reliability analysis of flexural members designed as per

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 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 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 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 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 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 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 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


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


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 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 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 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 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




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 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


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


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


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


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 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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