reliability analysis of flexural members designed as per
TRANSCRIPT
Reliability analysis of flexural members designed as per Indianstandard code
TAWSEEF IQBAL and SAHIL BANSAL*
Department of Civil Engineering, Indian Institute of Technology Delhi, Delhi, India
e-mail: [email protected]
MS received 10 June 2021; revised 2 August 2021; accepted 12 August 2021
Abstract. The present study focuses on the reliability analysis of flexural reinforced concrete sections
designed as per Indian Standard code, and the reliability indices for the limit state of strength and ductility are
estimated. The reliability indices are estimated for different grades of concrete and reinforcement steel, different
sectional dimensions, different percentages of tension and compression reinforcement steel, combinations of
values of dead and imposed loads, and for singly and doubly reinforced flexural sections. The analysis results
show that with regard to the limit state of strength, the level of safety provided by the current design code is
acceptable, however, the provided reliability for the design to remain ductile is relatively low.
Keywords. Reinforced concrete; flexural section; reliability analysis; ductility; strength; probabilistic
approach.
1. Introduction
In any practical situation there are several parameters, such
as loadings, structural parameters, geometric parameters,
operation conditions, etc. which are uncertain. In the
presence of these uncertainties achieving absolute safety is
impossible. In this regard, the principles of probability and
its allied fields of statistics and decision theory offer the
mathematical basis for modelling uncertainty and the
analysis of its effect on engineering design [1]. The prin-
ciples of structural reliability have been developed to
compute the probability of failure, which is the complement
of reliability, as a quantitative measure of structural safety.
Using the principles of structural reliability, the level of
reliability of an existing structure, which is designed as per
the existing structural standards, can be evaluated. It can
also be used for developing a reliability-based design cri-
terion, in the form of code calibration to compute the partial
safety factors for an accepted level of reliability. Several
studies have been conducted in the past focusing on relia-
bility analysis of concrete structures, such as, reliability
analysis at serviceability limit state [2–5], reliability anal-
ysis at ultimate limit state [6–11], reliability analysis for
durability [12–15].
There are two major criteria that need to be satisfied
during the design of a flexural section at the ultimate limit
state: first, the design resistance should be greater than the
design load effects, and second, the section should be
ductile or under-reinforced. The resistance and ductility of
a flexural section depend on many variables that are related
to material properties and sectional dimensions. A lot of
research has already been done on the reliability analysis of
concrete structures considering the uncertainty in the
parameters affecting strength. One such study [16] includes
reliability analysis of reinforced concrete beams, slabs and
columns designed as per Indian Standard IS-456 [17].
However, very few reliability studies [18–20] can be found
in the literature on the reliability analysis considering the
uncertainty in the parameters affecting ductility, such as the
strain at peak stress, ultimate strain, and ultimate stress for
concrete. The present study focuses on the reliability
analysis of singly and doubly reinforced concrete (RC) pure
flexural sections subjected to a combination of dead load
(DL) and imposed load (LL), with regards to limit state of
strength and ductility, and designed as per IS-456.
2. Flexural design as per code
The design philosophy of IS-456 is based on the partial
safety factor format recommended by CEB-FIP Model
Code [21]. The design resistance is related to the nominal
DL, DLN , and nominal LL, LLN , specified by the IS
875:Part 1 [22], by
MuR;D � cDDLN þ cLLLN ; ð1Þwhere cD ¼ 1:5 and cL ¼ 1:5 are partial factors of safety forDL and LL at the ultimate limit state, and MuR;D is the
design ultimate moment of resistance of the section. The*For correspondence
Sådhanå (2021) 46:185 � Indian Academy of Sciences
https://doi.org/10.1007/s12046-021-01715-z Sadhana(0123456789().,-volV)FT3](0123456789().,-volV)
estimation of MuR;D is based on the characteristic stress-
strain relationship for concrete and reinforcement steel as
shown in figure 1, where for concrete fck is the charac-
teristic cube compressive strength, eco is the strain at
peak stress and ecu is the ultimate strain, and for rein-
forcement steel fyk is the characteristic yield strength and
Es is the Young’s modulus. IS-456 defines the charac-
teristic strength as that value of the strength below which
not more than 5% of the test results may be expected to
fall. For the stress-strain relationship for concrete, the
first part of the curve is a parabola, while the second part
is constant. The equivalent in-situ concrete strength dif-
fers from the cube strength. The possible factors
responsible for this deviation include the placing of
concrete, quality of curing, hardening conditions of in-
situ concrete, spatial variability of the compressive
strength, age of concrete, size effects, rate of loading
effects, etc [23]. Assuming that the cube strength is equal
to fck, the in-situ concrete strength is estimated as kfck,where k ¼ 0:67[17]. For reinforcement steel, usually, the
post yield strength for structural steel is neglected and is
replaced by a yield plateau. For mild steel (of Fe250
grade) a bilinear stress-strain relationship is assumed, and
for cold worked bars (of Fe415/Fe500 grade) a strain
offset of 0.2% is used to determine the yield strength.
The partial factors of safety for concrete and steel taken
are cc ¼ 1:50 and cs ¼ 1:15, respectively. The stress-
strain diagram of a typical rectangular RC section of
width b0 and effective depth d0 is shown in figure 2.
Based on the strain in the extreme compression fibre of
concrete and the strain at the centre of the tension
reinforcement, the section is classified as under-rein-
forced (UR), balanced, or over-reinforced (OR). If the
strain at the centre of tension reinforcement exceeds the
yield strain value, and the strain in the extreme
compression fibre of concrete does not exceed the ulti-
mate strain, then the section is classified as an UR sec-
tion. If the strain at the centre of tension reinforcement
does not exceed the yield strain value, and the extreme
compression fibre of concrete exceeds the ultimate strain,
then the section is classified as an OR section. Balanced
section corresponds to the simultaneous yielding of ten-
sion reinforcement and the attainment of ultimate strain
at the extreme compression fibre of concrete. To ensure
an UR section, IS-456 imposes a limit on the percentage
tension reinforcement, pt;lim, by limiting the depth of the
neutral axis (NA), xu. IS-456 specifies the limiting NA
depth ratio xu;lim=d0 as
xu;limd0
¼ 0:0035
0:0055þ fyk�csEs
ð2Þ
According to IS-456, the minimum percentage of tensile
reinforcement, pt;cr, to be provided, that approximately
equals the reinforcement requirement to resist the cracking
moment, is given by
pt;cr ¼ 0:85
fyk� 100 ð3Þ
3. Variability of main uncertain variables
The variables affecting the resistance of a RC flexural
section are related to the material properties, and the sec-
tional dimensions. In this study, the statistical distributions
of the uncertain variables are taken from the literature. A
summary about these uncertain variables is provided in
table 1, and a brief description is provided in the following
sections.
Strain
Str
ess
c0 cu
kfck
(0,0)
Strain
Str
ess
Fe 415 / 500
Fe 250
Es
0.002 0.002+fyk
/Es
fyk
Figure 1. Characteristic stress-strain relationship for concrete in structures (left) and reinforcement steel (right) as per IS-456.
185 Page 2 of 16 Sådhanå (2021) 46:185
3.1 Uncertainty of mechanical propertiesof concrete
For concrete, the mechanical properties that possess
uncertainty include the compressive strength fc, the elastic
modulus Ec, the tensile strength fct, the strain at peak stress
eco, and the ultimate strain ecu.fc in this study is defined in
terms of fck as
fc ¼ kfc fck ð4Þwhere kfc is an independent normal random variable.
Several researchers have studied the variability in the
compressive strength of concrete, and most of them have
suggested that the normal or lognormal distribution repre-
sents this variability adequately. In this study, a normal
distribution is assumed, and the values for bias, defined as
ratio of mean value to nominal value, are adopted from the
study conducted by Ranganathan [16], which are consistent
with the data presented in the literature [27–29]. The
coefficient of variation (cov), which is defined as the ratio
of standard deviation to mean value, remains saturated near
0.10 for all grades of normal concrete [16, 27–29].
Figure 2. Stress-strain diagrams of a typical rectangular section.
Table 1. Statistical parameters of random variables considered in this study.
Uncertain variable Distribution type Bias cov Uncertainty source Nominal values as per IS-456
kfc Normal* 1.46 0.10 M20 Compressive cube strength, fc [16] fck1.21 0.10 M25
1.21 0.10 M30
1.21 0.10 M35
kfct Normal* 1.00 0.10 Tensile strength, fct 0:7ðfckÞ0:50kEc
Normal* 1.01 0.17 Elastic modulus, Ec [24] 5000ðfckÞ0:50keco Log-Normal 1.04 0.17 Strain at peak stress, eco [24] 0.002
kecu Log-Normal 1.00 0.21 Ultimate strain, ecu [20] 0.0035
kfy Normal* 1.10 0.06 Yield strength, fy [16] fykkEs
Constant 1.00 0.00 Young’s modulus,Es 200 GPa
kD Normal* 1.00 0.05 Overall depth, D[25] D0
kb Normal* 1.00 0.05 Width, b [25] b0kd Normal* 1.00 0.05 Effective depth, d[25] d0kAst
Normal* 1.00 0.02 Area of tension steel, Ast [26] Ast;0
kAscNormal* 1.00 0.02 Area of compression steel, Asc [26] Asc;0
ke Normal* 1.03 0.06 Model error [19] -
kDL Normal* 1.05 0.10 Dead load, DL [16] DLNkLL Type 1 extremal 0.62 0.25 Imposed load (office buildings), LL [16] LLN
�Truncated at zero
Sådhanå (2021) 46:185 Page 3 of 16 185
Therefore, a cov value of 0.10 for kfc is adopted. Differentresearchers suggest slightly different estimate of the elastic
modulus of concrete in terms of the characteristic cube
strength [30, 31], and very few researchers have studied the
variation in the tensile strength, strain at peak stress, and
ultimate strain. In this study, these properties are estimated
as follows
Ec ¼ kEc5600ð0:8fcÞ0:48 ð5Þ
fct ¼ kfct0:7ðfcÞ0:50 ð6Þ
eco ¼ kec00:0012ð0:8fcÞ0:182 ð7Þecu ¼ kecu0:0037 ð8Þ
where kEc, kfct , keco and kecu are independent normal random
variables. The statistical properties for these parameters
have been obtained from literature [20, 24].
3.2 Uncertainty of mechanical properties of steel
A few statistical distribution types for yield strength of
reinforcement steel have been proposed such as normal
[29], log-normal [26], and beta distributions [32]. In this
study, normal distribution is adopted. The yield strength is
defined as
fy ¼ kfyfyk ð9Þwhere kfy is an independent normal random variable with
the value of statistical parameters adopted from literature
[16] and provided in table 1. Several studies indicate that
the variation in Es is minimal [26]. Hence, in this study, it is
taken as deterministic with a value equal to 200 GPa.
0b
DOFAN0d
0D
0b
,0stA
,0scA
,0stA
,0scA
Strain
Str
ess
k1kf
c
kfc
Ec
c0 cu
fct
Ets
(0,0)
Strain
Str
ess
fy
fy/E
s
Es
Figure 3. Figure showing the cross-section details of the flexural section (left), stress-strain relationship adopted for concrete (middle)
and reinforcing steel (right).
Figure 4. Simulated moment-curvature curves.0.8 0.9 1 1.1 1.2 1.3 1.4
M
-3
-2
-1
0
1
2
3
Sta
ndar
d n
orm
al v
aria
ble
M20
M25
M30
M35
Figure 5. Normal probability plots of kM for different grades of
concrete and pt ¼ 0:5.
185 Page 4 of 16 Sådhanå (2021) 46:185
3.3 Uncertainty of sectional dimensions
Geometric imperfection in RC elements is caused by
deviations from the specified values of the cross-sectional
shape and dimensions, the position of reinforcing bars,
ties and stirrups, the horizontality and verticality of the
concrete lines, and the alignment of columns and beams.
Most researchers recommend the use of normal distri-
bution to model the statistical variation of dimensions of
structural members [3, 29, 33]. In this study, the overall
depth, the effective depth, the width of the section, and
the area of reinforcement steel have been taken as
uncertain variables. The statistical models for the sec-
tional dimensions are taken from the literature [16]. The
designed dimensions are taken as the mean values and
their variations are assumed to follow independent nor-
mal distribution, given by
D ¼ kDD0 ð10Þb ¼ kbb0 ð11Þd ¼ kdd0 ð12Þ
As ¼ kAsAs0 ð13Þ
where kD, kb, kd and kAsare independent normal random
variables and subscript ‘0’ indicates the nominal values.
4. Resistance modelling
In order to predict the ultimate moment of resistance MuR;P
of the section, moment curvature analysis is performed
using nonlinear material stress-strain relationship and
0 1 2 3 4
pt
0
2
4
6
8
10
12
14
M
0
0.2
0.4
0.6
0.8
1M20
0 1 2 3 4
pt
0
2
4
6
8
10
12
14
0
0.2
0.4
0.6
0.8
1
P(OR)
M25
0 1 2 3 4
pt
0
2
4
6
8
10
12
14
M
0.2
0.4
0.6
0.8
1M30
0 1 2 3 4
pt
0
2
4
6
8
10
12
14
0
0.2
0.4
0.6
0.8
1
P(OR)
M35
,M
,M - ,M
,M + ,MP(OR)pt,crpt,lim
Figure 6. Variation of statistics of kM with respect to pt for different grades of concrete.
Sådhanå (2021) 46:185 Page 5 of 16 185
displacement control algorithm. For this, the flexural sec-
tion is modelled in Opensees [34] as a zero-length fibre
element fixed at one side, and free to rotate and displace in
axial direction on the other side, as shown in figure 3.
Figure 3 also shows the stress-strain relationship adopted
for concrete and reinforcing steel, and the parameters
defining the relationships. The stress–strain relationship
adopted for concrete is the idealized stress–strain relation-
ship by Hisham et al [35]. TMT bars (of Fe415/Fe500
grade), which are widely used at present, exhibit a clear and
distinct yield point [36]. Therefore, for reinforcing steel a
bilinear stress-strain relationship is adopted. It may be
noted that the stress-strain relationships adopted by IS-456
are idealized relationships and the assumed nonlinear
material stress-strain relationships are closer to the exper-
imental results.
4.1 Model uncertainty
The model uncertainty includes the uncertainties arising
due to idealizations and simplifying assumptions made in
the numerical modelling of the problem [37]. The model
uncertainty associated with a particular mathematical
model may be expressed in terms of the probabilistic dis-
tribution of a variable defined by
MuR ¼ keMuR;P ð14Þwhere the estimated statistical parameters of the model
error ke are obtained from literature [19]. Finally, the
probability distributions and statistics for the bias defined as
kM ¼ MuR=MuR;N are determined using a Monte Carlo
simulation (MCS) procedure, using the uncertainty char-
acterization discussed earlier. MuR;N is defined as the
0 1 2 3 4
pt
0
0.2
0.4
0.6
0.8
1
P(O
R)
M20
M25
M30
M35
pt,cr
pt,lim
0 1 2 3 4
pt
0
0.2
0.4
0.6
0.8
1
M
M20
M25
M30
M35
pt,cr
pt,lim
0 1 2 3 4
pt
0
0.05
0.1
0.15
0.2
0.25
0.3
M
M20
M25
M30
M35
pt,cr
pt,lim
0 1 2 3 4
pt
0
0.5
1
1.5
2
M
M20
M25
M30
M35
pt,cr
pt,lim
Figure 7. Figure showing variation of PðORÞ, cM , gM , and dM with respect to pt.
185 Page 6 of 16 Sådhanå (2021) 46:185
nominal moment of resistance obtained by setting the value
of partial factors of safety for materials equal to one.
4.2 Ductility measures
The ratio of strain in tension reinforcement, es, at the failureto the yield strain of steel, ey, can be used to define a limit
state to ensure adequate ductility. Because of the uncer-
tainty in the material properties and sectional dimensions,
there is uncertainty in the ductility measures. The ductility
of the section depends on several variables like the per-
centage of tensile reinforcement, compressive strength of
concrete, strain at peak stress, ultimate strain of concrete,
and yield stress of steel. Among these, percentage of tensile
reinforcement is the most influencing parameter. The per-
centage of tension reinforcement has an inverse relation
with section ductility. Hence, to ensure a certain minimum
level of ductility capacity, the IS-456 imposes limit on the
tensile reinforcement percentage by limiting the depth of
the NA. However, the presence of compression reinforce-
ment enhances the section ductility. Therefore, the worst
case with regards to ductility is a section with maximum
permissible tension reinforcement percentage, and mini-
mum compression reinforcement percentage. Because of
lack of accurate information, the effect of model error on
the ductility measures has been ignored.
5. Load model
In this study, the considered load effects include the
effects of DL and LL. The statistical parameters for the
load components have been taken from the literature
Figure 8. Surface plots showing the variation of mean values of kM with pt and pc, for different grades of concrete and Fe500 grade
reinforcement.
Sådhanå (2021) 46:185 Page 7 of 16 185
[16, 38]. The DL has been modelled as normally dis-
tributed. The LL consists of the sustained imposed load,
and the extraordinary imposed load. The mean duration
of the sustained imposed loads is often assumed to be
eight years, corresponding to the average period between
tenant changes in office building [39]. The present study
focusses on the ultimate limit state, and hence, the
parameters of imposed load corresponding to the lifetime
maximum imposed load have been considered. The
maximum total imposed load is modelled as a Type 1
extremal distribution [16, 29, 38]. The statistical param-
eters of the load variables are also summarized in
table 1.
6. Reliability analysis
For conducting the reliability analysis at the ultimate limit
state, the limit state of strength is written as
gs ¼ MuR � DL� LL ð15Þwhere MuR, DL and LL represents the variables of resis-
tance, dead load, and imposed load, respectively. The
above equation can be simplified by dividing it by the DLN :
g0s ¼MuR;N
DLN
� �MuR
MuR;N
� �� DL
DLN
� �� LLN
DLN
� �LL
LLN
� �
ð16ÞThe design resistance is related to the nominal loads
specified by the IS 875: Part 1 [22], as
MuR;D ¼ cDDLN þ cLLLN ð17ÞThe strength reduction factor, cM , which is the ratio of
design resistance MuR;D, to nominal resistance MuR;N , can
be simplified as
cM ¼ MuR;D
MuR;N¼ DLN
MuR;N
� �cD þ cL
LLNDLN
� �ð18Þ
Figure 9. Surface plots showing variation of PðORÞ, cM , gM , and dM with respect to pt and pc, for M20 grade concrete and Fe500 grade
reinforcement.
185 Page 8 of 16 Sådhanå (2021) 46:185
From the above equations, the limit state in simplified
form can be expressed as
g0s ¼ a1kM � kDL � a3kLL ð19Þwhere
a1 ¼ cD þ cLLLNDLN
� ��cM
and a3 ¼ LLN=DLN are the equation constants, and
kM ¼ MuR=MuR;N , kDL ¼ DL=DLN and kLL ¼ LL=LLN are
the random variables representing the uncertainty in the
resistance, DL and LL, respectively.
Non-ductile or brittle failure in the RC flexural members
is said to take place when the strain in the tension rein-
forcement at the failure does not exceed the yield strain.
The limit state corresponding to non-ductile failure is
defined as
gd ¼ es � eyey
¼ esey� 1 ð20Þ
where es and ey are the random variables representing the
uncertainty in the strain in the tension reinforcement at the
failure and yield strain, respectively.
The measure of reliability is conventionally defined by
the reliability index b, which is related to the probability of
failure, PF as
PF ¼ P g\0ð Þ ¼ Uð�bÞ ð21Þ
where U is the cumulative distribution function of stan-
dardized normal distribution. In this study, the reliability
indices for the limit state of strength have been estimated
using the Advanced first order second moment method
(AFOSM) [40], and the reliability indices for the limit state
for ductility have been evaluated using the MCS. It may be
noted that AFOSM has been used for the estimation of
reliability indices corresponding to the limit state of
strength because the computational effort and time required
decreases exponentially as compared to the MCS, without
much compromise in the analysis results.
7. Analysis and results
In the present study, the reliability analysis of singly and
doubly RC flexural sections has been conducted at limit
state of strength and ductility. For singly reinforced flexural
sections, b has been evaluated for different percentages of
tension reinforcement given by pt ¼ 100Ast;0=ðb0d0Þ, andfor doubly reinforced flexural sections, b has been evalu-
ated for different values of pt and percentage compression
reinforcement given by pc ¼ 100Asc;0=ðb0d0Þ. The sections
were subjected to different ratios of LLN=DLN ratios, i.e.,
from 0.25 to 1.5. The chosen range of values of LLN=DLNreflect the expected load combination for real-life situation.
The probability distributions for kM (defined as MuR
�b0d
20)
is determined next, which is followed by the reliability
analysis.
Table 2. Summary of the analysis results for resistance parameters (averaged values).
Member Concrete grade
Reinforcement grade
Fe415 Fe500
cM gM dM cM gM dM
Singly reinforced 300 9 600 M20 0.84 1.18 0.09 0.84 1.18 0.09
M25 0.84 1.17 0.09 0.84 1.16 0.09
M30 0.84 1.16 0.09 0.84 1.15 0.08
M35 0.84 1.15 0.09 0.84 1.15 0.09
Doubly reinforced 300 9 600 M20 0.86 1.15 0.09 0.84 1.16 0.09
M25 0.86 1.14 0.09 0.84 1.13 0.09
M30 0.86 1.14 0.09 0.84 1.13 0.09
M35 0.86 1.14 0.09 0.84 1.13 0.09
Singly reinforced 230 9 500 M20 0.84 1.19 0.09 0.84 1.18 0.09
M25 0.84 1.16 0.09 0.84 1.16 0.09
M30 0.84 1.16 0.09 0.84 1.16 0.09
M35 0.84 1.15 0.09 0.84 1.15 0.08
Doubly reinforced 230 9 500 M20 0.86 1.16 0.09 0.87 1.17 0.09
M25 0.86 1.15 0.09 0.87 1.16 0.09
M30 0.86 1.15 0.09 0.86 1.16 0.09
M35 0.86 1.15 0.09 0.86 1.16 0.09
Sådhanå (2021) 46:185 Page 9 of 16 185
7.1 Resistance parameters
To begin with, the dimensions of the RC flexural section
are taken as 300 mm 9 600 mm with an effective cover of
50 mm all around. Fe500 grade of steel is assumed. First, a
singly reinforced section is considered. For a singly rein-
forced section, to support the shear stirrups, hanger bars are
used. Thus, two 10 mm hanger bars are provided on the
compression side. The response of the section is studied by
analysing the moment-curvature relationship. Since the
main variables, i.e., the sectional dimensions, and the
material properties are considered uncertain, the response
of the section would be uncertain. Figure 4 depicts simu-
lated moment-curvature curves. Note that the points cor-
responding to the ultimate curvature, i.e., the curvature
corresponding to the crushing of concrete, (represented by
red circles) are much more dispersed as compared to the
points corresponding to yield curvature, i.e., curvature
corresponding to the yielding of tension steel reinforcement
(represented by blue circles).
The probability distribution of the resistance is deter-
mined by performing the MCS. MCS was performed for
different grades of concrete, M20, M25, M30 and M35, and
for different percentages of tension reinforcement varied
from 0 to 4%. The resistance data were fitted with normal
probability distribution. Figure 5 shows the normal proba-
bility plots kM for different grades of concrete, and
pt ¼ 0:5. Here, the normal probability plot is plotted with
standard normal variable (defined as
Z ¼ ðkM � lkM Þ�rkM ¼ U�1ðPÞ) on the vertical axis and
kM on the horizontal axis. It can be observed that the nor-
mal distribution fits the data for all the grades of concrete
quite well. A similar observation was made for other values
Figure 10. Variation of b for strength for singly reinforced flexural sections, with respect to pt and LLN=DLN , for different grades ofconcrete and Fe500 reinforcement.
185 Page 10 of 16 Sådhanå (2021) 46:185
of pt. Thus, the normal probability distribution model is
assumed for kM . Figure 6 shows the variation of statistics of
kM with pt. It shows, for kM , its mean value lkM , mean
value minus one standard deviation value lkM � rkM , andmean value plus one standard deviation value lkM þ rkM ,respectively. Figure 6 also shows pt;cr and pt;lim limits
specified by IS-456.
The probability of over-reinforced sections PðORÞ, hasbeen obtained for various reinforcement percentages, and is
also shown in figure 6. The variation of PðORÞ with ptfollows a sigmoidal curve. It is also observed that the
values of pt at which there is 50% probability of having an
OR section also correspond to the points at which there is a
significant change in the slope of kM ¼ MuR=MuR;N curves.
This also shows that having an OR section is not eco-
nomical as there is no significant increase in the moment
carrying capacity of the section.
Figure 7 shows the variation of strength reduction factor
cM ¼ MuR;D=MuR;N , bias factor of resistance
gM ¼ lMuR
�MuR;N , and cov of resistance dM ¼ rkM=lkM
with respect to pt for different grades of concrete. It can be
observed that for pt;cr\pt\pt;lim, cM decreases linearly
with increase in pt but the decrease is not significant. For
pt [ pt;lim, cM shows nonlinear decrease with increase in pt.gM shows a sharp decrease for pt\pt;cr, for all grades of
concrete. For pt;cr\pt\pt;lim, gM increases linearly with ptbut the increase is not significant. For pt [ pt;lim, gM shows
a non-linear increase with increase in pt. Similarly, dMshows a steep decrease for pt\pt;cr , for all grades of con-crete. For pt;cr\pt\pt;lim, dM remains nearly constant. For
pt [ pt;lim, dM shows a nonlinear increase with increase in
pt. All the plots show that in case of a singly reinforced
section, for all normal grades of concrete, the statistical
properties of various parameters involving the resistance of
Figure 11. Variation of b for strength for doubly reinforced flexural sections, with respect to pt and LLN=DLN , for different grades ofconcrete, Fe500 reinforcement, and pc ¼ 1%.
Sådhanå (2021) 46:185 Page 11 of 16 185
the section remain approximately constant in the range of ptdefined by pt;cr\pt\pt;lim.
For a doubly reinforced sections, a similar analysis was
performed by varying pt and pc. Figure 8 shows the vari-
ation of statistics of kM with pt and pc. It shows the vari-
ation of mean values for different grades of concrete and
Fe500 grade reinforcement. It can be observed that, for a
particular value of pt, the mean increases almost linearly
with increase in pc.PðORÞ has been computed for various combinations of pt
and pc, for M20 grade concrete and Fe500 grade rein-
forcement, and is shown in figure 9. The red planes in these
plots represent the limiting compression reinforcement
percentage p�c , which corresponds to the balanced condi-
tion. It can be observed that, for a particular value of pc, theprobability of over reinforced section, PðORÞ, shows a
sigmoidal variation, with increase in pt. As expected, for aparticular value of pt, PðORÞ decreases with increase in pc.Similar observations are made for other grades of concrete.
Figure 9 also shows the variation of cM , lM , and dM with
respect to pt and pc, for M20 grade of concrete and Fe500
grade reinforcement. These plots show that in case of a
doubly reinforced section, for all normal grades of concrete,
the statistical properties of various parameters involving the
resistance of the section remain approximately constant in
the range defined by pt;cr\pt\pt;lim and pc\p�c . A similar
analysis was done for a 230 mm 9 500 mm singly and
doubly RC flexural section, and with Fe415/ Fe500 grade
reinforcement. Table 2 presents the averaged analysis
results for resistance parameters.
7.2 Reliability indices for strength
The reliability indices for limit state of strength are esti-
mated for UR flexural sections defined as per IS-456. Fig-
ure 10 shows the variation of b for singly reinforced
flexural sections with respect to pt and LLN=DLN , for dif-ferent grades of concrete and Fe500 reinforcement. Simi-
larly, figure 11 show the variation of b for doubly
reinforced flexural sections with respect to pt and
LLN=DLN , for different grades of concrete, Fe500 rein-
forcement, and 1% pc. Table 3 summarizes the results of
reliability indices for both singly and doubly reinforced
Table 3. Range of b for strength for RC flexural sections.
Load Combination Member Concrete grade
Reinforcement grade
Remark
Fe 415 Fe 500
Range of b Average b Range of b Average b
DLN þ LLN Singly 300 9 600 M20 4.88–5.21 5.07 4.73–5.38 5.06 Range of LLN=DLN0.25 to 1.5M25 4.71–5.32 5.00 4.69–5.29 4.99
M30 4.69–5.30 4.99 4.66–5.26 4.96
M35 4.67–5.29 4.96 4.65–5.25 4.94
Doubly 300 9 600 M20 4.38–6.16 5.10 4.34–6.02 5.08
M25 4.36–5.87 5.11 4.36–5.83 5.10
M30 4.38–5.90 5.15 4.39–5.97 5.11
M35 4.43–5.67 5.15 4.39–5.86 5.11
Singly 230 9 500 M20 4.77–5.43 5.09 4.75–5.45 5.08
M25 4.69–5.27 4.98 4.68–5.31 4.98
M30 4.71–5.23 4.97 4.68–5.32 4.97
M35 4.75–5.59 4.94 4.64–5.27 4.94
Doubly 230 9 500 M20 4.36–6.38 5.18 4.32–6.52 5.19
M25 4.44–5.94 5.20 4.31–6.16 5.20
M30 4.45–6.12 5.21 4.41–6.09 5.20
M35 4.48–6.02 5.23 4.43–6.24 5.23
0 1 2 3 4 5 6 7 8 9 10
Normalized strain (s/
y)
0
0.02
0.04
0.06
0.08
0.1
0.12
Rel
ativ
efr
equen
cy
Figure 12. Typical distribution of normalized strain es�ey for
singly reinforced flexural section provided with pt;lim.
185 Page 12 of 16 Sådhanå (2021) 46:185
20 25 30 35
fck
(MPa)
0
1
2
3
D
20 25 30 35
fck
(MPa)
0
0.2
0.4
0.6
0.8
1
D
20 25 30 350
0.5
1
1.5
2
2.5
3
300mm x 600mm with Fe415
300mm x 600mm with Fe500
230mm x 500mm with Fe415
230mm x 500mm with Fe500
fck
(MPa)
Figure 13. Variation of mean value of es�ey (left), cov of es
�ey (middle), and reliability index for ductility (right) with respect to the
different grades of concrete and reinforcements, for singly reinforced sections provided with pt;lim.
Figure 14. Surface plots showing variation of ductility-based reliability index with respect to the pt and pc for different grades of
concrete and Fe415 reinforcement.
Sådhanå (2021) 46:185 Page 13 of 16 185
flexural sections. For singly reinforced sections, the average
value of varies from 4.94 to 5.09, and for doubly reinforced
section, the average value varies from 5.08 to 5.23. In a
previous study [16], the reliability indices for beams in
flexure with M20 and M25 grades of concrete and Fe415
grade of reinforcement steel varies from 4.3 to 5.5. It is
seen that the reliability indices estimated in this study and
the previous study are comparable. A slight difference in
the range of estimates of reliability indices is most likely
because of the use of the most recent values of the statis-
tical parameters defining the random variables, and con-
sideration of uncertainty in the parameters affecting
ductility, such as the strain at peak stress, ultimate strain,
and ultimate stress for concrete. Furthermore, the available
literature recommends the target reliability indices for limit
state of strength involving dead and live loads equal to 3.0
[16, 33, 41] or 3.5 [42]. The results show that the current
design code is adequately calibrated for the limit state of
strength.
7.3 Reliability indices for ductility
To determine the reliability index for ductility, the statistics
of the normalized strain es�ey have been studied. Figure 12
shows a typical histogram of es�ey for M25 grade concrete
and Fe500 steel reinforcement for a singly reinforced sec-
tion provided with pt;lim. It can be observed that the his-
togram has a clear positive skewness. Therefore, a
lognormal distribution is used to describe the probability
distribution of es�ey. The area under the lognormal
distribution corresponding to the region es�ey\1 (i.e., the
area hatched in black) gives the probability of non-ductile
failure, i.e., the probability of having an over reinforced
section PðORÞ.Figure 13 presents the statistics, i.e., the mean values and
cov of es�ey for singly reinforced sections provided with
pt;lim, for different grades of concrete and steel reinforce-
ment. It can be observed that the ductility measure exhibits
a large variability with cov values about 0.50. It may be
noted that the failure to provide ductile design for an RC
flexural section is an undesired phenomenon as it does not
provide adequate warning before failure of the sec-
tion. Figure 13 also shows the variation of reliability index
for ductility for singly reinforced sections provided with
pt;lim, for different grades of concrete and steel reinforce-
ment. Figure 14 shows the variation of reliability index for
a doubly reinforced 300 mm 9 600 mm cross-section with
respect to the pt and pc for different grades of concrete andFe415 reinforcement. It can be observed that the reliability
index decreases with increase in the percentage tension
reinforcement, and increases with increase in the com-
pression reinforcement for a given percentage tension
reinforcement. It is also observed that with increase in the
grade of steel, the ductility of the section reduces. This is
because, according to Eqn. (1), IS-456 specifies the limiting
neutral axis depth ratio, xu;lim=d0, which is inversely related
to the yield stress of reinforcement steel. Thus, the pt;limsuggested by IS-456 decreases with increase in the yield
stress of reinforcement steel. On the other hand, the use of
steel reinforcement with higher yield stress results in lower
values of normalized strain, as indicated by the mean values
Table 4. Summary of the parameters related to ductility (averages values).
Member Concrete grade
Reinforcement grade
Fe415 Fe500
gD dD bD gD dD bD
Singly reinforced 300 9 600 M20 2.03 0.59 1.96 1.72 0.53 1.87
M25 1.91 0.58 1.95 1.80 0.49 1.85
M30 1.91 0.53 1.97 1.92 0.47 1.89
M35 1.95 0.51 1.89 1.91 0.47 1.78
Singly reinforced 230 9 500 M20 1.87 0.56 1.79 1.95 0.52 1.71
M25 1.89 0.53 1.75 1.78 0.48 1.65
M30 2.01 0.51 1.91 1.79 0.45 1.77
M35 2.07 0.48 1.88 1.77 0.44 1.77
Doubly reinforced 300 9 600 M20 1.50 0.63 1.51 1.49 0.57 1.50
M25 1.44 0.60 1.67 1.42 0.55 1.64
M30 1.42 0.54 1.80 1.43 0.54 1.75
M35 1.40 0.53 1.87 1.38 0.53 1.82
Doubly reinforced 230 9 500 M20 1.50 0.61 1.50 1.52 0.55 1.49
M25 1.43 0.58 1.54 1.33 0.54 1.51
M30 1.42 0.53 1.74 1.29 0.49 1.69
M35 1.41 0.52 1.83 1.32 0.48 1.79
185 Page 14 of 16 Sådhanå (2021) 46:185
of es�ey, i.e., gD. Using steel reinforcement with higher
yield stress leads to reduction of both pt;lim and gD.Although the reduction in tension reinforcement percentage
tends to increase the ductility-based reliability index, the
lower values of gD, tend to decrease the ductility-based
reliability index. Therefore, the ductility-based reliability
indices depend on these two contradicting effects on
employing steel reinforcement with higher yield stress.
From table 4, it can be observed that with increase in the
yield strength of reinforcement steel, the ductility-based
reliability index decreases, however, the decrease is not
large enough to make a considerable impact.
Table 4 presents average statistics of parameters related
to ductility measure and reliability indices for ductility
evaluated for the balanced sections. For singly reinforced
sections, the average value of bD varies from 1.65 to 1.96,
and for doubly reinforced section, the average value of bDvaries from 1.49 to 1.87. The available literature recom-
mends the target reliability indices for limit state of duc-
tility involving dead and live loads equal to 2.3 [43]. The
results suggest that although the current design code is
adequately calibrated for the limit state of strength, the
provisions regarding the limit state of ductility need further
modification.
8. Conclusions
The present study focuses on the reliability analysis of a
singly and doubly reinforced concrete pure flexural section
subjected to a combination of dead load and imposed load,
with regards to limit state of strength and ductility, and
designed as per IS-456. According to the results of the
reliability analyses for strength and ductility, the overall
cov for the flexural strength is about 0.09, while for duc-
tility measure as a random variable the cov is about 0.50.
The reliability indices for the limit state of strength are
about 5.00, while for limit state of ductility are about 1.90.
From the available literature, it was found that the target
reliability indices for limit state of strength involving dead
and live loads are generally considered as 3.0 or 3.5, and for
limit state of ductility, the target reliability index is con-
sidered as 2.3. The current design code is calibrated for the
limit state of strength, and the results of this study confirms
adequacy of the currently in-place calibration. However,
the reliability indices for limit state of ductility are con-
siderably lower than those of the limit state of strength
which demands for definition of appropriate target safety
levels when dealing with the ductility as a limit state.
In this study, the reliability indices for the limit state of
strength have been calculated using AFOSM method.
Although AFOSM provides good estimates of the reliabil-
ity indices, Second Order Reliability Methods (SORM) can
also be used to provide more accurate values. Similarly, the
reliability indices for the limit state of ductility have been
calculated using MCS method for 1000 number of simu-
lations. More accurate results can be obtained by increasing
the number of simulations to 10000 or 100000. However,
this would increase the computational cost manifolds.
In future, load combination with wind and earthquake
loads will be considered.
Acknowledgement
The authors would like to gratefully acknowledge the New
Faculty Seed Grant received from the Indian Institute of
Technology Delhi, India, and Start-up Research Grant
received from Science and Engineering Research Board,
Government of India.
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