Reliability Analysis of Solid Concrete Slabs Reinforced with Carbon Fibre Reinforced Plastic (Cfrp)

International Journal of Advanced Engineering Research and Technology (IJAERT)Volume 2 Issue 3, June 2014, ISSN No.: 2348 – 8190

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Reliability Analysis of Solid Concrete Slabs Reinforced withCarbon Fibre Reinforced Plastic (Cfrp)

1OKORO, E. E., 2ABEJIDE, O. S., 3OLOWOSULU, A. T., 4ABUBAKAR, I.1,2,3,4 Department of Civil Engineering, Ahmadu Bello University, Zaria, Nigeria.

ABSTRACTThe reliability of CFRP reinforced concrete solid slabs isinvestigated in this paper. The procedure of the FirstOrder Reliability Method (FORM), is used for thecomputation of the reliability indices, with varying loadratios for bending, deflection and shear criteria with slabthicknesses in the practical range of 100mm--250mm.Typical results obtained for continuous slabs of equalspans indicate the intrinsic reliability and correspondingprobability of failure of the slabs designed to CAN/CSAS806-02 (2002) prediction. It is observed that thereliability level of the slab is non-uniform at any of thecriterion examined for the depth of slab and span chosen.In general, the safety of the slab is endangered withincreasing load ratios in all the criteria and thicknessesinvestigated. The safety criterion is violated in bendingfor all the thicknesses of slab examined by the safetyindex, which is less than 2.0. However, the criterion forsafety is satisfied in shear and deflection for all slabthicknesses up to a load ratio of 0.6, except in 225mmand 250mm thick slabs where it is violated fordeflection, being less than 2.0 at a load ratio of 0.2. Atnone of the load ratios considered is the safety indexnegative for deflection and shear criteria except forbending. This suggests that for CFRP reinforcedconcrete solid slabs, once the bending criterion for safetyis satisfied, deflection and shear criteria are metautomatically, when the current prediction is used.Hence, there is the need to optimize this prediction in astochastic environment in order to achieve reliable slabswith adequate and appropriate safety, and with possiblemodifications to the current formulation for design.

Keywords: CFRP, FORM, reliability, solid concreteslabs.

1. INTRODUCTION

When a structure is loaded it will respond to theload in a manner which depends on the type andmagnitude of the load, as well as on its strength and

stiffness. It is an axiom in engineering that the goal ofstructural design is to achieve a structure which fulfillsthe requirements of the client at reasonable cost andwhose resistance capacity is greater than the loaddemands placed on it. The reliability of a structure,therefore, is an important factor in the design proceduresince it quantifies the probability that a structure willfulfill its design requirements [1]. The basic requirementtherefore, for an efficient structural design, is that theresponse of the structure should be acceptable forvarious specifications. Therefore, whether the responseof the structure to the load is judged satisfactory or not,depends on the requirements which the structure mustsatisfy. However, the fulfillment of the requirements is atask that is fraught with uncertainties which are inherentin structural design. [2] opined that resistance and loadsinfluences cannot be described by deterministicquantities in the presence of uncertainties, beingstochastic. Because uncertainties create a range ofpossible resistance and load values and introduce thepossibility that the applied load will exceed the capacityof the structure, there is the risk of unacceptableperformance for structures, which is the probability thatfailure may occur [3][2]. Nearly every variableconsidered in design is uncertain to varying degrees. In adeterministic approach, design is based on specifiedminimum material properties and specified loadintensities. However, it has been recognized [4] thatspecifications of minimum material properties involvehigh degree of uncertainty. Also, detailed description ofreasonable load intensities is difficult and uncertain indeterministic method. These types of uncertainties,combined with several similar forms of uncertainty,result in an uncontrolled risk, which portends failurehazard for the structure. However, no structure orbuilding system can be absolutely risk-free or 100% safebecause of uncertainties in demands on the system, andin engineering properties of construction materials [5].There are values of probability of failure that areaccepted as being good enough depending on thehorizon of time that the structure is supposed to function,


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as well as its importance. This is defined in EurocodeBasis of Structural Design [6].

It is a fact that total structural safety cannot beachieved no matter what is done or the amount ofinvestment on the structure [4]. Furthermore, no measureof the safety or reliability of the structure can beobtained with deterministic approach. Risk, thoughimpossible to completely be eliminated, can be reducedto acceptable levels, through good engineering design,by accounting for the inherent uncertainties in design.Measuring the safety of a structure by its reliabilitymakes the reliability a useful decision parameter [6].

In this paper the performance functions for thelimit states of rectangular solid slabs singly-reinforcedwith Carbon Fibre-Reinforced Plastics (CFRP) weredeveloped. The failure modes considered are thebending, shear and deflection criteria. Each safetymargin is obtained by equating the resistance of the slaband the load effect calculated by elementary structuralanalysis, the basic variables are identified and theirstatistical characteristics obtained from literature. FirstOrder Reliability Method (FORM) was used to evaluatethe safety indices of the slab at varying values of thebasic variables.

2. FIRST ORDER RELIABILITYMETHOD (FORM)

Structural design is traditionally based ondeterministic analysis; however, due to many sources ofuncertainty, which is inherent in structural design, thereis the risk of unacceptable performance for structures,which is defined as the failure probability [6]. Instructural design, the reliability of a structuralcomponent is evaluated with respect to one or more limitstates. Since there are different limit states which can bedefined based on structural requirements, theperformance of an engineering structure or system canbe measured with regards to these limit states and thelevels of reliability associated with them [7]. The limitstate function returns a negative value under systemfailure conditions and a positive value when the systemis stable. Thus, reliability may be defined as theprobability of a performance function g(X) greater thanzero, i.e. P{g(X) > 0} [8]. In other words, reliability isthe probability that the random variables X = (X1,X2,…….., Xn) are in the safe region that is defined byg(X) > 0. The probability of failure is defined as theprobability P{g(X) < 0}. Or it is the probability that therandom variables X = (X1, X2,…….., Xn) are in thefailure region that is defined by g(X) < 0. A reliability

analysis entails the calculation of failure probability. Ifthe joint probability density functions pdf of X is ( ),the probability of failure is evaluated with the integral[8]:

Pf = P{g(X) < 0} = 0)(

)(xg

x dxxf (1)

The reliability is computed using:= 1 − = {g( ) > 0} = 0)(

)(xg

x dxxf (2)

A fundamental problem in structural reliabilitytheory is the computation of the multi-fold probabilityintegral of Equation (1). Difficulty in computing thisprobability has led to the development of variousapproximation methods, which are based on the fact thatthe failure condition is expressed as a function of severalrandom variables. Of these methods, the most commonlyused are the First-Order Reliability Method (FORM).The First-Order Reliability Method, in which the Taylorseries expansion of the performance function g(X) islinearized at some point, was employed in this paper.The idea is to ease the computational difficulties throughsimplifying the integrand of ( ) and approximatingthe performance function g(X). With the simplificationand approximation, solutions to Equations (1) and (2)will be easily obtained.

The FORM is an analytical approximation inwhich the Taylor series expansion of the performancefunction g(X) is lnearized at some point, say point (Y*1 ,Y*2 ,………Y*M) on the failure surface rather than at themean (Abejide, 2004). It aims at finding the ‘most likelyfailure point’ or ‘design point’ denoted as u*, which isthe closest point (see Figure 1) to the origin in thestandard space lying on the limit-state g(x) = G(u) = 0.The procedure is basically as follows [7][8]:(i) Achievement of the simplification throughtransforming the random variables from their originalrandom space into a standard normal space. Given thatthe limit state and its variables are:= ( , , ………… . , (3)

the random variables are transformed first to reduceduncorrelated variables with zero mean and unit variance.That is:

Yi

Yiii

YU

(4)

where, Yi and Yi are respectively, the mean and

standard deviation of the basic random variable, iY . The

transformation from Yi to Ui is based on the condition


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that the cumulative density functions (cdfs) of therandom variables remain the same before and after thetransformation. Thus, Yi and Ui can be related by theexpression:

)()( ii UYF (5)

in which ( . ) is the cdf of the standard normal

distribution and )( iYF is the random variable. This type

of transformation is called Rosenblatt. The transformedstandard normal variable is then given by:

ii YFU 1 (6)

For example, for a normally distributed random variableY ~N( µ,σ), Equation (6) yields:

YY

YFU i11

(7)

(ii) In order to further make the probabilityintegration easier to be evaluated, the integrationboundary ( ) = 0 is also approximated by a Taylorpower series expansion of Equation (6) incorrespondence with the checking point vector Y * andconsidering linear terms only. Thus:

*1

1ii

Y

iii YYYFY

YFU

(8)

where the notationYiY

1 indicates that the

derivatives are calculated at the point having coordinatesY*I. By the rules of derivative of the inverse function,Equation (8) becomes Equation:

*1

***1

i

iiiii YF

YYYfYFU

(9)where f and denote the probability densities

corresponding to the cumulative distributions F and

respectively. Rearranging Equation (10), weobtain:

*

*1

*

*1*1*

i

i

i

iiii

Yi

YF

YF

Yf

YFYFYY

U

(10)Comparison between Equation (4) and Equation (10)indicates that:

*

*1*1*

i

iiiiYi Yf

YFYFYYU

(11)and

*

*1

i

ii YF

YF

(12)

Lastly, in terms of ui the safety domain boundary can beexpressed in the form:

0,..........,, 321 nuuuug (13)

The parameter, , defined as the minimum distance tothe failure surface from the origin in the reducedcoordinate system (Fig 1) can be determined iteratively,assuming the transformed limit state is capable ofdifferentiation, by solving the following equations:

0............,).............,( 21**

2*1 nn guuug

(14)

where,

n

i ui

uii

U

g

U

g

1

2

*

*

; (15)

i = 1, 2, …………………, n


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Target reliability index is the most importantdesign factor in a reliability-based design method.Having obtained a satisfactory value of the minimumdistance, corresponding to convergence of theiterative process, the operational probability of failure Pf

can be computed from:

fP (16)

Or

fP1 (17)

Thus, the derivation has assumed independent randombasic variables. If Yi are correlated, they must betransformed to uncorrelated random variables.

3. RELIABILITY ANALYSIS

Reliability analysis is a tool that assists thedesign engineer to take into account all possibleuncertainties during the design and construction phasesand lifetime of a structure in order to calculate itsprobability of failure [1][9][10]. The overall aim is toquantify the reliability of structures under consideration,and the uncertainties associated with the resistances andloads [11]. The structural performance is assessed bymeans of models based on physical understanding andempirical data. Due to idealizations, inherent physicaluncertainties and inadequate or insufficient data, themodels themselves and the parameters entering themodels such as material parameters and loadcharacteristics are uncertain. Structural reliability theorytakes basis in the probabilistic modelling of theseuncertainties and provides methods for the quantificationof the probability that the structures do not fulfill theperformance criteria [12].

3.1 Limit State EquationsEquations (18), (19) and (20) are limit state

equations for bending, shear and deflection respectively,used for the computation of safety levels of a continuousCFRP solid slab:

2.354.23192002.002.1

0098.075.0)(

2

2''

LQ

bfbf

AEdAfXG k

cc

frpfrpfrpfrp

(18)

bd

LQ

S

dEAdbfXG xfrpfrp

wc88.0709.00015.0

12.0)( '

(19)

EI

LQ

d

L

EI

wL

d

LXG k

384

6.12..15

384

5)(

44

(20)

Where frpf --- tensile strength of FRP

frpA ---area of fibre-reinforced plastics (FRP)

frpE ---modulus of elasticity of FRP

'cf --- compressive strength of concrete

b---breadth of slabd---effective depth of slabL---effective span of slabQk --- live load on slabα--- load ratio (Gk/Qk)I---moment of inertiaE---modulus of elasticity of concrete.w---uniformly distributed load on slab.S----spacing of stirrups.

3.2 Statistical Properties of Design ParametersFrom literature, it was found that statistically,

dead load is typically treated as a Normal distributionvariable, and live load, Gumbel distribution variable(Extreme Type I) [13]. It was revealed also that allgeometric variables are normally distributed, whileconcrete strength and strength of steel are assumed to beLognormally distributed [7]. For FRP materials,research [13] has investigated the statistical distributionof FRP bars by testing several distribution types todetermine the most appropriate statistical representation.Chi-square statistical test conducted by [13] revealedthat the Weibull distribution (Extreme Event Type III)can adequately represent FRP bar material properties,which confirms previous findings for compositelaminates [14]. It is important, however, to note thatregardless of the distribution of the individual variate, itis the distribution of the safety margin SR , that isimportant in the calculation of the probability of failure,

fP [7]. It is also worthy of note that in the second

moment reliability method, the parameters of interest aremeans and variances. TABLE 1 shows values of relevantparameters for the basic variables used in this paper.


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TABLE 1: Essential Design Parameters Used in the Paper

S/N BASIC VARIABLE MEANSTANDARDDEVIATION

COEFFICIENT OFVARIATION

PROBABILITYDISTRIBUTION

1 Compressive strength of concrete, fcu 30N/mm2 4.5N/mm2 0.150 Lognormal2 Tensile strength of CFRP, ffrp 2250N/mm2 67.5N/mm2 0.030 Weibull3 Area of CFRP, Afrp 50.265mm2 0.75mm2 0.015 Normal4 Breadth, b 1000mm 50mm 0.050 Normal5 Effective depth, d 150mm 68.46mm 0.456 Normal6 Live load, Qk 2.25N/mm2 0.645mm 0.287 Gumbel7 Length, L 4500mm 1080mm 0.240 Normal8 Characteristic strength of links, ffrpv 692N/mm2 19.4N/mm2 0.003 Weibull9 Cross-sectional area of links, Afrpv 38.48mm2 3.36mm2 0.015 Normal10 Spacing of links, Sv 250mm 2.25mm 0.009 Normal11 Elastic modulus of CFRP, Efrp 147GPa 4.41N/mm2 0.030 Weibull12 Modulus of elasticity of concrete Ec 26 x 103N/mm2 7.8 x102 N/mm2 0.030 Lognormal13 Moment of inertia, I 2.81 x 108 mm4 8.43 x 106 mm4 0.030 Normal

4. RESULTS AND DISCUSSION

4.1 ResultsThe First Order Reliability Method (FORM)

coded in a FORTRAN module [15], was used for thecalculation of the reliability indices of the CFRPreinforced concrete solid slabs with varying load ratiosfor the criteria and thicknesses considered. Typicalresults obtained for continuous slabs of 4500mm equalspans are shown in Figs 2 to 8, which clearly indicate theintrinsic reliability and corresponding probability offailure of the slabs designed to [16].

4.2 DiscussionIt is clear from Figs 2 to 8 that the reliability

levels of the slabs are non-uniform at any of the criterionexamined for the depth of slab and span chosen. Ingeneral, the safety of the slab is endangered withincreasing load ratios in all the criteria and thicknessesinvestigated. The safety criterion is violated in thebending criterion for all the thicknesses of slab examinedby the safety index, which is less than 2.0, as has beenpredicted by [16]. In fact, the safety indices for 100mm,125mm, 150mm, 175mm and 200mm thick slabs areeven less than 1.0 (Figs 2 to 6). The intrinsic safetyindex even has a negative value, thus implying that thereis no safety at all, which means that the continuous slabswith these thicknesses and 4500mm spans will collapse.

However, the criterion for safety is satisfied inshear and deflection criteria for all slab thicknesses up toa load ratio of 0.6, except for 225mm and 250mm thick

slabs where it is violated for deflection, beingless than 2.0 at a load ratio of 0.2. At none of the loadratios considered is the safety index negative fordeflection and shear criteria except for bending. Thissuggests that for CFRP reinforced concrete solid slabs,once the bending criterion for safety is satisfied,deflection and shear criteria are met automatically.


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5. CONCLUSIONThe work investigates the reliability of CFRP

reinforced concrete solid slabs. The criteria consideredin the investigation are bending, deflection and shear,and the First Order Reliability Method (FORM) has beenemployed for the calculation of the safety levels withvarying load ratios. It is noticed that the reliability levelsof the slabs are not uniform at any of the criteriainvestigated for the thicknesses of slab and spanselected. The span selected is typical of a residential andaverage commercial building. In general, the safety ofthe slab is jeopardized with increasing load ratios in allthe criteria and thicknesses investigated. The safetycriterion is not satisfied in bending for any of thethicknesses and span of slab examined. The intrinsicsafety index even has a negative value, thus implyingthat there is no safety at all, which means that thecontinuous slabs with these thicknesses and span willcollapse in flexure.

However, the criterion for safety is satisfied inshear and deflection for all slab thicknesses up to a loadratio of 0.6, except for 225mm and 250mm thick slabswhere it is violated for deflection, being less than 2.0 at


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a load ratio of 0.2. The load ratio of 0.6 means that theslab will support a live load that is 60% more than itsself-weight or dead load. At none of the load ratiosconsidered is the safety index negative for deflection andshear criteria except for bending. This suggests that forCFRP reinforced concrete solid slabs, once the bendingcriterion for safety is satisfied, deflection and shearcriteria are met automatically, when the currentprediction is used. Hence, there is the need to optimizethis prediction in a stochastic environment in order toachieve reliable slabs with adequate and appropriatesafety, and with possible modifications to the currentformulation for design.

REFERENCES[1] Tsompanakis, Y. and Papadrakakis, M. (2000),

Robust and Efficient Methods for Reliability-Based Structural Optimization, ComputationalMethods for Shell and Spatial Structures,Institute of Structural Analysis and SeismicResearch, National Technical University,Athens, Greece.

[2] Atadero, R. A. (2006), Development of Loadand Resistance Factor Design for FRPStrengthening of Reinforced ConcreteStructures, A dissertation for Doctor ofPhilosophy Degree, University of California,San Diego.

[3] Daloglu, A. and Togan, V. (2006), Reliabilityand Reliability-Based Design Optimization,Turkish Journal of Engineering andEnvironmental Science, 237-249, KoradenizTechnical University, Trabzon- Turkey

[4] Thoft-Christensen (1984), Structural Reliability,Proceedings of “Safety and Reliability inEurope”, Pre-launching Meeting of ESRA, Ispra,Italy. Island.

[5] Osorio, P., Odenbreit, C. and Vrouwenvelder, T.(2010), Structural Reliability Analysis of QuayWalls With Steel Sheet Piles, PIANC MMXCongress Liverpool, UK.

[6] DNV, (1992), Structural Reliability Analysis ofMarine Structures. Det Norske Veritas, Høvik.

[7] Abejide, O. S. (2004), Improving the UltimateResistance Moment of Singly Reinforced,Concrete Solid Slabs, Ph.D. Thesis, Department

of Civil Engineering, Ahmadu Bello University,Zaria, Nigeria.

[8] Du, X. (2005), First Order and SecondReliability Methods, Probabilistic EngineeringDesign,University of Missouri-Rolla.

[9] Eldred, M. S. and Bichon, B. J. (2006), Second-Order Reliability Formulations, inDAKOTA/UQ, 47thAIAA/ASME/ASCE/AHS/ASC Structures,Structural Dynamics, and Materials Conference,Newport, Rhode.

[10] Lee, Y. K. and Hwang, D. S. (2008), A Study onthe Techniques of Estimating the Probability ofFailure, Journal of Chungcheong MathematicalSociety, Vol 21, No 4. Republic of Korea.

[11] Fico, R (2007), “Limit States Design of ConcreteStructures Reinforced with FRP Bars”, Ph.DProgramme in Materials and StructuresEngineering XX Cycle, University of NaplesFederico II.

[12] Faber, M. H. and Sorensen, J. D. (2002),Reliability Based Code, Joint Committee onStructural Safety, Swiss Institute of Technology,Zurich, Switzerland.

[13] Kulkarni, S. (2006), Calibration of StructuralDesign of Concrete Members Reinforced withFRP Bars, Thesis for the Degree of Master ofScience in Civil Engineering; Department ofCivil and Environmental Engineering,University of Pune, India.

[14] Okeil A., El-Tawil S, and Shahawy M. (2002),Flexural Reliability of Reinforced ConcreteBridge Girders Strengthened with CFRPLaminates, Journal of Bridge Engineering,Vol.7, No. 5.

[15] Gollwitzer, S., Abdo, T. and Rackwitz, R.(1988), First Order Reliability Method (FORM)manual, RCP, Munich, Germany.

[16] CAN/CSA-S6-02, 2002, “Design andConstruction of Building Components withFibre-Reinforced Polymers,” CAN/CSA S806-02, Canadian Standards Association, Rexdale,Ontario, Canada.

[17] Ellingwood, B. (1978) Reliability Bases of Loadand Resistance Factor for Reinforced ConcreteDesign, National Bureau of Standards, BuildingScience Series 110, Washington, D. C.

Reliability Analysis of Solid Concrete Slabs Reinforced with Carbon Fibre Reinforced Plastic (Cfrp)

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