a probabilistic study on the ductility of reinforced concrete sections

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Probabilistic models for curvature ductility andmoment redistribution of RC beams

ARTICLE in COMPUTERS AND CONCRETE APRIL 2015

Impact Factor: 0.87 DOI: 10.12989/cac.2015.16.2.191

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2 AUTHORS:

Hassan Baji

RMIT University

21PUBLICATIONS 5CITATIONS

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H. R. Ronagh

Western Sydney University

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Retrieved on: 23 November 2015
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A probabilistic study on the ductility of reinforced concrete1

sections2

Hassan Bajia,*, Hamid Reza Ronagha3

School of Civil Engineering, University of Queensland, St Lucia, Queensland 4072, Australia4

5

Abstract:Although the current design codes apply reliability-based calibration procedures6

to evaluate safety factors for the strength based limit state, the safety factors used to ensure7

minimum ductility capacities are rather simple and are not resulted from a probability-based8

procedure. This study examines level of safety delivered by the current design codes with9

regards to providing minimum curvature ductility for reinforced concrete (RC) beams made10

with normal strength concrete. Reliability analysis results show that with regard to the11

strength limit state, the considered design codes are in good agreement with one another.12

However, there is considerable disparity in the level of safety provided for minimum13

curvature ductility amongst the codes. The provided reliability for the design to remain14

ductile is too low in some and just about acceptable in the others. This signifies the15

importance and the need to introduce reliability based methods of design for ductility.16

Keywords:Reinforced concrete beams, design codes, curvature ductility, ultimate concrete17

strain, Monte Carlo Simulation, Reliability18

19

20

21

*Corresponding Author: Tel.: +61-7-33661652; Fax: +61-7-3365459922

Email:[email protected]
mailto:[email protected]:[email protected]


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Introduction11

In the design of RC structures, checking the strength adequacy often takes priority over the2

deformation and the ductility, which are indirectly incorporated in the design process. Design3

codes prescribe some limits such as the rebar percentage limit to ensure sections would4

possess adequate ductility. Park et al. (1988)assessed the available ductility of doubly RC5

beam sections using the moment-curvature analysis. They concluded that the implementation6

of the general requirements of the American and New Zealand design codes will ensure a7

curvature ductility of more than 2.0, while the application of moment redistribution8

requirements will ensure curvature ductility larger than 4 for the sections. Ho et al. (2004)9

investigated the minimum flexural ductility design of high strength concrete beams. Their10

results showed that the current practice of providing minimum flexural ductility in existing11

design codes would not really provide a consistent level of minimum flexural ductility. Kwan12

and Ho (2010)studied the flexural ductility of high-strength concrete beams and columns by13

extensive parametric studies using nonlinear moment-curvature analysis. Based on their14

study, a minimum ductility design method for ensuring the achievement of a minimum15

ductility of 3.32 was proposed. In order to check the criterion of local ductility in the cross16

sections, Kassoul and Bougara (2010) have taken into account the recommendations of EC217

(2004) regarding the stress-strain relationship for concrete and steel. They developed a18

methodology for evaluating the available curvature ductility factor in RC beams. All of these19

studies have used a deterministic approach to assess the minimum ductility requirements of20

RC beams and columns. In this paper, a probabilistic framework for assessing the minimum21

ductility requirements of RC sections is proposed.22

Realistic description of strength and deformation requires probabilistic models and23

implementation of a reliability based analysis. There have been numerous studies performed24


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on the strength of RC members, resulting in code calibration, which has now been1

implemented in many design codes (Bartlett, 2007;Szerszen et al., 2003). In contrast, limited2

research can be found on the probabilistic aspects of the inelastic deformation and ductility3

(De Stefano et al., 2001;Kappos et al., 1999; Lu et al., 2004; Trezos, 1997). In some of these4

studies, probabilistic models for ductility related measures such as curvature ductility have5

been proposed. However, none of these studies has directly addressed the issue of minimum6

ductility requirements suggested by the current codes of practice. Ito and Sumikama (1985)7

study is amongst very few, if not the only one, that are directly related to the reliability8

analysis of code provisions with regards to the ductile design of RC beams. They examined9

the suitability of the reduction coefficient for the balanced steel ratio provided in ACI 318-83.10

Results of their study showed that using the ACI reduction factor of 0.75 results in high11

probability of producing over-reinforced cross-sections when concrete is placed in situ.12

This study aims at investigating the reliability of minimum ductility requirements in the13

current design codes using a comparative-based approach. The amount of tensile rebar has an14

inverse relation with section ductility. Therefore, design codes prescribe a maximum rebar15

percentage limit in order to ensure that RC sections would exhibit adequate ductility. On the16

other hand, the presence of compressive rebar enhances the section ductility. Noting these,17

the worst-case scenario for investigating the minimum ductility is to have the maximum18

tensile rebar at the tension side of the section, while the compression side is reinforced by the19

minimum rebar. This would result in a lower reliability level for ductility of RC sections, and20

as such, it is devised in this study.21

Moment-Curvature analysis222

In order to determine the load-deformation behavior of a cross section, moment curvature23


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analysis is performed using nonlinear material stress-strain relationship. In Figure 1, the1

typical stress-strain curves for concrete and steel bars are shown. Stress-strain curve for2

concrete is based on JCSS probabilistic model code (2012). This model is similar to the3

parabolic model proposed by the EC2 to be used in section analysis. The first part of the4

curve is a parabola, while the second part is constant. As is shown in Figure 1b, for5

reinforcing steel a bilinear relationship is employed.6

7

Figure 1: Material Models for Concrete and Steel8

Although the material stress-strain relationship is nonlinear, the strain variation across the9

height of the section can be assumed linear. This assumption seems to have adequate10

accuracy and the experimental results have proven its validity (Parket al., 1975). Fiber model11

is used to derive the moment-curvature of the cross section. In the limit state design, ductility12

of a member is usually defined as the ratio of the ultimate deformation to the deformation at13

first yield. The first yield is the state at which the tensile rebar yields. On the other hand, the14

ultimate curvature is the state at which either the concrete crushes or the tensile bar ruptures.15

In case of moment curvature analysis, ductility is the ratio of ultimate to yield curvature as16

shown in Equation 1.17
https://www.researchgate.net/publication/279416663_Reinforced_Concrete_Structure?el=1_x_8&enrichId=rgreq-14bb1238-ac09-41b1-b4b9-8960babf42ca&enrichSource=Y292ZXJQYWdlOzI3OTA1ODYwNDtBUzoyNzQ1MjM2MzUzMTg3ODRAMTQ0MjQ2Mjk0MzQ2MQ==https://www.researchgate.net/publication/279416663_Reinforced_Concrete_Structure?el=1_x_8&enrichId=rgreq-14bb1238-ac09-41b1-b4b9-8960babf42ca&enrichSource=Y292ZXJQYWdlOzI3OTA1ODYwNDtBUzoyNzQ1MjM2MzUzMTg3ODRAMTQ0MjQ2Mjk0MzQ2MQ==


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u

y

(1)

In Equation 1,y

, u and represent the yield curvature, the ultimate curvature and the1

curvature ductility respectively. Curvature ductility depends on factors such as tensile and2

compressive reinforcement ratios, concrete and steel material ductility, axial force and other3

parameters.4

Code provisions on ductility35

In this section, provisions of the American (ACI 318, 2011), Canadian (CSA A23.3, 2004),6

Australian (AS 3600, 2009), New Zealand (NZS 3101, 2006)and European (EC2, 2004)with7

respect to the important issue of ductility are reviewed.8

Figure 2 shows a typical stress-strain diagram of an RC beam cross section. In Table 1, the9

parameters of the equivalent stress block for different design codes are shown. Furthermore,10

this Table shows the limiting constraints for ductile design based on the aforementioned11

codes. In this Table, fc is the concrete compressive strength, t (used with the American12

code) denotes the strain at foremost tensile bar, Parameters 1 and 1 are the concrete stress13

block parameters,cu

is the ultimate strain of concrete and (used with the European code)14

refers to the moment reduction factor. Here, the effect of moment redistribution is not15

considered. Hence, the moment reduction factor is taken as 1.0. All other parameters are16

shown in Figure 2. Unlike other codes, the American code does not directly use the neutral17

axis parameter (limit the c/dratio) to ensure adequate ductility; rather it applies a minimum18

tensile bar strain (tensile strain >0.005). Nevertheless, its limit on minimum tensile strain19

could be transformed to this neutral axis parameter format if needed. In Table 1, the result of20

this transformation is shown. It is worth mentioning that Table 1 only covers normal strength21


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concrete.1

2

Figure 2: Stress-strain diagrams of a typical rectangular section3

Using force equilibrium and geometric compatibility, the limiting neutral axis parameter can4

be described as per Equation 2. In this Equation, effect of compressive rebar is neglected.5

The limiting c/dvalues for each design code are shown in Table 1.6

'

1 1

1

. .

y cu

Limitc cu y

fc c

d f d S F

(2)

S.F.shows the safety factor used to ensure adequate safety margin for a ductile design. This7

safety factor is different from code to code. As is noted in Table 1, the New Zealand and the8

Canadian codes follow exactly the mentioned format. The limiting ratio depends on yield9

strain of rebar steel as well as the ultimate concrete strain. Thus, for different steel grades, the10

limiting ratio varies. In Table 2, based on Equation 2, the relationship between the limiting11

neutral axis parameter and the safety factor for different design codes is shown.12

Table 1: Concrete stress block parameters of considered design codes13

Code cu 1 1 max/c d


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ACI 0.0030 0.85

for ' 30cf

0.85

for ' 30cf '1.09 0.008 cf

10.65 0.85

0.005t

0.375c

d

AS 3600 0.0030

'0.85 0.003 cf

10.67 0.85

'0.85 0.003 cf

10.67 0.85 0.360

c

d

NZS 3101 0.0030 0.85

for ' 30cf

0.85

for ' 30cf '1.09 0.008 cf

10.65 0.85

0.75/

cu

cu y s

c

d f E

CSA A23.3 0.0035

'0.85 0.0015 cf

1 0.67

'0.85 0.0025 cf

1 0.67

700

700 y

c

d f

EC2 0.0035 1.00 0.801

2

( 0.44)

( 1.25)

kc

d k

1

Design codes that apply the strength reduction factor to material properties rather than2

strength component (like the Canadian and European codes) already consider certain amount3

of safety margin required for the ductile design. To make results of these design codes4

consistence with the other codes, the limit shown in Equation 2 needs to be multiplied by5

s/cfactor, where sand care the steel and concrete material reduction factors. In Table 3,6

these material reduction factors are shown. It should be noted that in deriving the safety7

factors for each code, its own ultimate concrete strain is used, and the steel modulus of8

elasticity of 200GP is assumed.9

Table 2: Neutral axis limiting values and corresponding safety factor10

Code

yf (MPa)

300 400 500


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Limit

c

d

S.F.Limit

c

d

S.F.Limit

c

d

S.F.

ACI 318 (2011) 0.375 1.78 0.375 1.60 0.375 1.45

AS 3600 (2009) 0.360 1.86 0.360 1.67 0.360 1.50

NZS 3101 (2006) 0.500 1.33 0.450 1.33 0.409 1.33

CSA A23.3 (2004) 0.535 1.31 0.486 1.31 0.446 1.31

EC2 (2004) 0.343 2.04 0.343 1.86 0.343 1.70

1

When only the difference in ultimate strain of concrete is considered, the American,2

Australian and New Zealand design codes are more conservative in limiting the cross section3

ductility. This conservatism is not transparent in the safety factors shown in Table 2. This is4

because the available safety factors for the considered design codes depend on other factors5

such as the material reduction factors and the neutral axis parameter limit provided by those6

codes. It should be noted that the overall safety depends on other parameters of the equivalent7

rectangular stress block, e.g.1

and1

parameters as well. Only a complete moment-8

curvature analysis in which the curvature ductility is directly derived can reveal the level of9

safety in any of the mentioned design codes.10

In this study, in addition to investigating reliability of the curvature ductility of cross sections11

designed based on different design codes, the safety levels of strength limit state is also12

considered for the sake of comparison. Due to the different statistical load models used in the13

calibration of load and resistance factors in each design codes, the load combination that only14

includes the effect of dead load is considered. The considered design codes agree on the dead15

load combination. Furthermore, the statistical model for dead load is universally the same. In16

Table 3 the dead load factor and resistance reduction factors are shown for different design17

codes.18

Table 3: Safety factors of the considered design codes for dead load combination19


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Code DL

Resistance Reduction

Factors

Behavior Material

ACI 318 (2011) 1.40 0.90 -

AS 3600 (2009) 1.35 0.80 -NZS 3101 (2006) 1.35 0.85 -

CSA A23.3 (2004) 1.40 -0.65c 0.85s

EC2 (2004) 1.35 -1/1.50c 1/1.15s

1

Reliability analysis42

Limit states4.13

In the current study, two limit states are defined. The first one is a strength based limit state in4

which the bending capacity of the cross section is treated as strength. The second limit state is5

a deformation based limit state. The curvature ductility of the section is compared with 1.0,6

which is the boundary between ductile and brittle design for RC beams subjected to bending.7

In this limit state, failure is deemed to occur when curvature ductility exceeds 1.0. Equations8

5 and 6 show the considered limit state functions.9

1 R Qg M M (3a)

2 1.0u u u

y y yR Q R

g

(3b)

In Equation 3,MRandMQrepresent the bending capacity and bending resulted from demand,10

respectively. The parameter u

y Q

can be taken as 1.0 as any value smaller than 1.0 denotes11

a brittle failure. As previously stated, in this study, only the effect of dead load is considered.12


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10

Parameters y and u denote the yield and ultimate curvature respectively. In the previous1

sections, the yield and ultimate states were defined.2

Statistical models4.23

Two major types of uncertainties exist concerning RC member behavior, namely physical4

uncertainty, and model uncertainty. In this study, the majority statistical models for basic5

random variables are taken from the JCSS Probability Model Code (2012). Table 4 shows the6

statistical data for all considered random variables.7

JCSS describes the dimensional deviations of any dimension by statistical characteristic of its8

deviation from the nominal value. For concrete cover, two different models are suggested in9

the JCSS for top and bottom reinforcement. For simplicity only the model reported in Table 410

is employed in this study for both top and bottom reinforcement covers. In order to remain11

within the boundaries of practicality, a common 600400 mm cross section with 60 mm12

cover to rebar center is selected in this study. Although in the JCSS states that the Normal13

distribution seems to be satisfactory for dimensional parameters, in this study the lognormal14

distribution is used instead. For random variables having small coefficient of variation15

approximating normal distribution with lognormal one would not affect the results16

significantly (Benjamin et al., 1975).17

In the JCSS model code, all concrete properties are related to reference property of concrete,18

which is the compressive strength of the standard test specimens tested according to the19

standard conditions at the age of 28 days. The other concrete properties are related to the20

reference strength of concrete according to the following Equations.21

In situ compressive strength: ' '0 1( )c cf f Y (4a)

Tensile strength:' 2 /3

20.3( )cf Y (4b)


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Modulus of elasticity: 1/3

'

310500 cf Y (4c)

Strain at peak stress: 1/6

'

40.006cu cf Y

(4d)

The statistical model for parameters, Y1to Y4are given in Table 4. For all these parameters,1

the lognormal distribution is employed. For the strength of standard concrete specimen '0cf ,2

the student distribution is suggested in JCSS. However, this distribution can be approximated3

by a lognormal distribution. In this study, for the reliability analysis, the lognormal4

distribution is employed for the concrete compressive strength. The statistical information for5

C25, C35 and C45 ready mix concrete grades are presented in JCSS and are shown in Table 4.6

The Statistical model used for the concrete strength has a good agreement with findings of a7

recent statistical analysis on the concrete strength (Nowak et al., 2003).8

Reinforcing steel generally is classified and produced according to grades. In this study, in9

accordance with the European standard, grades S300, S400 and S500 are used. These grades10

are nearly equivalent to grades G40, G60 and G75 rebar steel materials according to the11

American standard. The statistical models for yield and ultimate strength, modulus of12

elasticity and ultimate strain of steel are shown in Table 4. For all these parameters,13

lognormal distribution is considered.14

Table 4: Statistical properties of random variables15

Variable Nominal /Bias /COVb 300 mm 1.003 4 mm+0.006Nominal

h 600 mm 1.003 4 mm+0.006Nominal

cover 60 mm Nominal+10 mm 10 mm

sA max n nb d 1.0 0.02

'

sA

min n nb d 1.0 0.02

sE 200GPa 1.0 0.04

yf 300/400/500MPa Nominal+2 30MPa


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uf - 1.08 yf 40MPa

su 0.05 Nominal+2 0.09

'

0cf 25/35/45MPa 1.55/1.35/1.20 0.17/0.12/0.07

Y1 - 1.0 0.06

Y2 - 1.0 0.30

Y3 - 1.0 0.15

Y4 - 1.0 0.15

l - 0.96 0.005

It is assumed that the ratio of ultimate to yield stress of steel material is 1.08. This ratio1

corresponds to minimum ratio required for Class A reinforcement in EC2. Also for this steel2

grade, the minimum ultimate strain should be greater than 0.05. Therefore, in this study this3

value is used as nominal ultimate strain of steel material. The correlation among steel rebar4

area, yield strength, ultimate strength and ultimate strain of steel material is considered in this5

research. Table 5 shows the correlation among these variables.6

Table 5: Correlation among rebar steel material properties7

sA yf uf su

sA 1.00 +0.50 +0.35 0.00

yf 1.00 +0.85 -0.50

uf 1.00 -0.55

su 1.00

The statistical model used for rebar properties in this research can be compared with those8

used in the available literature (Bournonville et al., 2004;Nowak et al., 2003). In Table 4, a9

summary of the statistical models used in this study is shown. All of the random variables10

used in this study are treated as lognormal distributed random variables. The joint probability11

density function is a multivariate lognormal distribution with correlated variables.12


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Method of Analysis4.31

Reliability analysis containing stochastic finite element such as the problem being studied2

here are almost universally performed using the MCS technique. In the MCS method, the3

probability of failure is calculated using random number generation (Melchers, 1999). In this4

study, a Latin Hypercube Sampling technique (Ayyub et al., 1984)is used. In order to obtain5

a more accurate estimate of the probability of failure, variance reduction methods including6

Antithetic Variates are to be used in conjunction with the method (Ayyub et al., 1991).7

The measure of reliability is conventionally defined by the reliability index , which8

is related to the probability of failurepfby Equation 5.9

( )fp (5)

In Equation 5, is the cumulative distribution function of standardized normal distribution.10

The reliability index corresponds to the design working life of the structure and it has one-to-11

one correspondence with failure probability.12

For the purpose of reliability differentiation, the European code (EC2, 2004) establishes13

reliability classes. According to this code, for the reliability class of RC2 and based on a 5014

years reference period, the recommended minimum reliability indices for ultimate and15

serviceability (irreversible) limit states are 3.8 and 1.5, respectively. The reliability class of16

RC2 could be corresponding to the consequences class CC2, which covers residential and17

office buildings. With respect to the strength limit state in this study, target reliability index18

of 3.8 could be used. However, when it comes to the ductility limit state, it is difficult to set19

an appropriate target reliability based on available literature. The limit state, which is here20

defined for satisfying adequate ductility could not be treated either as ultimate limit state or21

as serviceability limit state. Failure in curvature limit state means brittle collapse, which22


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14

comes without warning. On the other hand, this failure does not lead to structural collapse. In1

this study, two target reliability indices of 2.3 and 3.1 are selected and will be used in the2

calibration of safety factors for ductility limit state.3

Results and discussion54

Results of the reliability analysis of moment-curvature curve are presented here. Three5

different grades of steel (S300, S400 and S500) as well as three different types of concrete6

(C25, C35 and C45) are used in the analysis. The MCS method with Variance Reduction7

technique is used to derive the probability of failure and the reliability indices. In previous8

sections, the strength and ductility limit states were discussed. These limit state are namedg19

andg2. Before performing the reliability analysis for considered limit states, using available10

experimental results the statistical models of model uncertainty are derived.11

Model uncertainty5.112

The model uncertainty is used to quantify the uncertainties associated with assumptions and13

simplifications used in derivation of the theoretical model. The model uncertainty associated14

with a particular mathematical model may be expressed in terms of the probabilistic15

distribution of a variableXdefined in Equation 6.16

PrM

ActualX

edicted (6)

Model error covers the uncertainties in the modeling of a structure as a mathematical17

model where the uncertainties arise from idealization of different parts of the structure. In this18

study, both strength and deformation model are of interest. In what follows, the statistical19

model (mean, coefficient of variation and probability density function) for strength and20

curvature models are evaluated. Moment and curvature data for RC sections (beams with21


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15

normal strength concrete) have been collected from published literature (Corley, 1966;1

Debernardi et al., 2002;Mattock, 1965). Mattock (1965)and Corley (Corley, 1966)based on2

similar test programs investigated the rotation capacity of RC beams. In total, they tested 773

beams with different dimensions, material properties and rebar percentage. In the Debernardi4

and Taliano (2002)test program, which was on evaluation of the rotation capacity of concrete5

beams, 22 beams were tested. They used two different load arrangements in their6

experimental program. In this study, the results of all available 99 test specimens (22+77) are7

used to derive a statistical model for the strength and curvature of RC sections. Details of all8

these specimens can be found in the mentioned studies.9

In the moment-curvature analysis of the available test results, the theoretical model and10

assumptions made are similar to those used in the reliability analysis. However, the stress-11

strain model for the rebar steel is similar to those used in the corresponding studies. Model12

errors for yield moment, ultimate moment, yield curvature and ultimate curvature are13

evaluated. The mean and the standard deviation along with the best-fit lognormal distribution14

parameters for each set of the experimental data are found. Method of ordered statistics is15

used to find the best-fit lognormal distribution for model error. Table 6 shows the mean and16

coefficient of variation for different components of the model error.17

Table 6: Mean and coefficient of variation of model errors18

ComponentMattock(1965)

Corley (1966)Debernardi et

al. (2002)All

Mean COV Mean COV Mean COV Mean COV

y 1.16 0.09 1.41 0.17 1.15 0.26 1.26 0.20

u 1.03 0.21 0.80 0.16 0.83 0.23 0.90 0.23

yM 1.02 0.04 1.02 0.03 1.05 0.09 1.03 0.06

uM 0.89 0.13 0.87 0.09 1.03 0.06 0.91 0.12


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16

In Table 6,y

, u , yM and uM refer to yield curvature, ultimate curvature, yield moment and1

ultimate moment, respectively. Terms Mean and COV show the average and the2

coefficient of variation. As is expected, the uncertainty in evaluating yield point components3

(considering all the test results) is lower than that of ultimate components. Furthermore, as is4

seen, the theoretical procedure that is used in this study underestimates the yield curvature5

and bending moment, while overestimates the ones for ultimate state. In Figure 3, based on6

the experimental data and the theoretical results, the best-fit line for the model error is shown7

on a normal probability paper. The results show that the model error could be reasonably8

modeled by the lognormal distribution.9

a)

Yield curvature b)

Ultimate curvature

Figure 3: Best fit statistical models for yield and ultimate curvature model error10

The average and the coefficient of variation resulted from these fitted lognormal distributions11

are very close to the sample mean and coefficient of variation. Therefore, in this study the12

statistical model for model error of all components is modeled using lognormal distribution13

with mean and coefficient of variation shown in Table 5. The statistical data for ultimate14

-0.80

-0.40

0.00

0.40

0.80

-3.0 -2.0 -1.0 0.0 1.0 2.0 3.0

Lo

g(ModelError)

Standard Normal Variable

Best Fit Line Mattock (1964)

Corley (1966) Debernardi et. Al (2002)

-0.80

-0.40

0.00

0.40

0.80

-3.0 -2.0 -1.0 0.0 1.0 2.0 3.0

Log(M

odelError)

Standard Normal Variable

Best Fit Line Mattock (1964)

Corley (1966) Debernardi et. al (2002)


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17

bending moment are only shown for comparison, and they will not be used in the reliability1

analysis.2

Level of safety in the current design codes5.23

In this part, the MCS technique is used to simulate the required samples for the reliability4

analysis. Based on the simulated samples, moment-curvature curves are developed. Then, the5

yield and ultimate curvatures as well as the yield moments are derived from the moment-6

curvature graph. Figure 4 depicts typical moment-curvature curves obtained from one of the7

considered cases. In this Figure, statistical properties of the normalized yield strength and8

curvature ductility (with respect to nominal yield strength and curvature ductility) are shown.9

It should be noted that by substituting the nominal values of random variables, nominal yield10

strength and curvature ductility are derived. The statistical properties of flexural capacity is11

comparable with those of Szerszen and Nowak study (2003). For the ordinary cast-in-place12

concrete, their results showed 1.19 and 0.089 as the bias factor and the coefficient of13

variation of the flexural strength of RC beams.14


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18

Figure 4: Sample set of simulate moment-curvature graphs1

Figure 4 shows that the disparity of the ultimate curvature is much higher than that of the2

yield curvature. Furthermore, for many of the considered cases the coefficient of variation of3

the curvature ductility is about triple that of the strength. The reason behind this considerable4

difference is that the curvature ductility depends on all of the concrete stress block parameters5

including the ultimate strain of concrete while the strength depends on fewer random6

parameters. The concrete stress block parameters depend on the concrete compressive7

strength. Therefore, dependence of the ductility to all concrete stress block parameters makes8

the reliability of the ductility limit state highly sensitive to the concrete compressive strength.9

Figure 5 shows the sensitivity of the reliability of ductility limit state to the concrete strength.10

The moment-curvature curves are used to derive the flexural capacity (yield strength) and the11

curvature ductility of the cross section. Then, using limit statesg1 andg2, reliability indices12

for the strength and the ductility limit states are derived. Figure 5 shows the reliability indices13

of strength and curvature ductility limit states for different values of steel and concrete14

strengths and based on different design codes. Results in Figure 5 show that the reliability of15

strength limit state is higher than that of ductility limit state. Furthermore, using different16

design codes in the design procedure almost results in about the same level of safety for17

strength. The European and ACI code have lower strength reliability indices in comparison18

with the other design codes. In contrast, safety level of ductility limit state has high disparity19

for different design codes. The American and Australian codes provide safer design for the20

ductility based limit state while the Canadian code provides the lowest safety level.21


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19

a) Ductility, 400yf MPa b) Strength, 400yf MPa

c) Ductility, 500yf MPa d) Strength, 500yf MPa

Figure 5: Reliability indices for strength and ductility limit states1

For a specific concrete cross section, using a higher concrete strength allows for higher rebar2

percentage and accordingly less ductility and this in turn results in a lower reliability index3

for the ductility based limit state. Figures 5a and 5c show that in some cases the reliability of4

ductility based limit state could drop to less than 1.5 for some design codes such as the5

Canadian standard, while the corresponding strength based limit state shows high reliability6

0.0

1.5

3.0

4.5

15 25 35 45

Reliabili

tyIndex

Concrete Compressive Strength (MPa)

ACI 318-11 AS 3600-09

NZS 3101-06 CSA A23.3-04

EC2-04

0.0

1.5

3.0

4.5

15 25 35 45

Reliabili

tyIndex


ACI 318-11 AS 3600-09

NZS 3101-06 CSA A23.3-04

EC2-04

0.0

1.5

3.0

4.5

15 25 35 45

ReliabilityIndex


ACI 318-11 AS 3600-09

NZS 3101-06 CSA A23.3-04

EC2-04

0.0

1.5

3.0

4.5

15 25 35 45

ReliabilityIndex


ACI 318-11 AS 3600-09

NZS 3101-06 CSA A23.3-04

EC2-04


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20

index of near 4.0.1

According to Equation 2, design codes allow lower rebar percentage for higher rebar yield2

stress. On the other hand, using higher rebar yield stress results in lower nominal ductility3

levels. Using higher rebar yield steel leads into a reduction of both the maximum rebar4

percentage and the curvature ductility. As can be seen in Table 7, for all of the considered5

design codes, using higher rebar yield steel leads into a reduction of both the maximum rebar6

percentage and the curvature ductility. Although reduction in rebar percentage is in favor of7

increasing the reliability index of the ductility limit state, the lower nominal curvature8

ductility leads to a reduction in the reliability index of the ductility limit state. Therefore, the9

final reliability indices for the ductility limit state depend on these two contradicting effects10

of implementing the high yield stress rebar. The statistical models shown in Table 4 indicate11

that, despite the relatively lower uncertainty in the 500MPa steel, the gap is not large enough12

to make a considerable impact on the end reliability index comparing to the 400MPa steel.13

Table 7: Maximum allowable rebar percentage and corresponding minimum curvature14

ductility for different design codes15

yf

(MPa)

'

cf

(MPa)

ACI 318-11 AS 3600-09NZS 3101-

06

CSA A23.3-

04EC2-04

max max

max max

max

300

25 0.0226 2.90 0.0217 3.19 0.0301 2.02 0.0329 1.86 0.0229 3.53

35 0.0296 2.81 0.0303 3.01 0.0402 1.94 0.0440 1.78 0.0321 2.99

45 0.0346 2.78 0.0390 2.84 0.0465 1.93 0.0539 1.72 0.0412 3.12

400

25 0.0169 2.35 0.0163 2.57 0.0203 1.90 0.0224 1.74 0.0172 2.84

35 0.0222 2.29 0.0228 2.45 0.0271 1.83 0.0300 1.67 0.0240 2.67

45 0.0260 2.27 0.0293 2.32 0.0314 1.82 0.0367 1.63 0.0309 2.55

500

25 0.0135 1.99 0.0130 2.17 0.0148 1.81 0.0164 1.65 0.0137 2.38

35 0.0178 1.94 0.0182 2.07 0.0197 1.76 0.0220 1.60 0.0192 2.26

45 0.0208 1.93 0.0234 1.98 0.0228 1.75 0.0269 1.57 0.0247 2.16

16


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21

According to the results shown in Figures 5a and 5c, the difference in the reliability indices1

resulted from 400MPa and 500MPa steel materials is not considerable. As mentioned2

previously, this is due to the fact that the effects of lower nominal ductility and lower rebar3

percentage on the reliability index act opposite to each other.4

Calibrating the ductility safety factors5.35

Level of target reliability has a big influence on the calibration of safety factors. As6

previously discussed, target reliability generally depends on the cost of safety measure and7

the consequences of failure. Here, two target reliability indices of 2.3 and 3.1 will be used to8

calibrate appropriate safety factors for the ductility based limit state. Equation 7 shows the9

relation between the safety factor and the maximum rebar percentage.10

'

1 1

1

. .

c cu

y cu y

f

f S F

(7)

Now, instead of using the code requirements for making a ductile design, Equation 7 is used11

to calculate the maximum rebar percentage. A wide range of safety factors is used to evaluate12

the maximum rebar percentage. Then, using reliability analysis, the safety factor13

corresponding to the desired target reliability is evaluated. In Figure 6, based on two different14

target reliability indices, the normalized safety factors for different design codes are shown.15

To derive the normalized safety factor, the safety factor resulted from reliability analysis is16

divided by the available safety factors currently used by each design code. Therefore, a17

normalized safety factor greater than 1.0 shows that the considered code does not provide18

adequate safety margin for that particular case. Results in Figure 6 are based on rebar yield19

stress of 400MPa.20


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22

(a) arg 2.30t et (b) arg 3.10t et

Figure 6: Normalized safety factors for different design codes1

Results for the target reliability index of 2.3, except for the Canadian code, almost all2

available design codes provide adequate margin of safety for a ductile design. However,3

when the target reliability index is increased to 3.1, none but the American code, provide4

sufficient safety margin for ductile design in many of the considered cases. Due to the5

allowance for higher rebar percentage, using the Canadian design code leads to the highest6

normalized safety factor, and even for the low target reliability of 2.3, the provided safety7

level in not acceptable. The only safety factor that the Canadian standard is relying on is a8

factor that indirectly comes from strength safety factors. Besides, this design code introduces9

the ultimate concrete strain of 0.0035 instead of 0.0030.10

Conclusions611

The probabilistic analysis of RC members with respect to strength and ductility limit states at12

the sectional level are investigated for different design codes. Base on the results, the salient13

features of this study are summarized as follows:14

0.00

0.25

0.50

0.75

1.00

1.25

1.50

15 25 35 45

Normalizedsafetyfactor


ACI 318-11 AS 3600-09

NZS 3101-06 CSA A23.3-04

EC2-04

0.00

0.25

0.50

0.75

1.00

1.25

1.50

15 25 35 45

Normalizedsafetyfactor


ACI 318-11 AS 3600-09

NZS 3101-06 CSA A23.3-04

EC2-04


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23

1. Current literature does not adequately address appropriate probabilistic models for1

evaluating ductility related parameters and thus special attention should be paid to2

this area.3

2. Setting a target safety level is critical when calibrating safety factors. Current4

design codes worldwide are calibrated for the strength limit states and the results5

of these study confirms adequacy of the currently in-place calibration. On the6

other hand, there is a need for definition of appropriate target safety levels when7

dealing with the ductility as a limit state.8

3.

RC beam sections designed based on different standards show almost uniform9

reliability for the strength based limit state. However, with respect to the curvature10

ductility reliability, the results exhibit great disparity. This is somewhat expected,11

as the minimum ductility requirements of these design codes are different. Except12

in a few cases, the reliability indices for ductility limit state are considerably lower13

than those of the strength limit state. The results confirm the understanding that14

the statistical properties of the flexural capacity of reinforced cross sections does15

not depend on concrete and rebar strengths; on the other hand, the ductility16

capacity of RC sections depends on the concrete and steel strengths as well as the17

equivalent rectangular concrete stress block parameters.18

4. Current design codes provide different minimum requirements for curvature19

ductility that are generally not close to each other. These minimum requirements20

aim to provide a minimum level of curvature ductility. Lower bound values along21

with a rudimentary safety factor are used for evaluating these requirements. Based22

on the findings of this study, apart from not being rational, this procedure would23

not guarantee a minimum safety level.24


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24

References1

Aci 318 (2011) "Building Code Requirements for Structural Concrete and Commentary".2

Farmington Hills, MI, USA, American Concrete Institute.3

As 3600 (2009) "Australian Concrete Structures Standard", Standard Australia.Sydney,4

Australia.5

Ayyub, B. M. and Haldar, A. (1984) "Practical Structural Reliability Techniques",Journal of6

Structural Engineering,vol. 110, no. 8, pp. 1707-1724.7

Ayyub, B. M. and Lai, K.-L. (1991) "Selective Sampling in Simulation-Based Reliability8

Assessment",International Journal of Pressure Vessels and Piping,vol. 46, no. 2, pp. 229-9

249.10

Bartlett, F. (2007) "Canadian Standards Association Standard A23. 3-04 Resistance Factor11

for Concrete in Compression", Canadian Journal of Civil Engineering,vol. 34, no. 9, pp.12

1029-1037.13

Benjamin, J. and Cornell, C. (1975) "Probability, Statistics and Decision for Civil Engineers",14

New York: McGraw-Hill.15

Bournonville, M., Dahnke, J. and Darwin, D. (2004) "Statistical Analysis of the Mechanical16

Properties and Weight of Reinforcing Bars", University of Kansas Report.17

Corley, W. (1966) "Rotational Capacity of Reinforced Concrete Beams",Journal of the18Structural Division,vol. 92, pp. 121-146.19

Csa A23.3 (2004) "Design of Concrete Structures". Canadian Standards Association.20

De Stefano, M., Nudo, R., Sar, G. and Viti, S. (2001) "Effects of Randomness in Steel21

Mechanical Properties on Rotational Capacity of Rc Beams",Materials and Structures,vol.22

34, no. 2, pp. 92-99.23

Debernardi, P. G. and Taliano, M. (2002) "On Evaluation of Rotation Capacity for24

Reinforced Concrete Beams",ACI Structural Journal,vol. 99, no. 3, pp. 360-368.25

Ec2 (2004) "Design of Concrete Structures: Part 1: General Rules and Rules for Buildings".26

Brussels, Belgium, European Committee for Standardization.27

Ho, J., Kwan, A. and Pam, H. (2004) "Minimum Flexural Ductility Design of High-Strength28

Concrete Beams",Magazine of Concrete Research,vol. 56, no. 1, pp. 13-22.29

Ito, K. and Sumikama, A. (1985) "Probabilistic Study of Reduction Coefficient for Balanced30

Steel Ratio in the Aci Code",ACI Structural Journal,vol. 82, pp. 701-709.31

Jcss (2012) "Probabilistic Model Code", The Joint Committee on Structural Safety.Technical32


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25

University of Denmark.1

Kappos, A., Chryssanthopoulos, M. and Dymiotis, C. (1999) "Uncertainty Analysis of2

Strength and Ductility of Confined Reinforced Concrete Members",Engineering Structures,3

vol. 21, no. 3, pp. 195-208.4

Kassoul, A. and Bougara, A. (2010) "Maximum Ratio of Longitudinal Tensile Reinforcement5

in High Strength Doubly Reinforced Concrete Beams Designed According to Eurocode 8",6

Engineering Structures,vol. 32, no. 10, pp. 3206-3213.7

Kwan, A. K. and Ho, J. C. (2010) "Ductility Design of High-Strength Concrete Beams and8

Columns",Advances in Structural Engineering,vol. 13, no. 4, pp. 651-664.9

Lu, Y. and Gu, X. (2004) "Probability Analysis of Rc Member Deformation Limits for10

Different Performance Levels and Reliability of Their Deterministic Calculations", Structural11

safety,vol. 26, no. 4, pp. 367-389.12

Mattock, A. H. (1965) "Rotational Capacity of Hinging Regions in Reinforced Concrete13

Beams",ACI Special Publication,vol. 12, pp. 143-181.14

Melchers, R. E. (1999) "Structural Reliability Analysis and Prediction".15

Nowak, A. S. and Szerszen, M. M. (2003) "Calibration of Design Code for Buildings (Aci16

318): Part 1-Statistical Models for Resistance",ACI Structural Journal,vol. 100, no. 3, pp.17

377-382.18

Nzs 3101 (2006) "Concrete Structures StandardPart1the Design of Concrete Structures",19

Wellington: Standards New Zealand.20

Park, R. and Paulay, T. (1975) "Reinforced Concrete Structures / R. Park and T. Paulay",21

New York, Wiley.22

Park, R. and Ruitong, D. (1988) "Ductility of Doubly Reinforced Concrete Beam Sections",23

ACI Structural Journal,vol. 85, no. 2, pp. 217-225.24

Szerszen, M. M. and Nowak, A. S. (2003) "Calibration of Design Code for Buildings (Aci25

318): Part 2-Reliability Analysis and Resistance Factors",ACI Structural Journal,vol. 100,26

no. 3.27

Trezos, C. (1997) "Reliability Considerations on the Confinement of Rc Columns for28

Ductility", Soil Dynamics and Earthquake Engineering,vol. 16, no. 1, pp. 1-8.29

30


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26

Notations1

sA Tensile rebar area, mm22

'

sA Compressive rebar area, mm

23

b Width of the rectangular section, mm4

c Neutral axis depth, mm5

d Effective depth of the rectangular section, mm6

'd Distance from extreme compression fiber to centroid of compression reinforcement,7mm8

cE Secant modulus of concrete, mm9

sE Modulus of steel, mm10

'

0cf Concrete compressive strength of a standard speciment, MPa11

'

cf Concrete compressive strength, MPa12

tf

Concrete tensile strength, MPa

13

uf

Reinforcement ultimate strength, MPa14

yf

Reinforcement yield strength, MPa15

1, 2g g Limit states16

h Height of the rectangular section, mm17

l A parameter used in defing concrete compressive strength18

QM Bending moment resulted from loads, N-mm19

RM Nominal bending capacity, N-mm20

fp Probability of failure21

Q Load random variable22

R Resistance random variable23

iX Random variables24

MX Model error random variable25

1 4Y to Y Random variables related to concrete properties26

1 ,

1

Equivalent stress block parameters27

Reliability index28

DL Safety factor for dead load29


28/28

27

0c Concrete strain at peak stress, mm/mm1

cu Extreme fiber concrete ultimate strain, mm/mm2

su Rebar steel yield strain, mm/mm

3

t

Extreme tensile rebar strain, mm/mm4

y

Rebar steel yield strain, mm/mm5

Curvature ductility6

Tensile rebar percentage7

c Concrete material resistance reduction factor8

s Steel material resistance reduction factor9

u Ultimate curvature10

y

Yield curvature11

a probabilistic study on the ductility of reinforced concrete sections

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