A CONFINEMENT MODEL OF HIGH STRENGTH CONCRETE

Transcription

1 3 th World Coferece o Earthquake Egieerig Vacouver, B.C., Caada August -6, 24 Paper No. 873 A CONFINEMENT MODEL OF HIGH STRENGTH CONCRETE Nobutaka NAKAZAWA, Kazuhiko KAWASHIMA 2, Gakuho WATANABE 3, Ju-ichi SAKAI 4 SUMMARY A empirical formulatio of the lateral cofiemet effect o high stregth cocrete is proposed based o a series of uiaxial loadig tests o cocrete cyliders. A emphasis is placed to develop uloadig ad reloadig paths as well as the evelop of stress vs. strai relatios. The proposed formula ca be used to evaluate stress vs. strai relatios of the laterally cofied cocrete with a wide rage of stregth betwee 3-9 MPa. INTRODUCTION The lateral cofiemet effect o cocrete has bee studied by may researchers to evaluate the ductility capacity of reiforced cocrete colums [for example,, 2]. However the past cofiemet models were developed o the ormal stregth cocrete ad their applicatio to high stregth cocrete was ot verified. This paper presets a ew cofiemet model o high stregth cocrete with the target cylider stregth of as high as 9 MPa. A series of uiaxial compressio tests was coducted o 36 cocrete cyliders. A emphasis was placed to develop a empirical model for uloadig ad reloadig paths with various degrees of uloadigs ad reloadigs. TEST SPECIMENS AND EXPERIMENTAL SET-UP Thirty six cocrete cyliders as show i Fig. were costructed for the test. They were 6 mm tall with a diameter of 2 mm. Target cocrete stregth c was 3, 6 ad 9 MPa. We used deformed bars with a 6 mm diameter ad the yield stregth ad the stregth of 32 MPa ad 492 MPa, respectively (SD295) as both the logitudial ad ties bars. Four logitudial bars were provided i all cyliders. Tie bars were set at every 5, ad 5 mm so that the volumetric tie reiforcemet ratio ρ s is.36,.68 ad.45 %. Oly the top ad bottom of the cyliders were cofied by ties with smaller spacig. To measure compressio strais of a cylider, a acrylic bar with strai gauges at every 5 mm was embedded at the ceter [3]. This avoided the error ivolved i the strai measuremet due to imperfect cotacts or deformatio of plaster betwee a cocrete cylider ad a loadig cell. By accumulatig the Graduate Studet, Tokyo Istitute of Techology, Tokyo, Japa 2 Professor, Tokyo Istitute of Techology, Tokyo, Japa. kawasima@cv.titech.ac.jp 3 Research Associate, Tokyo Istitute of Techology, Tokuyo, Japa. Gappo@cv.titech.ac.jp 4 Postdoctoral Researcher, PEERC, Uiversity of Califoria, Berkeley, CA, USA

2 Fig. Test Specimes cc ul. ul.. 2 ul 3 Stress Stress pl. 2 cc (a) Mootoic Loadig pl. pl. 3 (b) Full Uloadigs ad Full Reloadigs Stress ul. ul.. 2 ul 3 Stress ul. i. 2 i. 3 pl. 2 ( rl, rl ) pl. pl. 3. i (c) Partial Uloadigs ad Full Reloadigs (d) Full Uloadigs ad Partial Reloadigs Fig. 2 Loadig Hysteresis strais alog the height, a averaged cylider strai was estimated by dividig the total compressio of the cylider by the etire height. The cyliders were loaded uder the displacemet cotrol by a 5 MN loadig machie. Four types of loadig were used i the test as show i Fig. 2: () mootoic loadig, (2) full uloadigs ad full reloadigs, (3) partial uloadigs ad full reloadigs, ad (4) full uloadigs ad partial reloadigs [4]. Uloadig ad reloadig were repeated -5 times at specific strais.

3 STRESS VS. STRAIN RELATIONS Failure Modes As the itesity of loadig icreased, failure of the cocrete progressed gradually i the cyliders with the target stregth c of 3 ad 6 MPa, while the cyliders with target stregth of 9 MPa failed i a brittle maer. However the cyliders with target stregth of 9 MPa did ot fail i the brittle maer whe uloadigs ad reloadigs were repeated several times just before the cocrete stress reaches the stregth cc. The post-yield behavior of the cyliders described hereiafter was evaluated uder this coditio. Stress vs. Relatios Fig. 3 shows a example of stress c vs. strai c relatio of cocrete with the target stregth c of 3 ad 9 MPa. Although the lateral cofiemet effect is ot sigificat prior to the peak stregth, it is sigificat i the post-yield rage. It is importat however to ote that such a cofiemet effect becomes ρ s =% ρ s =.45% ρ s =.68% ρ s =.36% (a) 3 MPa (b) 9 MPa Fig. 3 Stress vs. Relatio of Cocrete with a Target Stregth of 9 MPa co MPa 6MPa 9MPa by Eq.() (MPa) co Fig. 4 Stregth ad at the Stregth of Ucofied Cocrete less sigificat as the cocrete stregth icreases. Fig. 4 shows how the strai at the stregth of ucofied cocrete c depeds o the stregth c. It is importat to ote that c icreases as c icreases. Although there are may studies which reported that c was idepedet of c, it seems that this assumptio was derived from tests o the cofied cocrete with a arrow bad stregth. From Fig. 4, oe obtais

4 5 3 c =.54 c +.4 () The predicted value by Eq. () well correlates the experimetal relatio as show i Fig. 4. Fig. 5 shows the stregth cc vs. strai at the stregth cc relatio of the cofied cocrete. Both cc ad cc icrease as the lateral cofiemet ρ s sy icreases. They may be expressed as ρs sy.22 cc = c ρ s sy ρs c sy c c ρs sy.23 cc = c ρ s sy ρs c sy c c (2) (3) cc / co MPa 6MPa 9MPa by Eq. (2) cc / co MPa 6MPa 9MPa by Eq. (3) ρ s sy / c ρ s sy / c (a) Peak Stregth cc (b) at the Peak Stregth cc Fig. 5 Stregth cc ad at the Stregth cc of Cofied Cocrete Formulatio of Stress vs. Relatio of the Evelops As show i Fig. 3, the cocrete stress c approaches to a certai level of stress u (residual stress) at high strai after reachig the peak stregth. The evelop of stress c vs. strai c relatio must satisfy the followig requiremets. c = at c = d c d c = E c at c = c = cc at c = cc (4) d c d c = at c = cc c = u at c = i which, E c is the elastic modulus. As a fuctio which satisfies Eq. (4), the followig stress c vs. strai c relatio is proposed.

5 where, I this aalysis, estimated as c = u + Ec c u + Ec cc cc + cc u c cc E c = cc (6) cc u u is defied as the stress at c =.2. Based o the test results, the residual stress u cc.4.35 ρs (5) u is = (7) Fig. 6 compares the predicted ad experimetal stress c vs. strai c relatios of the cofied cocrete with the target stregth of 3 ad 9 MPa. Eq. (5) provides a good approximatio for the c vs. c relatios of the cofied cocrete with a wide rage of stregths betwee 3 ad 9 MPa. FULL UNLOADING AND FULL RELOADING PATHS Normalized Stress vs. Normalized Relatios If a full uloadig occurs from a poit at a strai ad stress ul. o a skeleto curve, the strai reaches pl. at zero stress as show i Fig. 2 (b). The, ul. ad pl. are called uloadig strai, st uloadig stress, ad st plastic strai, respectively. If the cocrete is reloaded from pl. to, the stress at ( ul. 2 ) may be smaller tha ul.. If we repeat a set of full uloadig ad full reloadig times, the -th uloadig stress ad the plastic strai become ad pl., respectively. ul. A ormalized stress ad a ormalized strai are defied here as = c ; c pl. = (8) ul. ad deterioratio rates of stress ad strai, β ad γ respectively, are defied as β ul. + ul pl. = ; γ = (9) ul. ul pl. Fig. 7 shows how the full uloadig ad full reloadig paths deped o the uloadig strai. Sice the similar results are obtaied i other cases, oly the case for c =.6 MPa ad ρ s =.36% is preseted here. It is obvious that the vs. relatio is almost liearly proportioal at small cc (.-.63), while a certai oliearity occurs at cc >.. As show i Fig. 8, the depedece of vs. relatio o ρ s is less sigificat i the rage of.45% ρ s.36%. Figs. 9 ad show the depedece of vs. relatios o the cocrete stregth c ad the umber loadig, respectively. Both the cocrete stregth c ad the umber of loadig are less sesitive to the vs. relatios. ul pl.

6 Experimet Experimet (a) c =9 MPa (b) c =3 MPa () ρ s =.45% Experimet 6 Experimet (a) c =9 MPa (b) c =3 MPa (2) ρ s =.68% Experimet 6 Experimet (a) c =9 MPa (b) c =3 MPa (3) ρ s =.36% Fig. 6 ad Experimetal Stress ad Relatio of Cofied Cocrete Based o the above clarificatio o the experimetal results, a full uloadig path of the cocrete stress c vs. strai c relatio may be represeted by disregardig ρ s, c ad depedece as ul = () where ul is a parameter which represets the degree of oliearity of stress c vs. strai c relatio, ad this may be expressed from the test results as

7 .8.6 / cc =. / cc =.324 / cc = / cc =. / cc =.324 / cc = (a) Full Uloadig Paths (b) Full Reloadig Paths () Prior to Peak Stregth ( cc <.).8 / cc =.2 / cc =.23.8 / cc =.2 / cc =.23.6 / cc =.68.6 / cc = (a) Full Uloadig Paths (b) Full Reloadig Paths (2) Post-Yield Rage ( cc >.) Fig. 7 Effect of Uloadig o the Full Uloadig ad Full Reloadig Paths.8 ρ s =.45% ρ s =.68%.8 ρ s =.45% ρ s =.68%.6 ρ s =.36%.6 ρ s =.36% (a) Full Uloadig Paths (b) Full reloadig Paths Fig. 8 Depedece of vs. Relatios o ρ s ( cc =.3) ul =.54 ta 2.2 ul cc.44 + ta () I a similar way, a full reloadig path is represeted as,

8 .8 c =9MPa c =3MPa.8 c =9MPa c =3MPa (a) Full Uloadig Paths (b) Full Reloadig Paths Fig. 9 Depedece of vs. Relatios o Cocrete Stregth c Cycle Cycle.8 2 Cycle.8 2 Cycle 3 Cycle 3 Cycle.6 4 Cycle.6 4 Cycle 5 Cycle 5 Cycle (a) Full Uloadig Paths (b) Full Reloadig Paths Fig. Depedece of vs. Relatios o the Number of Loadig where, rl c pl. = < ul. +.5 c ul pl. E ( ) rl c ul ul. (2).5 ul ul. E ul. rl = ( ul pl. ) (3) rl = ul cc ul / cc ul / cc.25 (4).25 / cc Deterioratio Coefficiets β ad γ The stress deterioratio ratio β by Eq. (9) represets the degree of deterioratio of ul. i by repeatig a set of full uloadig ad full reloadig. Fig. shows how β depeds o the uloadig strai. The stress deterioratio is sigificat whe uloadigs occur i the post-yield rage. This may be writte as

9 Stress Detorioratio Ratio β.2 9MPa ρ s =.45 9MPa ρ s =.68 9MPa ρ s =.36 3MPa ρ s =.68 by Eq. (6) by Eq (5) / cc / cc 4 5 (a) = (b) =3.2 Stress Detorioratio Ratio β 5 Stress Detorioratio Ratio β 3.2 9MPa ρ s =.46 9MPa ρ s =.7 3MPa ρ s =.7.8 9MPa ρ s = / cc 5 by Eq. (6) (c) =5 Fig. Depedece of Stress Deterioratio Ratio 9MPa ρ s =.46 9MPa ρ s =.7 3MPa ρ s =.7 9MPa ρ s =.4 β o Uloadig = 2.5 ul ul / cc cc β = ul ul / cc.25 (5) cc ul / cc ul / cc + 4(.27.33) ul. β = cc ul/ cc ul / cc (6) The values by Eqs. (5) ad (6) well correlate the experimetal results as show i Fig.. O the other had, Figs. 2 ad 3 show the depedece of plastic strai pl. ad the plastic strai ratios γ o the uloadig strais. From the results, pl. ad γ may be expressed as

10 pl. / cc 2.5 9MPa ρ s =.45 9MPa ρ s =.68 9MPa ρ s =.36 3MPa ρ s =.68.5 Plastic Ratio γ / cc Fig. 2 Depedece of Uloadig o the Plastic pl..2 9MPa ρ s =.46 9MPa ρ s =.46 9MPa ρ s =.7 9MPa ρ s =.7 3MPa ρ s =.7 3MPa ρ s =.7 9MPa ρ s =.4 9MPa ρ s =.4 by Eq. (9) / cc / cc (a) =3 (b) =5 Plastic Ratio γ 5 by Eqs. (9) Fig. 3 Depedece of Plastic Ratio γ o Uloadig =2 3 pl. cc 2.62 ul ul = cc cc (7).889 ul.827 ul cc cc ul / cc γ = ul 2.34 ul / cc.25 cc / cc (8) ul / cc + 4(.33.23) ul γ = cc ul / cc ul / cc (9)

11 Cycle 2 Cycle 3 Cycle 4 Cycle 5 Cycle by Eq. (2)..5.2 (a) Partial Uloadig ad Full Reloadig Paths PARTIAL UNLOADING AND FULL RELOADING PATHS Sice a partial uloadig is a part of a full uloadig path util a reloadig occurs, the partial uloadig path may be represeted by a full uloadig path by Eq. (). Cosequetly, it is eeded to formulate a full reloadig path from a reloadig poit at a stress of rl o a uloadig path. To represet where a reloadig occurs after uloaded, a parameter β ul (partial uloadig ratio) is defied as ul. β rl ul = (2) Fig. 4 (a) shows a example of the partial uloadig ad full reloadig paths with β ul =.5. Usig the ormalized stress ad strai, ad, by Eq. (8), a full reloadig path from a poit correspodig to β ul =.5 becomes as show i Fig. 4 (b). Sice the effect of repeatig partial uloadigs ad full reloadigs is less sigificat, a full reloadig path may be idealized as where ul. ( + (2) c = Erl c rl ) rl. (22) E ul rl = + ul (b) Normalized Reloadig Paths Fig. 4 Full Reloadig Paths whe Reloaded at β ul =.5 Predicted full reloadig paths by Eq. (2) agree well with the experimetal results (refer to Fig. 4(b)). rl rl FULL UNLOADING AND PARTIAL RELOADING PATHS If a cofied cocrete is subjected to a full uloadig, it reaches the plastic strai pl. at zero stress. Whe it is subjected to a reloadig from this plastic strai pl., it may follow a full reloadig path by Eq. (2). If a full uloadig occurs at a strai. i ( ul. i ul ) o this path before reachig the uloadig strai, the uloadig path has to be determied. Sice the uloadig above is the 2d uloadig if we cout the first uloadig from the uloadig strai, it is called here as the 2d iteral uloadig. The stress where the 2d uloadig occurs is called the 2d iteral uloadig stress i. 2. After the 2d iteral uloadig, the cocrete strai c reaches the 2d plastic strai pl. 2. Similarly, the stress at a iteral uloadig strai. i ad the plastic strai after a set of times full uloadig ad iteral

12 c / ul Cycle 2 Cycle 3 Cycle 4 Cycle 5 Cycle..2.3 (a) Full Uloadig ad Partial Reloadig Paths partial reloadig are defied i. + ad pl. +, respectively. To represet a poit util where a iteral reloadig cotiues, a parameter γ rl (partial reloadig ratio) is defied as γ RL ul. i pl. = (23) If a set of full uloadig ad partial reloadig are repeated five times after fully uloaded, the stress c ad strai c hysteresis become as show i Fig. 5 (a) i the cofied cocrete with the target stregth c of 9 MPa ad ρ s =.68%. Other combiatios of c ad ρ s have the similar relatios. The plastic strai pl. ad the iteral uloadig stress i. do ot sigificatly deteriorate durig the repeated loadigs. This is differet to the cyclic full uloadigs ad full reloadigs, i which the plastic strai pl. ad the uloadig stress ul. deteriorate due to load reversals. This meas that the deterioratio of plastic strai pl. ad uloadig stress ul. is limited if uloadigs occur before reachig the uloadig strai. By defiig a ormalized stress i ad strai i as c i = ; i. ul c / (b) Normalized Uloadig Paths Fig. 5 Partial Uloadig Paths whe Uloaded at Iteral Uloadig Poit with i. ad i. ( γ rl =.5) pl. c pl. i = (24) ul. i pl. the uloadig paths preseted i Fig. 5 (a) becomes as show i Fig. 5 (b). The first uloadig path from a uloadig strai is preseted here as the st cycle for compariso. The ormalized uloadig paths are very close to the ormalized uloadig path from the uloadig strai. Cosequetly, by replacig ul ad i Eq. (2) by i ad. i, respectively, oe obtais rl c pl. = < i. + i.5 c ul. i (25) pl. E ( ) rl c ul. i i. i

13 Experimet (a) c =9 MPa (b) c =3 MPa Fig. 6 Applicatio of Model to Full Uloadigs ad Full Reloadigs ( ρ =.68% ) s 8 Experimet 8 Experimet Fig. 7 Applicatio of Model to Full Uloadigs ad Partial Reloadigs..2.3 Fig. 8 Applicatio of Model to Partial Uloadigs ad Full Reloadigs APPLICATION OF THE LOADING AND UNLOADING MODEL Fig. 6 shows a applicatio of the proposed model to a series of full uloadigs ad full reloadigs of the cocrete with the target cocrete stregth of 9 ad 3 MPa, ad ρ =.68 %. The proposed model provides a good estimate of the lateral cofiemet effect o the cocrete with a wide rage of stregth betwee 3 ad 9 MPa. Figs. 7 ad 8 show applicatios of the proposed model to a series of full uloadigs ad partial reloadigs ad a series of partial uloadigs ad full reloadigs, respectively, of the cocrete with =9 MPa. The models show a good agreemet with the test results. c CONCLUSIONS A empirical costitutive model for the lateral cofiemet of cocrete cyliders with a wide rage of stregth betwee 3 ad 9 MPa was developed based o a series of loadig test. Based o the results preseted herei, the followig coclusios may be deduced:

14 . The strai at the peak stregth c of the ucofied cocrete depeds o the peak stregth c, ad this depedece of c ad c ca be represeted by Eq. (). 2. The peak stregth cc ad the strai at the peak stregth cc of the cofied cocrete ca be represeted by Eqs. (2) ad (3), respectively. 3. The evelops of the stress vs. strai relatio of the cofied cocrete ca be represeted by Eq. (5). 4. Uloadig ad reloadig paths for a combiatio of full uloadigs ad full reloadigs, full uloadigs ad partial reloadigs, ad partial uloadigs ad full reloadigs ca be represeted by Eqs. (), (2), (2), ad (25). REFERENCES. Mader, J.B., Priestley, M.J.N. ad Park, R. Theoretical stress-strai model for cofied cocrete. Joural of Structural Divisio, ASCE, 988: 4(8), Hoshikuma, J., Kawashima, K., Nagaya, K. ad Taylor, A. Stress-strai model for cofied reiforced cocrete i bridge piers. Joural of Structural Egieerig, ASCE, 997: 23(5), Lertsrisakulrat, T., Wataabe, K., Matsuo, M. ad Niwa, J. Experimetal study o parameters i localizatio of cocrete subjected to compressio. Proc. JSCE, 2: 669(5), Sakai, J. ad Kawashima, K. A uloadig ad reloadig stress-strai model for cocrete cofied by tie reiforcemets. 2 WCEE, 2: Paper No. 432 (CD-ROM), New Zealad.