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
166PUBLICATIONS 692CITATIONS
SEE PROFILE
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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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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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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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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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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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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
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
ReliabilityIndex
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
ReliabilityIndex
Concrete Compressive Strength (MPa)
ACI 318-11 AS 3600-09
NZS 3101-06 CSA A23.3-04
EC2-04
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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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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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(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
Concrete Compressive Strength (MPa)
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
Concrete Compressive Strength (MPa)
ACI 318-11 AS 3600-09
NZS 3101-06 CSA A23.3-04
EC2-04
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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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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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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
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Park, R. and Ruitong, D. (1988) "Ductility of Doubly Reinforced Concrete Beam Sections",23
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30
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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
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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