American J. o Engineering and Applied Sciences (1): 31-38, 009 ISSN 1941-700 009 Science Publications Reliability o Axially Loaded Fiber-Reinorced-Polymer Conined Reinorced Concrete Circular Columns Venkatarman Nagaradjane, Pulipaka Narasimharao Raghunath and Kannan Suguna Department o Civil Structural Engineering, Annamalai University, India Abstract: Monte-Carlo simulation based procedure was applied or assessing the reliability o Fiber Reinorced Polymer (FRP) conined reinorced concrete columns. Assessment o reliability provides a quantitative measure o eectiveness o repair. Problem statement: The procedure presented in the paper could be used or assessing the reliability o reinorced concrete columns beore and ater the application o FRP wrap, thus helping to quantitatively assess the eectiveness o repair. The equations provided in ACI 440.R were used or ormulating the reliability equations Approach: Java based computer program was developed to assess the reliability o circular concrete columns strengthened using FRP composites. The inluence o diameter, thickness o wrap, longitudinal steel and elasticity modulus o wrap on reliability was studied. Results: The results indicated that the diameter o column and wrap thickness strongly inluenced the reliability o columns Conclusion/Recommendations: The inluence o internal steel reinorcement area and elasticity modulus o iber reinorced polymer on reliability was very limited. Computationally eicient ormulation or estimating reliability was proposed. The new equation estimated reliability by up to 1575.314 times aster than the common equation, thus resulting in the saving o time and computational power. Key words: Coninement, FRP, monte-carlo simulation, normal distribution, reliability, reliability index, random number INTRODUCTION Repair and rehabilitation o reinorced concrete columns is necessitated by several actors like corrosion o reinorcement, damage due to vehicular collision, deterioration o core concrete due to environmental actors or an increase in the service loads o bridge like structures caused by ever expanding vehicular traic. FRP is a composite material ormed out o a variety o small diameter ibers (made o glass, carbon and aramid held in place using polymer resins. The application o FRP on the surace o concrete columns provides additional axial strength due to the resistance to lateral expansion oered by the FRP wraps thus reducing the axial deormation, consequent axial strain and resulting in more strength or core concrete. FRP improves the compressive ductility o concrete columns enormously due to the act that FRP takes large strain beore ailure thus increasing the lateral dilation o concrete maniold. Incidentally, FRP arrests the ingress o moisture and other degrading agents, resulting in more durability or the core concrete. Measurement o the level o repair and rehabilitation o structural members is essential to justiy the amount o money and materials spent on repair and to ensure that the repair has resulted in measurable improvement in perormance. Reliability is a statistical tool suitable or measuring perormance o any structural system against perormance requirements. Reliability is associated with the statistical probability that a structural member will survive the demand or perormance imposed under service conditions. Probability o ailure is the complementary part o the reliability o a structural element, accounting or the possibility that the element will ail under service conditions. Hence, the objective o repair or rehabilitation is to maximize reliability or to minimize probability o ailure ater repair or rehabilitation is work is carried out. One o the techniques used or estimating reliability o structural systems is Monte Carlo simulation. This technique permits modeling o structural systems using the parameters aecting their behavior. Monte Carlo is general purpose method and its applications are ound in wide ranging areas o social science, traic low, population growth, inance, genetics, quantum chemistry and radiation sciences [1]. Corresponding Author: Venkatarman Nagaradjane, Department o Civil Structural Engineering, Annamalai University, India 31
Assessment o reliability provides a quantitative tool or strengthening, repair and rehabilitation o reinorced concrete columns with FRP wraps. The present study is signiicant in that a ramework or computing reliability is proposed based on the basic relationships or compressive strength o conined concrete provided in ACI 440.R. The study also attempts to investigate the relationship between reliability and the column parameters like diameter o conined column, thickness o FRP wrap, area o longitudinal steel and elasticity modulus o FRP sheets. Ruiz and Aguiler evaluated reliability indices or reinorced concrete short slender columns using Monte- Carlo simulation technique []. The study revealed that the slenderness ratio, longitudinal steel, eccentricity and load levels were the signiicant parameters aecting reliability o reinorced concrete columns. Dinz and Frangopal analyzed the bases or reliability assessment o high strength reinorced concrete columns [3]. The eect o load and unmagniied moments on slender high strength concrete columns. The study proposed a reliability based method or the assessment o high strength concrete columns. Dinz and Frangopal studied the reliability o high strength concrete columns designed according to the American Concrete Institute provisions [4]. A combination o irst order reliability method and Monte-Carlo simulation was adopted to evaluate the reliability o short as well as slender columns. The study ound that short high strength concrete columns have lower reliability than normal strength concrete columns. Longitudinal steel reinorcement played a signiicant role on the reliability o slender high strength concrete columns. Val studied the reliability o FRP conined, short, circular shaped reinorced concrete columns considering reinorcement ratio, live load to dead load ratio and eccentricity as the major parameters [5]. The study concluded that the load reduction actors were to be urther reduced to ensure the level o reliability o unconined column is comparable to that o FRP conined column. The study identiied compressive strength o conined concrete, uncertainty coeicient or FRP conined concrete, error o the structural model and live load as the most inluential parameters. Tools and methods: Nominal strength o FRP conined concrete cylinders is inluenced by the presence o internal steel ties, their spacing and coniguration. ACI committee report 440.R speciies that the compressive strength o concrete cylinders with spiral steel ties should be calculated using the expression [6,7] : Am. J. Engg. & Applied Sci., (1): 31-38, 009 3 ' P n = 0.85 0.85 cc(ag A st ) + ya st (1) P n = The nominal axial compressive strength = The strength reduction actor = The additional FRP strength reduction actor (recommended value is 0.95) cc = The apparent compressive strength o conined concrete A g = The gross cross - sectional area A st = The total area o longitudinal reinorcement y = The yield strength o non-prestressed steel reinorcement For conined concrete with circular steel ties, the nominal compressive strength can be estimated using Eq. : ' P n = 0.80 0.85 cc(ag A st ) + ya st () ACI 440.R recommends that the value o the additional reduction actor be taken 0.95. As a general case, the two equations can be combined into a third equation, by introducing an additional, which stands or the eectiveness o the internal steel ties in providing coninement and is 0.85 or spiral ties and 0.80 or circular ties: P = 0.85 (A A ) + A ' n cc g st y st (3) Further, the value o strength reduction actor should be taken at 0.7 or compression members without spiral ties and at 0.75 or compression members with spiral ties satisying the condition: g s = 0.45 1 Ac ' A c (4) y s = The ratio o volume o spiral reinorcement to the core volume A c = The core area o conined concrete measured up to the outside o the spiral ties The conined compressive strength o concrete ( cc ) is to be evaluated using the model proposed by Mander et al. [8] :,, l l cc = c.5 1+ 7.9 1.5 ' ' c c (5)
Am. J. Engg. & Applied Sci., (1): 31-38, 009 The parameter l stands or the eective coninement pressure generated by the passive conining mechanism o FRP wrap. The value o l can be estimated using the expression: κ ρ ε E a e l = (6a) ε = 0.004 0.75ε (6b) e 4nt D u ρ = (6c) where, a is the coninement eectiveness actor which depends on the shape o cross-section, value being unity or a circular section and less than unity or noncircular sections, is the area o composite to area o concrete per unit height o the column section, n is the number o layers o FRP, e is the eective strain in FRP, E is the elasticity modulus o FRP, u is the ultimate strain or FRP, t is the thickness o FRP and D is the diameter o the concrete column or circular sections. Hence, the actual process o estimation o the compressive strength o FRP conined reinorced concrete columns begins with the calculation o maximum conining pressure using Eq. (6) ollowed by computation o the strength o conined plain concrete using Eq. (5). The conined plain concrete strength is substituted in the appropriate Eq. 1 or recommended by ACI, based on the type o tie provided or column (whether circular or spiral). In case the column is provided with spiral ties, the volume o ties to core volume should satisy the condition suggested by ACI (as represented by Eq. 4). The deterministic ormulation o strength estimation or FRP conined reinorced concrete columns orms the basis or the probabilistic simulation o strength using Monte Carlo technique. The basic equations and procedure remaining the same as that or deterministic evaluation o strength, Monte Carlo simulation simply permits random variations in all properties and parameters involved in the equations within allowable limits. This situation mimics the result o a real world experiment which is also subject to reasonable levels o uncertainty resulting rom the act that that material properties deviate rom designated mean values. Reliability as a measure o structural perormance: All design codes and guidelines provide a way or ensuring that the structural member will survive the 33 speciied loading conditions. Design procedures o dierent codes lead to dierent levels o economy. This is because saety level is taken as a qualitative actor where a structural element is permitted to have arbitrary levels o excess strength. But Reliability (R) provides rational approach to saety by quantiying the level o saety associated with a particular combination o materials. Hence, reliability can be used to assess the degree o conservatism associated with the deterministic analysis [9]. Reliability might be deined as the probability that a structural system will survive the given load level. There is a counterpart to reliability called probability o ailure (P ). It is deined as the probability that a structural system will ail under the given loading conditions. Hence, reliability and probability o ailure orm two extremes related to the saety o structural systems. Probability theory states that the sum o reliability and probability o ailure is always equal to unity. This rule makes it easy to evaluate one quantity i the other one is known. In the normal probability distribution space the requency (y) associated with each numeric value (x) is calculated using the expression: x / e y = (7) The plot o which is shown in Fig. 1. The probability o ailure is represented by the shaded area and the reliability is represented by the un-shaded area. The negative o value at which the areas o reliability and probability o ailure meet is called the reliability index (). Reliability index () value zero is associated with reliability 0.5. Fig. 1: Relation between reliability, probability o ailure and reliability index in normal distribution space
Am. J. Engg. & Applied Sci., (1): 31-38, 009 The relationship being clear, knowing any one o the three parameters R, P or, calculation o the other two parameters is possible. Knowing reliability index (), the probability o ailure and reliability can be ound out using the relations: x e P = dx (8) R = 1 P (9) Sometimes, the probability o ailure is known beore hand. Reliability index can be evaluated by carrying out the integration rom negative ininity up to such a point on the normal distribution curve that the area equals the known probability o ailure. The negative o the value on the x - axis is the reliability index. Alternatively, reliability index can be visualized as the ratio between mean level o saety and standard deviation in saety and calculated using the expression: r a = r + a (10) values or non-deterministic input parameters, estimating the strength using the randomly chosen values and checking whether the perormance meets structural demand. I the estimated perormance o the structure does not meet the perormance demand it is counted as a ailure. Probability o ailure is estimated as the ratio between the number o ailures and the total number o runs: P = N < Sa Sr > i=1 N (11) S r = The simulated resistance S a = The simulated action or load acting on the member N = The number o runs o the Monte Carlo simulation process The triangular brackets (<>) stand or McAuley bracket unction, which yields one i the result is positive and zero otherwise. Hence, the summation counts only the number o ailures (or the number o simulated runs which resulted in less resistance than required). The random parameter value o each parameter X is generated using: r = The mean resistance o the system a = The mean action or load on the system r = The standard deviation in resistance o the system a = The standard deviation in action or load on the system The three major parameters o saety being deined, estimation o these parameters is possible using Eq. 8-10. Monte carlo simulation or assessment o structural reliability: Reliability o FRP structural systems can be estimated using techniques like First Order Reliability Method (FORM) [10], First Order Second Moment Reliability Method (FOSMRM) [11] and Monte Carlo Simulation [10]. The FORM and FOSMRM employ Taylor s series expansion or estimating mean and standard deviation in member strength and perormance demands, thus permitting calculation o reliability index (Eq. 10). Knowing the reliability index, probability o ailure and reliability can be estimated using Eq. 8 and 9 respectively. The present work employs monte carlo simulation or estimating reliability o structural members. Monte carlo simulation procedure involves generating random 34 X = X + kr x (1) +k x e / c = dx (13) k X = The mean value o the parameter K = The spread constant which corresponds to the given conidence level R = The generated random value lying between -1 and +1 x = The standard deviation in the population o X c = The conidence level The simulation process takes conidence level as input and evaluates the constant k using the relationship 13. Implementation o monte carlo simulation algorithm: Monte Carlo Simulation algorithm or estimating the reliability o FRP conined reinorced concrete columns is presented as ollows: Generating random parameter values or column geometry Generating random parameter values or steel reinorcement properties
Generating random parameter values or FRP properties Estimating the load carrying capacity o reinorced concrete column using Eq. 1-6 as necessary Generating random load on the column using mean load, standard deviation in load and the conidence level used I the load carrying capacity turns out to be less than the load placed on the column, the situation is counted as a ailure. The above steps are repeated or predetermined number o runs. The larger the number o runs, the lesser is the error associated with the results. Probability o ailure is calculated as the ratio between number o ailures and total number o runs. The algorithm was implemented in Java programming language. The sotware was called Monte Carlo Simulator or Fiber Reinorced Polymer Conined Column (MCS-FRPCC) and was developed on Fedora Core Linux operating system using Java Development Kit (JDK) version 1.6. The source can be obtained rom the corresponding author ree o cost. Optimization or aster perormance: The sotware was tested or time consumption against computation o random numbers, simulation o column parameters and computation o reliability index. It was ound that a lot o time was consumed or evaluating the reliability index using Eq. 8. Although Eq. 8 is mathematically correct, it suggests numerical integration rom negative ininity up to the point where desired probability o ailure is reached. It takes a long time to perorm numerical integration rom an extremely negative value. Hence, an alternative equation was ormulated to evaluate reliability index. x β ( β) (14a) 1 e R = + dx or β 0 β π 0 1 R = or β 0 (14b) This ormulation uses the basic property that the area under normal distribution curve rom zero to positive ininity is ½. This proposition shits the lower limit o integration to zero point rather than positive ininity or negative ininity. Hence, reliability index () is ound out by numerical integration rom zero to until the summation equals desired reliability. The summation rom integration is added to ½ when reliability is greater Am. J. Engg. & Applied Sci., (1): 31-38, 009 35 Table 1: Time Consumptions or alternative evaluation schemes o reliability index Average time Average time Sl. Interval on taken by Eq. 8 taken by Eq. 14 Time No. X-axis (dx) (milli sec) (milli sec) ratio 1 0.1000 7.7143 7.5918 9.5780 0.0100 513.6531 6.5510 78.4081 3 0.0010 4984.8163 78.5510 63.4596 4 0.0001 47650.7500 30.500 1575.314 than 0.5 and subtracted rom ½ when reliability is less than ½, since the value o ()/ simply gives the sign (whether positive or negative) o the expression being integrated. Table 1 shows the time averages or calculating reliability index () using Eq. 8 and 14. The measurements were made on a Fedora core Linux operating system running on AMD Sempron processor clocked at 1400 MHz with 56 MB o RAM. While the actual times may vary on other systems, the time ratio is expected to provide an estimate o how ast the new algorithm works, especially when the interval or numerical integration (dx) is reduced rom larger values to smaller values. MATERIALS AND METHODS The present parametric study was perormed or 150 mm (5.906 inches) diameter reinorced concrete circular column conined with 3 mm (0.118 inch) thick GFRP wrap having elasticity modulus o 1000 MPa (5006.94 lb/sq. in). The internal steel reinorcement was 315 mm (0.488 sq. inch) and steel ties were spaced at 115 mm (4.58 inch) c/c. The compressive strength o concrete was 3.64 MPa (657 lbs/sq. inch), yield strength o steel was 415 MPa (46643.61 lb/sq. in), ultimate tensile strain in FRP was 0.016. The basic requirements or the computation o reliability index o FRP wrapped reinorced concrete column are an equation which describes the load carrying capacity o the column or given material properties (concrete, steel and FRP), the load level. The mean values and their standard deviations or all material properties and loading values are also required or simulation. The inluence diameter o column and load carrying capacity on reliability was studied by choosing diameters ranging rom 100 mm (3.94 inches) to 1000 mm (39.37 inches), keeping all other parameters constant. The imposed compressive load was varied rom the lower bound value (corresponding to 100 mm (3.94 inches) diameter) to the upper bound value (corresponding to 1000 mm (39.37 inches) diameter).
Am. J. Engg. & Applied Sci., (1): 31-38, 009 Fig. : Reliability as a unction o column diameter and imposed compressive load Fig. 4: Reliability as a unction o area o steel and imposed compressive load Fig. 3: Reliability as a unction o thickness o FRP and imposed compressive load RESULTS The inluence o parameters like diameter o column, thickness o FRP wrap, area o longitudinal steel and elasticity modulus o FRP wrap was studied using the sotware ramework. The reliability values were plotted in contour orm as a unction o the given parameter and imposed load. The reliability contour plotted as a unction o diameter and the demanded load carrying capacity is shown in Fig.. Figure 3 shows the reliability contour plotted as a unction o thickness o FRP and imposed compressive load. The indings are in agreement with those reported by Val [5]. Figure 4 shows the reliability contour as a unction o area o longitudinal reinorcement and compressive load applied on the column. Figure 5 shows the reliability contour as a unction o elasticity modulus and imposed compressive loads. The contour boundaries or reliability have a tendency to drit rom their mean path. 36 Fig. 5: Reliability as a unction o elasticity modulus o FRP and imposed compressive load DISCUSSION The results indicated that reliability was signiicantly aected by diameter o column, thickness o FRP, area o internal steel reinorcement and elasticity modulus o FRP. The imposed loads were calculated or the lower bound and upper bound values o the parameter being studied. When a single parameter was subjected to variation, all others were kept constant. With increasing diameter o column, the reliability went on increasing and reached the maximum value 1.0. Increasing the applied load led to a reduction in the reliability o the column, reaching a minimum value o 0. Increasing thickness o FRP wrap improved the reliability o conined RC column. Without any wrap, the reliability value was around 0.5 or lower-bound compressive load
o 60 kn (58450.33 lb). For the same imposed load, reliability reached 0.8 or 4 mm (0.16 inch) wrap and 0.9 or 7 mm (0.8 inch) wrap. For higher imposed loads, the increase in reliability was lower. At loads around 450 kn (101164.0 lb), reliability reached 0.8 or 4 mm thick wrap and 0.5 or 10 mm thick wrap. Area o longitudinal reinorcing steel aected the reliability values only in the range o 0.3-0.7. This meant that FRP wrap itsel played a role towards improving the reliability o RC column. Increasing the presence o steel up to 400 mm (0.60 sq. in.) led to increase in reliability value up to 0.7. Hence, the internal steel reinorcement might be inerred as a signiicant parameter, but its role was complementary in the presence o FRP wrap. Elasticity Modulus o FRP was varied rom 8000 MPa (40789.5 lb/sq. in) to 50000 MPa (54938.15 lb/ sq. in) and the corresponding imposed compressive loads varied rom 30 kn (71938.86 lb) to 515.83 kn (115963.0 lb). For the imposed load o 30 kn (7193.86 lb) the reliability started at 0.5 or elasticity modulus o 8000 MPa (40789.5 lb/sq. in), reached a value o 0.8 or elasticity modulus o 31000 MPa (1580583.44 lb). For the high end load o 515.83 kn (115963.0 lb), the reliability starts near 0 or FRP elasticity modulus o 8000 MPa (40789.5 lb/sq. in) and reaches about 0.5 or that o 50000 MPa (54938.15 lb/sq. in). CONCLUSION Assessment o reliability o reinorced concrete columns provides a rational means o scheduling repair and rehabilitation works, deciding the type o repair and amount o repair. In case o columns on which FRP is installed as a wrapping or strengthening or rehabilitation works, the improvement in reliability provides a quantitative measure o the amount o improvement in strength. The ollowing conclusions are drawn rom the investigation: The diameter o reinorced concrete column was a vital parameter or the reliability o reinorced concrete columns conined by FRP wraps. The closely spaced contour lines indicated that even small variations in diameter o RC column with FRP wrap resulted in measurable change o reliability Thickness o FRP wrap was also identiied as a major parameter to assess the reliability o conined concrete columns Area o internal steel reinorcement played a moderate role in improving the reliability o FRP Am. J. Engg. & Applied Sci., (1): 31-38, 009 37 conined column. The variations in area o internal steel did have a low contribution to reliability o FRP conined reinorced concrete columns Elasticity modulus o FRP wrap was studied in a widely varying range o values. The increase in reliability was not relective o the increase elasticity modulus. Increasing elasticity modulus rom 5000 MPa (174664.07 lb/sq. in) to 50000 MPa (54938.13 lb / sq. in) shited the reliability rom 0.7-0.8 Computationally eicient ormulation or estimating reliability was proposed. The new equation estimated reliability by up to 1575.314 times aster than the common equation, thus resulting in the saving o time and computational power The revised procedure proposed in this work might be used in uture works or aster estimation o the reliability parameters. The Graphical User interace (GUI) based sotware developed as part o this work was suitable or assessing reliability o FRP conined concrete columns based on ACI 440.R ACKNOWLEDGEMENT The researchers acknowledge the inancial support oered by University Grants Commission (UGC) o India, New Delhi, or carrying out investigations on FRP conined concrete columns. REFERENCES 1. Kamiski, M. and T. Trapko, 006. Experimental behavior o reinorced concrete column models strengthened by CFRP materials. J. Civil Eng. Manage., 1: 109-115. http://www.jcem.vgtu.lt/upload/civil_zurn/kaminsk i%0and%0trapko.pd. Ruiz, S.E. and J.C. Aguiler, 1994. Reliability o short and slender reinorced concrete columns. ASCE J. Struct. Eng., 10: 1850-1865. DOI: 10.1061/(ASCE)0733-9445(1994)10:6(1850) 3. Dinz, S.M. and D.M. Frangopal, 1997. Reliability bases or high strength concrete columns. ASCE J. Struct. Eng., 13: 1375-1381. DOI: 10.1061/(ASCE)0733-9445(1997)13:10(1375) 4. Dinz, S.M. and D.M. Frangopal, 1998. Reliability assessment o high strength concrete columns. ASCE J. Eng. Mechanics, 14: 59-536. DOI: 10.1061/(ASCE)0733-9399(1998)14:5(59)
Am. J. Engg. & Applied Sci., (1): 31-38, 009 5. Val, D.V., 003. Reliability o ibre-reinorced polymer conined reinorced concrete columns. ASCE J. Struct. Eng., 19: 11-1130. DOI: 10.1061/(ASCE)0733-9445(003)19:8(11) 6. ACI Committee 318, 1999. Building Code Requirements or Reinorced Concrete, American Concrete Institute, Detroit, USA. http://www.concrete.org/pubs/newpubs/318-08.htm 7. ACI Committee 440.R 00. Guide or the Design and Construction o Externally Bonded FRP Systems or Strengthening Concrete Structures. http://www.concrete.org/bookstorenet/productdeta il.aspx?sacode=4400 8. Mander, J.B., M.J.N. Priestley and R. Park, 1988. Theoretical stress-strain model or conined concrete. J. Struct. Eng., ASCE, 114: 1804-186. 9. Sarveswaran, V., M.B. Robers and J.A. Ward, 000. Reliability assessment o deteriorating reinorced concrete beams. Proc. Instn. Civil Eng., Struct. Build., 39-48. DOI: 10.1680/stbu.15.1.83.40887 10. Rao, S.S., 199. Reliability-Based Design. McGraw-Hill, New York, pp: 1-569. 38