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Engineering, Technology & Applied Science Research Vol. 12, No. 5, 2022, 9155-9159 9155 
 

www.etasr.com Mohammed & Al-Zaidee: Deflection Reliability Analysis for Composite Steel Bridges 
 

Deflection Reliability Analysis for Composite Steel 

Bridges 
 

Duaa Mohammed 

Department of Civil Engineering 

College of Engineering  

University of Baghdad  

Baghdad, Iraq  

d.rasol1901m@coeng.uobaghdad.edu.iq 

Salah Al-Zaidee 

Department of Civil Engineering 

College of Engineering  

University of Baghdad  

Baghdad, Iraq 

salah.r.al.zaidee@coeng.uobaghdad.edu.iq
 

Received: 18 June 2022 | Revised: 6 July 2022 | Accepted: 17 July 2022 

 

Abstract-Reliability methods offer a very efficient serviceability 

assessment of structures with randomness due to geometry, 

material, and loading. Al-Awsej composite bridge in Diyala-Iraq 

with a span of 33.2m has been studied and its deflection 

reliability index for three lifespans was estimated and compared 

with the reliability target index. The reliability indices of the 

bridge have been evaluated through the First-Order Reliability 

Method (FORM) and Monte Carlo Simulation (MCS) method. 

MCS has adopted Matlab functions to generate pseudo-random 

numbers for the considered parameters, but it requires large 

sample sizes to estimate the small probabilities of failure. That 

leads to the use of the reduction variance methods such as the 

Importance Sampling (IS) method. Four cases of random loading 

were included: dead load and three cases of live loads, i.e. 

uniformly distributed load with knife-edge load, military load, 

and sidewalk load. Some assumptions are needed to assess the 

system behavior, where the bridge is represented as a parallel 

system with uncorrelated and perfect correlated girders. The 

reliability index of the composite bridge in the two cases was 

investigated for lifespans of 1, 10, and 50 years. For the 

uncorrelated case, the system shows the reliability index in the 

range of 5 and 4. In contrast, the correlated case offers a range 

between 4 and 2. With these assumptions, the results show that 

no failure occurs, hence the reliability index of the system is still 

within range of the target. 

Keywords-steel girder bridge; statistical characteristics; FORM; 

MCs; importance sampling; parallel system; reliability 

I. INTRODUCTION  

Structure design is associated with a substantial level of 
uncertainty due to the limited information in the estimation of 
structural parameters. In practice, most structural engineering 
designs are based on deterministic parameters where structural 
performance is determined using a deterministic model for 
simplification and often ignore the variations in material 
properties, structure geometry, and applied loads. In previous 
studies, structural reliability methods that rationally evaluate 
the safety of complex structures or systems with unusual 
designs have been developed [1]. In this study, these methods 
will be used to assess the reliability of a bridge by determining 
the reliability indices of the steel girders considering the bridge 

is a parallel system and comparing it with target reliability that 
ensures that structures meet the specified safety level. Thus, 
reliability is an approach to determine the relationship between 
an element and system reliability [2]. 

II. COURBON'S METHOD OF ANALYSIS 

Courbon's method is one of the earliest rational analyses of 
bridges and is very popular due to its simplicity. The reaction 
factor for individual longitudinal girders is given by [3]:  

�� = �� �1 + 	
���

∑ �
� ��    (1) 
where ��  is the reaction load distribution factor of the girder, P 
the total concentrated live load on the span, N the number of 
longitudinal girders, e the eccentricity of live load to the axis of 
the bridge, and ��  is the distance from the girder to the central 
axis of the bridge. 

III. LOAD APPLICATION 

The applied loads on the composite bridge have been 
determined based on the Iraqi code specifications. The self-
weight of the pavement, deck, sidewalk, supporting girders, 
and bracing have been determined depending on the cross-
section dimensions and material densities. The own weight of 
the handrail has been estimated based on Iraqi standard 
specifications for road bridges [4]. Two cases, namely lane live 
and military load have been considered and applied according 
to the Iraqi standard. For a span of 33.2m, the lane load has a 
value of 28.7kN/m per lane with a Knife-Edge Load (KEL) of 
120kN per lane. The military load consists of 900kN (Class 
100) tracked vehicles.Two tracked vehicles have applied for a 
carriageway width of 9m [4]. 

IV. RELIABILITY ANALYSIS OF THE SYSTEM 

Structural reliability analysis is based on the limit state 
design �(�), where � = (��, ��, . . . , ��) represents the set of 
random variables that have some statistical information. The 
limit �(�) = 0  separates the failure domain (�(�) < 0)  and 
the safety domain (�(�) > 0) [7]. The probability of failure 
can be defined by: 

Corresponding author: Duaa Mohammed



Engineering, Technology & Applied Science Research Vol. 12, No. 5, 2022, 9155-9159 9156 
 

www.etasr.com Mohammed & Al-Zaidee: Deflection Reliability Analysis for Composite Steel Bridges 
 

 ! =  [�(�) ≤ 0]    (2) 
The major source of uncertainty is actions with statistically 

independent maximums each year. The values of the 
probability of failure,  !%  and of the reliability index, '�  for 
each reference period may be estimated using [5]: 

 !% = 1 − (1 −  !) )�     (3) 
and 

'� = −*+�,− !% -    (4) 
where * , and *+�  are the functions of the standard normal 
cumulative distribution. 

The target reliability is a design constraint that ensures that 
structures meet the specified safety level. Table I shows the 
target values of reliability indices for various limit states and 
periods [6]. 

TABLE I.  TARGET RELIABILITY INDICES 

Reference period 

T (years) 

Target reliability index β for 

serviceability 

Target reliability index 

β for moment 

1 2.9 4.7 

10 2.2 4.2 

50 1.5 3.8 

100 1.1 3.6 

200 0.6 3.5 
 

V. RELIABILITY ANALYSIS METHODS 

The evaluation of the failure probability of the structure in a 
closed-form is difficult and almost impossible. As a result, 
various analytical and numerical approaches have been 
developed, e.g. Taylor-series-based approaches, such as the 
First-Order Reliability Method (FORM), and simulation-based 
methods, such as Monte Carlo Simulation (MCS). The FORM 
approximation approaches are efficient for simple issues with 
few random variables, but for more complex problems with 
many random variables, MCS seem to be more reliable [7]. 

A. First-Order Reliability Method 

It is possible to expand the original model into an infinite 
Taylor Series (TS) around the mean values: 

�(�) = �(./) + (� − ./ ) 010/ +
�
� (� − ./ )�

0�1
0/� + ⋯ +�

�! (� − ./ )�
0%1
0/%    (5) 

where the function and derivatives are evaluated at ./ . It is 
common to include only linear terms, assuming that random 
input variables are independent. A function � (�)  of N 
independent random variables can be approximated by linear 
terms of the TS, which are [8]: 

4(5) ≈ �,./) , ./� , … , ./% -    (6) 
and 

89:(5) ≈ ∑ ;/
� 	 <10/
�
���=�     (7) 

The limit state function is: (�) = � − >, �, and > both are 
random variables uncorrelated and assumed to be normally 

distributed. The reliability index  ' is evaluated as a function of 
mean and standard deviations of resistance �  and load >  as 
given by [8]: 

' = ?@+?A
BC@�DCA�

    (8) 

B. Monte Carlo Method 

The failure probability in MCS is the ratio of the number of 
samples in the failure domain to the total number of samples: 

 ! = �E� =
�
� ∑ F[�(�) ≤ 0]��=�     (9) 

where the sampling points �  are generated according to the 
probability density function, G! is the number of sampling 
points such that �(�) ≤ 0, G is the total number of sampling 
points, �(�) is the limit state, F[�(�)] is an indicator function 
taking values of unity when are �(�) ≤ 0 and zero otherwise. 
With increasing N, the accuracy of this estimation improves, 
and more simulations are needed to predict a smaller failure 
probability [9]. The required number of trials G  of MCS is 
approximated by: 

G ≈ (�+�E)(HIJ �×�E)    (10) 
where KL8  is the coefficient of variation of the response 
estimates smaller than 0.1 and  !  is usually between 10+�  to 
10+M . The total number required for that simulation is 
determined by: 

G ≈ (�+�NO�)(N.��×�NO�) ≈ 9900    (11) 
C. Importance Sampling 

The main idea of IS is to distribute the sampling points in 
the most important area so that the failure probability 
evaluation may be completed faster. The failure probability can 
be estimated as (9) is rewritten as follows: 

 ! = �� ∑ F(�� )
!�(�Q)
R�(�Q)

�S=�     (12) 

where the sampling points �S , T = 1, . . . , G  are generated 
according to the distribution  ℎ�  instead of V� . The 
effectiveness of IS is dependent on choosing an adequate ℎ� (�) 
so that the probabilistic sampling in (12) may be prioritized for 
the most important area, resulting in a higher convergence rate. 
Although no overall conclusion has been reached on the best 
choice of ℎ� (�), it has been suggested that the Most Probable 
Point (MPP) and its surroundings can be a suitable selection for 
the most important area [10]. 

VI. SYSTEM RELIABILITY 

The system reliability is affected by component reliability 
and several other parameters, such as the correlation among the 
component resistances and the system type. Structural system 
reliability is determined by considering the system failure 
rather than a single component failure [11]. The series system 
means that the entire system fails. The formula of the failure 
system for the statistically independent elements is given by: 

 ! = 1 − ∏ (1 −  !
 )��=�     (13) 



Engineering, Technology & Applied Science Research Vol. 12, No. 5, 2022, 9155-9159 9157 
 

www.etasr.com Mohammed & Al-Zaidee: Deflection Reliability Analysis for Composite Steel Bridges 
 

where  !
  and  !  are the probability of failure for the element X 
and the system respectively. The equation of the system failure 
for perfectly correlated elements can be defined as: 

 ! = Y9�Z !
 [    (14) 
A parallel system is an overall system that fails after all its 

elements have failed. The probability of failure for the entire 
system when the elements are statistically independent and 
perfectly correlated respectively is defined as [11]: 

 ! = ∏  !
��=�     (15) 
 ! = YX\Z !
 [    (16) 
VII. CASE STUDY 

The Al-Awsej composite bridge is constructed in Awsej 
valley at Diyala town in Iraq. It has 33.2m length and 14m 
width and consists of three longitudinal steel girders connected 
by steel bracings (2L100×10). The I-steel plate girders are non-
symmetrically with an upper flange of 400×30mm and two 
lower flanges of 510×30mm and 490×30mm respectively. The 
floor bridge consists of a concrete deck slab with a thickness of 
250mm and pavement bitumen of 60mm extending above a 
carriageway width of 9m, with a 2.5m sidewalk width and 
handrail of 1m on each side. 

 

(a) 

 

(b) 

 

Fig. 1.  The composite steel bridge. 

VIII. STATISTICAL CHARACTERISTICS OF VARIABLES 

In this article, the applied loads, beam span, cross-section 
dimensions, and modulus of elasticity are considered random 
variables. The statistical characteristics of these variables have 
been collected based on previous studies and are summarized 
in Table II. 

TABLE II.  STATISTICAL CHARACTERISTICS OF VARIABLES 

Uncertainty Mean/Nominal COV 
Distribution 

type 

Weight of the deck slab 1.05 0.10 Normal 

Weight of the girder 1.03 0.08 Normal 

Weight of the pavement 1.00 0.25 Normal 

Weight of the military 1.10 0.18 Lognormal 

Uniform distribution with 

knife-edge live loads 
1.10 0.18 Lognormal 

Sidewalk live loads 1.10 0.18 Lognormal 

Length of the girder 1.00 0.000502 Normal 

Width of the Flange girder 1.00 0.00403 Normal 

Thickness of the flange and 

web 
1.05 0.044 Lognormal 

Depth of the web girder 0.996 0.000486 Normal 

Modulus of elasticity 0.993 0.034 Normal 

 

IX. STATISTICAL PROPERTIES FOR THE DEFLECTION OF 
GIRDERS 

Two methods have been used to analyze the exterior and 
interior girders: First-Order approximation and MCS. 

A. First-Order Approximation Method 

The determination of statistical characteristics of girder 
deflection has been determined due to the uncertainties in the 
moment of inertia F, span ], modulus of elasticity 4, and loads. 
Self-weight and superimposed loads ^]  are considered 
uncertain due to the randomness in the weight of concrete deck 
slab, girders, and pavements. Randomness in the own weight of 
bracing, sidewalk, and handrail has been neglected. Live Load 
(LL) uncertainties have been considered for the cases of 
military, sidewalk, uniform distributed, and knife-edge loads. 
Table III provides the mean and standard deviation of 
deflection. 

TABLE III.  STATISTICAL CHARACTERISTICS OF THE GIRDER 
DEFLECTION - FORM 

Loads 
Nominal 

(mm) 

Mean 

(mm) 

Standard 

deviation (mm) 
COV 

E
x

te
r
io

r
 G

ir
d

e
r
 Self-weight and 

superimposed loads 
70.44 70.72 7.10 0.10 

Military with sidewalk 

live loads 
52.54 56.11 10.48 0.18 

Uniformly distributed 

with knife-edge and 

sidewalk live loads 

61.26 65.45 12.22 0.18 

In
te

r
io

r
 G

ir
d

e
r
 Self-weight and 

superimposed loads 
51.95 51.60 6.50 0.13 

Military and sidewalk live 

loads 
52.54 56.11 10.48 0.18 

Uniformly distributed 

with knife-edge and 

sidewalk live loads 

49.11 52.45 9.80 0.18 
 

B. Monte Carlo Simulation 

The MCS was utilized for deflection of the exterior and 
interior girders. A Matlab code has been used to generate 
pseudo-random sampling. Six samples, each with a size of 
1000, have been generated to include the randomness in the 
moment of inertia F, the span of the girder ], and the modulus 
of elasticity 4.  



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www.etasr.com Mohammed & Al-Zaidee: Deflection Reliability Analysis for Composite Steel Bridges 
 

  

Fig. 2.  Deflection sample for the case of dead and superimposed loads. 

  

Fig. 3.  Deflection sample for the case of military and sidewalk live loads. 

  
Fig. 4.  Deflection sample for the case of uniform lane with knife-edge and 
sidewalk live loads. 

Three samples have been considered for each of the exterior 
and interior girders. For each girder, the samples represent the 
randomness in dead loads, military with sidewalk loads, and 
uniform lane with knife-edge and sidewalk loads respectively. 
Table III shows that each random variable has been generated 

based on preselected statistical parameters and probability 
density function. The sample data are presented in the 
histogram Figures 2-4, and the statistical characteristics for the 
deflection of the girders are shown in Table IV. 

TABLE IV.  STATISTICAL CHARACTERISTICS OF THE GIRDER 
DEFLECTION - MCS 

Loads 
Mean 

(mm) 

Standard 

deviation 

(mm) 

COV 
Distribution 

type  

E
x

te
ri

o
r 

G
ir

d
e
r 

Self-weight and superimposed 

loads 
70.76 5.74 0.08 Normal 

Military and sidewalk live loads 56.59 8.74 0.15 Lognormal 

Uniformly distributed with knife-

edge and sidewalk live loads 
65.37 9.10 0.13 Lognormal 

In
te

ri
o

r 
G

ir
d
e
r Self-weight and superimposed 

loads 
51.99 4.77 0.09 Normal 

Military and sidewalk live loads 56.12 8.21 0.15 Lognormal 

Uniformly distributed with knife-

edge and sidewalk live loads 
52.46 6.71 0.13 Lognormal 

 

X. RELIABILITY ANALYSIS FOR STEEL GIRDER DEFLECTION 

FORM and MCS methods have been adopted to estimate 
the reliability associated with girder deflection. Some 
assumptions have been adopted in the FORM, assuming all 
independent variables have a normal distribution. The mean 
and standard deviation of girders deflection in Table III have 
been used to evaluate the reliability index  '  and the 
probability of failure,  !  by: 

' = ∆`aabcdeaf gdh+i(∆)j∆     (17) 
where ∆kllmnopl� qo�  represented a threshold limit of 80mm 
that was adopted as a camper in the design of the Al-Awsej 
bridge case study. 

TABLE V.  RELIABILITY INDEX AND PROBABILITY OF FAILURE BY FORM 

Loads 
1 year 10 years 50 years 

rs t rs t rs t 

E
x

te
r
io

r
 

G
ir

d
e
r
 

Self-weight and superimposed loads 0.09 1.31 0.63 −0.34 0.99 −2.47 
Military with sidewalk live loads 0.001 3.09 0.009 2.33 0.048 1.66 

Uniform distributed with knife edge and sidewalk live loads 0.03 1.89 0.26 0.65 0.77 −0.75 

In
te

r
io

r
 

G
ir

d
e
r
 

Self-weight and superimposed loads 6 × 10+M 4.37 6 × 10+v 3.84 0.0003 3.42 
Military and sidewalk live loads 0.001 3.09 0.009 2.33 0.048 1.66 

Uniformly distributed with knife edge and sidewalk live loads 0.0001 3.68 0.001 3.04 0.005 2.52 

TABLE VI.  RELIABILITY INDEX AND PROBABILITY OF FAILURE BY IS METHOD 

Loads ws 1 year 10 years 50 years rs t rs t rs t 

E
x
te

ri
o

r 

G
ir

d
e
r Self-weight and superimposed loads 39 0.039 1.76 0.33 0.44 0.86 −1.09 

Military and sidewalk live loads ≈0 ≈0 / ≈0 / ≈0 / 
Uniformly distributed with knife edge and sidewalk live loads 9 0.009 2.36 0.09 1.36 0.36 0.35 

In
te

ri
o

r 

G
ir

d
e
r Self-weight and superimposed loads ≈0 / / ≈0 / ≈0 / 

Military and sidewalk live loads ≈0 / / ≈0 / ≈0 / 
Uniformly distributed with knife-edge and sidewalk live loads ≈0 / / ≈0 / ≈0 / 

 

Regarding the live load deflection, � has been taken as the 
limit ratio of L/375 according to The American Association of 
State Highway and Transportation Officials (AASHTO) [12]. 
In the MCS method, importance sampling techniques have 

been used to reduce the number of simulations. These methods 
rely on random sampling from random variable distributions to 
obtain the response uncertainty and the numerical estimate 
probability of failure. The assessment of the probability of 



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www.etasr.com Mohammed & Al-Zaidee: Deflection Reliability Analysis for Composite Steel Bridges 
 

failure is based on the samples that exceed the threshold limit 
for girder deflection. The reliability index and probability of 
failure for exterior and interior girders for the three loading 
cases and through a lifetime of 1, 10, and 50 years are 
summarized in Tables V and VI. The results show that the first-
order approximate analytical method has the ability to show the 
values of the probability of small failures, unlike MCS. Also, 
these Tables noted that the probability of failure increases with 
the passage of time thus, the reliability indices of the system 
decrease. 

XI. RELIABILITY ANALYSIS FOR COMPOSITE BRIDGE 
DEFLECTION 

The composite bridge has been simulated as a system of 
elements (girders). The live load was distributed to the girders 

through a flexible medium and the deck slab and bracing were 
represented as the applied load to the girders. 

In this case, the composite bridge can be considered as a 
parallel system. Hence, the reliability has been estimated with 
two instances of correlation: uncorrelated and perfect 
correlated. The composite bridge reliability depended on the 
probability of failure and the reliability index of the steel 
girders deflection. The results of MCS show that the 
probability of failure approaches zero, so the reliability of the 
parallel system was estimated based on the outcome of FORM 
(Table VII), which showed that the probability of failure 
increases with the passage of time, thus the reliability indices 
of the system decrease. 

TABLE VII.  RELIABILITY INDEX AND PROBABILITY OF FAILURE FOR PARALLEL BRIDGE DEFLECTION 

Loads 
One year Ten year fifty year 

rs t rs t rs t 
Uncorrelated, xyz=0 

Self-weight and superimposed loads 5.42 × 10+~ 5.31 5.42 × 10+� 4.88 2.70 × 10+M 4.55 
Military and sidewalk live loads 0.1 × 10+~ 5.99 0.1 × 10+� 5.61 0.05 × 10+M 5.33 

Uniformly distributed with knife-edge and sidewalk live loads 8.41 × 10+~ 5.23 8.41 × 10+� 4.79 4.20 × 10+M 4.45 
Perfect correlated, xyz=1 

Self-weight and superimposed loads 6 × 10+M 4.37 6 × 10+v 3.84 0.0003 3.42 
Military and sidewalk live loads 0.001 3.09 0.009 2.33 0.048 1.66 

Uniformly distributed with knife edge and sidewalk live loads 0.0001 3.68 0.001 3.09 0.005 2.52 
 

XII. CONLCUSION 

A reliability analysis of Al-Awsej composite bridge has 
been performed in this paper, considering the randomness in 
geometry, material, and applied loads. The bridge is located in 
Diyala-Iraq and has a span of 33.2m. The main conclusions of 
the current study are: 

• The first-order reliability method has estimated two 
moments (mean and standard deviation) of the girder 
deflection. At the same time, the MCS method determined 
four moments (mean, standard deviation, skewness, and 
peakedness) in addition to the distribution type. 

• The MCS with IS method allowed the generation of 1000 
pseudo-random samples instead of a large number of 
samples to estimate the probability of failure of a girder’s 
deflection by distributing the failure sampling in the most 
crucial area. Hence, the failure probability evaluation was 
completed while saving cost and time. 

• With an increase in the correlation between girders, the 
reliability decreases and the probability of failure increases.  

• The reliability index of the deflection with two cases of 
correlation (uncorrelated and perfect correlated) was greater 
than the target reliability of the deflection. 

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