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Barambu USEP: Journal of Research Information in Civil Engineering, Vol.14, No.1, 2017
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1275
Reliability-Based Code Calibration for Load and
Safety Factors for the Design of a Simply Supported
Steel Beam
A.U. Barambu1, O.A.U. Uche2, and M.T. Abdulwahab3
1Department of Civil Engineering Bayero University Kano,
Kano, Nigeria, habubarambu@yahoo.com
2Department of Civil Engineering Bayero University Kano,
Kano, Nigeria, okoauche@yahoo.co.uk
3Department of Civil Engineering Bayero University Kano,
Kano, Nigeria, amtaytechnology@yahoo.com
Abstract
The reliability-based calibration of safety factors for the design of a simply
supported steel beam, based on BS5950 (2000) was presented in this research
work. The calibration was undertaken using a specialized computer program
in Microsoft Excel environment developed by the Joint Committee for
Structural Safety (JCSS) CODE-CAL 2001. The design variables considered
were modeled using the CODE-CAL software, and the safety factors for the
material, dead and live load were calibrated by varying the safety index.
From the results obtained, mathematical prediction models were developed
using least square regression analysis for bending, shear and deflection
modes of failure considered in the study. The results showed that the safety
factors for material, dead and live load are not unique, but they are influenced
by safety index and it was also shown that the safety factors for material,
dead and live load varies from 0.61 to 1.15, 1.44 to 1.91 and 1.40 to 1.65
respectively for both bending and shear mode of failure. The deflection mode
of failure results showed that the safety factors for material, dead and live
load varies from 1.08 to 1.56, 1.10 to 1.17 and 0.83 to 1.25 respectively for
target safety index (βt) of 2.0 to 4.5. The mathematical prediction models
developed for both bending and shear modes of failure are the same.
Therefore, it was recommended that the mathematical prediction models
developed in this study for bending and deflection modes of failure could be
used when designing a simply supported steel beam to BS 5950 (2000).
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Keywords
Reliability, code calibration, load factor, safety factor, design, steel,
beam
1. Introduction
According to Faber and Sorensen (2003) Structural design codes are
established for the purpose of providing a simple, safe and economically
efficient basis for the design of ordinary structures under normal loading,
operational and environmental conditions. They reiterated that Design codes
not only greatly facilitate the daily work of structural engineers but also
ensure certain standardization within the structural engineering profession
which in the end enhances an optimal use of the resources of society for the
benefit of the individual.
Structural design codes take basis in design equations/calculations from
which the reliability verification of a given design may be easily performed
by a simple comparison of resistances and loads and/or load effects. Due to
the fact that loads and resistances are subject to uncertainties, design values
for resistances and load effects are introduced in the design equations to
ensure that the design is associated with an adequate level of reliability
(Andre and Antonio 2010).
Structural reliability methods therefore involves choosing a rationale safety
formats for the design codes considering the design equations, characteristics
values of material strength and the uncertainties inherent within. This format
ensures that design codes are homogenous and independent of the choice of
material and the prevailing loadings, operational and environmental
conditions. This also ensures that the desired level of reliability or target
reliability commonly referred to as code calibration is achieved (Faber and
Sorensen 2003).
This study aims at reliability-based code calibration for load and safety
factors for the design of a simply supported steel beam in accordance with
BS5950 (2000).
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2. Theoretical Conception
The uncertainties in structural parameters such as material properties,
external loads, geometry, etc., have caused serious attentions to reliability in
structural design and analysis (Ghasemi and Yousefi, 2011). Thus, reliability
theory, as a branch of theory of probability, provides a firm framework which
can introduce a proper factor of safety when required (Reynolds, 2008; ISSC,
2006; Faber and Sorensen, 2003). Thus, any system made of a satisfied
reliability index, may be referred to as safe.
Uche and Afolayan, (2008) described reliability as the probability or
likelihood of structure performing its purpose adequately for a period of time
intended under the operating conditions encountered. The problem associated
with the traditional method of ensuring safety can be resolved by rendering
broad, general concepts, such as uncertainties and risks, into precise
mathematical terms that can be operated upon consistently. This approach
essentially forms the basis of reliability-based design. Uncertain engineering
quantities (e.g. loads, capacities) are modelled by random variables, while
design risk is quantified by the probability of failure.
Ditlevsen and Madsen (2005) considered structural reliability as a method
that attempt to treat rationally, the various sources of uncertainties.
According to Tsompanakis and Papadrakakis (2000) structural reliability
analysis is a tool that assists the design engineer to take into account all
possible uncertainties during the design and construction phases and the
lifetime of a structure in order to calculate its probability of failure.
Structural element will be considered to have failed if its Resistance (R) is
less than the Stress Resultant (S) acting on it. Once the uncertainties in the
Resistance (R) and the Stress Resultant (S) have been modeled as random
variables, the probability of failure (
) can be evaluated as follows
(Melchers, 1999).
(1)
where
= Probability Density Function of S and = Probability
Distribution Function of R
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2.1 Target Reliability
According to the two steps procedure in Figure 1, the designer should first
establish target reliability. The JCSS model Code-Cal gives a table presented
in Table 1. the relative cost to increase safety.
Table 1: Target Reliabilities for some Selected Countries
3.1 3.5 4.0
4.5 5.0
Argentina
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
Canada
X
X
X
X
X
X
X
X
X
X
X
X
X
X
China
X
X
X
X
X
X
X
X
X
X
X
Denmark
X
X
X
X
X
X
X
X
X
X
X
Estonia
X
Germany
X
Holland
X
X
X
X
X
X
South
Africa
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
Spain
X
Sweden
X
X
X
X
X
X
X
X
X
X
X
X
X
UK
X
USA
X
(Source: Bartlett et al. 2003; Andre and Antonio, 2010)
Figure 1: Two step procedure for code calibration
Code calibration: two step procedure
target reliability t
- values
calibration optimization
Standardised ’s Min wj (j - t)2
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The point is that in economic optimization the target depend on the
consequences of failure as well If the cost, to increase the safety are small
one should do so and vice versa. Of course this may be impracticable in some
cases but that is another matter. In many countries, present target reliability
has been found by looking at an existing code. The idea is that application of
new approach should not lead to large difference from the existing code.
2.2 First Order Reliability Method
First order reliability method (FORM) is a convenient tool for assessing the
Reliability of structural elements. It also provides a means for calculating the
partial safety factor. FORM uses combination of analytical and approximate
methods and comprises the stages. Firstly, independent of whether each
parameter has been defined as Normal, Log-Normal, or Gumbel distribution;
all variables are first transformed into equivalent normal space with zero
mean and unit variance. The original limit state surface is then mapped onto
the new limit state surface. Secondly, the shortest distance between the origin
and the limit state surface, termed the reliability index β, is evaluated; this is
known as the design point, or point of maximum likelihood, and gives the
most likely combination of basic variables to cause failure. Finally, the
probability associated with this point is then calculated. FORM can be easily
extended to non-linear limit states and has a reasonable balance ease of use
and accuracy.
First order reliability method (FORM) is one of most common basic
techniques and is applicable to all probabilistic problems. It is usually
preferred method, because it does not depend on the number of simulations to
be carried out.
In reliability-based concept, the performance function of a structural system
according to a specified mission is given by:
M = performance criterion – given criterion limit
= g(X1,X2, . . . . . ,Xn) (2)
Where the Xi (i = 1, . . . . .,n) are the n basic random variables (input
parameters), with g( ) being the functional relationship between the random
variables and the failure of the system. The performance function can be
defined such that the limit state of failure surface, is given by M = 0. The
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failure event is defined as the space where M > 0. Thus a probability of
failure can be evaluated by the following integral.
Pf = ∫∫ . . . .∫ fx(x1, . . . . ,xn)dxi . . . dxn (3)
Where fx is the joint density function of x1, x2, . . .,xn and the integration is
performed over the region where M < 0. Because each of the basic random
variables has a unique distribution and they interact, the integral cannot be
easily evaluated. Use is made of approximate method.
Traditionally, the concerns of researchers were on the evaluation of structural
reliability of steel and concrete structures and/or components. (Rosowsky et
al, 2002; Chinwedu, 2002; Afolayan and Abubakar, 2003; Jinquan and
Baidurya, 2007; Afolayan and Opeyemi, 2008; Uche and Ahmed, 2013).
3. Methods
Deterministic design was carried out on a simply supported symmetrical I-
Beam subjected to a dead load of 3.53kN/m and an imposed load of 4.0kN/m
in accordance with BS5950 (2000), and 457x152x67UB to satisfy the code
design criteria for bending, shear and deflection. The free-body diagram of
the beam as well as beam cross-section are presented in Figures 2 and 3
respectively.
Figure 3.1 Free body diagram of simply supported steel beam
(b)
(a)
Figure 2: Free body diagram simply supported steel beam
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Figure 3: Cross-section of the simply supported steel beam
The reliability analysis was then conducted using FORM5 (Gollwitzer et al.,
1988), which involves certain approximate interactive calculation procedures
to obtain an approximation to the failure probability of structure or structural
system. It generally requires an idealization of failure domain and often
associated with a simplified representation of the joint probability
distribution of the variables. The measures of reliability are based on
reliability index.
Reliability-based analysis and calibration were made, considering ultimate
limit state only. For the simply supported steel beam under study, the
identified modes of failures are bending, shear and deflection. The safety
margin for each mode can be expressed as equation 4
Z= R- S (4)
Where R is the resistance model and S is the load action.
For the reliability analysis, resistances (R) are obtained from the BS5950
steel design code, while load (S) values are obtained from structural analysis.
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The limit state function for bending, shear and deflection failure mode were
then developed and given by equations 5, 6 and 7.
Bending mode of failure
(5)
Shear mode of failure
(6)
Deflection mode of failure
(7)
Where is the dead load, is the imposed load,, is the ultimate
moment, is the applied moment, n is the design yield strength, is the
section modulus, is the length of the beam.
4. Results and Discussions
The reliability analysis ad calibration of safety factors for materials, dead and
live loads were performed using FORM5 and Code-Cal (2001) respectively
for bending, shear and deflection modes of failure and the results were
presented in Tables 2, 3 and 4 respectively.
Table 2: Calibration of load safety factors for bending mode of failure
S/N
1
2.0
10-2
0.63
1.85
1.40
2
2.5
10-3
0.66
1.89
1.55
3
3.0
10-4
0.76
1.77
1.58
4
3.8
10-5
0.97
1.54
1.58
5
4.0
10-5
1.02
1.50
1.59
6
4.5
10-6
1.15
1.44
1.65
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Where is target safety index,
is the probability of failure, is safety
factor for material, is safety factor for dead load and is safety factor
for live load.
Table 3: Calibration of load safety factors for shear mode of failure
S/N
1
2.0
10-2
0.61
1.91
1.44
2
2.5
10-3
0.66
1.89
1.55
3
3.0
10-4
0.76
1.77
1.58
4
3.8
10-5
0.97
1.54
1.58
5
4.0
10-5
1.02
1.50
1.59
From Tables 2 and 3, safety factors value of 0.97, 1.54 and 1.58 for
materials, dead and live loads respectively for both bending and shear mode
of failures were recorded at a target safety index of 3.8 as recommended by
the Joint Committee of Structural Safety Code (JCSS 2001) for ultimate limit
state design. Comparing the obtained safety factors value for materials, dead
and live loads with those prescribed by BS 5950 (2000)
, a difference of -3%, +14% and -2% were observed
respectively.
Table 4: Calibration of load safety factors for deflection mode of failure
S/N
1
2.0
10-2
1.08
1.10
0.83
2
2.5
10-3
1.14
1.11
0.92
3
3.0
10-4
1.22
1.13
1.0
4
3.8
10-5
1.32
1.17
1.19
5
4.0
10-5
1.4
1.14
1.19
6
4.5
10-6
1.56
1.11
1.25
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It was also recorded from Table 4 that the safety factors value for materials,
dead and live loads for deflection mode of failure are approximately 1.08,
1.10 and 0.83 respectively corresponding to a target safety index of 1.8
recommended by the JCSS (2001) for serviceability limit state design.
Comparing the obtained safety factors value for materials, dead and live
loads with the safety factors recommended by BS 5950 (2000)
, a difference of +8%, +10% and -17% were
observed respectively.
4.1 Variation of Safety Index with Importance Factor
Importance factor is the relative contribution of dead and live loads in
calibration and is designated as α. It was multiplied by dead load to know the
contribution of dead load and live load to obtain the contribution of live load
(1- α). It was discovered that α equals to one (1) when the contribution of
dead load is full and that of live load is zero (0), while it becomes zero (0)
when the contribution of dead load equals to zero (0) and that of live load is
full.
It was also observed that the dead load contribution increases as live load
contribution decreases as α approaches one (1), while the reverse is the case
as α tends to zero (0). The two load contributions become equal when α is
0.5. The variation of safety index with importance factor was computed for
target βt =2.0, 3.0 and 4.0 for the bending mode of failure as shown in Figure
4, 5 and 6 respectively.
Figure 4: Safety index against importance factor
with βt=2 (Bending)
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Figure 5: Safety index against importance factor
with βt =3.0 (Bending)
From Figures 4 and 5, it was recorded that the target safety index was only
achieved when the importance factor is between 0.5 and 0.95. The maximum
achieved safety index was 2.2 for βt =2.0 and 3.3 for βt =3.0 corresponding to
0.8 and 0.9 value of importance factor respectively.
Figure 6: Safety index against importance factor
with βt =4.0 (Bending)
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Figure 6 shows that the beam meets the target safety index between 0.53 and
0.95 value of the importance factor for bending. Hence, the maximum safety
index achieved was 4.5 at βt =4.0 which correspond to 0.9 value of
importance factor.
The relationship between the safety index and the importance factor for shear
mode of failure at βt =2.0, 3.0 and 4.0 were presented in Figures 7, 8 and 9
accordingly.
Figure 7: Safety index against importance factor
with βt =2 (Shear)
Figure 8: Safety index against importance factor
with βt =3.0 (Shear)
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Figure 9: Safety index against importance factor
with βt =4.0 (Shear)
It was observed from Figures 7, 8 and 9 that the target safety index can only
be achieved at a range of 0.5 and 0.95 value of the importance factor. The
maximum safety index was 2.2, 3.3 and 4.5 for βt =2.0, 3.0 and 4.0
respectively which correspond to importance factor value of 0.8 and 0.9.
Figure 10: Safety index against importance factor
with βt =2.0 (Deflection)
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Figure 11: Safety index against importance factor
with βt =3.0 (Deflection)
Figure 12: Safety index against importance factor
with βt =4.0 (Deflection)
It was also discovered that the target safety index achieved was at a range of
0.6 and 0.95 value of the importance factor as shown in Figures 10, 11 and
12 for the deflection mode of failure. Therefore, a maximum achieved safety
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index was 2.29, 3.39 and 4.49 for βt =2.0, 3.0 and 4.0 respectively which
correspond to importance factor value of 0.9.
5. Conclusions
In conclusion, the reliability-assessment of structures requires both safety and
economical design, and based on the reliability-based calibration results in
this study, the following conclusions were made:
(i) The safety factors for material, dead and live loads are 0.97, 1.54 and
1.58 respectively at target βt =3.8 for both bending and shear modes of
failure which fulfill JCSS Code Calibration (2001) recommendation for
ultimate limit state design.
(ii) In deflection mode of failure, 1.08, 1.10 and 0.83 were obtained as the
safety factors for material, dead and live loads respectively which is in
satisfaction with the JCSS Code Calibration (2001) recommendation for
serviceability limit state design.
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