RELIABILITY ANALYSIS OF RC BEAMS EXPOSED TO

ELEVATED TEMPERATURE

A THESIS

Submitted in partial fulfillment of the

Requirements for the award of the degree

Of

MASTER OF TECHNOLOGY

In

STRUCTURAL AND CONSTRUCTION ENGINEERING

By

SACHIN H

(Roll No. 13217023)

DEPARTMENT OF CIVIL ENGINEERING

Dr. B R AMBEDKAR NATIONAL INSTITUTE OF TECHNOLOGY

JALANDHAR – 144011, PUNJAB

May 2015

i

Dr. B R AMBEDKAR NATIONAL INSTITUTE OF TECHNOLOGY

JALANDHAR

DEPARTMENT OF CIVIL ENGINEERING

CANDIDATE’S DECLARATION

I here certify that the work which is being presented in this dissertation entitled

“RELIABILITY ANALYSIS OF RC BEAMS EXPOSED TO ELEVATED

TEMPERATURE”, in partial fulfillment of the requirements for the award of the degree of

Master of Technology (Structural & Construction Engineering) submitted in the department of

Civil Engineering of the institute is an authentic record of my own work carried out during a

period January-May 2015 under the supervision of Dr. S.P Singh.

Date........ Signature of the candidate

This is to certify that the above statement made by candidate is correct to the best of my

knowledge.

Date… …. (Dr. S.P Singh)

Professor

The viva voce examination of Mr. SACHIN H has been held on ……….……………..

Signature of supervisor Signature of External examiner Signature of HOD

ii

ACKNOWLEDGEMENT

I express my deep sense of gratitude to Dr. S P Singh (Professor), Department of Civil

Engineering, Dr. B R AMBEDKAR NATIONAL INSTITUTE OF TECHNOLOGY,

JALANDHAR, for his excellent guidance and whole hearted involvement during the course of

my Study without whose invaluable suggestions, meticulous efforts, versatility and untiring

guidance, this report would not have been feasible. I am also indebted to him for his

encouragement and moral support and sparing their valuable time in giving me concrete

suggestions and increasing my knowledge through fruitful discussions throughout the course of

my study.

I owe thanks to library staff of, Dr. B R AMBEDKAR NATIONAL INSTITUTE OF

TECHNOLOGY, JALANDHAR, for their full cooperation in providing the necessary literature.

I will be failing in my duties, if I don’t acknowledge my sense of gratitude to God Almighty and

my parents, the blessing of whom have made me reach my destination.

(SACHIN H)

iii

ABSTRACT

In general, because of the unpredictable nature of fire and the various uncertainties related, for

example, to material properties at elevated temperature, the reliability of structural fire design

can be justifiably questioned.

A procedure for conducting reliability analysis of reinforced concrete beams subjected to a fire

load is presented. This involves identifying relevant load combinations, specifying critical load

and resistance random variables, and establishing a high-temperature performance model for

beam capacity. Based on the procedure, an initial reliability analysis is conducted using currently

available data. Significant load random variables are taken to be dead load, sustained live load,

and fire temperature. Resistance is in terms of moment capacity, with random variables taken as

steel yield strength, concrete compressive strength, and placement of reinforcement, beam width,

and thermal diffusivity. A semi-empirical model is used to estimate beam moment capacity as a

function of fire exposure time, which is calibrated to experimental data available in the literature.

The effect of various beam parameters were considered, including cover, beam width, aggregate

type, compressive strength, reinforcement ratio and other parameters. Using the suggested

procedure, reliability was estimated from zero to four hours of fire exposure using First Order

Second moment method. It was found that reliability decreased nonlinearly as a function of time,

while the most significant parameters were concrete cover, mean fire temperature and

compressive strength of concrete.

iv

CONTENT

Page No.

CANDITATE’S DECLARATION (i)

ACKNOWLEDGEMENT (ii)

ABSTRACT (iii)

CONTENT (iv)

LIST OF FIGURES (viii)

LIST OF TABLES (xi)

CHAPTER

1. INTRODUCTION

1.1 GENERAL …………………………………………………………….....…1

1.2 OBJECTIVES…………………………………………………………......…2

1.3 METHODOLOGY……………………………..…………………………….2

1.4 FIRE RESISTANCE OF CONCRETE STRUCTURES …………………....3

1.5 PROBLEMS IN CONCRETE STRUCTURE DUE TO FIRE ……..……….4

1.5.1 Spalling of concrete…………………………………………….……6

1.5.2 Colour……………………………………………………….……….7

1.5.3 Crazing…………………………………………………………….....8

1.5.4 Cracking ……………………………………..……….…………..….8

1.5.5 Distortion……………………………………………………...….….8

2. LITERATURE REVIEW

2.1 GENERAL…………………………………………………………………...9

2.2 LITERATURE REVIEW……………………………………………………9

2.3 SCOPE OF THE STUDY…………………………………………………...12

3 RELIABILITY THEORY

3.1 GENERAL ……………………………………….………………….…..….13

3.2 BASIC RELIABILITY THEORY…………………..……………..…..…....13

3.3 SPACE OF STATE VARIABLES …………………...……………….…....14

v

3.4 BASIC PROBLEM FORMULATION …….……...…………………..……15

3.5 FIRST ORDER RELIABILITY METHODS …………………...………….17

3.5.1 First Order Second moment method…………………………...……17

3.5.2 Hasoffer and Lind’s method (AFOM)……………………………....19

3.5.3 Rackwitz and Fiesseler Algorithm…………………………………..22

3.6 COMENTS ON FOSM MEAN VALUE INDEX ………………………….23

4 FIRE RESISTANT DESIGN OF BEAM

4.1 GENERAL…………………………………………………………………..24

4.2 BEHAVIOUR OF FIRE ………………………………...………………….25

4.3 PROCESS OF FIRE DEVELOPMENT ……………………………………27

4.4 SIMPLIFIED CALCULATION METHODS ………………………………28

4.4.1 500°c isotherm method………………………………..………….…28

4.4.2 Zone method …………………………………...........……………...29

4.4.3 Advanced calculation methods ………………………………...…...31

4.5 BREIF REVIEW OF CODAL PROVISION ………………………………31

4.6 TEMPERATURE PROFILE………………………………………………..32

5 THERMAL ANALYSIS USING FE SOFTWARE

5.1 SIMPLE METHOD FOR PREDICTING TEMPERATURES

IN REINFORCED CONCRETE BEAMS

EXPOSED TO A STANDARD FIRE …..…………………........................35

5.2 FIRE RESISTANT DESIGN……… …………………...……….…………36

5.3 EXISTING METHODS FOR TEMPERATURE PREDICTIONS………...37

5.3.1 Wickstrom’s Method………………………………………………..37

5.3.2 Desai’s Method……………………………………………………...38

5.3.3 Abbasi and Hogg’s Method…………………………………………39

5.3.4 Kodur Et Al.’S Method……………………………………………..39

5.3.5 Fe Method Analysis………………………………………………....40

5.4 MATERIAL BEHAVIOUR AT ELEAVATED TEMPERATURE………..42

5.4.1 Thermal Properties……………………………………….……...….42

5.4.1.1 Thermal Conductivity……………………………………...42

vi

5.4.1.2 Specific Heat……………………………………………….44

5.4.2 Mechanical Properties…………………………………………….…46

5.4.3 Deformation Properties……………………………..……………….48

5.5 THERMAL ANALYSIS USING ANSYS 14……………………………....49

5.5.1 Ansys14 Finite Element Model……………………………….….....50

5.5.2Development of Temperature Profile…………………………….........52

6 RELIABILITY ANALYSIS

6.1 GENERAL ………………..…………………………....…………………..55

6.2 EXAMPLE PROBLEM-NORMALLY DISTRIBUTED

UN-CORRELATED VARIABLES …………...…………………………...55

6.3 EXAMPLE PROBLEM-NON-NORMALLY DISTRIBUTED

UN- CORELATED VARIABLES …………..………..………………..…..57

6.4 VALIDATION OF JOURNAL PROBLEM ………………..……………...58

6.4.1 Reliability Analysis………………………………………………....59

7 RELIABILITY ANALYSIS OF RC BEAMS UNDER FIRE- RESULTS

AND DISCUSSIONS

7.1 GENERAL……………………………………………………………….....65

7.2 DESIGN OF RC BEAM WITH UDL………………………………….......65

7.3 LIMIT STATE OF FLEXURE………………………………………….….67

7.3.1 Strength Degradation of Reinforcement with Respect

to Fire Exposure…………………………………………………....69

7.3.2 Parametric Study……………………………………………….…..72

7.3.2.1 Characteristic Strength of Concrete……………………...73

7.3.2.2 Yield Strength of Steel…………………………………..74

7.3.2.3 Effective Cover……………………………………….....75

7.3.2.4 No of Distribution bars……………………………….…76

7.3.2.5 Type of aggregate used……………………………….....77

7.4 LIMIT STATE OF DEFLECTION………………………………………..78

7.5 LIMIT SATE OF COLLAPSE AGAINST SHEAR………………….…...80

7.5.1 Parametric Study…………………………………………………...80

vii

7.5.1.1 Characteristic Strength of Concrete……………………...81

7.5.1.2 Yield Strength of Steel………………………………..….81

7.5.1.3 No of Distribution bars……………………………….....82

7.5.1.4 Type of aggregate used……………………………….....82

7.6 TARGET VALUES FOR NOMINAL FAILURE PROBABLITY…………...83

8 CONCLUSION

9 MATLAB CODING

9.1 EXAMPLE PROBLEM-NORMALLY DISTRIBUTED

UN-CORRELATED VARIABLES……………………………………….86

9.2 EXAMPLE PROBLEM-NON-NORMALLY DISTRIBUTED

UN CORRELATED VARIABLES………………………………….…....87

9.3 LIMIT STATE OF FLEXURE-BEAM-250MMX450MM

2 HOUR EXPOSURE,FY =415, FCK =30 ,

EFFECTIVE COVER=50MM,2-25MM DIA BARS……………………..88

9.4 LIMIT STATE OF DEFLECTION-BEAM…………………………….…89

9.5 LIMIT STATE OF SHEAR-BEAM…………………………………...….91

10 REFERENCES …………………………………………………………………..93

viii

LIST OF FIGURES

Figure no. Title

Page no.

Fig 1.1 The spalling mechanism of concrete cover is visualized in 7

Fig 1.2 Spalling in the concrete due to elevated temperature 7

Fig 1.3 Colour change in the concrete due to elevated temperature 7

Fig 1.4 Cracks formed in concrete due to elevated temperature 8

Fig 3.1 Safe domain and failure domain in two dimensional state spaces 15

Fig 3.2 Probability density functions of random (basic) variables R, S 16

Fig 3.3 Concept of β 18

Fig 3.4 Reliability index defined as the shortest distance in the space of

reduced variables

19

Fig 3.5 Hasofer and Lind reliability index 21

Fig 3.6 Mean value second - moment formulation 23

Fig 4.1 Time temperature curve for fire development 27

Fig 4.2 Reduced section after fire 29

Fig 4.3 Reduction of cross section of slab exposed to fire 29

Fig 4.4 Division of slab with both sides exposed to fire, into zones for use in

calculation of strength reduction and az value

30

Fig 4.5 Area of cross-section for which the temperature profiles presented in

Eurocode2

33

Fig 4.6 Temperature profile of 160mm x 300mm beam (Anne: A of

Euro code2)

34

Fig 4.7 Temperature profiles for 90 min exposure of a beam, h x b = 300 x

160

34

Fig 4.8 5OO°C isotherms for a beam, h x b = 300 x 160 34

Fig 5.1 Thermal conductivity of concrete as a function of temperature 43

Fig 5.2 Thermal conductivity of steel as a function of temperature 44

Fig 5.3 Specific heat Cp(θ) as function of temperature at moisture content u 44

ix

of 1.5 by weight for siliceous concrete

Fig 5.4 Stress-strain curves for concrete at various temperatures 46

Fig. 5.5 Beam modelled in ANSYS14 49

Fig. 5.6 Reduction factors as per Euro code for fy 51

Fig. 5.7 Contours of temperature gradient obtained from thermal analysis of

simple beam and a RC beam using Ansys15

52

Fig. 5.8 Temperature profile corresponding to 30 minutes 53

Fig. 5.9 Temperature profile corresponding to 60 minutes 53

Fig. 5.10 Temperature profile corresponding to 90 minutes 53

Fig. 5.11 Temperature profile corresponding to 120 minutes 53

Fig. 5.12 Temperature profile corresponding to 150 minutes 54

Fig. 5.13 Temperature profile corresponding to 180 minutes 54

Fig. 5.14 Temperature profile corresponding to 210 minutes 54

Fig. 5.15 Temperature profile corresponding to 240 minutes 54

Fig. 6.1 Cross section of a reinforced concrete beam 55

Fig. 6.2 Quarter portion of a beam showing the position of reinforcement in

terms of x and y

59

Fig. 6.3 Validated curve (FOSM) 63

Fig. 6.4 Reliability index degradation curve (Christopher D. Eamon, Elin

Jensen,(2013), Reliability Analysis of RC Beams Exposed to Fire,

Journal of Structural Engineering, 139, pp. 212-220.)

63

Fig. 7.1 Cross section details of the beam 67

Fig. 7.2 Cross section of the beam considered 68

Fig. 7.3 ANSYS- thermal analysis contour plot (the numbers in figure

indicate the node numbers)

70

Fig. 7.4 Time dependent strength degradation of reinforcement with fire

exposure

70

Fig. 7.5 Comparison of reliability degradation curve using Wick storm’s

model and Ansys method

71

Fig. 7.6 Effect of concrete’s compressive strength in reliability indices 73

x

Fig. 7.7 Effect of grade of steel in reliability indices 74

Fig. 7.8 Effect of cover provided in beams 75

Fig. 7.9 Effect of number of bars in reliability indices 76

Fig. 7.10 Effect of type of aggregate in reliability indices 77

Fig. 7.11 Effect of strength of concrete in reliability indices 81

Fig. 7.12 Effect of yield strength of steel in reliability indices 81

Fig. 7.13 Effect of distribution bars in reliability indices 82

Fig. 7.14 Effect of type of aggregate in reliability indices 82

xi

LIST OF TABLES

Table no. Title

Page no.

Table 3.1 Reliability index β and probability of failure Pf 20

Table 4.1 Reduction factors for yield strength of steel as per EC – 2 31

Table 4.2 Minimum dimension and nominal cover to meet specified period of

fire resistance for RCC beam (IS456:2000)

32

Table 5.1 Values for the Parameters of the Stress-Strain Relationship of

Reinforcing Steel at Elevated Temperature

47

Table 5.2 Development of Temperature Profile 51

Table 6.1 Random variables considered 56

Table 6.2 Results of Iteration 56

Table 6.3 Random variables considered 57

Table 6.4 Results of Iteration 57

Table 6.5 Random variables considered 62

Table 6.6 Results of iteration 63

Table 6.7 Validated data’s 64

Table 7.1 Statical distribution of random variables 68

Table 7.2 Reliability indices using Wick storm’s method 71

Table 7.3 Reliability indices using Ansys method 71

Table 7.4 Reliability indices calculated using Wick storm’s Method. 72

Table 7.5 Reliability indices calculated using Ansys Method 72

Table 7.6 Details of reinforcement 73

Table 7.7 Comparison of β 73

Table 7.8 Details of reinforcement 74

Table 7.9 Comparison of β 74

Table 7.10 Details of reinforcement 75

Table 7.11 Comparison of β 75

Table 7.12 Comparison of β 76

xii

Table 7.13 Comparison of β 77

Table 7.14 Statical distribution of random variables 79

Table 7.15 Reliability indices for the first 4 hours 79

Table 7.16 Statical distribution of random variables 80

Table 7.17 Social criteria factor 83

Table 7.18 Activity and warning factors 84

1

Chapter - 1

INTRODUCTION

1.1 GENERAL

Fire remains one of the serious potential risks to most buildings and structures. The

extensive use of concrete as a structural material has led to the need to fully understand t h e

effect of fi re on concrete. Reinforced concrete (RC) structural systems are quite frequently

used in high-rise buildings and other built infrastructure owing to a number of advantages they

provide over other materials. When used in buildings, the provision of appropriate fire safety

measures for structural members is an important aspect of design since fire represents one

of the most severe environmental conditions to which structures may be subjected in their

lifetime. Fire resistance means the ability of building components and systems to perform

their intended fire separating and/or load bearing functions under fire exposure. Fire

resistant building components and systems are those with specified fire resistance ratings

based on fire resistance tests. These ratings, expressed in minutes and hours, describe the

time duration for which a given building component or system maintains specific functions

while exposed to a specific simulated fire event.

In 2005 alone, fires caused 3,762 deaths, 17,925 civilian injuries, and $10.7 billion in property

damage in the United States (NFPA 2011). In addition to fire prevention techniques, various

means of fire damage mitigation are used. Some of these include providing the proper architectural

planning of exits and escape routes; the use of active fire protection techniques such as sprinklers

to reduce the number of severe fires; and providing structural fire protection to achieve a minimum

fire resistance rating, with the intent to allow structural members to maintain their integrity

throughout the escape and firefighting phases.

Recognizing these problems, the fire engineering community has become interested in adopting a

more systematic and rational way to assess and achieve a consistent level of fire safety (SFPE

2002). The general framework that allows the achievement of this goal is performance based

design, which in regard to fire engineering, is a robust method allowing probabilistic assessment

that is founded on the principles of fire science, heat transfer, and structural analysis. A 2008

position paper by the International FORUM of Fire Research Directors on the application of

performance-based design for fire code application identified five research priorities needed to be

2

fulfilled to achieve inclusive PBD (Croce 2008). One of these priorities is of interest to the topic of

this study: the estimation of uncertainty and means to incorporate it into (structural) risk analyses

when considering fires. Over the last several decades, there has been limited research on the

probabilistic analysis of structures exposed to fire, although diverse types of analyses have been

considered. Currently, however, there exists no systematic assessment of the reliability of RC

beams exposed to fire that have been designed to current (ACI 318; ACI 2011) standards

considering both load and resistance uncertainties, nor has there been an examination of the

changes in reliability as various important beam parameters change. As a step toward PBD

(performance based design), this study presents a procedure that can be used to estimate the

reliability of RC beams exposed to fire. Using the suggested procedure, currently available high-

temperature performance models and random variable data are incorporated to estimate safety

levels of RC beams designed according to ACI 318 code (ACI 2011) exposed to a standard fire.

The potential effect of changing various design parameters on beam reliability when exposed to

fire is also investigated.

1.2 OBJECTIVES

Reliability analysis of RC beams exposed to fire using First Order Second

Moment Method.

Establishing reliability degradation with respect to exposure time.

1.3 METHODOLOGY

Literature survey.

Development of MATLAB code for reliability analysis by First Order Second

Moment Method.

Identifying relevant load combinations.

Specifying critical load and resistance random variables.

Establishing a high-temperature performance model for beam capacity

Establishing reliability degradation with respect to exposure time.

The effects of various beam parameters are considered.

3

1.4 FIRE RESISTANCE OF CONCRETE STRUCTURES

Building codes require designers to provide fire protection for buildings by combining “active” fire

protection systems with “passive” fire protections systems. Active fire protection systems include

smoke detectors, sprinklers and other systems that activate in the presence of smoke or fire. Passive

fire protection uses the building components and layout to reduce the risk and spread of fire by

providing non-combustible fire rated walls, floors and roofs. These building components help to

compartmentalize the building so a fire that starts in one part of a building does not spread to other

parts of the building. The concept of combining active and passive fire protection systems is called

balanced fire protection design. Implementing balanced fire protection design provides the highest

achievable level of protection. Before the advent of technology based active systems, fire

protection for buildings relied almost exclusively on passive fire protection. However, as active fire

protection systems were developed, the relative importance of passive systems for reducing smoke

and fire spread through compartmentalization and fire resistant components has been slowly

diminished in the building codes.

The primary goal of building codes is to protect public safety (life). While reliable active fire

protection can achieve this goal, the additional advantage of passive systems in a balanced design

is to minimize damage to the owner’s property. Without the balanced design approach, one relies

solely on the effectiveness and reliability of a mechanical system to provide the needed fire

protection for the building. Fire containment provides a reliable method to reduce the spread of fire

and smoke even in the presence of mechanical system failure. For the building designer, fire

containment can be provided through the use of fire resistive concrete walls, floors and ceilings.

Although fire resistive containment and construction includes all structural members, here we will

be focusing on fire resistant concrete walls.

The fire resistance of concrete walls is directly impacted by the choice of aggregate. As mentioned

earlier, concrete has performed well in large structural fires due to its non-combustibility and low

thermal conductivity. The most common method of determining a structural member’s

performance in a fire is by a series of tests leading to a fire resistance rating. The most common test

method for determining fire resistance in the United States is the ASTM Standard E 119, Test

Methods for Fire Tests of Building Construction and Materials. ASTM Standard method E 119 is a

fire test that exposes the structural member to a standard fire on one side of the wall. For the

structural member to pass the test three criteria must be met – structural stability, integrity and

4

temperature rise on unexposed face. Concrete structural members tend to perform well in the

ASTM E 119 test. However, unlike steel, the concrete fire resistance cannot be determined by

calculating a single critical temperature. The temperature within the concrete member cross section

is not uniform throughout the fire exposure; therefore, the thermal and mechanical properties of the

concrete vary with time and location of fire exposure within the section. The calculation of fire

resistance in concrete is further complicated by the wide range of aggregates and other properties

of concrete used in the concrete member. Results of fire tests and fire ratings are very specific to

the assemblies tested.

Aggregate can amount to 60-80% of the total volume of concrete; therefore, the use of structural

lightweight concrete can significantly improve the fire resistance of concrete walls directly impacts

the performance of concrete during a fire. As the temperature rises in a concrete wall, the strength

of the wall is diminished. If we are taking the case of different types of aggregates such as siliceous

aggregate, carbonate and light weight aggregate, the siliceous aggregate concrete strength is

reduced by half at temperatures of 1200ºF, the carbonate and lightweight aggregate concrete

maintains near 100% of its original strength.

In lieu of performing standard fire tests on walls, building codes permit designers to calculate the

fire resistance rating using analytical methods. These would generally be more conservative than

fire ratings obtained from fire tests. Two methods exist for determining the fire resistance of

concrete walls: empirical or the more complicated analytical process. ACI 216.1-97, Standard

Method for Determining Fire Resistance of Concrete and Masonry Construction Assemblies,

provides a simplistic empirical method for determining fire resistance of concrete walls. Table 2.1

from ACI 216, reproduced here, provides the minimum equivalent thickness required of a concrete

wall based on aggregate type to achieve a fire resistance rating of one hour to four hours. For solid

flat concrete walls, the actual thickness is equal to the equivalent thickness. For walls that are more

complex, such as cast-in-place walls that are not flat (varying thickness) and concrete masonry

walls, the equivalent wall thickness is determined using formulas provided in ACI 216.1-97.

1.5 PROBLEMS IN CONCRETE STRUCTURE DUE TO FIRE.

Concrete does not burn – it cannot be ‘set on fire’ like other materials in a building and it does not

emit any toxic fumes when affected by fire. It will also not produce smoke or drip molten particles,

unlike some plastics and metals, so it does not add to the fire load. For these reasons concrete is

5

said to have a high degree of fire resistance and, in the majority of applications, concrete can be

described as virtually ‘fireproof’. This excellent performance is due in the main to concrete’s

constituent materials (i.e. cement and aggregates) which, when chemically combined within

concrete, form a material that is essentially inert and, importantly for fire safety design, has a

relatively poor thermal conductivity. It is this slow rate of heat transfer (conductivity) that enables

concrete to act as an effective fire shield not only between adjacent spaces, but also to protect itself

from fire of temperature through the cross section of a concrete element is relatively slow and so

internal zones do not reach the same high temperatures as a surface exposed to flames. A standard

ISO 834/BS 476 fire test on 160 mm wide x 300 mm deep concrete beams has shown that, after

one hour of exposure on three sides, while a temperature of 600°C is reached at 16 mm from the

surface, this value halves to just 300°C at 42 mm from the surface–a temperature gradient of 300

degrees in about an inch of concrete! Even after a prolonged structural capacity and fire shielding

properties as a separating element.

Fires are caused by accident, energy sources or natural means, but the majority of fires in buildings

are caused by human error. Once a fire starts and the contents and/or materials in a building are

burning, then the fire spreads via radiation, convection or conduction with flames reaching

temperatures of between 600°C and 1200°C. Harm is caused by a combination of the effects of

smoke and gases, which are emitted from burning materials, and the effects of flames and high air

temperatures. Concrete does not burn – it cannot be ‘set on fire’ like other materials in a building

and it does not emit any toxic fumes when affected by fire. It will also not produce smoke or drip

molten particles, unlike some plastics and metals, so it does not add to the fire load. For these

reasons concrete is said to have a high degree of fire resistance and, in the majority of applications,

concrete can be described as virtually ‘fireproof’.

Concrete is one of the best fire resistant materials due to its low thermal diffusivity and high

specific heat. However, it is the reinforcing bars which are most affected when a fire occurs in

reinforced concrete structure. Though IS: 456-20002 and IS: 1649-19623 Codes have specified a

minimum concrete cover protection to reinforcement in reinforced concrete structures and flues

respectively, the reinforcing steel bars may still be exposed to elevated temperature once the

concrete cover spalls. The strength of all engineering materials reduces as their temperature

increases. Concrete is considered as one of the best fire resistant material. But when exposed to fire

beyond say 600oC, it may undergo an irreversible degradation in mechanical strength and spalling.

6

Concrete is a composite material that consists mainly of mineral aggregates bound by a matrix of

hydrated cement paste. The matrix is highly porous and contains are relatively large amount of free

water unless artificially dried. When exposing it to high temperatures, concrete undergoes changes

in its chemical composition, physical structure and water content. These changes occur primarily in

the hardened cement paste in unsealed conditions. Such changes are reflected by changes in the

physical and the mechanical properties of concrete that are associated with temperature increase.

Deterioration of concrete at high temperatures may appear in two forms:

(1) Local damage (Cracks) in the material itself

(2) Global damage resulting in the failure of the elements.

One of the advantages of concrete over other building materials is its inherent fire resistive

properties; however, concrete structures must still be designed for fire effects.

1.5.1 SPALLING OF CONCRETE

One of the most complex and hence poorly understood behavioral characteristics in the reaction of

concrete to high temperatures or fire is the phenomenon of “spalling”. This process is often

assumed to occur only at high temperatures, yet it has also been observed in the early stages of a

fire, and at temperatures as low as 200 Cº. If severe, spalling can have a deleterious effect on the

strength of reinforced concrete structures; due to enhanced heating of the steel reinforcement.

Spalling may significantly reduce or even eliminate the layer of concrete cover to the

reinforcement bars, thereby exposing the reinforcement to high temperatures, leading to a reduction

of strength of the steel and hence a deterioration of the mechanical properties of the structure as a

whole. Another significant impact of spalling upon the physical strength of structures occurs via

reduction of the cross-section of concrete available to support the imposed loading, increasing the

stress on the remaining areas of concrete. This can be important, as Spalling may manifest itself at

relatively low temperatures, before any other negative effects of heating on the strength of concrete

have taken place. Spalling of concrete is generally categorized as: pore pressure induced spalling,

thermal stress induced spalling or a combination of the two.

Spalling of concrete surfaces may have two reasons:

(1) Increased internal vapour pressure (mainly for normal strength concretes).

(2) Overloading of concrete compressed zones (mainly for high strength concretes).

7

Fig. 1.1: The spalling mechanism of concrete cover is visualized in.

1.5.2 COLOUR.

The colour of concrete may change as a result of heat due to fire may give an idea of the maximum

temperature attained. Due to fire discoloration takes place and the possible change in concrete is

normal, pink, whitish grey and puff.

Fig. 1.2 Fig. 1.3

Fig. 1.2: Spalling in the concrete due to elevated temperature

Fig. 1.3: Colour change in the concrete due to elevated temperature.

8

1.5.3 CRAZING

The development of fine cracks on the surface of the concrete due to sudden cooling of surface

with water is termed as crazing. These fine cracks are restricted to surface layer and no structural

significance on material has been accounted for.

1.5.4 CRACKS

Concrete members exposed to high temperature during fire may develop severe cracks which may

extend across the body of the member. Cracking is classified as minor and major with the recording

of length of cracks.

Fig 1.4: Cracks formed in concrete due to elevated temperature

1.5.5 DISTORTION

The extent of distortion of the structural members affected by fire in the form of deformations

(deflections, twisting etc.) are also recorded into three categories; none, slight but insignificant and

severe and significant.

9

Chapter – 2

LITERATURE REVIEW

2.1 GENERAL

There are many researches going on the topic of reliability analysis of structural elements

exposed to fire, which include both numerical and experimental works. This chapter

discusses a number of researches and previous studies which were conducted to study the

reliability of reinforced concrete beams under fire conditions.

2.2 LITERATURE REVIEW

R.Ranganathan et al. (1990), had done the reliability analysis using modern reliability methods,

in which the formulation of the limit-state functions is consistent with the underlying design

criteria. It was found that the reliability indices are most sensitive to live load, model

uncertainties, and material strengths. For the failure modes considered, the reliability indices

were found to be rather insensitive to design parameter values, indicating that the ACI

Building Code achieves its desired objective of uniform reliability across a wide range of

design situations. He explained the basics of structural reliability and reliability methods in his

publishing. Here the discussion is based on both normal as well as non-normal random

variables.

Renjian Lu et al. (1994) had done the reliability evaluation of reinforced concrete beams. The

purpose of this paper is to evaluate the time-invariant reliability of reinforced concrete beams

designed under the provisions of the ACI Building Code. A wide range of practical design

situations was considered. The beams were subjected to bending, shear, and torsion. The

interaction between shear and torsion is considered via an elliptical failure surface defined in the

shear-torsion stress space. No interaction was assumed between flexural resistance and resistance

in both shear and tors ion .

A. M. Arafah et al. (1997) had done studies on Reliability of reinforced concrete beam section as

affected by their reinforcement ratio. This paper presented a reliability-based analysis for

10

reinforced concrete beam sections at their flexural limit states. These results indicate that

reliability of beam section is highly sensitive to variation in the compression and tension

reinforcements even when the design safety factors are kept constant.

Andre J. Torii and Roberto D. Machado (2010) studied the reliability analysis of nonlinear

reinforced concrete beams. Failure w a s assumed to occur when the structure presents

displacements bigger than a prescribed limit. A First Order Reliability Method (FORM) was used,

and the results were compared to the ones given by Monte Carlo simulation. Since the structural

model is nonlinear, special techniques were used for sensitivity analysis. These techniques allow

one to consider almost any variable as probabilistic in this problem. Finally, two examples were

presented in order to validate the proposed approach.

Zhenqing Wang et al. (2010) had done the reliability analysis of reinforced concrete beams under

high temperature for the mechanical properties of reinforced concrete under high temperature with

large deterioration, the reliability of reinforced concrete beams had been largely discounted. The

plastic zone resistance of concrete under high temperature had been considered in this paper. The

action of ISO 834 temperature rising curve on the reliability index of different specifications of

concrete beams at different time has been analyzed. The results had shown that the increase in the

reinforcement ratio and concrete cover thickness appropriately has an effective measure to

improve the fire resistance limit of reinforced concrete beams.

Christopher D. Eamon and Elin Jensen (2012) studied reliability analysis of pre stressed

concrete beams exposed to fire. This involved identifying relevant load combinations, specifying

critical load and resistance random variables, and establishing a high-temperature performance

model for beam capacity. The effect of various beam parameters were considered, including cover,

aggregate type, concrete compressive strength, dead to live load ratio, reinforcement ratio, end

restraints, fire exposure, and proportion of end strands to total strands. Using the suggested

procedure, reliability was estimated from zero to four hours of fire exposure using Monte Carlo

simulation. It was found that reliability decreased non-linearly as a function of time, while the

most significant parameters were concrete cover, load ratio, fire type, end restraints, and

proportion of end strands to total strands.

11

Osvaldo Luiz de Carvalho Souza et al. (2013) studied reliability analysis of RC beams

strengthened for torsion with carbon fibre composites. The analyses were performed for different

torsional moment ratios, defined as the ratio of the torsional moment due to live loads to the total

torsional moment. The examples showed that despite the constant value of the total

torsional moment, the increase in the torsional moment ratio leads to a decrease in the system

reliability levels. The fact that the values of the reliability indexes obtained in both sets of

reliability analyses are very similar validates the efficiency of the sensitivity analyses.

Christopher D. Eamon and Elin Jensen (2013) studied reliability analysis of reinforced concrete

columns exposed to fire. From an evaluation of load frequency of occurrence, load random

variables are taken to be dead load, sustained live load, and fire temperature. Resistance was

developed for axial capacity, with random variables taken as steel yield strength, concrete

compressive strength, placement of reinforcement, and section width and height. A rational

interaction model based on the Rankine approach was used to estimate column capacity as a

function of fire exposure time. Reliability was computed from 0 to 4 h of fire exposure using

Monte Carlo simulation. It was found that reliability decreased nonlinearly as a function of time,

while the most significant parameters were fire type, load ratio, eccentricity, and reinforcement

ratio.

Christopher D. Eamon and Elin Jensen (2013) had analysed the reliability of RC beams exposed

to fire, suggested a procedure. Using that procedure, an initial analysis was conducted for various

RC beams designed according to ACI 318 (ACI 2011) that are exposed to fire. Based on the load

and resistance models used, it was found that most beams had a cold-strength reliability index of

approximately 5.4 while exposed to dead load and sustained live load. Reliability rapidly

decreased as a function of time for the first 1-2 h after fire exposure and continued to decrease at a

slower rate thereafter to become asymptotic to a minimum reliability index that ranges between 0

and 21 for most cases.

12

2.3 SCOPE OF THE STUDY

Recent years have seen a gradual transition from the prescriptive approach to the performance-

based approach in the fire resistance design of reinforced concrete (RC) structures; the latter

provides a more rational and flexible design tool. In the performance-based design approach, the

fire resistance of the structure needs to be accurately evaluated, which requires the accurate

prediction of temperature fields in the structure.

Over the last several decades, there has been limited research on the probabilistic analysis of

structures exposed to fire, although diverse types of analyses have been considered. Only a few

studies were identified in the technical literature that considered the failure probabilities of RC

structural elements exposed to fire. Currently, however, there exists no systematic assessment of

the reliability of RC beams exposed to fire that have been designed to current (ACI 318; ACI 2011)

standards considering both load and resistance uncertainties, nor has there been an examination of

the changes in reliability as various important beam parameters change. As a step toward PBD

(performance based design), this study presents a procedure that can be used to estimate the

reliability of RC beams exposed to fire. Using the suggested procedure, currently available high-

temperature performance models and random variable data are incorporated to estimate safety

levels of RC beams designed according to ACI 318 code (ACI 2011) exposed to a standard fire.

The potential effect of changing various design parameters on beam reliability when exposed to fire

is also investigated. In performance-based fire safety design, the fire performance of a structure

needs to be accurately evaluated, which requires the accurate prediction of temperatures in the

structure. While a finite-element or a finite-difference analysis may be carried out for this purpose.

13

Chapter - 3

RELIABILTY THEORY

3.1 GENERAL

One of the principal aim of any engineering design is the assurance of each structural member

with respect to safety, serviceability and economy. But often uncertainties are encountered in the

information about the input variables of the analysis and design. These uncertainties arises from

the inherent randomness in many design variables and can be broadly classified into two

categories.1) random variables corresponding to response of the structure, and 2) random

variable corresponding to effect of loading on the structure. Since this variables exhibit

statistical regularity, "probability theory" can be used for studying this. This theory deals with

the uncertainties involved in the design and provides mathematical model for these uncertainties

and thus be able to determine the probability of failure of the structure. By taking in to account

above of the uncertainties, response of a structure can be considered as satisfactory, only when

some performance criteria, such as safety against collapse, limitation on damage or deflection

etc. are satisfied. Each of these requirements may be termed as 'limit state'. The violation of

limit state can thus be treated as failure state for structures. The study of structural reliability is

concerned with the calculation and prediction of limit state violation for engineering structures at

any stage during their life. The probability of occurrence of an event such as limit state violation

is a numerical measure of chances of its occurring. The measure may be obtained either from

measurements of the long term frequency of occurrence or may be simply subjective estimation

of numerical value.

3.2 BASIC RELIABILTY THEORY

There are different levels of reliability analysis, which can be used in any design methodology

depending on the importance of the structure. The term 'LEVEL' is characterized by the extent of

information about the problem that is used and provided. The methods of safety analysis proposed

currently for the attainment of a given limit state can be grouped under four basic “levels” (namely

levels IV, III, II, and I) depending upon the degree of sophistication applied to the treatment of the

various problems.

14

1. In LEVEL I methods, the probabilistic aspect of the problem is taken into account by

introducing into the safety analysis suitable “characteristic values” of the random variables,

conceived as fractile of a predefined order of the statistical distributions concerned. These

characteristic values are associated with partial safety factors that should be deduced from

probabilistic considerations so as to ensure appropriate levels of reliability in the design. In this

method, the reliability of the design deviate from the target value, and the objective is to minimize

such an error. Load and Resistance Factor Design (LRFD) method comes under this category.

2. In LEVEL II, safety checking is done at finite number of points on the failure surface (as

defined by appropriate limit state equation in the space of the basic variables) and often the

checking is done at one point only. So this does not require the multi-dimensional integration.

Random variables are characterized by the mean and variance with relevant distribution functions.

The reliability is expressed in terms of safety index, β. This is an approximate method, involving

iteration.

3. In LEVEL III methods encompass complete analysis of the problem and also involve

integration of the multidimensional joint probability density function of the random variables

extended over the safety domain. Reliability is expressed in terms of suitable safety indices, viz.,

reliability index, β and failure probabilities.

4. In LEVEL IV methods are appropriate for structures that are of major economic importance,

involve the principles of engineering economic analysis under uncertainty, and consider costs and

benefits of construction, maintenance, repair, consequences of failure, and interest on capital, etc.

Foundations for sensitive projects like nuclear power projects, transmission towers, highway

bridges, are suitable objects of level IV design.

3.3 SPACE OF STATE VARIABLES

For analysis, we need to define the state variables of the problem. The state variables are the basic

load and resistance parameters used to formulate the performance function. For ‘n’ state variables,

the limit state function is a function of ‘n’ parameters.

If all loads (or load effects) are represented by the variable Q and total resistance (or capacity) by

R, then the space of state variables is a two-dimensional space as shown in Figure 3.1 Within this

space, we can separate the “safe domain” from the “failure domain”; the boundary between the two

domains is described by the limit state function g(R, Q)=0.

15

Since both R and Q are random variables, we can define a joint density function fRQ

(r, q). A

general joint density function is plotted in Figure 3.2. Again, the limit state function separates the

safe and failure domains. The probability of failure is calculated by integration of the joint density

function over the failure domain [i.e., the region in which g(R, Q) <0]. As noted earlier, this

probability is often very difficult to evaluate, so the concept of a reliability index is used to

quantify structural reliability.

Fig 3.1: Safe domain and failure domain in two dimensional state spaces.

3.4 BASIC PROBLEM FORMULATION

The first step in evaluating the reliability or probability of failure of a structure is to decide on

specific performance criteria and the relevant load and resistance parameters, called basic

variables, and the functional relationship among them corresponding to each performance

criterion. The basic reliability problem consider only one load effect S (bending moment, shear

force etc.) resisted by one resistance R (capacity or strength), which are distributed by known

density functions fS(s) and fR(r).The probability of the event (R>S) will give the measure of

reliability of structure. Now probability of failure of the structural element can be stated in any of

the following ways

16

Pf = P ( R < S )

= P ( R – S < 0 )

Pf = P ( R/S < 1 )

In general, Pf = P ( G ( R,S ) ≤ 0 ), where G ( ) is the limit state function.

Fig 3.2: Probability density functions of random (basic) variables R, S.

If R and S are statistically independent, then probability of failure can be determined by the

following equation

But R and S being functions of several variables, may not be independent .In this case failure

function may be expressed in terms of the set of n basic variable X. Let X = {x1, x2,….xn ) T is the

set of basic variables which controls the performance of structure. The limit state function G(X) is so

formulated that G(X) ≤ 0 represents failure state and G(X) > 0 , safe state of structure. If fx(X) is the

joint probability density function of X, then

17

Very often, the joint probability PDF of X may not be available in practice and if at all, it is

available, computation of multidimensional integration equation above is too complex. In the

present work the reliability work the probability of failure (or reliability) is computed using First

Order Second Moment Method and Monte Carlo Simulation Technique (Halder and Mahadevan

2000 and Ranganathan 1990).

3.5 FIRST ORDER RELIABILTY METHODS.

These are level 2 reliability methods in which failure equation is simplified by approximating the

distribution of random variables as normal and by linearizing the performance function at some

point. The name First Order derivatives from the fact that the performance function is linearized

based on first order Taylor series approximation. First Order Second Moment Method and Lind's

Method (Advanced First Order Second Moment Method) are some of the first order reliability

methods.

3.5.1 FIRST ORDER SECOND MOMENT METHOD (FOSM)

In this method linearization is done at the mean values of random variables. So this method is also

known as Mean Value First Order Second Moment Method. Here reliability is computed as a

function of first and second moment (mean and variance) of basic variables.

The first definition is given by Cornell (1969) using simple two variable approach. He defined a

term reliability index, which is margin of safety as

where μG and σG are the mean and standard deviation.

Also the probability of failure can be evaluated as

Φ is the standard normal distribution function.

18

The concept of β is illustrated in fig. below, which shows the PDF of G( ) for the fundamental case

of two variable problems. The safety condition is defined by Condition G >0 and failure by

G≤0.The reliability index may be thought of as the distances from the origin (G=0) to the mean

measured in standard deviation units.

Fig 3.3: Concept of β

In case of non-linear failure functions, Taylor series expansion is used about mean retaining only

the linear terms. Thus,

The mean and standard deviation of G(X) are calculated from the above equation and the failure

probability is estimated. It has realized that significant were introduced at increasing distance from

the linearizing point by neglecting higher terms. The reliability index value changes, when

different but equivalent non-linear functions are us .This lack of inconvenience can be removed in

Hansofer and Lind reliability index definition.

19

3.5.2 HASOFER AND LIND’S METHOD (AFOSM)

A version of the reliability index was defined as the inverse of the coefficient of variation The

reliability index is the shortest distance from the origin of reduced variables to the is illustrated in

Figure 3.4, line g(ZR, Z

Q) = 0 .This definition, which was introduced by Hasofer and Lind (1974)

following formula:

Using geometry we can calculate the reliability index (shortest distance) from the following

formula:

where β is the inverse of the coefficient of variation of the function g(R, Q) = R-Q When R and Q

are uncorrelated for normally distributed random variables R and Q, it can be shown that the

reliability index is related to the probability of failure by

Fig 3.4: Reliability index defined as the shortest distance in the space of reduced variables.

20

Pf β

10-1

1.28

10-2

2.33

10-3

3.09

10-4

3.71

10-5

4.26

10-6

4.75

10-7

5.19

10-8

5.62

10-9

5.99

Table 3.1: Reliability index β and probability of failure Pf

The definition for a two variab1e case can be generalized for n variables as follows. Consider a

limit state function g(X1, X

2…… X

n), where the X

i variables are all uncorrelated. The Hasofer-Lind

reliability index is defined as follows:

In the Hasofer and Lind reliability Method, the basic variables are normalized using the

relationship,

where Z is the reduced or normalized variate.

Because of normalization of basic variable μzi=0, σzi=1, the new failure surface equation is in

normalized z-coordinate system. The position of failure surface with respect to the origin in z-

coordinate system determines the measure of reliability. Hasofer and Lind reliability index is the

shortest distance from the origin O to the failure surface in normalized coordinate system. For a

two variable problem this concept is depicted in fig. below

21

Fig 3.5: Hasofer and Lind reliability index

If the safety margin is linear and Xi are normally distributed the reliability index can be connected

to the true value of probability of failure of structure using equation above. But in practical

situation basic variable may non-normal. In such case β is obtained using equivalent normal

distributions at the design point. The transformation of non-normal variable at design point is as

follows.

At failure point xi*

(1) The ordinate of probability density of the original non-normal variable Xi is made equal

to the probability density of the equivalent normal variable Xi’

(2) The cumulative probability of the original non-normal variable Xi is made equal to the

cumulative probability of the equivalent normal variable Xi’.

(3) Fxi ( xi * ) = Fxi ( xi * )

Now, solving equations

The limit state equation needs to be solved to find the new design point. The algorithm

suggested by Rackwitz and Fiessler is used.

22

3.5.3 RACKWITZ AND FIESSLER ALGORITHM

The algorithm is suggested by Rackwitz and Fiessler (1976) (Madsen.H.O et al. 1986 and

Halder.A. and Mahadevan.S 2000) also linearized the performance function at each iteration point;

however instead of solving the limit state equation explicitly for β, it use derivatives to find out the

next iteration point. The algorithm can described as follows,

(1). Define appropriate performance function.

(2). Assume initial values of the design point xi*, and compute corresponding value of

performance function G ( ).In the absence of any other information, the initial design point

can be taken as the mean values of the random variables.

(3). Compute the mean and standard deviation at the design point of the equivalent normal

distribution for those variables that are non-normal.

(4). Compute the partial derivatives of G ( ) with respect to xi*at the design point.

(5). Now the new design point is obtained using the iteration rule in the original x space, in

matrix notations, as

Where E[x] is the mean values, Cx is the covariant matrix

(6). Design point in the normalized z space is obtained by

(7). Reliability index β is calculated as follows

(8). Check the convergence of β

This algorithm may fail to converge in some situations. It may converge very slowly, or

oscillate about the solution without convergence, or diverge away from the solution. In

such situations other possibilities such as simulations are to be used.

23

3.6 COMMENTS ON THE FOSM MEAN VALUE INDEX

In this research paper we are considering the First-Order Second-Moment Method. First order

because we use first-order terms in the Taylor series expansion. Second moment because only

means and variances are needed. Mean value because the Taylor series expansion is about the

mean values. The first-order second-moment mean value method is based on approximating non

normal CDFs of the state variables by normal variables, as shown in Figure 3.6 for the simple case

in which g(R, Q) = R - Q. The method has both advantages and disadvantages in structural

reliability analysis.

Fig 3.6: Mean value second - moment formulation.

Among its advantages:

1. It is easy to use.

2. It does not require knowledge of the distributions of the random variables.

Among its disadvantages:

1. Results are inaccurate if the tails of the distribution functions cannot be approximate by a

normal distribution.

2. There is an invariance problem: the value of the reliability index depends on the specific

form of the limit state function.

24

Chapter - 4

FIRE RESISTANT DESIGN OF BEAMS

4.1 GENERAL

In performance-based fire safety design, the fire performance of a structure needs to be accurately

evaluated, which requires the accurate prediction of temperatures in the structure. While a finite-

element or a finite-difference analysis may be carried out for this purpose, structural engineers

generally prefer a simpler method. Current design codes and standards provide a choice of three

methods for determining the fire resistance of RC beams: (a) tabulated data (tables or charts), (b)

simplified method, and (c) advanced method. The tabulated data offered by BS 8110-2 (1985),

FIP/CEB (2004), EN 1992-1-2 (2004), ACI 216.1 (2007) and AS 3600 (2009) provide the fastest

and most direct way of determining the minimum dimensions and concrete cover depth of an RC

beam for a required fire resistance rating. However, these tables were established on the basis of

empirical relationships obtained from limited fire resistance test results (Narayanan and Beeby

2005), and as a result, many significant factors are not properly considered in this prescriptive

approach. More importantly, it has been shown that this approach does not always provide a

conservative fire resistance evaluation for RC beams (Kodur and Dwaikat 2011). By contrast, both

the simplified and the advanced methods are based on numerical calculations, and therefore they

provide a more rigorous and flexible way for fire resistance design. The advanced method

recommended by Eurocode 2 (EN 1992-1-2 2004) involves detailed thermal and mechanical

analyses of RC beams exposed to fire (i.e., a heat transfer analysis to determine temperatures

within the cross-section, followed by an accurate mechanical response analysis to determine the

strength degradation of the RC beam).The advanced method requires detailed descriptions of

constitutive laws for various materials at elevated temperatures as well as complex computations.

The simplified method specified in EN 1992-1-2 (2004) and BS 8110-2 (1985) provides a

reasonably easy and accurate analytical approach for the evaluation of fire resistance: it involves a

simple structural analysis based on the “500oC isotherm method” (Anderberg 1978) or the “Zone

method” (Hertz 1981, 1985).

25

4.2 BEHAVIOUR OF FIRE

Fires behave differently. Some burn slowly and evenly; others are extremely hot, burning fiercely

and quickly. Different fires have different coloured flames. Some fires start easily; others don’t.

Some fires produce deadly gases that could kill you if not ventilated. The behaviour of the fire

often depends on the fuel. Other factors or variables may include where the fuel is situated and how

near it is to other fuels, the weather (especially wind and relative, humidity), oxygen concentration

and, in the case of outside fires, the shape of the terrain.

Fuel Type: Different fuels catch fire at different temperatures. It takes a certain amount of heat

energy to change any particular material into a gas (if it is not already). Then it takes more heat

energy to trigger the reaction with oxygen. The amount of heat produced depends on the

molecules that make up the fuel. The most flammable fuels are hydrocarbons (contain carbon

and hydrogen) that recombine with oxygen quite easily to form carbon dioxide, water and other

gases.

Size: How quickly a fuel catches fire and burns relates to the surface area or the size of the fuel.

For example, large pieces of wood take a lot longer to absorb heat energy to ignition

temperature. A twig catches fire easily because it heats up easily.

Surface area: The bigger the area of the surface of the fuel, the more oxygen molecules

cancollide with the surface. The more oxygen molecules that collide per second with the fuel,

the faster the combustion reaction is.

You can increase the surface area of a solid by breaking it up into smaller pieces. If you chop or

break up wood into small pieces, it will ignite and burn more quickly than larger pieces of

wood. People often start fires with kindling (small pieces of wood) that they criss-cross to

allow greater surface area and lots of oxygen getting in and around. A powder has the largest

surface area and will have the fastest reaction rate.

Heat produced: How much energy is released in the reaction and how quickly the fuel burns

depends on what the fuel is made up of. Different compounds react with oxygen differently –

some contain lots of heat energy while others produce a smaller amount. The reaction with the

oxygen may happen very quickly or more slowly.

26

Amount: The amount of fuel available to burn is known as the fuel load. The bigger the fuel

load, the more intense the fire will be in terms of heat energy output.

Moisture content: If the fuel isn’t dry enough, it won’t burn. The less moisture in the fuel, the

more likely it will ignite and burn.

Oxygen availability: The amount of oxygen available will affect the rate of burning. A low

concentration of oxygen will slow the burning right down. An example of dangerous fire

behaviour that can occur in a situation where there is a low concentration of oxygen is called

back draught. This is when an enclosed fire has used up most of the oxygen and is just

smouldering. If there is a sudden influx of oxygen (like someone opening a door or window),

the fire will immediately explode into flame.

Relative humidity: This reflects the amount of moisture in the air. If relative humidity is low,

it will contribute to the drying of fuels. If it is high, fuels will absorb moisture from the air,

making ignition more difficult.

Wind: This is a major factor in determining fire spread. Wind affects the rate of oxygen supply

to the burning fuel (controlling combustion) and it tilts the flame forward so that unburned fuel

receives energy by radiation and convection at an increased speed. Wind can also dry out the

fuel.

Rainfall: This also has an effect on wetting fuels, but absorption of moisture is dependent on

fuel size. Fine fuels absorb moisture more quickly than coarse fuels. Lack of rain (precipitation)

is the biggest factor affecting the drying process of fuels.

Increased temperatures: These will dry out potential fuel so that there will be less preheating

of fuels to reach ignition temperature.

Terrain: The terrain (shape of the land) has significant influence on wild fire behaviour. Steep

slopes may increase fire speed because fuels (scrub and vegetation) are preheated ahead of the

fire through convection and radiation.

Rugged terrain with narrow valleys, sharp ridges and irregular slopes affect the direction and

rate of fire spread. For example, narrow valleys can funnel winds, increasing the rate of spread

of a fire due to convection. The direction a slope is facing will depend on how much sun it gets.

This will affect the amount of drying the fuels get. The drier the fuel, the faster it will burn.

27

4.3 PROCESS OF FIRE DEVELOPMENT

Fig. 4.1 shows a typical time temperature curve for the complete process of fire development inside

a typical room, assuming no fire suppression by sprinklers or fire fighters. Not all fires follow this

development because some fires go out prematurely and others do not reach flashover, especially if

the fuel item is small and isolated or if there is not enough air to support continued combustion. If a

room has very large window openings, too much heat may flow out the windows for flash over to

occur.

Fig.4.1 Time temperature curve for fire development

The load carrying capacity of structural member under fire is of major concern. Members are

usually designed for normal loads and when a fire breaks out, it causes an extra load on the

structure. This may lead to instability of the structure and successive failure. Concrete structures

have a good resistance to fire. The behaviour of reinforced concrete structures in the event of a fire

attack is generally satisfactory. However, due to sudden spalling, failure may occur prematurely

and fire resistance may be reduced substantially. Different approaches are used to design a concrete

structure under fire viz. tabulated data, simplified calculation methods based on reduced cross-

section, experimental tests and advanced calculation methods like numerical methods. Simple

analytical methods required to predict the capacity of structural elements are recommended in

28

Eurocode 2 (EN 1992-1-2:2004). They are 5000 C isotherm method and zone method for various

structural elements. Here we are using 500°C isotherm method.

4.4 SIMPLIFIED CALCULATION METHODS

Simplified calculation methods are mainly of two kinds i.e., 500°C isotherm method and zone

method. Brief reviews of both methods are given below

4.4.1 500°C ISOTHERM METHOD

This method is applicable to a standard fire exposure and any other time heat regime, which cause

similar temperature fields in the fire exposed members. Concrete with temperatures lower than

500°C is assumed to have full strength and the rest is disregarded. It should be noted that, the 500

oC isotherm method was originally devised for RC sections subjected to pure bending, where

failure is generally controlled by the yielding of steel tension reinforcement (Anderberg 1978; fib

2007). Rigberth (2000) has however demonstrated that the simplified method generates slightly

conservative estimates of fire resistance than the exact numerical model (i.e., the advanced

method). The basic design procedure as per 500°C isotherm method is given below.

Step 1: The 500°C isotherm for the specified fire exposure is calculated using standard fire or

parametric fire.

Step 2: A reduced width bf and effective depth of the cross-section is obtained by excluding the

concrete outside the 500°C isotherm. The temperature of the individual reinforcing

bars is evaluated from the temperature profiles as given in Annex A of EN 1992-1-

2:2004(E).Those reinforcing bars which fall outside the reduced cross section may also

be included in the calculation of the ultimate load carrying capacity of the fire exposed

cross section.

Step 3: The reduced strength of the reinforcement due to the elevated temperature is determined

according to EN 1992-1-2:2004 (E) [4].

Step 4: The conventional method for the determination of the ultimate strength based on limit-

state design as specified in IS 456:2000 [7] is used to find the ultimate load-carrying

capacity for reduced cross-section with strength of the reinforcing bars.

29

Step 5: The ultimate load-carrying capacity is compared with the design capacity or,

alternatively, the estimated fire resistance with the required fire resistance.

Fig 4.2: Reduced section after fire

4.4.2 ZONE METHOD

In this method, the cross section is divided into a number of parallel zones of equal thickness

(rectangular elements) where the mean temperature and the corresponding mean compressive

strength and modulus of elasticity (if applicable) of each zone is assessed. This method although

more laborious, provides a more accurate method than the 500°C isotherm method. The method is

applicable to the standard temperature curve only. The reduction of the cross-section is based on a

damaged zone of thickness az (Fig 4.3) at the fire exposed surfaces which is calculated as follows:

Fig 4.3: Reduction of cross section of slab exposed to fire

30

1) The half thickness of the wall is divided into n parallel zones of equal thickness as shown in Fig

4.4, where n≥ 3 for slabs exposed to two side fire.

2) The temperature is calculated for the middle of each zone.

3) The corresponding reduction factor for compressive strength, kc( θi) is determined.

4) The mean reduction coefficient for a particular section, incorporating a factor (1- 0.2/n) which

allows for the variation in temperature within each zone, may be calculated by expression.

n

i

icmc kn

nk1

, )()2.01(

Where:

n is the number of parallel zones in width w

w is half the total width

m is the zone number

5) The width of the damaged zone for beams, slabs or members in plane shear is calculated using

Expression

])(

1[,

mc

mc

Zk

kwa

Where kc(θm) denotes the reduction coefficient for concrete at point M.

6) After the determination of reduced cross section, the fire design follows the normal temperature

design procedure.

Fig 4.4: Division of slab with both sides exposed to fire, into zones for use in calculation of

strength reduction and az value

31

4.4.3 ADVANCED CALCULATION METHODS

This method simulating the behaviour of structural members, parts of the structure or the entire

structure. In the code, only principles are given and no detailed design rules are provided.

Table 4.1 Reduction factors for yield strength of steel as per EC - 2

4.5 BRIEF REVIEW OF CODE PROVISIONS

IS456:2000 provides minimum width of beam and separate nominal cover requirement for simply

supported and continuous beams required for different fire resistance which is shown in Table

below.

32

Fire

Resistance

(min)

Minimum

beam width

(mm)

Nominal cover(mm)

Simply

supported

Continuous

30 200 20 20

60 200 20 20

90 200 20 20

120 200 40 30

180 240 60 40

240 280 70 50

Table 4.2: Minimum dimension and nominal cover to meet specified period of fire

resistance for RCC beam (IS456:2000)

Euro code 2, Part 1–2: Structural fire design also gives a choice of advanced, simplified or tabular

methods for determining the fire resistance of beam similar to slabs. The tables have been

developed on an empirical basis and are confirmed by experience and theoretical evaluation of tests

and which gives recognized design solutions for the standard fire exposure up to 240 minutes. The

values given in the tables can be apply to normal weight concrete (2000 to 2600 kg/m3), made with

siliceous aggregates. Code specifies to reduce the minimum dimension of the cross-section of

beams and slabs by 10% for calcareous aggregates or lightweight aggregates concrete. No further

checks are required concerning shear, torsion capacity, anchorage details and spalling, except for

surface reinforcement.

4.6 TEMPERATURE PROFILE

Calculated temperature profiles are provided in Annex A of Eurocode2 for the fire resistant

design of beams. Temperature profile gives temperature distribution across quarter cross section of

beam by taking symmetry into account. Fig 4.5 shows how the temperature profiles represent the

33

temperature in the cross section of beams and columns by taking symmetry into account. In

figure, area 2 represents full cross section of beam and area1 represents area for which

temperature profiles provided in Eurocode2. As an illustration temperature profile of 160mm

x 300mm beam for 30 and 60minute fire exposure is given below.

Fig 4.5: Area of cross-section for which the temperature profiles presented in Eurocode2

34

Fig 4.6 Temperature profile of 160mm x 300mm beam (Anne: A of Euro code2)

Fig 4.7 Fig 4.8

Fig 4.7: Temperature profiles for 90 min exposure of a beam, h x b = 300 x 160

Fig 4.8: 5OO°C isotherms for a beam, h x b = 300 x 160

35

Chapter – 5

THERMAL ANALYSIS USING FE SOFTWARE

5.1 SIMPLE METHOD FOR PREDICTING TEMPERATURES IN REINFORCED

CONCRETE BEAMS EXPOSED TO A STANDARD FIRE

In performance-based fire safety design, the fire performance of a structure needs to be accurately

evaluated, which requires the accurate prediction of temperatures in the structure. While a finite-

element or a finite-difference analysis may be carried out for this purpose, structural engineers

generally prefer a simpler method. Recent years have seen a gradual transition from the

prescriptive approach to the performance-based approach in the fire resistance design of reinforced

concrete (RC) structures; the latter provides a more rational and flexible design tool. In the

performance based design approach, the fire resistance of the structure needs to be accurately

evaluated, which requires the accurate prediction of temperature fields in the structure. A generic

approach for temperature field analysis is to employ the finite-element (FE) or the finite-difference

(FD) method using an appropriate software package. However, structural engineers often prefer a

simpler approach as they may not have the expertise to deal with the complexity involved in such

numerical computations within the tight timeframe of a design task.

Attempts have been made in the past to develop simple design-oriented methods for the prediction

of temperature profiles in RC members subjected to one dimensional heat transfer (Hertz 1981;

Harmathy 1993; Kodur et al. 2013) or two-dimensional heat transfer (Wickstrom 1986; Desai

1995, 1998; Abbasi 2003; Abbasi and Hogg 2005; Kodur et al. 2013). Wickstrom (1986) presented

a method for predicting temperatures in fire-exposed RC structures. For the ISO 834 standard fire

curve and normal weight concrete, this method needs only two input parameters: the fire-exposure

time and the concrete depth. Wickstrom’s (1986) method was used by Eamon and Jensen (2012) to

determine the 500 oC isotherm and the temperature history of steel reinforcement in the fire

resistance analysis of pre stressed RC beams. Desai (1995, 1998) assumed that the isotherms of a

rectangular RC beam are parallel to the exposed surfaces of the beam, and then proposed a simple

equation to determine these isotherms. Despite the existence of these methods, there is still

considerable uncertainty with the prediction of temperatures in RC beams exposed to a standard

fire:

36

(a) The above-mentioned methods have all been validated using only a limited test database, so

their ability to provide close predictions of other experimental results is uncertain.

(b) The differences between the predictions of these models are unclear and may be substantial.

5.2 FIRE RESISTANCE DESIGN

Current design codes and standards provide a choice of three methods for determining the fire

resistance of RC beams: (a) tabulated data (tables or charts), (b) simplified method, and (c)

advanced method. The tabulated data offered by BS 8110-2 (1985), FIP/CEB (2004), EN 1992-1-2

(2004), ACI 216.1 (2007) and AS 3600 (2009) provide the fastest and most direct way of

determining the minimum dimensions and concrete cover depth of an RC beam for a required fire

resistance rating. However, these tables were established on the basis of empirical relationships

obtained from limited fire resistance test results (Narayanan and Beeby 2005), and as a result,

many significant factors are not properly considered in this prescriptive approach. More

importantly, it has been shown that this approach does not always provide a conservative fire

resistance evaluation for RC beams (Kodur and Dwaikat 2011). By contrast, both the simplified

and the advanced methods are based on numerical calculations, and therefore they provide a more

rigorous and flexible way for fire resistance design. The advanced method recommended by

Eurocode 2 (EN 1992-1-2 2004) involves detailed thermal and mechanical analyses of RC beams

exposed to fire (i.e., a heat transfer analysis to determine temperatures within the cross-section,

followed by an accurate mechanical response analysis to determine the strength degradation of the

RC beam). The advanced method requires detailed descriptions of constitutive laws for various

materials at elevated temperatures as well as complex computations.

The simplified method specified in EN 1992-1-2 (2004) and BS 8110-2 (1985) provides a

reasonably easy and accurate analytical approach for the evaluation of fire resistance: it involves a

simple structural analysis based on the “500 oC isotherm method” (Anderberg 1978) or the “Zone

method” (Hertz 1981, 1985).The former method is based on the assumption that the part of

concrete with temperatures exceeding 500 oC has completely lost its strength, whilst the rest of the

concrete retains its full initial strength. In the latter method, the cross-section is divided into several

zones of equal thickness and the reduced strength of each zone is evaluated. It should be noted that,

the 500 oC isotherm method was originally devised for RC sections subjected to pure bending,

where failure is generally controlled by the yielding of steel tension reinforcement (Anderberg

1978; fib 2007). Rigberth (2000) has however demonstrated that the simplified method generates

37

slightly conservative estimates of fire resistance than the exact numerical model (i.e., the advanced

method). The above review of current design methodology clearly indicates that, when fire

resistance is assessed using a calculation-based method (i.e., the simplified or the advanced

method), the first step is to determine temperatures in the beam cross-section. Therefore, an

explicit, design-oriented method for predicting temperatures in RC beams is always needed by

design engineers. The availability of such a temperature prediction method, especially in

combination with the simplified method for structural analysis, facilitates a quick yet sufficiently

accurate approach for the fire resistance evaluation of RC beams under a standard fire exposure.

5.3 EXISTING METHODS FOR TEMPERATURE PREDICTIONS

The temperature fields of an RC beam exposed to a standard fire depends on several factors, such

as the fire-exposure time, the beam width, the aggregate type and the moisture content of concrete.

Many studies have been carried out to determine the temperature fields of RC beams. In total, four

simple methods have been found in the published literature for predicting the temperature fields of

RC beams under a standard fire exposure as summarized below.

5.3.1 WICKSTROM’S METHOD

For an RC beam made of normal weight concrete and exposed to the ISO 834 standard fire,

Wickstrom’s (1986) method predicts the temperature rise ∆T(x, y) at a given point (x, y) in the

concrete at the fire-exposure time t by the following equation:

fyxyxyxw nnnnnnnyxT ])2([),(

Where ∆θf is temperature rise of the standard ISO 834 fire curve; nw is the ratio between the

temperature rise of the beam surface to that of the fire, which depends on the fire-exposure time

and is given by:

88.00616.01 tnw

In Eqn above, t (in hours) is the fire exposure time; nx (or ny) is a function of fire-exposure time

and the ratio between the thermal diffusivity of the RC beam (i.e., α) and a reference value ac (i.e.,

αc = 417 × 10–9 m2 s–1):

38

81.0)ln(18.02

x

tXn

c

x

Where x (in m) is the distance between the point under consideration and the fire-exposed surface

in the beam width direction. A similar equation is used to calculate ny by replacing x with y in the

beam height direction. It should be noted that the above equation only holds if

5.0)0015.0(6.32)( thyx

Where h (in m) is the dimension of the RC beam in the direction under consideration.

5.3.2 DESAI’S METHOD

Desai (1995, 1998) adopted a simple equation to predict the temperature profiles of rectangular RC

beams based on his own experimental results and those of some others (e.g., Lin et al. 1988; Wade

1991). The temperature T, (oC) of a contour at a distance x (in mm) from the fire-exposed surfaces

of the beam was assumed to be influenced by the following factors:

(a) t, the fire-exposure time (in min);

(b) b, the width of beam cross-section (in mm); and

(c) r, the ratio between beam height and beam width.

Desai’s (1998) equation is as follows

Where

B=0.085

C=0.000221

The applicability of above equation is limited to beams satisfying the following two conditions:

100 mm < b <300 mm and 1 < r < 3. If r ≤ 1.5, then it is assumed that r = 1.5 in the calculation.

The initial temperature calculated by above equation varies from negative values to positive ones

depending on the beam geometry and the distance from the fire-exposed surfaces, which is an

39

undesirable feature. In the comparisons presented later in the paper, the temperature rise is taken as

the temperature predicted using equatn minus the initial temperature of a standard fire (i.e., 20oC).

5.3.3 ABBASI AND HOGG’S METHOD

Abbasi and Hogg’s (2005) method was developed only for predicting the temperatures of FRP

rebars in beams subjected to the standard ISO 834 fire. In this method, the difference between the

rebar temperature and the fire temperature, after 30 minutes of fire exposure, is assumed to be in

the following exponential form:

θ – T = A′ exp (–β * t)

where θ and T are the fire temperature and the rebar temperature, respectively; A′ is an empirical

constant and = 767; t is the fire exposure time (in min); β, being the gradient of versus t curves,

was derived from fire test data by regression analysis and has the following exponential form:

where c′ (in mm) is the concrete cover thickness; a′, b′ and d′, being empirical constants

determined from regression of test data, are as follows:

a′ = 0.001, b′ = 7.602, d′ = –23.623

By the combined use of above equations, the rebar temperature in the beam subjected to the

standard ISO fire can be obtained as follows:

As this method was derived on the basis of fire test data after 30 minutes of fire exposure, the

initial temperature calculated by above equation (i.e., t = 0) is always equal to –747oC, which is

unreasonable. Again for comparison purposes in the paper, the temperature rise is taken as the

predicted temperature minus the initial temperature of a standard fire (i.e., 20oC).

5.3.4 KODUR ET AL.’S METHOD

More recently, Kodur et al. (2013) modified Wickstrom’s (1986) equations and proposed a simple

method for predicting temperatures over cross-sections of fire-exposed RC members. The method

was derived from temperature data generated from a FE parametric study on RC members exposed

40

to a standard fire. Based on a trial–and-error process as well as regression analysis, the following

formulae were developed for calculating temperatures in RC members:

1-D heat transfer (for RC slabs):

2-D heat transfer (for RC beams and columns):

where at n is an approximation of the standard fire curve; and c1 and c2 are coefficients whose

values depend on the aggregate type (i.e., siliceous or calcareous) and the strength (i.e., high-

strength or normal-strength) of concrete. nx is a function of the fire-exposure time and the distance

from the point under consideration to the fire exposure surface and is given by:

where t is the fire exposure time (in hours); and x is the distance from the point under consideration

to the fire exposure surface in the member width direction. A similar equation was also proposed

for determining ny in the member height direction. For comparison purposes in the present paper,

the temperature rises of all points of the member section are assumed to be the predictions of the

model minus 20oC.

5.3.5 FE METHOD ANALYSIS

In the FE analysis, the ISO 834 standard fire curve was adopted as the thermal load. Heat fluxes

flow to the bottom and two side surfaces of RC beams and exchange heat with them through

convection and radiation, whereas heat transmission occurs within concrete through conduction.

The time-dependent temperature distribution in an RC beam is described by Fourier’s differential

equation for heat conduction (Purkiss 2007):

where k, ρ and c denote the temperature-dependent thermal conductivity, density and specific

heat capacity, respectively; Q is the rate of heat generated internally per unit volume; and t is the

time variable. For heat transfer analysis of an RC beam exposed to fire, internal heat generation is

inactive (i.e., Q = 0) (fib 2007). To solve the above differential equation, the initial temperature

41

distribution and proper boundary conditions are required. The initial temperature distribution in the

RC beam at t = 0 is described by:

The free boundary condition is applied to the unexposed surface (i.e., the top surface) of the beam

specimens. The heat fluxes exchange heat with the fire- exposed surfaces of the RC beam via

convection and radiation, which can be depicted by means of Robin’s boundary condition (Purkiss

2007):

where n represents the outward normal direction of the beam surface; hc is the convective heat

transfer coefficient; Tf denotes the fire temperature in degree Celsius; Tz is the absolute zero

temperature and is equal to -273.15 oC; φ is a configuration parameter; εm and εf are the emissivity

coefficients of the exposed surfaces and of the fire, respectively; and σ is the Stephan- Boltzmann

constant. The values recommended by EN 1991-1-2 (2002) were adopted to define the boundary

conditions and the corresponding initial temperature:

(a) Coefficient of convective heat transfer of the exposed surfaces: hc = 25 W/(m2⋅K);

(b) Configuration factor for radiation: φ = 1.0;

(c) Emissivity of the exposed surfaces: εm = 0.8;

(d) Emissivity of the fire: εf = 1.0;

(e) Stephan-Boltzmann constant: σ = 5.67×10–8 W/m2.K4;

(f) Initial temperature: T0(x, y) = 20oC.

The thermal conductivity of concrete made of siliceous aggregate or calcareous aggregate shown in

Figure 5.1 is determined according to ENV 1992-1-2 (1995) because in the updated version (EN

1992-1-2 2004) the lower and upper limits rather than specific values are provided for thermal

conductivity of concrete (Capua and Mari 2007). The effect of the moisture content of concrete is

taken into account by adjusting the specific heat capacity to represent the latent heat of water

evaporation. An early study conducted by Harmathy (1965) showed that the presence of moisture

in building components is beneficial to their fire resistance if it is not so excessive as to trigger the

spalling of concrete. Therefore, a slightly lower-than- normal moisture content of 1.5% by weight

(i.e. u =1.5%) was assumed in the FE analysis to ensure conservative predictions (Hertz 1981;

42

Biondini and Nero 2011). According to EN 1992-1-2 (2004), a peak value (i.e., cc,peak) of 1.470

kJ/kg was adopted for the specific heat capacity of concrete to implicitly consider the latent heat of

evaporation component [see in Figure 5.1].The thermal properties of steel are not considered in the

FE analysis since the effects of steel reinforcement are usually negligible in heat transfer analysis

(Rodrigo et al. 2010; Biondini and Nero 2011). In other words, the temperature of internal steel

reinforcement is taken to be equal to that of concrete at the same location. It should be noted that

the thermal properties of concrete at elevated temperatures and the thermal boundary conditions of

RC beams in heat transfer analysis were both determined according to the appropriate Eurocodes

(EN 1991-1-2 2002; EN 1992-1-2 2004). These code provisions have been extensively validated by

previous standard fire tests in the literature.

5.4 MATERIAL BEHAVIOUR AT ELEVATED TEMPERATURES

Thermal properties, mechanical properties and deformation properties are the input material

properties of concrete and steel in ANSYS13 for thermal analysis.

In order to make calculations of temperatures in fire exposed structures, it is necessary to know the

thermal properties of the material. Specific heat, thermal conductivity and density are the thermal

properties needed for ANSYS13 analysis.

5.4.1 THERMAL PROPERTIES

Thermal conductivity is the quantity of heat transmitted through a unit thickness in a direction

normal to a surface of unit area, due to a unit temperature gradient under steady state conditions.

The thermal conductivity of concrete (λc) is temperature dependent and varies in a broad range

depending on the type of aggregate.

5.4.1.1 THERMAL CONDUCTIVITY

As per Eurocode 2 the thermal conductivity of concrete may be determined between lower and

upper limit values. The variation of upper limit and lower limit of thermal conductivity with

temperature is as shown in Fig 5.1. Eurocode2 provision for thermal conductivity is used for

ANSYS13 analysis

43

Fig 5.1: Thermal conductivity of concrete as a function of temperature

For steel, thermal conductivity varies according to temperature reducing linearly from 54W/mK at

00C to 27.3W/mK at 8000C as shown in Fig 5.2.For simple calculations, the thermal conductivity

can be taken as 45W/mK but it is more accurate to use the equations given below. The thermal

conductivity of concrete may be determined between lower and upper limit values, given below.

λs= 54-0.0333T 200C≤T<8000C

λs= 27.3 8000C≤T≤12000C

Where, λc is the thermal conductivity in W/mK and T is the steel temperature

44

Fig 5.2: Thermal conductivity of steel as a function of temperature.

5.4.1.2 SPECIFIC HEAT

The specific heat(Cp) is the amount of heat required to heat a unit mass of the material by one

degree(J/kg/K).The specific heat of concrete varies in a broad range depending on the moisture

content as shown in Fig 5.3 (Eurocode2). The peak between 1000C and 2000C allows for water

being driven off during the heating process. Approximate design values are 1000J/kgK for

siliceous and calcareous aggregate concrete and 840J/kgK for light weight concrete.

Fig 5.3: Specific heat Cp(θ) as function of temperature at moisture content u of 1.5 by

weight for siliceous concrete

0

10

20

30

40

50

60

0 500 1000 1500

Th

erm

al

con

du

ctiv

ity(W

/mK

)

Temperature(0C)

45

The specific heat Cp(8) of dry concrete (u = 0%) may be determined from the following:

Siliceous and calcareous aggregates having a moisture content of 1.5%

Where the moisture content is not considered explicitly in the calculation method, the

function given for the specific heat of concrete with siliceous or calcareous aggregates may

be modelled by a constant value, Cp.peak, situated between 100°C and 115°C with linear

decrease between 115°C and 200°C.

Cp.peak = 900 J/kg K for moisture content of 0 % of concrete weight

Cp.peak 1470 J/kg K for moisture content of 1,5 % of concrete weight

Cp.peak = 2020 J/kg K for moisture content of 3,0 % of concrete weight.

For steel, specific heat varies according to temperature. For simple calculation the specific heat can

be taken as 600J/kgK. But it is more accurate to use the following equation.

Cp= 425+0.773T-1.6910-3T2+2.2210-6T3 200C≤T<6000C

= )738(

13002666

T 6000C≤T<7350C

= )731(

17820545

T 7350C≤T<9000C

= 650 7350C≤T<12000C

The density of concrete depends on the aggregate and the mix design. Typical dense concrete has a

density of about 2300kg/m3. When heated to 10000C the density of most concretes will be reduced

by up to 100kg/m3 due to the evaporation of free water, which has a minor effect on thermal

response. Other than moisture change, the density of concrete does not change much at elevated

46

temperature, except for lime stone (calcareous) aggregate concrete which decomposes above 8000C

with a corresponding decrease in density.

The emissivity related to the concrete surface 0.7

Convection factor is 25 W/m2K.

5.4.2 MECHANICAL PROPERTIES

The stress strain curves for concrete with a compressive strength of f’c at various temperatures are

shown in Fig 5.4. For mechanical properties, as a conservative estimate, concrete is assumed not to

recover any strength in the cooling phase. Thus its stress strain diagram is determined based on the

maximum temperature the concrete attains.

Fig 5.4 Stress-strain curves for concrete at various temperatures

It can be seen from the figures that the properties of concrete vary significantly with temperature,

with large decrease in strength (stress) once the temperature exceeds 5000C. The equations that

describe these curves are as follows:

Stress-strain relationship

max

2

max,

max,

,

max

2

max,

max,

,

,3

1

,1

T

T

Tc

T

T

Tc

c

f

f

Str

ess

Strain

2000C

4000C

6000C

8000C

47

TC

CTCT

f

CTCf

f c

c

Tc

0

00

00

,

8740

8744501000

20353.2011.2

45020

Where T = temperature (0C), f’c and f’c,T = concrete compressive strength at room temperature and

high temperature, respectively, σc stress in concrete, ε = strain in concrete, and εmax,T = strain at

peak stress in the stress-strain curve of concrete.

Modulus of elasticity of steel decreases with increase in temperature. Eurocode2 provide values for

the parameters (ratio of modulus of elasticity at elevated temperature, ESθ to that at room

temperature, ES) of the stress-strain relationship of hot rolled and cold worked reinforcing steel at

elevated temperature in Cl.3.2.3. These parameters are given in Table 5.1. These relations include

the effect of creep at elevated temperatures and were obtained at heating rates approximately the

same as those that occur in a fire in actual practice.

Steel Temperature(0C)

Es,θ/Es

Hot rolled steel Cold worked steel

20 1.0 1.0

100 1.0 1.0

200 0.9 0.87

300 0.8 0.72

400 0.7 0.56

500 0.6 0.4

600 0.31 0.24

700 0.13 0.08

800 0.09 0.06

900 0.07 0.05

1000 0.04 0.03

1100 0.02 0.02

1200 0 0

Table 5.1: Values for the Parameters of the Stress-Strain Relationship of

Reinforcing Steel at Elevated Temperature

48

The relations have been generalized for other structural steels by assuming that, for a given

temperature, the stress-strain curves are same for all steels, but the stress below which the stress-

strain relation is linear, is proportional to the yield strength of steel. The equations that describe the

relations between the stress in steel (fy), the strain (εs) and the temperature of the steel (T) are as

follows:

For ps

Sy

Tff

001.0

)001.0,(

Where εp = 4 x 10-6 fyo

9.6)001.0()03.030(exp)04.050()001.0,( TTTf

For ps

)001.0,()001.0(,001.0

)001.0,(TfTf

Tf pspy

Where,

5.4.3 DEFORMATION PROPERTIES

The thermal expansion of material can be related to its temperature by a coefficient of expansion

(α), which can be defined as the expansion of a unit length of the steel when it is raised one degree

in temperature.

For siliceous and carbonate aggregate concrete coefficient of thermal expansion is

α=(0.008T+6)10-6 /0C

and α=1610-6 /0C for expanded shale aggregate concrete.

9.6001.0)03.030(exp1)047.050()001.0(, psps TTTf

49

Co-efficient of thermal expansion of steel is

α = (0.004T+12)10-6 /0C T≤10000C

α=1610-6 /0C T≥10000C

Fig 5.5: Beam modelled in ANSYS14

5.5 THERMAL ANALYSIS USING ANSYS 14

Generally there are many FE software through which we can carry out the temperature analysis of

the structure. Out of these we are using ANSYS 14 for our research. There are other programs such

as TASEF which is a Computer program for temperature analysis of structures exposed to fire.

TASEF stands for Temperature Analysis of Structures Exposed to Fire. The program is based on

the finite element method. It is developed for temperature analysis of two dimensional and axi-

symmetrical structures

TASEF is developed for calculating temperature in fire exposed structures and has a number of

features that makes it particularly suitable for that purpose. Structures may contain several

materials with thermal properties varying with temperature. Latent heat, for instance due to

evaporating water, may be considered. Heat flux to boundaries by convection and radiation from

fires may conveniently be specified. The fire impact is expressed as time-temperature relations.

Heat transfer across internal voids by radiation is calculated considering view factors. Heat transfer

by convection may be approximated assuming appropriate heat transfer parameters. Computed

nodal temperatures are printed at specified times, and when the analysis is terminated maximal

nodal temperatures are printed. All output data are saved on the specified output file.

50

5.5.1 ANSYS14 FINITE ELEMENT MODEL

ANSYS14 software is used for the development of finite element model. It is general purpose finite

element analysis software having many finite element analysis capabilities ranging from simple

linear static analysis to very complex nonlinear dynamic analysis. In general, a finite element

solution may be broken into the following three stages.

(a) Pre-processing

In pre-processing the problem is defined using 3 steps. First key points/lines/areas or

volumes are defined, then element type and material/geometric properties. As a third step

mesh lines/areas/volumes as required are given. The amount of detail required will depend

on the dimensionality of the analysis (i.e. 1D, 2D, axi-symmetric, 3D).

(b) Solution

Here the loads (point or pressure), constraints (translational and rotational) are specifies

and finally solve the resulting set of equations.

(c) Post processing

Post processing means further processing and viewing the results. Results may include lists

of nodal displacements, element forces and moments, deflection plots and stress contour

diagrams.

The FE model was developed using (ANSYS, 2015). Finally, a 3D transient thermal analysis is

conducted to simulate the applied ISO 834 fire curve. Different element types were selected from

the ANSYS element type selection library. The thermal element implemented to model the

concrete material was SOLID 70 respectively. SOLID70 has eight nodes with a single degree of

freedom at each node, defined as temperature as well as 3-D thermal conduction capability.

SOLID70 has 2×2×2 integration scheme for both conductivity and specific heat matrices. In

addition, the element is applicable to conduct 3-D, steady-state and/or transient thermal analysis

(ANSYS, 2015).

51

Time (min) Temperature(0C)

0 20

5 576

10 678

30 842

60 945

120 1050

150 1082

240 1153

480 1257

Table 5.2: Development of Temperature Profile

Fig 5.6: Reduction factors as per Euro code for fy

52

5.5.2 DEVELOPMENT OF TEMPERATURE PROFILE

Beams are meshed into 25 mm × 25mm grid. Finite element mesh for 300 × 600 mm beam is

shown below in Fig 5.7. Here actually we have modelled two beam one a simple beam and the

other a reinforced concrete beam. The values that we got after applying temperature were quite

similar as result. For validation in W.Y. Gao, J.G. Dai and J.G. Teng paper its said that the thermal

properties of steel are not considered in the FE analysis since the effects of steel reinforcement are

usually negligible in heat transfer analysis (Rodrigo et al. 2010; Biondini and Nero 2011). In other

words, the temperature of internal steel reinforcement is taken to be equal to that of concrete at the

same location. It should be noted that the thermal properties of concrete at elevated temperatures

and the thermal boundary conditions of RC beams in heat transfer analysis were both determined

according to the appropriate Eurocodes (EN 1991-1-2 2002; EN 1992-1-2 2004).

.

Fig 5.7: Contours of temperature gradient obtained from thermal analysis of simple beam

and a RC beam using Ansys15

Some of the resulting temperature profiles are shown below in Fig 5.8, Fig 5.9, Fig 5.10, Fig 5.11,

Fig 5.12, Fig 5.13, Fig 5.14 and Fig 5.15 corresponding to 30 minutes, 60 minutes, 90 minutes, 120

minutes, 150 minutes, 180 minutes, 210 minutes, 240 minutes.

53

30 minutes exposure 60 minutes exposure

Fig 5.8 Fig 5.9

90 minutes exposure 120 minutes exposure

Fig 5.10 Fig 5.11

54

150 minutes exposure 180 minutes exposure

Fig 5.12 Fig 5.13

210 minutes exposure 240 minutes exposure

Fig 5.14 Fig 5.15

55

Chapter -6

RELIABLITY ANALYSIS

6.1 GENERAL

Reliability analysis for problems involving both normally and non-normally distributed un-

correlated variables were carried out. Details of work done for the present work is discussed

below. With these problems we will get good idea about how we can calculate the reliability index

of different beams using First order second moment method.

6.2 EXAMPLE PROBLEM-NORMALLY DISTRIBUTED UN-CORRELATED

VARIABLES

Fig 6.1: Cross section of a reinforced concrete beam

The sectional bending moment is MB. The ultimate bending moment is

Where AS is the area of reinforcement, TS the yield stress of reinforcement, TC the

maximum compressive strength of concrete, B width of the beam, D the effective depth of

the reinforcement, and K is a factor related to the stress-strain relation of concrete.

The set of basic variables are Z= (MB, D, TS, AS, K, B, TC). The mean values and standard

deviations are given in table below. Safety margin M is the difference between MU and MB:

56

The set of normally uncorrelated variables X is simply obtained by the relations

Variable Symbol Mean Value Standard deviation

MB ZI 0.01 MNm 0.003 MNm

D Z2 0.30 m 0.015 m

TS Z3 360 MPa 36 MPa

AS Z4 226x10-6

m2 11.3x10

-6 m

2

K Z5 0.5 0.05

B Z6 0.12 m 0.006 m

TC Z7 40 MPa 6 MPa

Table 6.1: Random variables considered

Iteration no. 1 2 3 4 5

:I 0 2.4018 2.6284 2.6150 2.6157

:2 0 -0.9771 -0.8319 -0.8328 -0.8311

:3 0 -1.8437 -1.8488 -1.8669 -1.8673

:4 0 -0.9219 -0.7904 -0.7922 -0.7907

:5 0 0.0552 0.0371 0.0372 0.0370

:6 0 -0.0276 -0.0187 -0.0187 -0.0186

Β 0 3.3141 3.4131 3.4131 3.4131

Table 6.2: Results of Iteration

The iteration is stopped at x(5),and the reliability index is 3.4131.

57

6.3 EXAMPLE PROBLEM-NON-NORMALLY DISTRIBUTED UN- CORELATED

VARIABLES

A cantilever steel beam (ISLB 450) of span l is subjected to a load P at the free end. The resisting

moment capacity of a section is taken as Fy, Z, where Fy is the yield stress and Z is the section

modulus. Hence at the limit state of collapse in flexure, the safety margin can be written as

M= Fy Z-Pl

Variable Symbol Mean value Standard deviation

Fy ZI 0.32 kN/mm2 0.032 kN/mm

2

Z Z2 1400x103

mm3 70x10

3 mm

3

p Z3 100 kN 40 kN

Table 6.3: Random variables considered

Fy and Z are normally distributed and P is log normally distributed. Calculate β if l=2m.

Iteration no. 1 2 3 4

x1 -1.3943 -0.6343 -0.5815 -0.5775

x2 -0.6971 -0.2828 -0.2762 -0.2758

x3 2.3980 2.0908 2.0945 2.0955

Β 2.8601 2.2031 2.1912 2.1911

Table 6.4: Results of Iteration

The iteration is stopped at x(4),and the reliability index is 2.1911.

58

6.4 VALIDATION OF JOURNAL PROBLEM

The base beam for consideration is taken as a rectangular section with b=305mm(12 in.), h=610

mm(24 in.), fc’=28MPa (4 ksi) with siliceous aggregate, four #9 tension steel bars, and 38-mm

(1.5-in.) cover to a #3 stirrup on the sides and bottom[total cover to tension bar, 48 mm (1.875

in.)]. The base beam simply spans 4.5 m (15 ft) and is uniformly loaded with a D/D+L ratio of

0.50. Variations of this beam are reported in the results section. All beams are minimally designed

according to ACI 318 (ACI 2011) in terms of moment capacity (ɸ Mn =Mu), with the design load

combination relevant to this study as discussed previously: 1.2D+1.6L. All beams are tension

controlled, with ɸ=0.90.

A more difficult effect to model is the change in temperature throughout the section as external

temperature and time changes. This is a function of section geometry, material density, specific

heat, and other factors. If conduction is the only heat transfer mechanism and if thermal

conductivity is constant, two-dimensional heat transfer and resulting temperature T with respect to

time t and coordinate directions x and y within a section is governed by the following relationship:

However, this expression often becomes difficult to solve analytically. Thus, various models have

been proposed to approximate this behavior, including finite-element approaches (Bratina et al.

2005; Dwaikat and Kodur 2008; Wang et al. 2011) and semi empirical approaches (Wickstrom

1986; Hertz 1981).For the reliability analysis used in this paper, a large number of simulations are

needed. This practically precludes the use of involved finite-element analysis (FEA) approaches,

because the required computational effort becomes too great. However, FEA is generally only

needed for complex nonstandard cases, whereas the empirical approaches available can often

provide good results for regularly shaped sections subjected to standard fires, which are of interest

in this study. To determine how internal temperature changes in the section as a function of time (t;

hours) and external temperature (T), a specially calibrated version of Wickstrom’s model

(Wickstrom 1986) is used. The Wickstrom model was developed by conducting a series of FEAs of

RC sections exposed to fire and determining the resulting concrete and reinforcement bar

temperatures as a function of time (Wickstrom 1985). The analyses included a RC material model

that considered varying thermal conductivity, the influence of water evaporation, and nonlinear

thermal boundary conditions. From the results of the analysis, curves were constructed to fit to the

59

temperature data as a function of the fire time-temperature curve, individual rebar placement within

the section, and thermal diffusivity. Similar approaches have been developed for different materials

and design scenarios in the form of tables and charts by ACI (1989), PCI (Gustaferro and Martin

1989), ASCE (2006), and CEN (2002a, b), among others. For the Wickstrom model, excellent

agreement to the FEA results was reported for regular section shapes (Wickstrom 1986).

6.4.1 RELIABILITY ANALYSIS

In this study, Hasofer and Lind’s algorithm is used to calculate reliability. For a given beam design

and time after fire initiation for which reliability is to be computed, the iteration process becomes

as follows:

Load and resistance RVs are sampled based on the statistical parameters given in Table 6.1, and

basic beam parameters (Mn, cover, width, etc.) are calculated.

Using the results from Wickstrom model, the reduced moment capacity Mn(T) of the sampled

beam at time t at which reliability is to be computed is determined. Time t ranges from 0 to 4 h.

Evaluate the limit state function g=Mn(T)-DM -LsM, substitute in Hasofer and Lind’s algorithm

and get the value of reliability index at each time increment t.

For example at time t=1 hour the calculation of reliability index is as follows, From Wick

storm’s model.

Fig 6.2: Quarter portion of a beam showing the position of reinforcement in terms

of x and y

60

The mean value of fire temperature

T(0C)= 0

279553.3 241.170)1(750 Te

Where t=2 hour

T0=ambient temperature, taken as 200C

T(0C)= 0

279553.3 241.170)1(750 Te =1056.70C

nw=1-0.0616*t(-0.88)

t=2

nw= 0.9665

ns=0.18xlog(α/(0.417x10-6)x t/s2)-0.81

Here s=X=Y=effective cover= (48+28.65/2) mm (Since the cover on both sides are same in this

case ).

α=0.417x10-6

ns= 0.3139

Tr= (nw*(ns+ns-2*ns*ns) + (ns*ns))*T

= (0.9384 x (0.3720x0.3720-2x0.3720x0.3720) + ( 0.3720x0.3720))* 1056.7= 518.7202 0C

X500=

)18.0

4805.4exp(

)10417.0/10417.0( 66

xTxn

xtxx

w

=

)1.10649384.018.0

4805.4exp(

1)10417.0/10417.0( 66

xx

xxx

= 0.0379m

r= (720-(Tr+20))/470= 0.3857 470

))207724.532(720(

470

20))+(Tr-(720

The effective width of the compression block as a function of concrete temperature (Tc)

becomes b(Tc)=b-2xX500=0.250-2x0.0379=0.1742m

Reduced characteristic strength of steel=r x fy = 0.3857x474=182.8218 MPa

Hence the reduced moment capacity is given by (nominal moment capacity as a function of

temperature, Mn(T), can be computed as (National Institute of Standard and Technology 2009).

Mn(T)=Asfy(Tr)

2

)( cTad

61

Where a(Tc)=bTf

TfA

rc

rys

)(85.0

)('

Therefore reduced Mn(T)=

174.23485.02

8218.18265.284/45628218.18265.284/4

22

xxx

xxxxxxx

=242.875kN-m

Reliability index calculation

g=R-S

Here R= Mn(T)

S=external moment= (DL+LL)l2/8

S= (65.6+10)x4.52/8=191.3625 kN-m

g= 242.875-191.3625=51.5125 kN-m

g= As rZ1[Z2-rAsZ1/(1.7Z3(Z4-2X500)]-(Z5+Z6)l2/8

where r=(720-(Tr+20))/470

Tr= (nwx(ns+ns-2xnsxns)+(nsxns))x Z7

ns= 0.18xlog(Z8/(0.417x10-6)x t/s2)-0.81

X500=

)18.0

4805.4exp(

)10417.0/10417.0(

7

66

xZxn

xtxx

w

dg= [1Z

g

;

2Z

g

;

3Z

g

;

4Z

g

;

5Z

g

;

6Z

g

;

7Z

g

;

8Z

g

]

x= (z-u)/sd, where u is the matrix of mean values and sd is the matrix of standard deviations

β= 2

8

2

7

2

6

2

5

2

4

2

3

2

2

2

1 XXXXXXXX

Values for next iteration is given by z’=u +c*dg*((z-u)'*dg-g)/ (dg'*c*dg)

62

Where, C =

(sd1)^2 Cov(Z1,Z2) Cov(Z1,Z3) Cov(Z1,Z4) Cov(Z1,Z5) Cov(Z1,Z6) Cov(Z1,Z7) Cov(Z1,Z8)

Cov(Z2,Z1) (sd2)^2 Cov(Z2,Z3) Cov(Z2,Z4) Cov(Z2,Z5) Cov(Z,Z6) Cov(Z2,Z7) Cov(Z2,Z8)

Cov(Z3,Z1) Cov(Z3,Z2) (sd3)^2 Cov(Z3,Z4) Cov(Z3,Z5) Cov(Z3,Z6) Cov(Z3,Z7) Cov(Z3,Z8)

Cov(Z4,Z1) Cov(Z4,Z2) Cov(Z4,Z3) (sd4)^2 Cov(Z4,Z5) Cov(Z4,Z6) Cov(Z4,Z7) Cov(Z4,Z8)

Cov(Z5,Z1) Cov(Z5,Z2) Cov(Z5,Z3) Cov(Z5,Z4) (sd5)^2 Cov(Z5,Z6) Cov(Z5,Z7) Cov(Z5,Z8)

Cov(Z6,Z1) Cov(Z6,Z2) Cov(Z6,Z3) Cov(Z6,Z4) Cov(Z6,Z5) (sd6)^2 Cov(Z6,Z7) Cov(Z6,Z8)

Cov(Z7,Z1) Cov(Z7,Z2) Cov(Z7,Z3) Cov(Z7,Z4) Cov(Z7,Z5) Cov(Z7,Z6) (sd7)^2 Cov(Z7,Z8)

Cov(Z8,Z1) Cov(Z8,Z2) Cov(Z8,Z3) Cov(Z8,Z4) Cov(Z8,Z5) Cov(Z8,Z6) Cov(Z8,Z7) (sd8)^2

Where sd1, sd2 etc. corresponds to standard deviation of Z1, Z2.etc.respectively.

Repeat the procedure with new values of Z until we get a constant value for β.

Variable Symbol Mean Value C.V

fy Z1 474 0.05

d Z2 562 0.04

fck Z3 34 0.145

b Z4 308 0.04

WDL Z5 65.6 0.05

WLL Z6 15 0.65

T Z7 1056.70C 0.45

α Z8 0.417x10-6 0.06

Table 6.5: Random variables considered

63

Iteration no. 1 2 3

z1 474 473.9050 473.9182

z2 562 561.9107 561.9241

z3 34 33.9956 33.9967

z4 308 307.9959 307.9969

z5 65.6 65.6102 65.6101

z6 15 12.9425 13.0234

Z7 1056.70C 1056.70C 1056.40C

Z8 0.417x10-6 4.1717x10-7 4.1718x10-7

β 0.1094 0.1085 0.1085

Table 6.6: Results of iteration

Fig 6.3 Fig 6.4

Fig 6.3: Validated curve (FOSM)

Fig 6.4: Reliability index degradation curve (Christopher D. Eamon, Elin Jensen,(2013),

Reliability Analysis of RC Beams Exposed to Fire, Journal of Structural Engineering,

139, pp. 212-220.)

64

time of exposure(hours) β

0 5.9177

0.5 5.6218

0.75 3.1631

1 1.7481

1.5 0.6037

2 0.1085

3 -0.3653

3.4 -0.478

Table 6.7: Validated data’s

Obtained similar curve as in the paper. It was difficult to calculate the reliability after 3.5 hr

exposure. It may be due to the difference between the methods of calculation of reliability index

used in the journal paper.

65

Chapter – 7

RELIABILITY ANALYSIS OF RC BEAMS UNDER FIRE- RESULTS AND

DISCUSSIONS

7.1 GENERAL

In this study, reliability index is used to measure safety level. Most components designed by

LRFD have calculated reliability indices between 3.5 and 4.5, with 3.5 and 4.0 being code target

levels for beams and columns in ACI 318 (ACI 2011), respectively. However, it should be

emphasized that, because of modelling simplifications and limited statistical data to characterize

RVs, the usually obtained from reliability analysis is generally not used to represent failure

probabilities of actual structures, which are typically significantly higher than the theoretically

calculated values. Rather, β is more practically used as a tool to allow consistent comparison of

safety level rather than direct assessment. Reliability indices (β) as a function of time are given

below.

In the following section reliability analysis of RC beam designed with IS 456:2000, in terms of

flexure, deflection and shear has been carried out.

7.2 DESIGN OF RC BEAM WITH UDL

Design problem

A rectangular concrete beam located inside a building in a coastal town is simply supported on two

230-mm thick and 6-m apart masonry walls (centre-to-centre). The beam has to carry in addition to

its own weight, a distributed live load of 10kN/m and a dead load of 5kN/m. Design the beam

section for maximum moment at mid span. Assume Fe415 steel, the grade of concrete may be taken

as M25 and the effective cover as 37.5mm.

Solution

Determining Mu for design

Assume a trial cross-section b=250mm, and D=450mm.

Let d=D-50=400mm.

66

Therefore effective span (cl.22.2.2 of code)

l= 6.0 m (distance between supports) or (6.0-0.23) + 0.55=6.32m (clear span + d)

Taking the lesser value (as per code),l=6.0m.

Distributed load due to self weight

ΔWDL=25 kN/m3x0.25x0.45=2.8125kN/m

Therefore total WDL=5+2.8125=7.8125kN/m.

Factored load (as per code)

Wu=1.5(WDL +WLL)=1.5(7.8125+10.0)=26.71875kN/m

Factored Moment (maximum at mid span)

Mu=Wul2/8=26.7875x6.02/8= 120.234375 kNm

Calculation of Ast using SP-16

22 250x412.5

120.234375

bd

Mu = 2.826446281

From SP-16, Table3

We get pt =0.926

926.0100 xbd

Ast

Ast=954.9375 mm2

So provide 2-25mm diameter bars.

Design of shear

Factored shear, Vu=1.5x xl2

)W+ (W LLDL =6

2

)108125.7(5.1 xx

=80.15625 kN

Nominal shear stress, Ʈv= 777273.05.412250

15625.80

xbd

Vu kN/m2

Which is less than 3.1 MPa for M25 concrete.

Design shear strength of concrete:

From Table 19, IS 456-2000, for 1005.412250

254/2100

2

xx

xxx

bd

Ast =0.952

Ʈc= 0.626 N/mm2

Ʈv> Ʈc

67

Shear reinforcement shall be provided to carry a shear equal to Vus= Vu- Ʈcbd

Vus=80.15625-0.626x250x412.5=15.6 kN

Using 2 legged 6 mm diameter stirrup spacing,

Sv=usV

87.0 xdxAxf svy=

mmS

or

xxdS

mmx

xxxxx

v

v

300

375.3095.41275.075.0

2509.26910006.15

5.41264/241587.0 2

Fig 7.1: Cross section details of the beam

Therefore Sv=250mm

So provide 2 legged stirrups @250mm c/c

7.3 LIMIT STATE OF FLEXURE

Failure function for limit state of collapse against flexure can be written as

g=R-S

S= w

l wd l 2

8

68

Where S is the moment due to the external loading, fy is the yield strength of steel, fck is the

characteristic strength of concrete, b is the breadth of beam, d is the effective depth, wl is the live

load, wd is the dead load and Ast is the area of steel reinforcement. Random variables are

identified as fy, fck, b, d, wl and wd.

Statical distribution of random variables are shown below.

variables disribution type parameters

fy normal µ=415 ,cv=0.05

fck normal µ=30,cv=0.145

wl Extreme largest, Type-1 µ=10,cv=0.3

wd normal

µ=5+self weight

,cv=0.05

b normal µ=250,cv=0.04

d normal µ=450,cv=0.04

Table 7.1: Statical distribution of random variables

We are analyzing a beam with span 6m and other remaining specifications as above.

Fig 7.2: Cross section of the beam considered

69

On application of fire according 5000C isotherm method the effective section of concrete will be

reduced. In this method we are neglecting the portion of concrete above 5000C.In order to

determine the portion of concrete to be neglected thermal transient analysis was conducted in

ANSYS-15 by modelling the beam we have already designed. Meshing was carried out and

required thermal properties of concrete were assigned to the material. Temperature loading

(according to ISO834) was given by convection on three sides of the beam. The resulting contours

were obtained for different time period of fire exposure from which we can get the depth of

penetration of 5000C contour from the face of the beam. Thus we can obtain the reduced cross

sectional area of the beam for which we are calculating the reliability indices. The reduced yield

strength of steel reinforcement was obtained by getting the temperature of reinforcement which is

obtained from the above specified thermal analysis in ANSYS-15 from which we are getting the

temperature of reinforcement from the nodal temperature at the corresponding position of the

beam. After getting the temperature value we can get the reduction factor for fy from the Table 3.2a

(EN 1992-1-2). After getting the reduction factors for both concrete and steel we can easily get the

values of reliability indices at corresponding fire exposure time by Hasofer and Lind algorithm

(FOSM method).

7.3.1 STRENGTH DEGRADATION OF REINFORCEMENT WITH RESPECT TO

FIRE EXPOSURE

In the validation problem the depth of temperature contour and strength reduction factor for steel is

obtained from Wick storm’s model. But in the problem we are dealing with is based on Thermal

analysis results of ANSYS-15.The figure below show the variation in strength degradation of

reinforcement in accordance with the method chosen to get the temperature profile.

In Wick storm’s model the depth of penetration of 5000c isotherm and reduction factor for fy is

found out as explained before.

Using ANSYS-THERMAL, after applying temperature load and doing transient analysis we will

get a temperature profile as shown below.

70

Fig 7.3: ANSYS- thermal analysis contour plot (the numbers in figure indicate the node

numbers)

From the above figure we will the depth of penetration of 5000C contour. The temperature in each

node can be obtained from this analysis. So that the temperature of steel bars can be obtained by

the temperature of corresponding nodes from which we will get the reduction factors from the table

3.2a, EN 1992-1-2:2004 (E).

Fig 7.4: Time dependent strength degradation of reinforcement with fire exposure

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5

red

uct

ion

fac

tor

for

fy

fire exposure time(hours)

Time dependent strength degradation of reinforcement with fire exposure

from wickstormmodel

from ANSYS-thermal

71

Time of

exposure(hours) β

0.05 1

0.25 1

0.5 1

0.75 0.809

1 0.6235

1.5 0.3588

2 0.1692

3 0

4 0

Table 7.2 Table 7.3

Table 7.2: Reliability indices using Wick storm’s method

Table 7.3: Reliability indices using Ansys method

For a beam 250x450mm size with 37.5mm effective cover reinforced with 2 number of 25mm

diameter bars ,a comparison of reliability degradation curve using Wick storm’s model(and also

moment capacity equation suggested by NIST) and ANSYS-THERMAL(moment capacity

equation suggested by IS 456-2000) is shown below.

Fig 7.5: Comparison of reliability degradation curve using Wick storm’s model and Ansys

method.

-2

-1

0

1

2

3

4

0 0.5 1 1.5 2relia

bili

ty in

de

x

fire exposure time (hours)

Reliability degradation curve

wickstorm's model

ANSYS-THERMAL

Time of

exposure(hours) β

0 1

0.5 1

1 1

1.5 1

2 0.9511

2.5 0.7836

3 0.5948

3.5 0.4298

4 0.2744

72

fire exposure

time (hours) β

0.05 3.5731

0.25 3.4876

0.5 1.0174

0.75 0.1882

1 -0.1881

1.5 -0.5549

1.75 -0.6619

Table 7.4 Table 7.5

Table 7.4: Reliability indices calculated using Wick storm’s Method.

Table 7.5: Reliability indices calculated using Ansys Method.

Here we can see that Wick storm’s model presents a faster degradation of reliability index

compared to ANSYS-THERMAL. Initially the curves are almost matching, but before half an hour

itself the trend of the curve changes. We can see that Wick storm’s model under estimate the

moment capacity of the beam. The above graph show that in Wick storm’s method the Reliability

index value reaches zero about half an hour before as that in case of ANSYS-THERMAL. We can

also see a steep degradation of reliability after about 1 hour in case of ANSYS-THERMAL. It is

clear from the equation Pf=φ (-β) that, for a β value of 1.5 itself the Pf is 0.9332.That means the

beam is almost failed. Here for 1.75 hour itself the β value is -1.5375.That means the beam is

almost failed. So calculation in further exposure is quite unnecessary. But in case of Wick storm’s

model at 1.75 hour the β value is -0.6619, Pf is 0.7460, and it’ s degradation curve is almost flat

with slight inclination at these times.

7.3.2 PARAMETRIC STUDY

The effect of various beam parameters were considered, including cover, compressive strength,

aggregate type, distribution of bars, grade of steel.

fire exposure

time (hours) β

0 3.455

0.5 3.455

1 2.9122

1.5 0.7678

1.75 -0.9528

73

7.3.2.1 CHARACTERISTIC STRENGTH OF CONCRETE

For the same beam, studies under different grade of concrete was done. Fig.7.6 shows the effect of

fck value. The values in the fig.7.6 are based on a beam 250mmx450mm size, with 4-20mm & 2-25mm

dia bars of fy 415 MPa and 37.5mm effective cover. As can be seen, increasing the characteristic

strength of concrete generally increases reliability across all times. It is obvious from the moment

capacity equation that, as the characteristic strength of concrete increases, its moment capacity

increases. The reliability index values are almost same for all grades of concrete. The table 7.7 shows

the values of reliability indices up to 210 minutes fire exposure for different grades of concrete.

Table 7.6: Details of reinforcement Table 7.7: Comparison of β

Fig 7.6: Effect of concrete’s compressive strength in reliability indices

-2

-1

0

1

2

3

4

5

0 1 2 3 4

relia

bili

ty in

de

x

fire exposure time(hours)

Reliability degradation curve

fck=30MPa

fck=25MPa

fire exposure

time (hours)

β

fck

25 30

0 4.1306 3.4244

0.5 4.1306 3.4244

1 4.1306 3.4244

1.5 3.9956 3.3044

2 3.4839 2.7586

2.5 2.6013 1.7742

3 1.0866 0.0344

3.5 -1.4645

fck

number-bar

diameter in

mm

25 4-20mm

30 2-25mm

74

Here also we expect that the reliability index of M30 concrete will be more than that of M25.But

here the variation may also be due to the additional reinforcement provided than the required one.

The change in reinforcement in accordance with the change of grade of steel is tabulated below.

7.3.2.2 YIELD STRENGTH OF STEEL

For 250x450mm beam of span 6m variation under different yield strength values are considered.

Different yield strengths and their corresponding reinforcement are listed below.

fy

number-bar

diameter in

mm

415 4-20mm

250 4-25mm

Table 7.8: Details of reinforcement

Table 7.9: Comparison of β

Fig 7.7: Effect of grade of steel in reliability indices

-2

-1

0

1

2

3

4

5

0 1 2 3 4

relia

bili

ty in

dex

fire exposure time(hours)

Reliability degradation curve

fy=250MPa

fy=415MPa

Fire exposure

time (hours)

β

fy

250 415

0 4.2806 4.1306

0.5 4.2806 4.1306

1 4.2806 4.1306

1.5 4.1463 3.9956

2 3.6465 3.4839

2.5 2.7919 2.6013

3 1.352 1.0866

3.5 -1.0052 -1.4645

75

Since the moment capacity depends on the yield strength of steel, as the yield strength increases the

reliability index also should increase. But here we can see some controversy from that. This may

be due to the additional reinforcement given than the required one.

7.3.2.3 EFFECTIVE COVER

For 250x450mm beam of span 6m variation under different effective values are considered.

Different covers and their corresponding reinforcement are listed below.

Table 7.10: Details of reinforcement

Table 7.11: Comparison of β

Fig 7.8: Effect of cover provided in beams

-2

-1

0

1

2

3

4

5

0 1 2 3 4

relia

bili

ty in

dex

fire exposure time (hours)

Reliability degradation curve

37.5mm

50mm

75mm

fire exposure

time (hours)

β

cover

37.5mm 50mm 75mm

0 3.455 4.1306 3.8333

0.5 3.455 4.1306 3.8333

1 2.9122 4.1306 3.8333

1.5 0.7678 3.9956 3.7525

1.75 -0.9528 3.4839

2 2.6013 3.5749

2.5 1.0866 3.0856

3 -1.4645 2.45

3.5 1.3769

4 0.5017

effective cover in

mm

number-bar

diameter in mm

37.5 2-25

50 4-20

75 4-20

76

It is obvious that as cover to the reinforcement increases the failure probability decreases since the

strength reduction of steel bars will be less. It is clear from the graph. The initial deviation in case

of 75mm effective cover may be due to the additional reinforcement given than the required one.

7.3.2.4 NUMBER OF DISTRIBUTION BARS

Keeping the total area of steel bars almost constant the beam was analyzed under different number

of reinforcement bars. Here the comparison was done for 2 number of 25 diameter and 5 number of

16 diameter bars.

fire exposure

time (hours)

β

Distribution bars

5-16mm

2-25mm

0 3.5072 3.4244

0.5 3.5072 3.4244

1 3.5072 3.4244

1.5 3.3955 3.3044

2 2.9126 2.7586

2.5 1.9218 1.7742

3 0.2593 0.0344

Table 7.12: Comparison of β

Fig 7.9: Effect of number of bars in reliability indices

0

0.5

1

1.5

2

2.5

3

3.5

4

0 1 2 3 4

relia

bili

ty in

de

x

fire exposure time(hours)

Reliability degradation curve

5-16mm dia bars

2-25mm dia bars

77

Here we can see that as the number of reinforcement bars increases for same total reinforcement

area the reliability index value increases. The reason for this may be, as number of reinforcement

bars increases the total area of steel exposed to fire will be reducing. This will lead to increased

value of reliability index.

7.3.2.5 TYPE OF AGGREGATE USED

In beam construction we may be using siliceous and carbonate aggregate based on which the

failure probability may be changing. Different type of aggregate got different thermal conductivity.

fire exposure

time (hours)

β

aggregate type

carbonate siliceous

0 3.4244 3.4244

0.5 3.4244 3.4244

1 3.4244 3.4244

1.5 3.2284 3.2122

2 2.4144 2.2466

2.5 1.211 0.785

3 -0.8394

Table 7.13: Comparison of β

Fig 7.10: Effect of type of aggregate in reliability indices

-3

-2

-1

0

1

2

3

4

0 1 2 3 4

relia

bili

ty in

de

x

fire exposure time(hours)

Reliability degradation curve

carbonate

siliceous

78

The thermal conductivity of siliceous aggregate is higher than that of carbonaceous aggregate.

Obviously the temperature in steel bar as well as in concrete at a particular location will be higher

for siliceous aggregate. This will lead to lesser reliability index value in siliceous aggregate as

shown in figure.

7.4 LIMIT STATE OF DEFLECTION

For the purpose of calculation short-term deflections in reinforced concrete flexural members

equations based on elastic theory may be made use of.

Δ=EI

wl 3

384

5

An important parameter that needs to be considered in these calculations is the flexural rigidity EI ,

which is the product of the modulus of elasticity of concrete E= Ec, and the second moment of area,

I, of cross section. For loading up to service load levels static modulus of elasticity (Ec=5000 ckf )

is satisfactory.

Failure function for limit state of serviceability against deflection can be written as

g=R-S

R=EI

lww ld

384

)(5 4

S=l/20 (from BS 476 part 20)

Where l is the span of the beam, fck is the characteristic strength of concrete, wd the dead load, wl is

the live load, E is the modulus of elasticity of concrete,

E=5000 ckf

I is the second moment of area given by

I=12

3bd

The random variables are identified as wd, wl, b, d and fck. Statical distribution of random variables

are shown below.

79

variables distribution type parameters

fck normal µ=30 ,cv=0.145

d normal µ=450 ,cv=0.04

wl

Extreme largest,

Type-1

µ=10 ,cv=0.3

wd normal

µ=5+self

weight ,cv=0.05

b normal µ=250 ,cv=0.04

Table 7.14: Statical distribution of random variables

The values of reliability index was found that in first iteration results it is about 300 in case of

absence of fire and reaches only up to 20 in 240 minutes exposure. So we can say that it irrelevant

to calculate the reliability index in case of deflection.

time of fire

exposure

reliability

index(first

iteration

results)

0 266.6665

0.5 254.8709

1 175.0765

1.5 115.7644

2 77.7301

2.5 42.4335

3 22.7631

3.5 19.4408

4 12.5742

Table 7.15: Reliability indices for the first 4 hours.

80

7.5 LIMIT STATE OF COLLAPSE AGAINST SHEAR

Failure function for limit state of collapse against shear can be written as

g=R-S

R= τcbd + Vs

Where τc is the shear strength of concrete obtained from IS 456:2000.

Vs is the shear resistance given by shear reinforcement, calculated as per IS 456-2000

Vs=v

svy

s

dAf

S=V, shear to be resisted and is calculated as

V= lww dl

2

Where sv is the spacing of stirrups. Random variables were identified as fy, b, d, wl and wd .

variables disribution type parameters

fy normal µ=415 ,cv=0.05

wl

Extreme largest,

Type-1

µ=10 ,cv=0.3

wd normal

µ=5+self weight

,cv=0.04

b normal µ=250 ,cv=0.04

d normal µ=450 ,cv=0.04

Table 7.16: Statical distribution of random variables

7.5.1 PARAMETRIC STUDY

The effect of various beam parameters were considered, compressive strength, aggregate type,

distribution of bars and grade of steel.

81

7.5.1.1 CHARACTERISTIC STRENGTH OF CONCRETE

Here also we expect that the reliability index value will be more for higher grade of concrete. The

contradiction may be due to the amount of reinforcement given in addition to the required one.

Fig 7.11: Effect of strength of concrete in reliability indices

7.5.1.2 YIELD STRENGTH OF STEEL

Here we can see that reliability is higher for Fe 415 steel than Fe250 steel. The reason for this also

may be the extra reinforcement provided than the required one.

Fig 7.12: Effect of yield strength of steel in reliability indices

0

1

2

3

4

5

0 1 2 3 4 5

relia

bili

ty in

de

x

fire exposure time (hours)

reliability degradation curve

fck=30MPa

fck=25MPa

0

1

2

3

4

5

6

0 1 2 3 4 5

relia

bili

ty in

de

x

fire exposure time (hours)

reliability degradation curve

fy=415MPa

fy=250MPa

82

7.5.1.3 DISTRIBUTION OF BARS

Fig 7.13: Effect of distribution bars in reliability indices

Here we can see that if we provide small diameter bars in large numbers the total area of steel

exposed to fire will be less. So that the reliability of the same will be increasing.

7.5.1.4 TYPE OF AGGREGATE

Fig 7.14: Effect of type of aggregate in reliability indices

Siliceous aggregate is more conductive than carbonaceous aggregate. So that temperature inside the

beam will be higher for beam with siliceous aggregate, which will lead to less reliability index for

beam with siliceous aggregate than carbonaceous aggregate.

0

1

2

3

4

5

0 2 4 6

relia

bili

ty in

de

x

fire exposure time (hours)

reliability degardation curve

2-25mm dia bars

5-16mm dia bars

0

1

2

3

4

5

0 2 4 6

relia

bili

ty in

de

x

fire exposure time (hours)

reliability degradation curve

siliceousaggregate

carbonaceousaggregate

83

7.6 TARGET VALUES FOR NOMINAL FAILURE PROBABLITY

Just as in traditional design codes a factor of safety having a value of about 1.7-2.0 appears to be

appropriate, depending on the structure, materials, consequences and so on, the nominal probability

of failure might be expected similarly to have a target value. For structural design code writing

purpose is often convenient and appropriate to back –calculate it target value from existing

practice, and to use a similar value for modified or new code. The proposals or determining target

value pfN have been given on purely empirical grounds.

A proposal is to have nominal failure probability given by:

14* 10 ntp LfN

Where tL is the structural design life in years, n is the average number of people within or near the

structure during the period of use and µ is a social criteria factor.

Nature of structure µ

Places of public assembly, dams.

Domestic, office, trade, industry

Bridges

Towers, masts, offshore structures

.005

0.05

0.5

5

Table 7.17: Social criteria factor

A somewhat different proposal is

2/115* 10 ntAWp LfN

Where tL and n have the same meaning as above and where A and W are ‘activity and warning

factors’ respectively.

84

Activity factors A Warning factors W

Post- disaster activity 0.3 Fail safe condition 0.01

Normal activities:

Buildings

Bridges

1.0

3.0

Gradual failure with some warning likely 0.1

High exposure structures

(construction, offshore)

10.0 Gradual failure hidden from view 0.3

Sudden failure without previous warning 1.0

Table 7.18: Activity and warning factors

From the second proposal the pfN of typical structure can be calculated as follows.

A=1, for normal activities (buildings).

W=0.1, gradual failure with some warning likely.

tL =50 years.

n, the average number of people within or near the structure during the period of use,(considering a

16 storey building with 5 apartments and 5 persons in each apartment)=16*5*5=400

Therefore,

pfN=10-5x1x0.1-1x50x400-1/2

=2.5x10-4

We have pf =ɸ(-β)

β=- ɸ-1(pf)

Hence, β=3.40809.

This value mainly depends on the ‘n’ value, and we can fix a target reliability index almost equal to

3.5.If a structure exceeds this target reliability index it may be assumed as safe.

85

Chapter – 8

CONCLUSION

Mechanical & Physical properties of concrete changes with increase in temperature.

Many of the mechanical and physical properties of concrete changes and above 600 C

these changes are irreversible.

Reliability analysis of RC beams exposed to fire using First FOSM method has been

established & parameters are checked.

Different parameters are checked using FOSM method and it out of these all parameters

nominal cover provided to the beam came out to be most decisive factor.

Introduction of Balanced Fire Protection Design.

Balanced Fire Protection Design is a combination of both Active Fire Protection

Systems and Passive Fire Protection system. Active fire protection systems include

smoke detectors, sprinklers that activate in the presence of smoke. Passive fire

protection uses building components to reduce the risk and spread of fire by providing

non-combustible fire rated walls, floors and roofs. By introducing these techniques we

can actually reduce the chances of fatal accidents.

People should be made aware of the dangers caused due to fire so that the collateral

damage to life and property can be reduced.

86

Chapter – 9

MATLAB CODING

9.1 EXAMPLE PROBLEM-NORMALLY DISTRIBUTED UN-CORRELATED

VARIABLES

u=[ .01 .3 360 226*10^(-6) .5 .12 40];

u=u.';

z=u;

sd=[.003 .015 36 11.3*10^(-6) .05 .006 6];

sd=sd.';

B=0;

Bdiff=10;

for i=1:7

for j=1:7

if i~=j

c(i,j)=0;

end

end

end

c(1,1)=.003^2;

c(2,2)=.015^2;

c(3,3)=36^2;

c(4,4)=(11.3*10^(-6))^2;

c(5,5)=.05^2;

c(6,6)=.006^2;

c(7,7)=6^2;

while abs(Bdiff)>.0001

dg=[-1, z(3,1)*z(4,1),

z(2,1)*z(4,1)- 2*z(3,1)*z(4,1)^2*z(5,1)/(z(6,1)*z(7,1)) ,

z(2,1)*z(3,1)-2*z(4,1)*z(3,1)^2*z(5,1)/(z(6,1)*z(7,1)),

z(3,1)^2*z(4,1)^(2)/(z(6,1)*z(7,1)),

z(3,1)^2*z(4,1)^2*z(5,1)/(z(6,1)^2*z(7,1)),

z(3,1)^2*z(4,1)^2*z(5,1)/(z(7,1)^2*z(6,1))];

dg=dg.';

g(1,1)=z(2,1)*z(3,1)*z(4,1)-

z(3,1)^2*z(4,1)^2*z(5,1)/(z(6,1)*z(7,1))-z(1,1);

z=u+c*dg*((z-u)'*dg-g(1,1))/(dg'*c*dg);

x=(z-u)./sd;

B1=(x(1,1)^2+x(2,1)^2+x(3,1)^2+x(4,1)^2+x(5,1)^2+x(6,1)^2+

x(7,1)^2)^0.5;

87

Bdiff=B-B1;

B=B1;

B

x

end

B

9.2 EXAMPLE PROBLEM-NON-NORMALLY DISTRIBUTED UN CORRELATED

VARIABLES

u=[ .32 1400000 100];

u=u';

z=u;

sd=[.032 70000 40];

sd=sd';

B=4;

Bdiff=2;

c=zeros(3);

c(1,1)=.032^2;

c(2,2)=70000^2;

c(3,3)=40^2;

u3=u(3,1);

s3=sd(3,1);

slnx3=sqrt(log(((s3/u3)^2)+1));

X31=u3*exp(-0.5*(slnx3)^2);

while abs(Bdiff)>0.001

u31=(1-log(z(3,1))+log(X31))*z(3,1);

s31=z(3,1)*(slnx3);

u(3,1)=u31;

sd(3,1)=s31;

u3=u(3,1);

s3=sd(3,1);

c(3,3)=(s3)^2;

dg=[z(2,1) z(1,1) -2000];

dg=dg';

g=z(1,1)*z(2,1)-2000*z(3,1);

z=u+c*dg*((z-u)'*dg-g)/(dg'*c*dg);

x=(z-u)./sd;

B1=sqrt((x(1,1))^2+(x(2,1))^2+(x(3,1))^2);

Bdiff=B-B1;

88

B=B1;

B

end

9.3 LIMIT STATE OF FLEXURE-BEAM-250MMX450MM-2 HOUR XPOSURE,

FY =415, FCK =30 , EFFECTIVE COVER=50MM,2-25MM DIA BARS

B=1;

Bdiff=1;

fy=468.9;

fck=38.25;

b=.250;

D=.450;

l=6000;

cf=50;

As=2*pi/4*25^2;

bf=16.5;

df=37.5;

u=[fy;D*1000-cf;fck;b*1000-2*bf;5+25*b*D;10];

z=u;

sd=[34.2;(D*1000-cf)*0.04;5;(b*1000-

2*bf)*.04;(5+25*b*D)*0.05;10*0.3];

c=zeros(6);

c(1,1)=(sd(1,1))^2;

c(2,2)=(sd(2,1))^2;

c(3,3)=(sd(3,1))^2;

c(4,4)=(sd(4,1))^2;

c(5,5)=(sd(5,1))^2;

c(6,6)=(sd(6,1))^2;

a=sqrt(pi^2/(6*sd(6,1)^2));

u2=u(6,1)-.5772/a;

while abs(Bdiff)>0.001

A=a*(z(6,1)-u2);

AA=exp(-A);

fx=a*exp(-A-AA);

Fx=exp(-AA);

u1=Fx;

if(u1>=0.5)

u1=1-u1;

t=(abs(log(1/u1^2)))^0.5;

89

an=2.515517+t*(0.802853+t*0.010328);

ab=1+t*(1.432788+t*(0.189269+t*0.001308));

z11=t-an/ab;

else

t=(abs(log(1/u1^2)))^0.5;

an=2.515517+t*(0.802853+t*0.010328);

ab=1+t*(1.432788+t*(0.189269+t*0.001308));

z11=t-an/ab;

z11=-z11;

end

P=(1/(sqrt(2*pi)))*exp(-0.5*((z11))^2);

sd(6,1)=P/fx;

u(6,1)=-sd(6,1)*z11+z(6,1);

c(6,6)=(sd(6,1))^2;

dg=[As*z(2,1)*(1-(.42*As*z(1,1)*0.7309/(.54*z(4,1)*z(3,1)*z(2,1)

)))+As*z(1,1)*0.7309*z(2,1)*(-.42)*As/(.54*z(4,1)*z(3,1)*z(2,1));

As*z(1,1)*0.7309*(1-(.42*As*z(1,1)*0.7309 /(.54*z(4,1)* z(3,1)*

z(2,1))))+As*z(1,1)*0.7309*z(2,1)* (.42)*As*z(1,1)* 0.7309/(.54*

z(4,1) *z(3,1)*z(2,1)^2);

As*z(1,1)^2*0.7309^2*.42*As/(.54*z(3,1)^2*z(4,1));

As*z(1,1)^2*0.7309^2*.42*As/(.54*z(4,1)^2*z(3,1));-l^2/8;-^2/8];

g=As*z(1,1)*0.7309*z(2,1)*(1-(.42*As*z(1,1)*0.7309/(.54*z(4,1)*

z(3,1)*z(2,1))))-(z(5,1)+z(6,1))*l^2/8;

z=u+c*dg*((z-u)'*dg-g)/(dg'*c*dg);

x=(z-u)./sd;

B1=sqrt((x(1,1))^2+(x(2,1))^2+(x(3,1))^2+(x(4,1))^2+(x(5,1))^2+

(x(6,1))^2);

Bdiff=B-B1;

B=B1;

B

end

9.4 LIMIT STATE OF DEFLECTION-BEAM

B=1;

Bdiff=1;

d1=36;

L=6000;

B=.250;

D=.450;

90

bf=16.5;

df=25;

fck=38.25;

u=[5+25*B*D;10;B*1000-2*bf;D*1000-df;fck];

z=u;

sd=[(5+25*B*D)*0.05;10*0.3;(B*1000-2*bf)*0.04;(D*1000-df)*0.04;5];

c(1,1)=(sd(1,1))^2;

c(2,2)=(sd(2,1))^2;

c(3,3)=(sd(3,1))^2;

c(4,4)=(sd(4,1))^2;

c(5,5)=(sd(5,1))^2;

a=sqrt((pi^2)/(6*sd(2,1)^2));

u2=u(2,1)-0.5772/a;

while abs(Bdiff)>0.001

fx=a*exp(-a*(z(2,1)-u2)-exp(-a*(z(2,1)-u2)));

Fx=exp(-exp(-a*(z(2,1)-u2)));

u1=Fx;

if(u1>=0.5)

u1=1-u1;

t=(abs(log(1/u1^2)))^0.5;

an=2.515517+t*(0.802853+t*0.010328);

ab=1+t*(1.432788+t*(0.189269+t*0.001308));

z1=t-an/ab;

else

t=(log(1/u1^2))^0.5;

an=2.515517+t*(0.802853+t*0.010328);

ab=1+t*(1.432788+t*(0.189269+t*0.001308));

z1=t-an/ab;

z1=-z1;

end

P=(1/(sqrt(2*pi)))*exp(-0.5*((z1))^2);

sd(2,1)=P/fx;

u(2,1)=-sd(2,1)*z1+z(2,1);

c(2,2)=(sd(2,1))^2;

dg=(-5/(384))*L^4*[(1)*12/(0.4361*5000* sqrt(z(5,1))*z(3,1)

*(z(4,1)) ^3); (1)*12/(0.4361*5000*sqrt(z(5,1)) *z(3,1)

*(z(4,1))^3);-(z(2,1)+z(1,1)) *12/(0.4361*5000* sqrt(z(5,1))

*z(3,1)^2*(z(4,1))^3);-3*(z(2,1)+z(1,1))*12/(0.4361*5000*

sqrt(z(5,1))*z(3,1)*(z(4,1))^4);-(z(2,1)+z(1,1))*12/ (z(3,1) *

((0.5846)*(5000*(z(5,1))^1.5)*(z(4,1))^3))];

g=-(5*L^4/(384*((0.4361*5000* sqrt(z(5,1))))))*(z(1,1)+z(2,1))

*12/(z(3,1)*(z(4,1))^3)+L/20;

91

z=u+c*dg*((z-u)'*dg-g)/(dg'*c*dg);

x=(z-u)./sd;

B1=sqrt((x(1,1))^2+(x(2,1))^2+(x(3,1))^2+(x(4,1))^2+(x(5,1))^2)

Bdiff=B-B1;

B=B1;

B

End

9.5 LIMIT STATE OF SHEAR-BEAM

B=5;

Bdiff=5;

B1=5;

fy=250;

fck=30;

b=.250;

D=.450;

l=6000;

cf=25;

As=3*pi/4*20^2;

bf=40;

df=60;

Tc=.6284;

u=[fy;D*1000-cf-df;fck;b*1000-2*bf;5+25*b*D;10];

z=u;

sd=[fy*.05;(D*1000-df-cf)*0.01;fck*.145;(b*1000-

2*bf)*.04;(5+25*b*D)*0.05;10*0.3];

c=zeros(6);

c(1,1)=(sd(1,1))^2;

c(2,2)=(sd(2,1))^2;

c(3,3)=(sd(3,1))^2;

c(4,4)=(sd(4,1))^2;

c(5,5)=(sd(5,1))^2;

c(6,6)=(sd(6,1))^2;

a=sqrt(pi^2/(6*sd(6,1)^2));

u2=u(6,1)-.5772/a;

while abs(Bdiff)>0.001

a1=z(1,1)*0.1095;

a2=z(2,1);

a3=z(3,1);

92

a4=z(4,1);

a5=z(5,1);

a6=z(6,1);

A=a*(z(6,1)-u2);

AA=exp(-A);

fx=a*exp(-A-AA);

Fx=exp(-AA);

u1=Fx;

if(u1>=0.5)

u1=1-u1;

t=(abs(log(1/u1^2)))^0.5;

an=2.515517+t*(0.802853+t*0.010328);

ab=1+t*(1.432788+t*(0.189269+t*0.001308));

z1=t-an/ab;

else

t=(abs(log(1/u1^2)))^0.5;

an=2.515517+t*(0.802853+t*0.010328);

ab=1+t*(1.432788+t*(0.189269+t*0.001308));

z1=t-an/ab;

z1=-z1;

end

P=(1/(sqrt(2*pi)))*exp(-0.5*((z1))^2);

sd(6,1)=P/fx;

u(6,1)=-sd(6,1)*z1+z(6,1);

c(6,6)=(sd(6,1))^2;

g=Tc*a2*a4+0.87*a1*Asv*a2/200-(a5+a6)*l/2;

dg=[.87*Asv*a2/200; Tc*a4+0.87*a1*Asv/200;0;Tc*a2;-l/2;-l/2];

z=u+c*dg*((z-u)'*dg-g)/(dg'*c*dg);

x=(z-u)./sd;

B1=sqrt((x(1,1))^2+(x(2,1))^2+(x(3,1))^2+(x(4,1))^2+(x(5,1))^2+

(x(6,1))^2);

Bdiff=B-B1;

B=B1;

B

end

93

Chapter – 10

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