Content uploaded by Muhammad Tayyeb Abdulwahab

Author content

All content in this area was uploaded by Muhammad Tayyeb Abdulwahab on Mar 01, 2017

Content may be subject to copyright.

Barambu USEP: Journal of Research Information in Civil Engineering, Vol.14, No.1, 2017

et al.

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).

Barambu USEP: Journal of Research Information in Civil Engineering, Vol.14, No.1, 2017

et al.

1276

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).

Barambu USEP: Journal of Research Information in Civil Engineering, Vol.14, No.1, 2017

et al.

1277

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

Barambu USEP: Journal of Research Information in Civil Engineering, Vol.14, No.1, 2017

et al.

1278

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

Barambu USEP: Journal of Research Information in Civil Engineering, Vol.14, No.1, 2017

et al.

1279

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

Barambu USEP: Journal of Research Information in Civil Engineering, Vol.14, No.1, 2017

et al.

1280

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

Barambu USEP: Journal of Research Information in Civil Engineering, Vol.14, No.1, 2017

et al.

1281

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.

Barambu USEP: Journal of Research Information in Civil Engineering, Vol.14, No.1, 2017

et al.

1282

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

Barambu USEP: Journal of Research Information in Civil Engineering, Vol.14, No.1, 2017

et al.

1283

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

Barambu USEP: Journal of Research Information in Civil Engineering, Vol.14, No.1, 2017

et al.

1284

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)

Barambu USEP: Journal of Research Information in Civil Engineering, Vol.14, No.1, 2017

et al.

1285

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)

Barambu USEP: Journal of Research Information in Civil Engineering, Vol.14, No.1, 2017

et al.

1286

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)

Barambu USEP: Journal of Research Information in Civil Engineering, Vol.14, No.1, 2017

et al.

1287

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)

Barambu USEP: Journal of Research Information in Civil Engineering, Vol.14, No.1, 2017

et al.

1288

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

Barambu USEP: Journal of Research Information in Civil Engineering, Vol.14, No.1, 2017

et al.

1289

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.

References

Afolayan, J. O. and Abubakar, I. (2003). Reliability-Based Design Program

for Reinforced Concrete One-Way Slabs in Accordance with BS 8110

(1985). Nigeria Journal of Engineering Ahmadu Bello University, Zaria.

11(2), 1-5.

Afolayan, J.O. and Opeyemi, D.A. (2008). Reliability Analysis of Static Pile

Capacity for Concrete in Cohesive and Cohensionless Soils. Medwell Online

Research Journal of Applied Sciences, 3(5), pp. 407 – 4011.

Andre, T. B. and Antonio, C. S. (2010). A First Attempt Towards Reliability-

Based Calibration of Brazilian Design Codes. J. of the Brazilian Society of

Mechanical Scice & Engineerin, 31(2). pp. 119-127.

Bartlett, F. M., Hong, H. P. and Zhuo (2003). Load Factor Calibration for the

Proposed 2005 Edition Of the National Building Code of Canada: Statistics

of Loads and Load Effects. Canadian Journal of Civil Engineering, 429-439.

Barambu USEP: Journal of Research Information in Civil Engineering, Vol.14, No.1, 2017

et al.

1290

British Standard 5950 (2000). Structural Use of Steelwork in Building. Part

2; Specification for Materials, Fabrication and Erection-Rolled and Welded

Sections.

Chinwedu, U. P. (2002). Efficient Use of Steel in Truss System. MSc. Thesis,

Department of Civil Engineering, Ahmadu Bello University Zaria.

Ditlevsen, O. and Madsen, H. O. (2005). Structural Reliability Methods. John

Wiley & Sons, Chichester.

Faber, M. H. and Sorensen, J. D. (2003). Reliability-based Code Calibration.

Joint Committee on Structural Safety.

Ghasemi, M. R. and Yousefi, M. (2011). Reliability-Based Optimization of

Steel Framed Structures Using Modified Genetic Algorithm. Asian Journal of

Civil Engineering (Buildings and Housing). 12(4), 449-475.

Gollwitzer, S; Abdo,T and Rackwitz, R.(1988). “First Order Reliability

Method (FORM) Manual”, RCP, GMBH, Munich, West Germany

ISSC (2006). Reliability-Based Structural Design and Code Development:

Special Task Committee. International Ship and Offshore Structures

Congress. Southampton, UK.

JCSS (2001). Joint Committee on Structural Safety-Probabilistic Model

Code. http://www.jcss.ethz.ch

Jinquan, Z. and Baidurya, B. (2007). A System Reliability-Based Design

Equation for Steel Girder Highway Bridges, Journal of Structural

Engineering. 34(4): 284-290.

Melchers, R.E. (1999). Structural Reliability Analysis and Prediction. John

Wiley, New York.

Reynoldo, M. P. (2008). Risk Assessment of Highway Bridges: A

Reliability-Based Approach Proceedings of the 2008 IAJC-IJME

International Conference.

Barambu USEP: Journal of Research Information in Civil Engineering, Vol.14, No.1, 2017

et al.

1291

Rosowsky, D. V. (2002). Reliability-Based Seismic Design of Wood Shear

Walls. Journal of Structural Engineering. 128 (11), 1439-1453.

Tsompanakis, Y. and Papadrakakis, M. (2000). Robust and Efficient

Methods for Reliability-Based Structural Optimization. IASS-IACM

International Conference 2000, Chania – Crete, Greece.

Uche, O.A.U. and Afolayan, J.O. (2008). Reliability Based Rating for

Reinforced Concrete Columns. A Journal of Engineering and Technology

(JET), 3(1).12p.

Uche, O.A.U and Ahmed, A. (2013). Reliability Design Format for Steel

Plate Girder to BS 5950 in Epistemics in Science, Engineering and

Technology, Vol. 3, No 1 PP. 215-223, Silver Sping Mary Land (MD) USA.

(International; www.epistemicsgaia.webs.com.