =4 MPa Fig. (9). Variation o parameter with spaing hoop or various values ( =.9) =.85 y = -.33x R = y = -.83x R = y = -.357x R = = MPa =3 MPa =4 MPa Fig. (3). Variation o parameter with spaing hoop or various values ( =.85). In Figs. (3-33) is plotted as a untion o unonined onrete strength or some values o.

9 An Analytial Formulation o Stress-Blok Parameters The Open Constrution and Building Tehnology Journal, 8, Volume =.95 S=5mm S=mm S=5mm S=mm S=3mm S=4mm S=5mm (MPa) Fig. (3). Variation o parameter with unonined onrete strength ( =.95) = (MPa) S=5mm S=mm S=5mm S=mm S=3mm S=4mm S=5mm Fig. (3). Variation o parameter with unonined onrete strength ( =.9). transversal spaing S and unonined onrete strength (unless S5 transversal reinorement type). Reerring to k s parameter, an be expressed by the ollowing expression: k = = + (5a) ( ) ( ) s S =, 5 (5b), 79 S = + 43,9 3,93 S = + 48,78 4,67 S = +,56 4 S = + 3, 7 (6a) For S5 setion is a untion o lateral length tie, too. 5 8,5 S = + b() L (6b) where L b() L = + 53, 56 (6) 3 In the previous expressions is in MPa and L is in mm =.85 S=5mm S=mm S=5mm S=mm S=3mm S=4mm S=5mm (MPa) Fig. (33). Variation o parameter with unonined onrete strength ( =.85). Aording to the obtained results parameter an be desribed as ( is in MPa and S is in mm): = a() + b() S + () (3) where or a, b, parameters an be adopted the ollowing expressions: a = b = = (4) Fig. (34). S setion type. Variation o strength o onined onrete with s. Strength o Conined Conrete (k s Parameter) As Figs. (34-37) show, the onined onrete strength is a linear untion o wd and is nearly independent both Fig. (35). S3 setion type. Variation o strength o onined onrete with s.

10 8 The Open Constrution and Building Tehnology Journal, 8, Volume Braga et al. Fig. (36). S4 setion type. Variation o strength o onined onrete with s. Fig. (37). S5 setion type. Variation o strength o onined onrete with s. CONCLUSIONS It has been studied the behaviour o RC setions with lexure and axial load onsidering the oninement eets. Utilizing the analytial model or onined onrete proposed by Braga, Gigliotti and Laterza [], it has been arried out analytial analysis aording to the lassi setion methods. The analysis done put on evidene the importane o the maximum onrete strain to utilize or evaluating the interation urves o setion. This strain depends either on the onrete strength or the onining level inside the setion. It has been also dedued that the only one parameter is not enough useul to desribe the onining state along the member. Varying the maximum onrete strain it is possible to alulate the setion strength in dierent limit states. For example it an be deined the maximum onrete strain guaranteeing the onrete over wholeness or the ultimate onrete strain to alulate the lexural setion strength in the last state o ollapse. In this paper, a new analytial ormulation o stress-blok parameters or onined onrete is presented. The obtained equations, relating to the onsidered ases, are simple to use. Later, these analytial evaluations an be enlarged to a wide range o ases in order to obtain mathematial equations o a more general value. REFERENCES [] NZS 3:995, Conrete Strutures Standard-The Design o Conrete Strutures, Part and Part, Standards Assoiation o New Zealand, Wellington. [] Euroode 8, Design o Strutures or Earthquake Resistane, CEN, 3. [3] Euroode, Design o Conrete Strutures, CEN, 3. [4] ACI Committee 38, Building Code Requirements or Strutural Conrete and Commentary, Amerian Conrete Institute, 5. [5] NZS 43: 994, Code o Pratie or General Strutural Design and Design Loadings or Buildings. Standards Assoiation o New Zealand, Wellington. [6] Dereto Ministeriale 4 settembre 5. Norme Tenihe per le Costruzioni, Rome, Italy. [7] CSA A and CSA S6, Design o Conrete Strutures, Canadian Standards Assoiation, Rexdale, Ontario, 994. [8]. The norvegian ounil or building standardization, Norvegian Standard or Design o Conrete Strutures, Oslo, Norway, 995 [9] CEB-FIB. Model ode 99. Thomas Telord Servie Ltd., London or omitè Euro-International du Bèton, Lausanne, 993. [] F. Braga, R. Gigliotti, M. Laterza, Analytial stress-strain relationship or onrete onined by steel stirrups and/or FRP jakets, Journal o Strutural Engineering ASCE, Vol. 3, No. 9, pp. 4-6, 6. [] Sheikh, S. A., & Uzumeri, S.M., Strength and Dutility o Tied Conrete Columns. Journal o the Strutural Division, ASCE, 6 (5), 79-, 98. [] Park R. and Paulay T., Reinored Conrete Strutures, John Wiley, New York, 975. [3] C. B. Evan and P. C. Mihael. Response-. Reinored Conrete Setional Analysis (ver...5). University o Toronto. [4] S.A. Sheikh, C.C. Yeh, S. Khoury (99). Conrete strength in tied olumns - ACI Strutural Journal, Vol. 87. Issue 4, 99, Reeived: Marh 3, 8 Revised: July 5, 8 Aepted: July 6 8 Braga et al.; Liensee Bentham Open. This is an open aess artile liensed under the terms o the Creative Commons Attribution Non-Commerial Liense ( lienses/by-n/3./) whih permits unrestrited, non-ommerial use, distribution and reprodution in any medium, provided the work is properly ited.