STRAIN LIMITS FOR PLASTIC HINGE REGIONS OF CONCRETE REINFORCED COLUMNS

13 th World Conerence on Earthquake Engineering Vancouver, B.C., Canada Augut 1-6, 004 Paper No. 589 STRAIN LIMITS FOR PLASTIC HINGE REGIONS OF CONCRETE REINFORCED COLUMNS Rebeccah RUSSELL 1, Adolo MATAMOROS, JoAnn BROWNING 3 SUMMARY Etablihing reliable acceptance criteria or deormation in tructural member i crucial or the ucceul ue o perormance-baed method in the deign and evaluation o tructure. Thi paper explore extending the applicability o expreion propoed or the ultimate compreive train in the concrete and or the platic hinge length o reinorced concrete beam to thoe or reinorced concrete column. The limiting concrete train wa evaluated baed on a conitent deinition or limiting drit o 153 reinorced concrete column tet, and compared with the value obtained uing the previouly propoed expreion. An optimized equation or limiting compreive train wa developed. Thi equation indicate a bae level o train in the extreme compreion iber o the concrete o 1% or the column ailure criterion ued in the evaluation. INTRODUCTION Etablihing reliable acceptance criteria or deormation in tructural member i crucial to the ucceul execution o perormance-baed method or deign and evaluation o tructure. Limiting value o compreive train in the concrete within platic hinge region are o particular igniicance to characterize the expected perormance o ductile member. W. Gene Corley [3] and A. L. L. Baker [4] propoed expreion or the limiting compreive train in the concrete and or the platic hinge length o reinorced concrete member. Thee expreion were developed on the bai o meaured platic rotation in beam. Becaue deign proviion or beam require lightly reinorced member that ail in a ductile manner, thee expreion were derived or and are applicable to member in which the abence o axial compreion and the relatively low amount o reinorcement reult in a neutral axi depth that i mall compared with the overall depth. The ocu o thi paper i to explore the applicability o thee expreion to reinorced concrete column, where the location o the neutral axi can extend beyond the cro ection, and conequently the gradient o lexural train can be igniicantly lower. 1 Graduate Aitant, Univerity o Kana, USA. Email: babyonyx@yahoo.com Aitant Proeor, Univerity o Kana, USA. Email: amatamor@ku.edu 3 Aitant Proeor, Univerity o Kana, USA. Email: jpbrown@ku.edu

The invetigation wa perormed uing a databae o 153 reinorced concrete column. All column included in thi analyi were ubjected to cyclic lateral diplacement, had obervable ailure, and apect ratio exceeding.5. An original databae o 184 pecimen wa obtained uing three main ource: a column databae rom the Univerity o Wahington, reearch and thei report, and publication preented in engineering journal. Detail o the column data, including dimenion and material propertie, a well a limiting drit ratio, can be ound in Brachmann [1]. Thi databae wa then reduced to 153 column due to unreported member propertie in ome o the tet. Limiting Strain in the Concrete CALCULATION OF LIMITING CONCRETE STRAIN The limiting train in the extreme compreion iber o each column pecimen wa calculated or the deormed coniguration at the limiting drit. The limiting drit wa deined a the lateral drit correponding to a reduction in hear trength o 0%. The concrete train in the extreme iber wa determined uing the procedure decribed below. The total lateral diplacement ( t ) wa deined a the um o the elatic ( el ) and platic diplacement ( pl ) o the member = + (1) t el pl where and l φ = θ = y l el y 3 3 () l p l p = φ pl = θ pl l pl l p l (3) The rotation θ y andθ pl are thoe correponding to the onet o yielding o the longitudinal reinorcement and the inelatic range o repone, repectively (Fig. 1). The ditance along the member rom the point o maximum moment to the point o contralexure i deined a l, and the platic hinge length i repreented by l p. The curvature at yield φ y and the curvature aociated with inelatic behaviorφ pl = φu φy, are geometric propertie deined uing the relationhip c φ = (4) c where c i the train in the extreme compreion iber o the concrete, and c i the neutral axi depth o the member. The limiting train in the extreme compreion iber o the concrete wa determined uing Eq. 1-4.

l θp lp φy φu Figure 1. Aumed curvature ditribution at the limiting drit Diplacement aociated with hear and lip o the reinorcement The limiting concrete train wa determined uing two dierent deinition or the total drit. For the irt cae the eect o deormation related to hear and lip o the reinorcement on the limiting drit o column were neglected. For the econd approach, the total limiting drit wa deined a the um o three component lim = + + b (5) where lim i the limiting drit, and,, and b are the component related to lexure, hear, and bond lip, repectively. The lexural component o the limiting drit wa calculated by ubtracting the component related to hear and lip o the reinorcement rom the total. The diplacement related to hear wa aumed to be (Matamoro [5]) 6V (1 + υ) = * *10 (6) 5bd E where V i the maximum meaured hear applied to the member, b i the width, d i the eective depth, υ i Poion' ratio, and E i the modulu o elaticity or the material. The diplacement due to lip o the reinorcement wa deined a y ydb b bl * l 8 ( d c) = θ = (7) µ where θ b i the rotation o the member due to lip o the reinorcement, y i the yield trength o the longitudinal reinorcement, d b i the diameter o the longitudinal reinorcing bar, and µ i the average bond tre. An average bond tre o 6 ' c wa aumed in thi analyi. Thi value wa uggeted a a lower bound or beam ubjected to monotonic loading [A imilar value ha provided adequate reult been ued in previou tudie to etimate the repone o reinorced concrete rame []. The lexural diplacement,, wa then determined a the remaining portion o a deined in Eq. 5. t

Theoretical Equation or Ultimate Concrete Strain The main goal o the tudy wa to determine whether two equation previouly propoed to calculate the limiting compreive train in the concrete in beam could be ued in the cae o column, and to determine how applicable thee equation would be or calculating the limiting compreive train at the limiting drit a deined by Brachmann [1]. The irt equation wa propoed by Corley [3] b ρv yh c lim =.003 +.0 + l 138 (8) where yh i the hoop teel yield trength in unit o megapacal, ρ v i the volumetric ratio o binding reinorcement, b i the width o the member, and l i the ditance rom the critical ection to the point o contralexure. The econd equation wa propoed by Baker and Amarakone [4] d clim = 0.0015 1+ 150 ρ + (0.7 10 ρ) c (9) which can expanded to d d clim = 0.0015 + 0.5ρ + 0.00105 0.015ρ (10) c c where ρ i the volumetric tranvere reinorcement ratio, c i the neutral axi depth o the member at ultimate moment, and d i the eective depth o the member. The relevance o the parameter ued in thee expreion wa analyzed with the goal o developing an alternative expreion to etimate the limiting train o the elected column tet. Theoretical Equation or the Platic Hinge Region The limiting concrete compreive train inerred rom a known diplacement value i dependent on the length o the platic hinge, l p. Corley [3] and Baker and Amarakone [4] propoed dierent expreion or the equivalent platic hinge length o beam ubjected to monotonic load. According to Corley the length o the platic hinge i given by c l p C = 0.8 k1 k3 l (11) d Baker and Amarakone propoed the ue o the ollowing expreion: l l p B = 0. 5 d + d (1) d where d i the eective depth o the member (mm), k 1 and k 3 are actor depending on the concrete and teel trength, c i the neutral axi depth at ultimate moment (mm), and l i the ditance rom the critical ection to the point o contralexure (mm). Deinition o Failure

It hould be noted that the deinition o limiting drit o the column analyzed in thi paper dier rom the deinition o limit tate adopted by Corley and Baker and Amarakone. In the tet perormed by Corley, load were removed ater the maintainable applied load dropped igniicantly below the maximum load impoed on the beam. The value o limiting train, curvature, and rotation were recorded at the irt viible ign o ditre in the compreion zone. Due to the contraint o the teting machine, load could not be maintained at a contant rate a the beam deormed. Baker and Amarakone alo ued a deinition o limiting train correponding to the point at which the member began to deorm greatly with increaing load, or at the irt viible ign o concrete cruhing at contant load. In the cae o the member analyzed in thi tudy, the limiting drit wa deined a that correponding to a 0% reduction o the maximum hear orce recorded in the member [1]. Limiting train were calculated baed on the limiting drit, according to Eq. 1 to 4. Normalization o Expreion ANALYSIS OF THE DATA Equation 1 to 4 how that the limiting compreive train in the concrete i dependent upon the limiting drit and the platic hinge length, l p that i adopted in the calculation. While Corley and Baker and Amarakone ued the expreion hown in Eq. 11 and 1, repectively, a imple etimate o l p equal to d/ wa ued in the analyi preented in thi paper. A review o the data howed that there wa no improvement in the accuracy o the etimated limiting drit obtained uing Eq. 11 or 1 in lieu o thi imple aumption. To acilitate an objective comparion between the dierent equation preented in thi paper, the limiting compreive train expreion propoed by Corley and Baker and Amarakone were normalized to account or the dierent platic hinge length aociated with thoe expreion derived. Limiting train or an equivalent platic hinge length l p equal to d/ were calculated by equating the etimated rotation or each o the expreion derived uing Eq. 11 and 1 over the repective platic hinge length to that in a platic hinge with a length o d/ (Eq. 13) θ p = θ (13) CB p inerred which lead to and conequently d φ p CB l p CB = φ p inerred (14) clim CB lp CB clim inerred ( d /) =. c c (15) Equation 16 wa then ued to normalize the expreion l = = d / p inerred clim norm clim CB clim inerred (16) Normalized value o the limiting compreive train etimated uing the expreion propoed by Corley and Baker and Amarakone were ued or the comparion preented in thi paper.

Figure and 3 how the ratio o the limiting compreive train in the concrete obtained rom the Corley and Baker and Amarakone expreion to the value inerred rom the limiting drit a deine by Brachmann [1] (including or hear and lip deormation). Figure 4 and 5 how thee ratio in the cae where the eect o lip and hear deormation are neglected and it i aumed that the entire rotation i attributed to lexural deormation only. The olid line in the igure repreent the trend line or the mean ratio, while the broken line correpond to one tandard deviation above and below the mean. The data are plotted eparately or axial load ratio (P/A g ' c, where P i the axial load, A g the cro ectional area, and c the compreive trength o concrete) le than and exceeding 0.1, to indicate diering trend or beam and column pecimen. The overall mean and tandard deviation or the beam and column pecimen are indicated on the igure. 6 cu- calc / cu-inerred 5 4 3 1 mean= 1.83 td dev=1.36 mean=0.9 t dev=0.98 0 0 0.1 0. 0.3 0.4 0.5 0.6 P/Ag'c Fig.. Ratio o meaured to inerred limiting compreive train in the concrete according to the expreion by Corley baed on the lexural deormation, without the eect o deormation due to hear and lip o the reinorcement. Evaluation o Fig. -5 indicate that the expreion propoed by Corley typically overetimated the value o train in the beam pecimen or total and lexural diplacement value. For column, the expreion overetimated much le or lexural diplacement, and appear to work quite well when total diplacement i utilized. The expreion propoed by Baker and Amarakone underetimate the train value at nearly all point. It i alo intereting to note that or both propoed expreion, the column pecimen data how greater catter than the beam data when lexural diplacement are conidered, but that thi obervation i revered when total diplacement are ued and the column data ha a lower tandard deviation than the beam data.

6 5 mean=0.64 td dev=0.44 mean=0.46 td dev=0.38 cu- calc / cu-inerred 4 3 1 0 0 0.1 0. 0.3 0.4 0.5 0.6 P/Ag'c Fig. 3. Ratio o calculated to inerred limiting compreive train according to the expreion by Baker and Amarakone baed on the lexural deormation, without the eect o deormation related to hear and lip o the reinorcement. 6 cu- calc / cu-inerred 5 4 3 1 mean=1.86 td dev=1.39 mean=1.3 td dev= 1.5 0 0 0.1 0. 0.3 0.4 0.5 0.6 P/Ag'c Fig. 4 Ratio o calculated to inerred limiting compreive train in the concrete according to the equation propoed by Corley baed on the total deormation, neglecting the eect o deormation aociated with hear and lip o the reinorcement.

6 5 cu- calc / cu-inerred 4 3 mean= 0.66 td dev= 0.46 mean= 0.70 td dev= 0.68 1 0 0 0.1 0. 0.3 0.4 0.5 0.6 P/Ag'c Fig. 5. Ratio o calculated to inerred limiting compreive train in the concrete calculated uing the expreion by Baker and Amarakone baed on total deormation, neglecting the eect o deormation related to hear and lip. Linear Regreion A linear regreion analyi wa perormed uing the limiting compreive train inerred rom the elected column data and the parameter ued in the equation propoed by Corley and Baker and Amarakone. The primary objective o thi analyi wa to determine the parameter with the lowet tandard error, and thereore the bet tatitical it to the data. The tandard error calculated or each coeicient i hown in Table 1. The regreion analyi indicated a igniicant level o catter in the data, a indicated by coeicient o variation on the order o 1. The expreion propoed by Corley reulted in value o limiting train that were cloer to thoe inerred rom the data baed on the deinition o limiting drit ued by Brachmann[1]. The equation propoed by Baker and Amarakone reulted in more conervative etimate o the limiting train, on the order o 65% o thoe obtained with the deinition o limiting drit ued by Brachmann. From the regreion analyi, it wa noted that the parameter ( ρ yh ) rom the equation propoed by Corley and the parameter d/c rom the expreion propoed by Baker and Amarakone had the lowet tandard error. Thee reult indicate that the train gradient and the amount o coninement repreented by the product o the amount o tranvere reinorcement and the yield trength o the reinorcement had

the mot igniicance on the limiting compreive train in the concrete. Thi i conitent with a coniderable body o literature indicating that coninement ha the eect o increaing the train at peak tre and reducing the lope o the decending branch o the tre-train curve or concrete. It i intereting to note that the volumetric reinorcement ratio ( ρ ), when evaluated a a eparate parameter reulted in a much higher tandard error than the parameter ( ρ yh ). Becaue thee two parameter provided the bet it, they were maintained or evaluation o later expreion. The remaining parameter (b/l, ρ, and ρ d/c) had relatively large tandard error value and were not ued to develop an improved equation in later analye. Although the contant portion o the expreion propoed by Corley and Amarakone had tandard error value on the order o 10 to 100 time lower than the parameter eliminated rom the tudy, the error value exceeded the other remaining parameter by a igniicant actor. Other variation on the exponent n or the parameter ( ρ ) n yh yielded no igniicant improvement in tatitical it. Table 1 Standard Error or Coeicient Determined or Linear Regreion Analyi Parameter Diplacement Coeicient Standard Error Corley Shear and lip Contant 0.00641 neglected b/l 0.0365 ( ρ v yh ) 0.00003 Shear and lip Contant 0.00561 conidered b/l 0.0064 ( ρ v yh ) 0.0000 Baker Shear and lip Cont 0.00435 neglected ρ 0.3580 d/c 0.0010 ρ d/c 0.13440 Shear and lip Contant 0.00477 conidered ρ 0.34831 d/c 0.00103 ρ d/c 0.13753 Optimization Analyi Ater perorming the linear regreion to evaluate the igniicance o the parameter in the expreion by Corley and Baker and Amarakone, an optimization baed on the mot igniicant parameter in thee equation wa perormed to attempt to reduce the amount o catter evident rom the irt linear regreion analyi. The ormat o the equation ued in the optimization included the parameter ( ρ yh ) and d/c, a well a a new parameter to invetigate the inluence o apect ratio l/d. Optimization wa achieved a an iterative proce by imultaneouly changing the value o the coeicient to produce the bet tatitical it to the elected data by minimizing the coeicient o variation. The optimization wa perormed twice. Firt, the proce wa executed with the goal o achieving normalized theoretical value

o limiting train that were cloet to the inerred value (a mean ratio o calculated to inerred limiting train equal to unity). A econd optimization wa perormed with no contraint on the mean value o the ratio o thee value. The reult o thi optimization are preented in Table, along with the normalized expreion propoed by Corley and Baker and Amarakone. When the optimization wa perormed without retricting the mean value o the ratio o theoretical to inerred value o train (labeled orce opt. in the table), it can be een that the coeicient o the parameter or the normalized expreion propoed by Corley were imilar to the coeicient ound through optimization, with the exception o the b/l parameter. The optimized equation had a much maller contribution or the term b/l than implied by the Corley expreion, and it had an oppoite ign (an unlikely relationhip to exit in true behavior), which i conitent with the relative large tandard error calculated in the previou regreion analyi. The tatitical igniicance o the parameter in the expreion by Corley, a well a the new et o parameter, did not change igniicantly when the eect o diplacement related to hear and lip o the reinorcement were conidered in the optimization o the limiting compreive train in the concrete. The optimized equation baed on the parameter o the equation by Baker and Amarakone howed a imilar trend, although the tatitical it wa not a good a that o the normalized equation. In addition, tatitical analye baed on the parameter by Baker and Amarakone were enitive to whether the eect o diplacement related to hear and lip were conidered or not. The analyi neglecting the eect o diplacement related to hear and lip o the reinorcement howed much higher contant value and a lower enitivity to the remaining parameter (maller coeicient), wherea the analyi that neglected thee eect howed a lower contant train value and wa more enitivity to the parameter (larger coeicient). The change in ign aociated with the d/c term in the optimized equation i not the likely to relect true behavior expected or reinorced concrete member. Table Reult o Optimization o Propoed Equation Parameter Type o dip Analyi Equation Mean St Dev COV d/c, l/d, (ρ * y ) lexural orce opt c lim = 0.013-0.0005(d/c) - 0.0001(l/d) + 6.6x10-5 (ρ * y ) 1.00 0.7 0.7 total orce opt Corley lexural opt b/l, (ρ * y ) Baker and Amarakone ρ, d/c, ρ * d/c total orce opt opt orce opt c lim = 0.016-0.001(d/c)+0.0003(l/d)+0.001(ρ * y ) 1.00 0.66 0.66 c lim = 0.006-0.001(b/l)+( y *ρ v /17) 0.67 0.51 0.76 c lim = 0.009-0.00(b/l)+( y *ρ v /103) 1.00 0.76 0.76 c lim = 0.006-0.0033(b/l)+( y *ρ v /108) 0.57 0.43 0.74 c lim = 0.010-.006(b/l)+( y *ρ v /81) 1.00 0.74 0.74 actual eqtn normalized c lim = 0.0047+0.031(b/l)+( y *ρ v /110) 0.68 0.71 1.05 lexural total actual eqtn expanded opt orce opt opt orce opt c lim = 0.006+0.317(ρ )-0.00(d/c)-0.0019(ρ *d/c) 0.64 0.48 0.75 c lim = 0.009+0.49((ρ )-0.0003(d/c)-0.03((ρ *d/c) 1.00 0.75 0.75 c lim = 0.0031+0.7((ρ )-0.0001(d/c)-0.0344((ρ *d/c) 0.7 0.17 0.64 c lim = 0.0113+0.99((ρ )-0.0003(d/c)-0.165((ρ *d/c) 1.00 0.64 0.64 normalized c lim = 0.0018+0.77((ρ )+0.0013(d/c)-0.018((ρ *d/c) 0.74 0.9 1.3

When the mean ratio o theoretical to inerred value o the limiting compreive train in the concrete wa orced to a value o unity, it i intereting to note that all optimization reult had a contant compreive train in the concrete to be approximately 1%. Thi wa true even or the optimized equation uing only the newly elected parameter. The coeicient o variation improved uing the optimized equation in every cae, rom 1.05 and 1.3 or expreion propoed by Corley and Baker Amarakone, repectively, to 0.66 in the bet cae uing the new parameter. The improvement in tatitical it wa imilar when identical parameter were ued in the comparion, leading to the concluion that the bae limiting compreive train in the concrete wa better repreented uing a value o 1%. The expreion hown in the irt two row o Table repreent the reult o an optimization with the new et o parameter. In ome intance, parameter that had a mall inluence on the data et had ign that are counterintuitive to oberved behavior. In thee intance it i likely that the eect o the parameter i o mall that catter in the data i coniderable more igniicant. A a reult, the ollowing two expreion are uggeted or the limiting train o the concrete. For the cae when the limiting train i applied to calculation that include the eect o diplacement related to hear and lip o the reinorcement: ρ yh c lim =.0013 + (17) 130 When it i aumed that the total deormation i related to lexure, the limiting train hould be ρ yh c lim =.0015 + (18) 30 Ratio o calculated to inerred limiting train in the concrete baed on Eq. 17 and 18 are preented in Fig. 6 and 7. Both igure how that the level o catter wa coniderably reduced, although it remain very igniicant.

6 e cu calculated e cu inerred 5 4 3 1 mean = 1.3 td dev =.73 mean =.87 td dev =.57 0 0 0.1 0. 0.3 0.4 0.5 0.6 P/Ag'c Fig. 6. Ratio o meaured to inerred limiting compreive train in the concrete according to the expreion by Corley baed on the lexural deormation, without the eect o deormation due to hear and lip o the reinorcement. 6 e cu calculated / e cu inerred 5 4 3 1 mean = 1.13 td dev =.63 mean =.96 td dev =.78 0 0 0.1 0. 0.3 0.4 0.5 0.6 P/Ag'c Fig. 7. Ratio o meaured to inerred limiting compreive train in the concrete according to the expreion by Corley baed on the lexural deormation, without the eect o deormation due to hear and lip o the reinorcement.

CONCLUSIONS The evaluation o an expanded databae o reinorced concrete member teted under lateral load allowed or a new tatitical analyi o the relationhip between the limiting compreive train in the concrete and relevant material and member parameter. The accuracy o the relationhip between limiting compreive train and limiting drit wa not improved by uing more detailed expreion that accounted or the eect o the pread o platicity at the end o the member in lieu o the imple relation d/. Independent o the parameter ued to etimate the limiting compreive train in concrete, the bae train needed to achieve a drit correponding to a reduction in hear trength o 0% wa on the order o 1%. REFERENCES 1. Brachmann, I., (00). "Drit Limit o Rectangular Reinorced Concrete Column Subjected to Cyclic Loading," Mater o Science Thei, Univerity o Kana, Lawrence, Kana.. Browning, J., (001). "Proportioning o Earthquake-Reitant RC Building Structure," ASCE Journal o Structural Engineering, vol. 17, no., pp. 145-151. 3. Sozen, M. A. and Moehle, J. P. "Development and Lap-Splice Length or Deormed Reinorcing Bar in Concrete". Report to The Portland Cement Aociation and The Concrete Reinorcing Steel Intitute, Augut 1990. 3. Corley, W. G., "Rotational Capacity o Reinorced Concrete Beam,: Journal o the Structural Diviion. ASCE, Vol.9, No ST5, Proc. Paper 4939, Oct., 1966, pp11-146. 4. Baker, A. L. L., and Amarakone, A. M. N., "Inelatic Hypertatic Frame Analyi," Proceeding o the International Sympoium on the Flexural Mechanic o reinorced Concrete, ASCE-ACI, Miami, Nov., 1964, pp 85-14 5. Matamoro, A. B., Drit Limit o High-Strength Concrete Column, PhD. Thei, Department o Civil Engineering, Univerity o Illinoi at Urbana Champaign, 1999.