Reliability Analysis of Reinforced Concrete Shallow Footings Designed Using BNBC 2006

RELIABILITY ANALYSIS OF REINFORCED CONCRETE

SHALLOW FOOTINGS DESIGNED USING BNBC 2006

SUKANTA KUMER SHILL

DEPARTMENT OF CIVIL ENGINEERING

DHAKA UNIVERSITY OF ENGINEERING AND TECHNOLOGY, GAZIPUR

FEBRUARY, 2015

i

RELIABILITY ANALYSIS OF REINFORCED CONCRETE

SHALLOW FOOTINGS DESIGNED USING BNBC 2006

A Thesis

by

SUKANTA KUMER SHILL

Submitted to the Department of Civil Engineering,

Dhaka University of Engineering and Technology (DUET), Gazipur-1700

in partial fulfillment of the degree of

MASTER OF SCIENCE IN CIVIL ENGINEERING

FEBRUARY, 2015

The thesis titled "Reliability Analysis of Reinforced Concrete Shallow

Footings Designed Using BNBC 2006" submitted by Sukanta Kumer Shill,

Student Number-092107P, and Session:2009-2010 has been accepted as satisfactory in

partial fulfillment of the requirement for the degree of Master of Science in Civil

Engineering on 20 February 2015.

BOARD OF EXAMINERS:

: Chairman

Department of Civil EngineeringDUET, Gazipur-1700.

: Member

ProfessorDepartment of Civil EngineeringDUET, Gazipur-1700.

: Member

Associate ProfessorDepartment of Civil EngineeringDUET, Gazipur-1700.

Member(Supervisor)

: Member (External)

Department of Civil Engineering

BUET. Dhaka-1000.

Dr. Md. Khasro MiahProfessor & Head

Dr. M6hammad A6dur Rashid

Dr. Md. Nazr[rl IslamProfessorDepartment of Civil Engineering

DUE\Gazipur-1700.

mmel Hoque

iii

CANDIDATE’S DECLARATION

It is hereby declared that this thesis or any part of it has not been submitted elsewhere for

the award of any degree.

Signature of the candidate

(Sukanta Kumer Shill)

iv

(Dedicated to my Parents)

v

ACKNOWLEDGEMENT

The author expresses his sincere appreciation to Supervisor, Dr. Md. Mozammel Hoque

Associate Professor, Department of Civil Engineering, Dhaka University of Engineering

& Technology (DUET), Gazipur, for his cordial guidance and support during my research

efforts and course work. His keen interest and valuable suggestions, constructive

criticisms and proper advice at every stage made this research valuable. I would like to

thank Prof. Dr. Mohammad Abdur Rashid for his guidance and for serving on the

committee. I would also like to thank Prof. Dr. Ganesh Chandra Saha for his guidance

inside and outside the classroom. Most importantly I would like to thank my wife

Anindita Roy for her continual support.

My sincere appreciation also goes to Prof. Dr. Mehedi Ahmed Ansary for his valuable

comments on the research work and serving as committee member. I would also like to

thank Prof. Dr. Khasro Miah, Head, Department of civil Engineering and Prof. Dr. Nazrul

Islam, Department of civil Engineering. Finally, the author expresses his gratitude and

appreciation to his beloved parents and family members, without my parents blessing the

research work was not possible successfully.

vi

ABSTRACT Uncertainties in designing structures and its supporting foundations are inevitable in

nature. Despite of the uncertainties, deterministic values are used in designing

structures and its supporting foundations. Due to presence of uncertainties in various

parameters accounting for the analysis and design, it is very difficult to measure safety

for any structure from deterministic analysis. So the main objective of this research is

to evaluate the safety in terms of reliability of RC shallow footings design using BNBC

2006. To achieve the goal, three model buildings (six, eight and ten storied) have been

designed following the provisions of the Bangladesh National Building Code (BNBC),

2006. The bearing capacity failure of footing, punching shear failure of concrete, one

way shear failure and flexural failure of footings are used as performance functions. In

reliability analysis, the statistical parameters of design variables are selected from

available literatures. Monte Carlo Simulation (MCS) method has been used in the

study. Finally, the reliability index for different failure modes such as flexural failure,

one way shear failure, punching shear failure, and bearing failure of soil are determined

and compared with standard expected values. From the analytical investigation, it is

found that the reliability of footings for different failure modes is different. The

reliability against soil bearing capacity is lower than the reliability against structural

failure of footing designed (BNBC), 2006. The reliability index against soil bearing

capacity varies from 2.29 to 2.46 for COV of soil of 40% using a factor of safety of

2.50 under earthquake load. The reliability index for punching shear varies from 2.52 to

3.26 under earthquake load. The failure probability of shallow footings due to one way

shear and flexural moment has been found satisfactory accounting for the gravity loads

in combination with the effect of seismic load. However, the performance of RC

shallow footing designed using BNBC, 2006 is below average under gravity loads only

as evaluated in the study. On the other hand, the performance of RC shallow footing

designed using BNBC, 2006 is poor under the earthquake load. On the basis of the

results obtained in the analytical investigation of the study, a factor of safety 3.50

instead of 2.50 is recommended for footing design, because, it is seen that the

performance of shallow footing designed using FS = 3.50 is above average. It is also

observed that the reliability of footings depends highly on the live load to dead load

ratio. The reliability of footings under earthquake loads decreases with the increase of

COV of earthquake load.

vii

CONTENTS

Page No.

TITLE PAGE i

CERTIFICATION PAGE ii

DECLARATION iii

DEDICATION iv

ACKNOWLEDGEMENT v

ABSTRACT vi

CONTENTS vii

LIST OF FIGURES x

LIST OF TABLES xiii

Chapter 1 INTRODUCTION

1.1 Background 1

1.2 Objectives of the research 3

1.3 Methodology 3

1.3.1 Basic variables 4

1.3.2 Limit state function 4

1.3.3 Monte Carlo Simulation 5

1.3.4 Reliability analysis 5

1.4 Scope and limitation 7

1.5 Contents of the study 7

Chapter 2 ANALYSIS of MODEL BUILDINGS

2.1 Introduction 9

2.2 Model building 9

2.2.1 Building geometry 10

2.2.2 Cross sectional dimension 11

2.3 Finite element modeling 12

2.3.1 Slab modeling 14

2.3.2 Beam modeling 15

2.3.3 Column modeling 15

viii

2.3.4 Foundation modeling 15

2.4 Loads and other considerations 16

2.4.1 Dead load 16

2.4.2 Live loads 17

2.4.3 Earthquake loads 17

2.4.4 Load combinations 18

2.4.5 Materials properties 18

2.5 Results of deterministic analysis 18

2.6 Design of isolated shallow footings 19

2.6.1 Allowable bearing capacity of soil using SPT 19

2.6.2 Determination of Footing Sizes 20

2.6.3 Structural Design of RC Shallow Footing 21

2.7 Results 24

Chapter 3 STATISTICS of LOAD and RESISTANCE

3.1 Introduction 29

3.2 General load models 29

3.2.1 Dead load model 29

3.2.2 Live load models 30

3.2.3 Earthquake load model 32

3.3 Model of resistance 33

3.3.1 Compressive Strength of Concrete 33

3.3.2 Yield Strength of Reinforcing Steel 34

3.3.3 Bearing Capacity of Soil 34

Chapter 4 RELIABILITY ANALYSIS of FOOTINGS

4.1 Introduction 36

4.2 Reliability analysis 36

4.2.1 Monte Carlo Simulation 37

4.2.2 Random variables 38

4.2.3 Geotechnical versus structural variability 41

4.2.4 Limit state function 41

4.3 Result and discussion 45

ix

4.3.1 Failure probability of footings considering gravity

load

45

4.3.2 Failure probability of footings considering gravity

plus earthquake Load

47

4.3.3 Reliability of footings considering gravity loads 49

4.3.4 Reliability of Footings considering gravity plus

earthquake loads

51

4.3.5 Live load to dead load ratio 52

4.3.6 Effect of COV of soil on the reliability of footings

under gravity loads

54

4.3.7 Effect of COV of soil on the reliability of footings

under earthquake load

57

4.3.8 Effect of Factor of Safety on the reliability under

gravity loads

59

4.3.9 Effect of Factor of Safety on the reliability under

earthquake loads

62

4.3.10 Effect of COV of live load on the reliability of

footings under gravity load

65

4.3.11 Effect of COV of live load on the reliability under

earthquake load

69

4.3.12 Effect of COV of earthquake load on the reliability

against soil bearing capacity of footings

73

4.4 Conclusions 76

Chapter 5 CONCLUSIONS and RECOMMENDATIONS

5.1 Introduction 78

5.2 Specific conclusion 78

5.3 Recommendation for further study 79

REFFERENCES 80

SYMBOLS and NOTATIONS 86

APPENDIX

x

LIST OF FIGURES

Fig. No. Title of Figures

Page No.

Fig. 1.1 Reliability and probability of failure of structure 6

Fig. 1.2 Relationship between reliability index β and probability of

failure ��

6

Fig. 2.1 Typical Plan of Model buildings 10

Fig. 2.2 Typical beam column grid of all model buildings 12

Fig. 2.3 Analytical 3D model of model building-1 13



Fig. 2.6 Different type of base restraints. 16

Fig. 2.7 The plan of a concentric loaded square footing 21

Fig. 2.8 Critical section for punching shear and beam shear of a

square footing

23

Fig. 2.9 Critical sections for bending moment of footing 24

Fig. 2.10 Footing detail of footing F1 of model building-1 25








xi


Fig. 3.1 Standard normal distribution curve 30

Fig. 3.2 The Extreme value type I distribution 32

Fig. 3.3 Lognormal distribution for SPT values 35

Fig. 4.1 Failure probability, load effect and resistance effect. 37

Fig. 4.2 Live load to dead load ratio on the failure probabilities of

footings under gravity loads only

46


footings under gravity loads only

46


footings under earthquake loads

48


footings under earthquake loads

49

Fig. 4.6 Live load to dead load ratio on the reliability of footings of

model building-1

53


model building-2

53


model building-3

53

Fig. 4.9 Influence of COV of soil on the reliability against soil

bearing of footings

56

Fig. 4.10 Influence of COV of soil on the reliability against soil

bearing of footings under earthquake loads

59

Fig. 4.11 Effect of factor of safety on the reliability against soil

bearing of footings

62

Fig. 4.12 Effect of factor of safety on the reliability against soil

bearing of footings

65

Fig. 4.13 Effect of COV of live load on the reliability against soil

bearing of footings

68

Fig. 4.14 Effect of COV of live load on the reliability against

punching shear of footings

68

Fig. 4.15 Effect of COV of live load on the reliability of footings against soil bearing capacity under earthquake loads.

71

xii

Fig. 4.16 Effect of COV of live load on the reliability of footings against punching shear under earthquake loads.

72

Fig. 4.17 Effect of COV of earthquake load on the reliability of

footings against soil bearing capacity for Model building-1.

74



75



76

xiii

LIST OF TABLES

Table No. Title of Tables

Page No.

Table 1.1 The range of geotechnical reliability index (US Army

Corps of Engineers 1997).

6

Table 2.1 Building geometries of three Model Buildings 11

Table 2.2 Cross sectional dimensions of structural members of the

model building

11

Table 2.3 The material strengths are considered in the design 18

Table 2.4 Axial loads of columns of model building-1 19



Table 2.7 Footing schedule of model building-1 24



Table 3.1 Statistical parameters of dead load 30

Table 3.2 Coefficient of variation of maximum 50-year live load 31

Table 3.3 Statistical parameters of maximum 50-year earthquake

load

32

Table 3.4 Statistical parameters of concrete strength 33

Table 3.5 Statistical parameters of yield strength of steel 34

Table 3.6 The statistical variation of SPT of soil 34

Table 4.1 Basic variables for reliability evaluation of footings 40

Table 4.2 Failure probabilities of footings of model building-1 under

gravity loads

45


gravity loads

45

xiv


gravity loads

46

Table 4.5 Failure probabilities of footings of model building-1

considering gravity loads plus earthquake loads

47



48



48

Table 4.8 Reliability indices of footings of model building-1under

gravity loads

50

Table 4.9 Reliability indices of footings of model building-2 under

gravity loads

50

Table 4.10 Reliability indices of footings of model building-3 under

gravity loads

50

Table 4.11 Reliability indices of footings of model building-1


51



51



52

Table 4.14 Influence of COV of bearing capacity of soil on the

reliability of footings of model building-01

54



54



55


reliability of footings of model building-3 (continued)

56

Table 4.17 Effect of COV of live load on the reliability of footings

considering earthquake load for model building-1

57



57



(continued)

58

xv



58

Table 4.20 Effect of Factor of Safety on the reliability against bearing

failure of soil of model building-1

60



60


failure of soil of model building-2 (continued)

61



61


failure of soil of model building-1under earthquake load

63


failure of soil of model building-2 under earthquake load

63



(continued)

64



64

Table 4.26 Influence of COV of live load on the reliability of footings

of model building-01

66



66


of model building-02 (continued)

67



67



69



70



70


considering earthquake load for model building-3 (cont.)

71

xvi

Table 4.32 Effect of COV of earthquake load on the reliability of

footings for model building-1

73



74



75

1

Chapter 1

INTRODUCTION

1.1 BACKGROUND

Probability-based design of structure became practically realizable in the 1970's and

its conceptual framework was developed by Ang and Cornell (Ang and Cornell,

1974), influenced by Freudenthal's pioneering work on structural safety (A.M.

Freudenthal, 1947). Applications of reliability concepts in geotechnical engineering

have been reported by Ang and Tang (1975), Vanmarcke (1977a & b), Whitman

(1984), Li and Lumb (1987), Oka and Wu (1990), Mostyn and Li (1993), Tang

(1993), Christian et al. (1994) and Chowdhury and Xu (1995), Morgenstern (1997),

Phoon and Kulhawy (1999a & b), Duncan (2000), Lacasse (2001), Phoon et al.

(2003b), Christian (2004),etc. Engineers face uncertainties at all phases of a project.

Uncertainties in designing structures and its supporting foundations are invariable in

nature. Due to presence of uncertainties in different parameters accounting for the

analysis and design of any structure, it is very difficult to measure absolute safety for

any structure using deterministic analysis. Therefore, one of the most important ways

to specify a rational criterion for ensuring the safety of a structure is its reliability or

probability of failure. Uncertainties are always inherent in loads and resistance of

structure. Because of the presence of uncertainty in the effect of loading and in the

effect of resistance, the structural members as well as their foundation are certainly

uncertain. Consequently, structures and their supporting foundation should be

designed to serve their functions with a definite reliability or a definite probability of

failure. However, engineering community, building users and owner of building

always expects any building or non building structure and its supporting foundation to

be designed with a reasonable margin of safety. In practices, these expectations are

considered by following code requirements. Code requirements have enveloped to

include design criteria that taken into account some of the sources of uncertainty in

design. Such criteria are often referred to as reliability based design criteria. The

reliability of a structure is its ability to fulfill its design purpose for some specified

design lifetime (Nowak and Collins, 2000). Reliability is often understood to equal

the probability that a structure will not fail to perform its intended function. The term

failure of structure does not necessarily mean catastrophic failure but is used to

2

indicate that the structure does not perform as desired. In structural reliability

calculation, the probability of failure is taken as quantitative measure of structural

safety. Probability of failure is calculated by subtracting the reliability value from

unity. In this case probabilistic concepts are used in reliability analysis of any

structure. Using structural reliability theory, the level of reliability of the existing

structures which are designed following any specific standard or code can be

evaluated. Structural reliability concept can be applied to the design of new structure.

Structural reliability theory also can be applied for calibrating codes, developing

partial safety factors with an accepted level of reliability in engineering fields.

Presently, Norway, Canada, United State of America, United Kingdom follows the

reliability based design of structure, and other countries which are in the process of

modifying their standards (R. Ranganathan, 1999).

So far, it is generally believed that compared to super structures, foundations and

geotechnical structures have more uncertainty in the resistance side than the load side.

Consequently, many design codes in various parts of the world are now under revision

from the allowable or the working stress design format (ASD or WSD) to the Load

and Resistance Factor Design format (LRFD) based on reliability. A RC isolated

column footing may fail either due to punching shear, flexural shear, and flexural

moment, are termed as structural failure, or bearing capacity and excessive settlement

of foundation, are termed as geotechnical failure. If any one of the aforesaid failure

occurs, ultimately the foundation fails. A foundation failure leads the complete

collapse of the structure. The failure probability or the reliability of shallow isolated

column footing depends on any of the possible aforementioned failure modes.

Therefore, it is essential to determine the reliability or margin of safety and

corresponding failure probability of shallow foundation taking all the probabilities of

foundation failure into considerations.

Recent design codes are based on probabilistic model of loads and resistances. As for

instance, American Institute for Steel Construction (AISC) uses Load and Resistance

Factor Design (LRFD) for steel construction (AISC, 1986, 1994), Ontaio Highway

Bridge design code for bridges (OHBDC, 1979, 1983, 1991), American Association

of State Highway for Transportation Officials LRFD code (AASHTO, 1994, 1998),

Canadian Highway Bridge Design Code (1998), and many European codes (e.g.,

3

CEC, 1984). So far the reliability of structure designed following Bangladesh

National Building Code (BNBC), 2006 has not yet been evaluated. So, the principal

aim of this research to evaluate the reliability of Reinforced Concrete (RC) shallow

footings for lightly loaded industrial building designed following BNBC, 2006.

1.2 OBJECTIVES OF THE RESEARCH

The main objective is to evaluate the reliability and corresponding failure probability

of RC isolated shallow footings for lightly loaded industrial buildings designed

following BNBC, 2006.

The research has following specific objectives:

(i) To evaluate the reliability of RC shallow footings of industrial

buildings designed following BNBC, 2006.

(ii) To evaluate the effect of factor of safety on the reliability of shallow

footings under earthquake load.

1.3 METHODOLOGY

First of all three model buildings are selected. Then the model buildings are analyzed

and designed following the provision of Bangladesh National Building Code (BNBC),

2006. After that, foundations of all model buildings are designed as isolated RC

shallow footing following the recommendation specified in BNBC, 2006 for footing

design. The size of footing is determined using SPT data which is commonly used in

Bangladesh. Statistical parameters of design variables are selected from established

literature. Then the failure probabilities of footings are calculated using Monte Carlo

Simulation (MCS) method. However, the most common methods are based on the

limit state function of reliability analysis are namely, First Order Reliability Methods

(FORM), Second Order Reliability Method (SORM) and Simulation techniques.

FORM and SORM reliability methods have been developed to approximately

evaluate the probability of failure or probability volume in the failure region. In this

research Monte Carlo Simulation (MCS) is used to evaluate the reliability of isolated

RC shallow footings. The Monte Carlo simulation is very powerful and useful

technique for performing probabilistic analysis. The Monte Carlo method is based on

the generating of some values numerically without actually doing any physical testing

4

for given distribution functions. The procedure used for obtaining reliability and

corresponding failure probability of RC shallow footing is briefly discussed in

following section.

1.3.1 Basic Variables

The analysis and design parameters of building such as different types of loads and

strength properties of materials which related to resistance are the random variables

considered in this study. The variability concerning sectional dimensions such as the

height and width of a section, the depth of concrete cover and the amount of

reinforcement are ignored due to the less significant effects (Frangopol et al., 1996).

The variability of the fundamental random variables belongs to loads and three basic

materials: concrete, reinforcing steel and soil are used in this research. The dead load,

live load and earthquake loads are considered as random variable of load. For

concrete, compressive strength and modulus of elasticity are considered as the

fundamental random variables. The fundamental random variables related to

reinforcing steel are yield strength and modulus of elasticity. The bearing capacity of

soil based on SPT N-value is used as the fundamental random variable.

1.3.2 Limit State Function

The loads �� and resistance �� are treated as random variables. A random variable is a

parameter that can take different values which are not predictable. However, the

distribution of the frequency of occurrence of those random values can be

characterized using a distribution function (e.g., normal, lognormal, etc.) and

statistical parameters such as the mean and standard deviation. The probability of

failure �� is represented in the reliability analysis by the reliability index β. The

reliability index � represents the distance measured in standard deviations between

the mean safety margin and the failure limit is shown in Fig.1.1. The load and

resistance factors are set such that the probability of failure (i.e., failure occurs when �� − �� is less than zero) as determined from the reliability analysis. The performance

function or limit state function is expressed as g, where � = ��-��, when� < 0, the

foundation fails and when� ≥ 0, the foundation safe.

5

1.3.3 Monte Carlo Simulation

Monte Carlo is the most robust simulation method in which samples are generated

with respect to the probability density of each variable ( Massih et al., 2008). For each

sample, the response of the system is calculated. An unbiased estimator of the failure

probability is given by

�� = ��∑ �(��)�� … … … … … .. … … … … … … … … … … … … … .. …. (1.1)

Where, N = Number of sample. The coefficient of variation of the estimator is given

by

��(��) = �(��)�� … … … … … … … … … … … … … … … … … … … (1.2)

1.3.4 Reliability Analysis

In reliability theory the uncertainties in loads and resistance parameter to be

quantified and manipulated consistently in a manner that is free from self-

contradiction. A simple application of reliability theory is shown in Fig. 1.1 to define

some of the key terms used in Reliability Based Design (RBD). Uncertain design

quantities, such as the loads �� and resistance �� are treated as random variables,

while design risk is quantified by the probability of failure��. The basic reliability

problem is to evaluate �� from some pertinent statistics of load and resistance which

typically include the mean and the standard deviation, and possibly the probability

density function. A simple closed-form solution for�� is available if both �� and

resistance �� are normally distributed. For this condition, the safety margin (� = �� −��) also is normally distributed with the following mean and standard deviation: For

normally distributed R and the probability of failure ��and reliability index � are

given below: �� = �(� ≤ 0) = ф(−�) = 1 − ф(�) … … … … … … … … … … … … …. .(1.3)

� = !"�!#�$"%&$#%'… … … … … … … … … … … … … … … . … … … … … …(1.4)

Where, �( = )* = mean value of resistance

�( = )+ =mean value of Loads

6

Fig. 1.1: Reliability and probability of failure of structure

The reliability indices for most geotechnical components and systems lie between 1

and 5, corresponding to probabilities of failure ranging from about 0.16 to 3 × 10-7, as

shown in Fig. 1.2 and table 1.1 (US Army Corps of Engineers 1997).

Fig. 1.2: Relationship between reliability index β and probability of failure ��

Table1.1: The range of geotechnical reliability index (US Army Corps of Engineers

1997).

Reliability Index, β

Probability of failure �� = Φ(−�) Expected Performance level

1.0 0.16 Hazardous

1.5 0.07 Unsatisfactory 2.0 0.023 Poor

2.5 0.006 Below average

3.0 0.0001 Above average

4.0 0.00003 Good

5.0 0.0000003 High

7

1.4 SCOPE AND LIMITATION

Three model buildings are considered to evaluate the reliability of RC isolated

shallow footings. The coefficient of variation (COV) of loads such as dead load, live

load and earthquake load are considered in calculating reliability of footing. The COV

for earthquake load is used in this research which is in context of America. Due to

unavailability of variability of earthquake loads in respect of Bangladesh, the author

used uncertainty factor from established literature. Due to deficiency of wind data

(mean, standard deviation) for Bangladesh wind load is not considered in this

research. The statistical variation of materials such as variability of concrete and steel

is based on previous available literature which is not in context of Bangladesh. The

variability in bearing capacity of soil is also based on previous literature which is

published in different foreign journal.

1.5 CONTENTS OF THE STUDY

The principal aim of this research is to evaluate the reliability and corresponding

failure probability of isolated shallow footing designed following BNBC, 2006. In

order to maintain a systematic way and clarity in the presentation of this research, the

contents of the study is summarized as follows:

Chapter II deals with a brief description of model building. The typical plan of model

building and their geometry, cross sectional dimensions of structural elements of

model buildings are presented in this chapter. The Finite Element Modeling (FEM) of

model buildings and different loads on model buildings are also presented in this

chapter. The materials properties of RC structure and deterministic analysis of axial

force of columns of all model buildings are discussed in this chapter. Design of

shallow footings and bearing capacity of shallow footings are also discussed in this

section.

Chapter III deals with the loads and resistance model. Different probabilistic loads

models for instance dead load, live load and earthquake load are presented in this

chapter. The probabilistic models of yield strength of steel, ultimate strength of

concrete and bearing capacity based on SPT value of soil are presented as resistance

model in this chapter.

8

Chapter IV presents the analysis of reliability and corresponding failure probability of

footings. The chapter deals with the model uncertainty factors, basic variable and their

coefficient of variation (COV), Monte Carlo Simulation (MCS), performance function

or limit state function, reliability theory of footing, results and the effect of live load

to dead load ratio on the reliability of footings. The chapter also deals with the

relation between factor of safety and reliability index of shallow footings design

following BNBC, 2006. Effect of COV of live load on the reliability of footings

considering earthquake loads are also presented in this chapter.

Chapter V presents the conclusion of this research. The chapter also presents

recommendation for further study. Some specific conclusions based on objective of

this research are presented in this chapter.

9

Chapter 2

ANALYSIS OF MODEL BUILDING

2.1 INTRODUCTION

Structure that enclose a space and are used for various occupancies shall be called

building structure (BNBC, 2006). The chapter deals with information regarding the

model buildings. For simplicity of analysis model buildings are considered in this

study. However, lightly loaded manufacturing building (occupancy G) is considered

as model buildings. Three model buildings are analyzed considering gravity loads as

well as lateral loads following BNBC, 2006 and presented in the chapter. The analysis

of all model buildings are performed considering finite element models using ETABS

program.

2.2 MODEL BUILDINGS

The plan of beam column grids of the model buildings along with three dimensional

finite element model is presented in this section. The author considered model

buildings instead of real building to make the analysis simplicity. The model building

having 03 bays in x direction and 03 bays in y direction of six storey as shown in

Fig. 2.3 is considered as model building-1. The model building having 03 bays in x

direction and 03 bays in y direction of eight storey as shown in Fig. 2.5 is considered

as model building-2. The model building having 03 bays in x direction and 03 bays in

y direction of ten storey as shown in Fig. 2.7 is considered as model building-3. The

span length of each panel of three model buildings is considered as 6.0 m which is

commonly used in industrial building of Bangladesh. Typical floor height is fixed at

3.50 m for all model buildings. The location of model building is considered at Zone

II in context of Bangladesh. However, a lightly loaded manufacturing building is

considered as model building. According to BNBC, 2006, the building is classified as

occupancy G. The structural form of model building is an intermediate moment

resisting frame system considering RC floor panel supported by beam all sides. In this

case all corner columns are grouped in C1, all exterior columns are grouped in C2 and

all interior columns are grouped in C3. The column numbers and dimension of panel

are shown in Fig. 2.1.

10

Fig. 2.1: Typical Plan of Model buildings

2.2.1 Building Geometry

Geometries of three model building are presented in Table 2.1. The depth of footing

below the grade is same for all footings. The floor height for all storey is kept

constant. The floor height of building and depth of footings are considered in context

of garments industries in Bangladesh. Model buildings are regular in plan. Three

model buildings are in same plan. Number of storey is the only variable.

11

Table 2.1: Building geometries of three model buildings

Building ID

No. of Span in x-Direction

No. of Span in y-Direction

Span Length in

both direction

Depth of Footing

Typical Storey Height

No. of Storey

(Nos.) (Nos.) (m) (m) (m) (Nos.) Model

building-1 3 3 6 2.44 3.5 6

Model building-2

3 3 6 2.44 3.5 8

Model building-3

3 3 6 2.44 3.5 10

2.2.2 Cross Sectional Dimension

The thickness of all floor slabs is determined considering serviceability criteria. The

thickness of floor slab also checked against flexural moment and flexural shear. The

entire cross sectional sizes of beam are calculated limiting the maximum deflection at

mid span and from flexural moment. The depth of beam also checked considering

flexural shear at critical section. The column dimensions are determined considering

the load combination specified in BNBC, 2006. The reduction of live load is used to

determine the cross sectional sizes of column. All columns are designed as RC tied

short column. The cross sectional dimension of all structural members of the model

building is presented in Table 2.2

Table 2.2: Cross sectional dimensions of structural members of the model building

Building ID thickness

of all

floor

Slabs

cross

section of

all floor

beams

cross

section of

all grade

beams

Cross section of Columns

C-1 C-2 C-3

(mm) (mm) (mm) (mm) (mm) (mm)

Model

building-1

150 300 x500 300 x500 375x375 450x450 500x500

Model

building-2

150 300 x500 300 x500 400x400 500x500 550x550

Model

building-3

150 300 x500 300 x500 500x500 550x550 600x600

2.3 FINITE ELEMENT MODEL

Prior to analysis, structural engineer needs to model the

is the process of creation of idealized and simplified representation of structural

behavior and it is an essential step in structural analysis and design. Errors and

inadequacies in modeling may cause serious design defects a

analysis. In this study, the buildings are modeled as finite element modeling using

ETABS nonlinear V9.6. The typical beam slab floor system is presented in Fig 2.2 for

all model buildings and the three dimensional finite element compute

presented in Fig 2.3 for model building

model building-3, respectively.

Fig.2.2 Typical beam column grid of all model buildings

FINITE ELEMENT MODELING

Prior to analysis, structural engineer needs to model the building. Structural modeling



inadequacies in modeling may cause serious design defects and difficulties in



all model buildings and the three dimensional finite element compute

presented in Fig 2.3 for model building-1, Fig 2.4 for model building-2, Fig 2.5 for

3, respectively.

Fig.2.2 Typical beam column grid of all model buildings

building. Structural modeling



nd difficulties in



all model buildings and the three dimensional finite element computer model is

2, Fig 2.5 for

Fig.2.3 Analytical 3D model of model building

Fig.2.4 Analyti

Fig.2.3 Analytical 3D model of model building-1

Fig.2.4 Analytical 3D model of model building-2

14

Fig. 2.5 Analytical 3D model of model building-3

2.3.1 Slab modeling

A proper modeling of the slab is very important for both linear and nonlinear analyses

of building structures. Reinforced concrete (RC) slabs are modeled utilizing different

finite elements like shell, membrane or plate elements. Shell type behavior of RC slab

means that both in-plane membrane stiffness and out-of-plane plate bending stiffness

are provided for the section. Membrane type behavior of RC slab means that only in-

plane membrane stiffness is provided for the section. Plate type behavior of RC slab

means that only out-of-plane plate bending stiffness is provided for the section. The

shell has six degrees of freedom at each node and an in-plane rotational degree of

freedom. Since the floor system is composed of two way slab panels, and a two way

slab is bent into dished shape, so, the shell element can be used efficiently for the

analysis of RC slabs. Therefore, in this study, all the RC floor slabs are modeled as

15

shell element. However, in case of one way slab and cantilever slab, one can use

membrane element to model the RC slab.

2.3.2 Beam modeling

The entire floor beams and all grade beams are modeled as line type beam element

considering shear deformation by using Integrated Building Design Software,

ETABS. All beams are continuous and producing Intermediate Moment Resisting

Frame (IMRF) by connecting columns.

2.3.3 Column modeling

All columns are modeled as line type column element and producing Intermediate

Moment Resisting Frame (IMRF) by connecting beams at different elevation.

2.3.4 Foundation Modeling

Base restraint conditions of column of any building structures can have significant

effect on the loads and behaviors of moment resisting frames. For purpose of

determining seismic loads, it is permitted to consider the structure to be fixed at the

base (ASCE7, 2005). Alternatively, where foundation flexibility is considered for the

linear analysis procedures, the linear load-deformation behavior of foundations shall

be represented by an equivalent linear stiffness using soil properties that are

compatible with the soil strain levels associated with the design earthquake motion

(ASCE7, 2005). Modeling of foundation using pinned restraints at the base of the

columns is shown in Fig. 2.6 (a), is typical for frames that do not extend through

floors below grade. This assumption results in the most flexible column base restraint.

Pinned restraints at the column bases will also simplify the design of the footing. One

drawback to the pinned base condition is that the drift of the frame, especially the

inter-story drift in the lowest story, is more difficult to control within code-allowable

limits. In addition, a pinned base may lead to development of soft or weak stories

(ASCE7, 2005). Fig. 2.6 illustrates four types of base restraint conditions that may be

considered in foundation modeling. If the drift of the structure exceeds acceptable

limits, then rotational restraint can be increased at the foundation by a variety of

methods, as illustrated in Fig. 2.6 (b), (c), and (d). Therefore, the foundation elements

must also be capable of delivering the forces to the supporting soil.

16

Fig. 2.6: Different type of base restraints

However, in this study the entire footings are modeled as fixed base to account the

seismic force in column and footing.

2.4 LOADS AND OTHER CONSIDERATIONS

To analyze and design the model building structure according to BNBC, 2006

occupancy G, the following loads are considered:

2.4.1 Dead load

Dead load is the gravitational load due to the self weight of structural and

nonstructural components of a building, e.g., Columns, beams, structural walls, floors,

ceilings, floor finishing, permanent partitions and fixed service equipment, etc.

Permanent dead loads are:

a) Self weight of structural members

b) Floor Finish = 1 kN/m2

c) Loads from permanent periphery wall (125mm thick wall) = 7.5 kN/m

d) Ceiling load = 250 N/m2

Unit weight of materials & the calculation of design dead load shall be according to

BNBC, 2006.

17

2.4.2 Live Loads

The weight of machines, furniture and workers are considered to fix the live loads on

the industrial floor. However, for lightly loaded industrial floor the live load has been

considered as (6.0 kN/m2). Live load on roof top has been considered as (1.5 kN/m2).

Live load on stair has been considered as (3.0 kN/m2). Reduction of live load for

column load calculation was considered as per BNBC code. Some non permanent

partition walls are considered on each floor as uniformly distributed live load

of1.20 kN/m2.

2.4.3 Earthquake Loads

The calculation of earthquake loads conforms to BNBC, 2006. The total design base

shear is calculated from the following equation.

� = -�� .

Where,

Z = 0.15 Seismic zone coefficient

I = 1.0 Structural importance coefficient

R = 8 Response modification coefficient

W = Total seismic dead load

C = Numerical coefficient given by the following relation

� = 1.25234/6

Where,

S = 1.2 Site coefficient for soil characteristics

T = Fundamental period of vibration in seconds of the structure for the direction under

consideration.

2.4.4 Load Combinations

The following Load combinations are considered as per Bangladesh National Building

Code to design the model building.

1) 1.4 DL

2) 1.4 DL + 1.7 LL

3) 0.9 DL + 1.3 (W or 1.1E)

4) 0.75 (1.4 DL + 1.7 LL + 1.7 (W or 1.1 E))

5) 1.4 (D+LL+E)

18

2.4.5 Materials properties

The material strength plays a vital role in design of building structure. And the

strength deformation characteristics of the building structures depend largely on the

stress strain characteristics of the materials. The material strengths are considered in

the design of the model building structures are as shown in the Table 2.3.

Table 2.3: The material strengths are considered in the design

Structural elements

For all Model buildings Unit weight of concrete

Concrete (Mpa) Reinforcing Steel (Mpa)

789 :8 7; :< kN/m3

Footing 24 23456 415 2x105 24

Column 24 23456 415 2x105 24

Beam 24 23456 415 2x105 24

Slab 24 23456 415 2x105 24

2.5 RESULTS OF DETERMINISTIC ANALYSIS

After the deterministic analysis of model building using Etabs V9.6 program, it is

seen that the building does not show any torsional irregularities. The slab of model

building is adequate considering serviceability and flexural moment. The entire floor

beam as well as grade beams is also adequate. The cross sectional dimensions of

columns under the group C1, C2 and C3 are sufficient using main steel ratio of 1.5%

to 2.0%. All structural elements of model building are safe and adequate. However,

the axial loads of different columns of different model buildings at footing level are

presented in table 2.4, table 2.5 and table 2.6, respectively.

19

Table 2.4: Axial loads of columns of model building-1

Column ID Column size

(mm) Gravity Loads (kN)

Earthquake Loads (kN)

DL LL EQ C-1 375 x 375 810 354 68

C-2 450 x 450 1261 635 84

C-3 500 x 500 1505 1150 1


Column ID Column size (mm) Gravity Loads (kN)

Earthquake Loads (kN)

DL LL EQ C-1 400 x 400 1146 506 94

C-2 500 x 500 1725 990 127

C-3 550 x 550 2005 1820 5


Column ID Column size

(mm)

Gravity Loads (kN) Earthquake Loads

(kN)

DL LL EQ

C-1 500 x 500 1491 657 137

C-2 550 x 550 2192 1271 180

C-3 600 x 600 2537 2271 10

2.6 DESIGN OF ISOLATED SHALLOW FOOTINGS

Footing design must meet three essential requirements: (1) providing adequate safety

against structural failure of the footing; (2) offering adequate bearing capacity of soil

beneath the footing with a specified safety against ultimate failure; and (3) achieving

acceptable total or differential settlements under working loads.

2.6.1 Allowable bearing capacity of soil using SPT

There are many bearing capacity equations for shallow foundation proposed by

different authors and different codes. However, in Bangladesh, Standard Penetration

Test (SPT) is widely used to determine the bearing capacity of soil. So, in this

research SPT data is used for determining the bearing capacity at footing level. It is

20

one of the most common methods for determining allowable soil bearing capacity

from standard penetration test (SPT) numbers. It is simply because SPT numbers are

readily available from soil boring. The equations that are commonly used were

proposed by Meryerhof (1956, 1974) based on 25mm of foundation settlement.

Bowles revised Meyerhof’s equations because he believed that Meryerhof’s equation

might be conservative.

qall = )33.01(04.070

B

DN+

′… … … … … … … … … … … … … … … (2.1)

when (1+0.33B

D)≤1.33 and B ≤ 1.2 m.

)33.01(1

06.0

2

70

B

D

B

BNqall +

+′= … … … … … … … … … … … … … …(2.2)

when 33.1)33.01( ≤+B

Dand B > 1.20 m

Where: qall = Allowable bearing pressure in kPa, for ∆H = 25 mm settlement.

D = Depth of foundation (m)

B = Width of footing (m).

Standard Penetration Tests (SPT) is conducted at the site to estimate the bearing

capacity of soil at (2.44m below the grade) footing level and the average allowable

bearing capacity considered in this study is 292 kPa.

2.6.2 Determination of footing sizes

The design load of footing is calculated by adding the self weight of footings with the

total un-factored load of column. Firstly, footing sizes are determined by dividing the

total un-factored axial load (dead load + live load) by the allowable bearing capacity

of underneath soil.

all1 q

LLDLA

+=

Secondly, footings sizes are determined considering earthquake load or wind load

whichever is greater in addition to gravity loads. According to BNBC, 2006, to

determine the footing sizes’ considering gravity plus lateral loads, the allowable

bearing capacity is increased by 1.33 times, that is

all2 1.33q

EQLLDLA

++=

21

Where, A= Area of Footing

DL+LL = Gravity loads

EQ or W = Lateral loads

qall = Allowable bearing capacity of soil

Hence, greater area is selected as footing area. Then for square footing, footing

dimension= = √?. Therefore a plan of square footing is presented in Fig 2.7

Fig.2.7: The plan of a concentric loaded square footing

2.6.3 Structural design of RC shallow footing

After the selection of plan dimensions (B and L) of a spread footing or width of square

footing, the footing is need to be designed for flexural moment , flexural shear and

punching shear which is called structural design. The pressure distribution beneath

footings is influenced by the interaction of the footing rigidity with the soil type,

stress–state, and time response to stress. However, it is common practice to use the

linear pressure distribution beneath rigid footings. The thickness of footing (t) for

spread footings is usually controlled by shear stresses. Two-way action shear always

controls the thickness for centrally loaded square footings. However, wide-beam shear

may control the depth for rectangular footings when the L/B ratio is greater than about

1.2 and may control for other L/B ratios when there is overturning or eccentric

loading. According to BNBC, 2006, the structural design of concentric loaded column

footing should have following essential steps:

i) Evaluate the net factored soil pressure.

Evaluate the net factored soil pressure by dividing the factored column loads by the

chosen footing area, or

@A� = �.BCD&�.EDDFGD − HI … … … … … … … … …. … … … … … … … … … .(2.3)

22

@A4 = J.EK(�.BCD&�.EDD&�.E(�.�L))FGD − HI … … … … … … … … …. … … … … …(2.4)

Where, @A�= net upward soil pressure due to gravity loads only. @A4= net upward soil pressure considering seismic loading

γ = unit weight of soil

D = depth of footing

ii) Check footing thickness for punching shear.

Since huge soil pressures are developed under the footing, high shear stresses are

produced and since shear reinforcement is not normally used in footing, shear rather

than moment commonly governed in determining the minimum required thickness of

footing. The thickness of the footing must be set so that the shear capacity of the

concrete equals or exceeds the critical shear forces produced by factored loads. The

critical section for punching shear is located at a distance M4 from the column faces and

usually takes the shape of the column. Footing thickness is adequate for resisting

punching shear once�A ≤ ∅�8. The critical punching shear force can be calculated

using the following method: �A,8P�Q = (1.4IS + 1.7SS) − @A(V� + W)(V4 + W)… … … … … … … … … ….(2.5)

The nominal punching shear strength provided by concrete shall be the smallest of the

following three equations:

�8 = 0.17 X1 + 4YZ[\78́ ^JW … … … … … … … …. … … … … … … … …(2.6)

�8 = 0.17 X1 + _`.Mab [\78́ ^JW … … … … … … …. … … … … … … … …(2.7)

�8 = 1/3\78́ ^JW … … … … … … …. … … … … … … … … … … … … (2.8)

Where, d< = 40 for interior column

d< = 30 for edge column

d< = 20 for corner column ^J = Perimeter of critical section of footing in mm.

�8 = Ratio of long side to short side of concentrated load or reaction area.

789 = Uniaxial cylinder (compressive) Strength of concrete in MPa.

d = Effective depth in mm.

c1 and c2 are the dimensions of columns as shown in fig.2.8

23

Fig. 2.8: Critical section for punching shear and beam shear of a square footing

iii) Check footing thickness for beam shear in each direction.

If �A ≤ ∅�8, thickness will be adequate for resisting beam shear without using shear

reinforcement. The critical section for beam shear is located at distance d from

column faces. Hence, the factored shear force is given by:

�A = @A= eXD�f4 [ − Wg… … …. … … … … … … … … … … … …… … … .. (2.9)

And, the nominal shearing force resisted by concrete is given as �8 = 0.17\789bd

iv) Compute the area of flexural reinforcement in each direction.

The critical section for bending is located at face of column, or wall, as specified by

BNBC, 2006. Figure 2.9 shows critical sections for flexure for footings supporting

concrete column.

24

Fig. 2.9: Critical sections for bending moment of footing

The bending moment at critical section for square footing is calculated by using the

following relation

hA,8P�Q = @A F4 XD�84 [4… …. … … … … … … … … … … … …… … … … (2.10)

Then the thickness of footing is checked against flexural moment. If the thickness is

adequate, the area of steel is calculated for both directions considering the critical

bending moment. The minimum amount of area of steel is calculated from the

following equation

?<,i�j = J.44B��Zk�l ^W ≥ �.6m�; … … … … … … … … … … … …… … … … …(2.11)

2.7 RESULTS

After the geotechnical design and the structural design of footings following BNBC,

2006, the following footing schedules are obtained for model buildings and presented

in Table 2.5, Table 2.6 and Table 2.7, respectively.

Table 2.7: Footing schedule of model building-1

Footing

ID

Footing Sizes Footing

Thickness

Depth of

Footing

Steel Reinforcement in both

Direction Width Length

(m) (m) (mm) (m)

F1 2.02 2.02 435 2.44 Φ16mm @ 165 mm c/c

F2 2.58 2.58 536 2.44 Φ16mm @ 127 mm c/c

F3 3.05 3.05 665 2.44 Φ16mm @ 100 mm c/c

25

Fig. 2.10: Footing detail of footing F1 of model building-1



The structural designs of the footings are shown in Fig. 2.10, Fig. 2.11 and Fig. 2.12.

for model building-1, Fig. 2.13, Fig. 2.14 and Fig. 2.15 for model building-2 and Fig.

2.16, Fig. 2.17 and Fig. 2.18 for model building-3.

16mm @165 mm c/c2.44m

2.02m x 2.02m

GL

435mm

16mm @127 mm c/c2.44m

2.58m x 2.58m

GL

536mm

16mm @100 mm c/c2.44m

3.05m x 3.05m

GL

665mm

26


Footing

ID


Thickness

Depth of

Footing



(m) (m) (mm) (m)

F1 2.41 2.41 500 2.44 Φ20 mm @ 212 mm c/c

F2 3.09 3.09 650 2.44 Φ20 mm @ 157 mm c/c

F3 3.67 3.67 778 2.44 Φ20 mm @ 130 mm c/c



20mm @ 212 mm c/c2.44m

2.41m x 2.41m

GL

500mm

20mm @ 157 mm c/c2.44m

3.09m x 3.09m

GL

650mm

27



Footing

ID


Thickness

Depth of

Footing



(m) (m) (mm) (m)

F1 2.74 2.74 550 2.44 Φ20mm @ 190 mm c/c

F2 3.48 3.48 725 2.44 Φ16mm @ 140 mm c/c

F3 4.10 4.10 875 2.44 Φ20mm @ 112 mm c/c


20mm @ 130 mm c/c2.44m

3.67m x 3.67m

GL

778mm

20mm @ 190 mm c/c2.44m

2.74m x 2.74m

GL

550mm

28



20mm @ 140 mm c/c2.44m

3.48m x 3.48m

GL

725mm

20mm @ 112 mm c/c2.44m

4.10m x 4.10m

GL

875mm

29

Chapter 3

STATISTICS OF LOAD and RESISTANCE

3.1 INTRODUCTION

To analyze and design of any structure, designers must have understanding about the

magnitudes and type of the loads that are expected to act on the structure during its

life time. This chapter discusses different types of loads commonly considered in the

analysis and design of building structures in Bangladesh and some probabilistic

models of these loads that are used in reliability based design methods. The chapter

also deals with the probabilistic models of yield strength of steel, ultimate strength of

concrete and bearing capacity of soil based on SPT value.

3.2 GENERAL LOAD MODELS

The accurate evaluation of gravity loads and lateral loads for proper assessment of

maximum loads that a structure will have to carry during its lifetime are very

important for safe and economical design. Normally many types of loads act on

structure. These loads can be classified into three categories based on the types of

statistical data available and the characteristics of the load phenomenon (Nowak and

Collins, 2000):

Type I: - dead load and sustained live load

Type II: - severe wind, snow load, and transient live load

Type III: - earthquake and tornadoes.

Basically actual loading is the combination of concentrated loads, non-uniformly

distributed loads and uniformly distributed loads. On the other hand the loading is

simplified for analysis purpose into uniformly distributed. However, to proceed with a

reliability analysis we need at least mean and variance or standard deviation or

coefficient of variation for loads (Nowak and Collins, 2000).

3.2.1 Dead Load Model

The dead load considered in design is usually the gravity load due to the self weight

of structural and non structural elements permanently connected to the structure. Dead

loads are typically treated as normal random variables. Usually it is assumed that the

total dead load, DL remains constant throughout the life of structure (Nowak and

Collins, 2000). Often there is a tendency on the part of designers to underestimate the

total dead load. Therefore, it is rec

for this rather than the lower values shown in Table 3.1 is recommended (Ellingwood

et al., 1980). Here, Table 3.1 lists some representative statistical parameters of dead

loads and Fig 3.1 presents the norm

loads.

Table 3.1: Statistical parameters of dead load

Structure

Type

Mean to

Nominal Ratio

Building 1.05

Building 1.05

Fig 3.1 Standard normal distribution curve

3.2.2 Live Load Models

Live load is the weight of the people and their possessions, furniture, moveable

partitions, and other portable fixtures and equipments. Usually live load is idealized as

a uniformly distributed load. The magnitude of live load depends on the type of

occupancy. The live load also depends on the expected number of peoples using the

structure and the effect of possible crowding. The statistical parameters of live load

depend on the area under consideration. The larger the area which contributes to the

live load, the smaller the magnitude of the load intensity (Nowak and Collins, 2000).


total dead load. Therefore, it is recommended to partially account a bias factor 1.05



loads and Fig 3.1 presents the normal distribution considered in the study for dead

Table 3.1: Statistical parameters of dead load

Mean to

Nominal Ratio

COV% Distribution

type

Reference

1.05 8-10 Normal Ellingwood et al.,1980

1.05 10 Normal Galambos et al.,1982

Fig 3.1 Standard normal distribution curve

Live Load Models Live load is the weight of the people and their possessions, furniture, moveable



ancy. The live load also depends on the expected number of peoples using the



ad, the smaller the magnitude of the load intensity (Nowak and Collins, 2000).


ommended to partially account a bias factor 1.05



al distribution considered in the study for dead

Reference

Ellingwood et al.,1980

Galambos et al.,1982

Live load is the weight of the people and their possessions, furniture, moveable



ancy. The live load also depends on the expected number of peoples using the



ad, the smaller the magnitude of the load intensity (Nowak and Collins, 2000).

31

The reduction factors for live load intensity as a function of influence area (ASCE7-

95). From the statistical point of view, it is convenient to consider two types of live

load: sustained live load and transient live load. The sustained live load is the load

that can be expected to exist as a usual situation (nothing extraordinary). Transient

live load is the weight of people and their possessions that might exist during an

unusual case, such as an emergency, when everybody gathers in one room, or when

all furniture is stored in one room. Since the load is infrequent and its occurrence is

difficult to predict, it is called transient load. For design purpose, it is necessary to

consider the expected combination of sustained live load and transient live load that

may occur during the building’s design lifetime (50-100years). The combined

maximum live load can be modeled by as extreme type I distribution (Ellingwood, et

al 1980). Table 3.2 presents the coefficient of variation of maximum 50-years live

load.

Table 3.2 Coefficient of variation of maximum 50-year live load

Influence area (m2) Distribution type COV% Reference

18.60

93

465

930

Extreme Type-I

14-23

13-18

10-16

9-16

Ellingwood et al.,1980

The National Bureau of Standards (NBS) has published the results of the first

extensive load survey of office buildings in the U.S., wherein the data on unit floor

loads were presented for various conditions. The live load in office buildings fit a type

I extreme value distribution (Ellingwood et al., 1980). The mean of the 50 year

maximum value is given by-

:[S] = SJ[0.25 + �K√p] … … … … … … … … … … … … … … … … … … .. (3.1)

in which SJ= basic unreduced live load;

A = influence (rather than tributary) area expressed in square feet.

However, in order to evaluate the reliability of the Code-based design, E[L] is varied

by specifying a live-to-dead load ratio in order to cover a wide range of possible live

loads. In this manner, numerous design situations and corresponding reliability

analyses are performed. If this ratio is denoted byq, then the design live load intensity

L is given by-

32

S = q.rℎ<… … … … … … … … … … … … … … … … … … … … … … (3.2)

Where, (rℎ<) is the dead load per unit floor area. The design live load intensity of

Eq. (3.1) is also used as the expected value, E[L], for the reliability analysis. A

coefficient of variation (COV) of 25 percent is assigned to L (Ellingwood et al.) using

a type I extreme value distribution to describe the probabilistic nature of the

maximum live load. Fig. 3.2 presents the Extreme value type-I distribution.

Fig. 3.2 The Extreme value type-I distribution.

3.2.4 Earthquake Load Model Earthquake load is well presented by Extreme Type I distribution with mean-to-nominal

ratio of 1.0 and coefficient of variation of 138 percent (Ellingwood et al., 1980).

Table 3.3 Statistical parameters of maximum 50-year earthquake load

Load Type Distribution

type

Mean to

Nominal

Ratio

COV% Reference

Earthquake

load

extreme type I 1.0 1.38 Ellingwood et

al.,1980

Earthquake

load

extreme type I 0.3 0.7 Nowak,1994

3.3 MODEL OF RESISTANCE

Resistance of RC structural member depends on the ultimate strength of concrete,

yield strength of steel reinforcement, cross sectional dimension of the elements, unit

weight of concrete, modulus of elasticity of steel and concrete. These strength

33

parameters are variable in nature. The variability of the fundamental random variables

belonging to three basic materials: concrete, reinforcing steel and soil are used in this

research. In the case of concrete, compressive strength and modulus of elasticity are

considered as the random variables. In the case of reinforcing steel, yield strength and

modulus of elasticity are considered as the random variables. For soil SPT or N-value

is used as fundamental random variable.

3.3.1 Compressive Strength of Concrete

The mean and coefficient of variation of concrete in compressive and tensile strength

depends highly on the specified strength of the concrete that is 789 of the mix (Nowak

and Collins, 2000). The coefficients of variation, COV of the in situ compressive

strength for concrete grades 35 and 20 MPa are estimated to be 15% and 18%,

respectively (Mirza et al. 1979). The concrete strength follows a normal distribution

(Mirza et al. 1979). Ellingwood estimated the COV to be 20.7% under average control

of concrete. For the compressive strength of concrete, normal probability distribution

has been found best suitable by many investigators (Mirza, 1996; Mirza et al, 1979).

In this study, the Coefficient of Variation (COV) is selected as 0.18 (Ellingwood et al,

1980) for 24 MPa concrete. Table 3.5 represents the mean and coefficient of variation

of concrete proposed by different authors.

Table 3.4 Statistical parameters of concrete strength

Property Mean COV% Reference

f′c= 20 MPa

f′c= 27MPa

f′c= 35 MPa

For average control

19 MPa

23 MPa

28 MPa

---

18

18

15

20.7

Mirza et al., 1979 and Ellingwood et al.,

1980

Ellingwood et al., 1980

Mirza et al., 1979 and Ellingwood et al.,

1980

Ellingwood, 1978

3.3.2 Yield Strength of Reinforcing Steel

Different statistical distribution for the yield strength of reinforcing steel has been

proposed by different researchers: Low and Hao (2001) (normal); Galambos and

Ravindra (1978) (lognormal), and Mirza and McGregor (1979) (beta distribution).

34

However, the normal distribution is more appropriate for yield strength of

reinforcement at 95% confidence level (Arafah, 1997). Hence, the normal distribution

for yield strength of reinforcing steel is used in this study. Galambos and Ravindra

(1978) recommended COV for yield strength of steel equal to 8-12%. The mean and

coefficient of variation of yield strength for 60 grade steel are 465 MPa and 9.8%

(Mirza and MacGregor, 1979). Ito and Sumikama studied typical statistics of yield

strength of grade 60 steel from several countries; results indicated that λ for yield

strength is between 1.08 and 1.19 whereas the COV is between 4.8 and 10.6 percents.

Considering the progress of manufacturer’s control over quality with time, a lower

value of COV i.e., 9.8% is selected for this study. The probabilistic parameters of

yield strength of steel reinforcement are listed in Table 3.6.

Table 3.5 Statistical parameters of yield strength of steel

Yield

strength

Distribution Mean value COV% Reference

40 grade

60 grade

Normal

Beta distribution

312 MPa

465 MPa

11.6

9.8

Ellingwood et al., 1980

Mirza et al. (1979)

3.3.3 Bearing Capacity of Soil

In a research study on Reliability Based Design (RBD) for foundations, (Phoon et al.

1995) estimated that the COV of the inherent variability COV for N were between

25% and 50% and the probability distribution for N is assumed to be lognormal

because: (1) most soil properties can be modeled adequately as lognormal random

variables (Spry et al. 1988); Phoon and Kulhawy 1999) and (2) negative values of N

are inadmissible. The resistance factors reported by Foye et al. (2006b) for the design

of rectangular shallow foundations on sand deposit at the ultimate limit state varied

from 0.45 to 0.30 based on SPT at a target reliability index of 2.50. However, in this

study, the COV of SPT is considered as 40% and the distribution of N value is

considered as lognormal. The statistical properties of bearing capacity based on SPT

are listed in Table 3.7 and the distribution of N value are presented in Fig. 3.6.

35

Table 3.6: The statistical variation of SPT of soil

Soil Property Distribution Mean COV% Reference

SPT

(clay and

Sand)

Lognormal 10-70

blows/ft

(25-50)% Phoon and Kulhawy,

1999a

Fig. 3.3: Lognormal distribution for SPT values

RELIABILITY ANALYSIS

4.1 INTRODUCTION

Reliability of the footing is expressed

reliability index is related to the probability of failure of the footing (

reliability are complementary terms. Risk is unsatisfactory performance or probability

of failure. On the other hand, rel

of success. The chapter deals with the random variables considered in the study, limit

state function, calculation of probability of failures and evaluation of reliability of

shallow footings.

4.2 RELIABILITY ANALYSIS

The objective of the reliability analysis is to determine the probability of failure. The

probability of failure pf

yield a point in the failure domain, i.e.

… … … … … … …

Where, x = vector of basic variable; and G(x) limit state function defined such that the

region G(x)≤0 corresponds with the failure mode of interest. The corresponding

reliability index β can be calculated from

Where, Φ�� inverse of the standard normal cumulative distribution function. In this

study, Monte Carlo simulations have been used to evaluate the reliability and

corresponding failure probability of footings.

index is shown in Fig. 4.1. Where, Q is the load effects and R is the effect of

resistance.

Chapter 4

RELIABILITY ANALYSIS OF FOOTINGS

INTRODUCTION

Reliability of the footing is expressed in the form of reliability index (

reliability index is related to the probability of failure of the footing (�reliability are complementary terms. Risk is unsatisfactory performance or probability

of failure. On the other hand, reliability is the satisfactory performance or probability



ELIABILITY ANALYSIS


is the probability that the realization of the basic variables

yield a point in the failure domain, i.e.

… … … … … … … … … … … … … … … … … … …


0 corresponds with the failure mode of interest. The corresponding

can be calculated from � = −Φ��t��u inverse of the standard normal cumulative distribution function. In this


corresponding failure probability of footings. The graphical presentation of reliability

x is shown in Fig. 4.1. Where, Q is the load effects and R is the effect of

in the form of reliability index (β). This ��). Risk and

reliability are complementary terms. Risk is unsatisfactory performance or probability

iability is the satisfactory performance or probability




is the probability that the realization of the basic variables

… … … … … … … … … … … … ... (4.1)


0 corresponds with the failure mode of interest. The corresponding

inverse of the standard normal cumulative distribution function. In this


The graphical presentation of reliability

x is shown in Fig. 4.1. Where, Q is the load effects and R is the effect of

37

Fig. 4.1: Failure probability, load effect and resistance effect.

4.2.1 Monte Carlo Simulation

A reliability problem is normally formulated using a failure

function�(v�, v4, …… . . vj), where v�, v4,…… .vj are random variables. Violation of

the limit state is defined by the condition �(v�, v4, …… . . vj) ≤ 0and the probability

of failure,�� is expressed by the following expression:

�� = x[�(v�, v4, ……………… . . vj) ≤ 0]………………………………………… (4.2) = y y. . . y 7z{,z%,…….z| (��, �4, …… . �j)W��,}(z{,z%,……..z|)~J W�4…W�j… … … … (4.3)

Where, (��, �4, …… .…………… �j) are values of the random variables and 7z{,z%,…………z|(��, �4, …… . �j)is the joint probability density function. The Monte

Carlo method allows the determination of an estimate of the probability of failure,

given by:

�� = ��∑ �(v�, v4, ………… . . vj)�� … … … … … … … … … … … … … … (4.4)

Where, �(v�, v4, ………… . . vj)is a function defined by:

�(v�, v4, ………… . . vj) = �1�7�(v�, v4, …… . . vj) ≤ 00�7�(v�, v4, …… . . vj) > 0� … … … … … …... (4.5)

According to (5.4), N independent sets of values (��, �4, …… . �j) are obtained based

on the probability distribution for each random variable and the failure function is

computed for each sample. Using MCS, an estimate of the probability of structural

failure is obtained by

�� = �� … … … … … .. … … … … … … … … … … … … … .. …. … … … (4.6)

Where, �� is the total number of cases where failure has occurred.

38

4.2.2 Random Variables

Dead load: Generally the total dead load, DL remains constant throughout the life of

structure (Nowak and Collins, 2000). In this study a coefficient of variation (COV) of

10 percent is assigned to dead load and distribution of dead load is considered as

normal distribution (Ellingwood, et al 1980).

Live load: The live load is considered as random variable in this study and that may

occur during the building’s design lifetime (50-100years). A coefficient of variation

(COV) of 25 percent is assigned to live load and distribution of live load is considered

as is Extreme type I distribution (Ellingwood, et al 1980).

Earthquake Load: The highly variable earthquake load is considered as random

variables in this research. A coefficient of variation (COV) of 138 percent is assigned

to earthquake load and distribution of live load is considered as is Extreme type I

distribution (Ellingwood, et al 1980).

Crushing strength of Concrete: The variability of concrete strength of cast in situ

normal weight concrete for loading rate similar to that of a cylinder test (1.68 MPa/s)

can be described by a normal distribution (Mirza et al.). The mean value of this

distribution is equal to 0.67578′ + 1100 ≤ 1.1578′ (psi) where, 78′ is the nominal

design strength. However, in this study the COV of 78′ is taken as 0.18 and

distribution for crushing strength of concrete is normal.

Yield strength of steel: The variability of static yield strength of reinforcing steel

based on nominal area of the bar cross section can be represent as beta distribution

(Mirza and MacGregor). The mean and coefficient of variation of yield strength for

60 grade steel are 465 MPa and 9.8% (Mirza and MacGregor, 1979). However in this

study only 60 grade steel is considered and corresponding mean and coefficient of

variation of yield strength is considered as 465 MPa and 9.8% respectively.

Bearing Capacity of Soil:

The COV of mixed soil is 0.41 (Reese et al.1974). The COV of sand is 0.41 with a

mean1.00 (Chen and Kulhawy, 1994). In a research study on Reliability Based Design

39

(RBD) for foundations, (Phoon et al. 1995) estimated that the COV of the inherent

variability COV for N were between 25 and 50% and the probability distribution for

N is assumed to be lognormal (Spry et al. 1988); (Phoon and Kulhawy 1999).

However, in this study, the COV of SPT is considered as 40% and the distribution of

N value is considered as lognormal

Structural model uncertainty factors. The structural model uncertainty is

characterized by the variability of the ratio B of the measured to predicted (or

nominal) resistance. Such an overall model factor can be incorporated in the

formulation of each limit state and treated as a random variable in the reliability

analysis. The normal distribution model is typically used to represent these modeling

factors whose mean and COV; depend on the limit-state considered.

Geotechnical Model uncertainty factors: A similar effort is underway to quantify

uncertainties associated with geotechnical calculation models. Although many

geotechnical calculation models are “simple”, reasonable predictions of fairly

complex soil-structure interaction behavior still can be achieved through empirical

calibrations. Because of our geotechnical heritage that is steeped in such empiricisms,

model uncertainties can be significant. Even a simple estimate of the average model

bias is crucial for reliability-based design. If the model is conservative, it is obvious

that the probabilities of failure calculated subsequently will be biased, because those

design situations that belong to the safe domain could be assigned incorrectly to the

failure domain, as a result of the built-in conservatism. Phoon and Kulhawy (2003)

presents a critical evaluation of model factors using an extensive database collected as

part of an EPRI (Electric Power Research Institute) research program on transmission

line structure foundations (Chen and Kulhawy, 1994). The basic random variables

considered in this study to evaluate the reliability are listed in table 4.1

40

Table 4.1: Basic variables for reliability evaluation of footings

Sl Xi Description Distribution Mean COV References

01 fy Yield strength of

steel

Normal Nominal 0.098 (Mirza et.al.

1979).

02 fcʹ Cylinder strength

of concrete

Normal Nominala 0.18 (Mirza et.al.,

1979).

03 \78′ concrete splitting

strength

Normal Nominal 0.18 (Mirza et.al.,

1979).

04 DL Dead Load Normal 1.05 0.1 (Ellingwood,

et al., 1980).

05 LL Live Load Extreme

Type I

Nominab 0.25 (Ellingwood,

et al., 1980).

06 EQ Earthquake Load Extreme

Type I

Nominal 1.38 (Ellingwood,

et al., 1980).

07 γ Unit weight of

soil

Normal Nominal 0.10 (Lee et

al.,1983)

08 Bf Flexural model

uncertainty

Normal 1.1 0.12 (MacGregor

et al., 1983)

09 Bv Shear model

uncertainty (ACI)

Normal 1.2 0.112 (R. Lu. et al.,

1994).

10 qu Soil capacity

based on N value

Lognormal 1.0 0.25-

0.50

Phoon and

Kulhawy

1999a.

11 Bv Punching shear

model uncertainty

for seismic loads

Normal 1.00 0.12 Luo et

al.,1995

12 Bv Punching shear

model uncertainty

for gravity loads

Normal 1.65 0.27 Luo et

al.,1995

Nominala = 0.67578′ + 1100 ≤ 1.1578′ in psi

Nominalb=:[S] = SJ[0.25 + �K√p] in which SJ= basic unreduced live load; A = influence (rather than

tributary) area expressed in square feet.

41

All variables are assumed to be mutually statistically independent. The nominal mean

values are obtained from the deterministic analysis of model building.

4.2.3 Geotechnical versus structural variability

The differences between geotechnical and structural variability’s have a significant

impact on the development of reliability-based design procedures for geotechnical

engineering. The uncertainties in structural resistances typically fall within a narrow

range of 10–25% for a wide range of materials (e.g., concrete, steel) and resistance

models (e.g., tension, flexure, shear). Note that the uncertainties in structural material

properties are even lower because the uncertainties in structural resistances also

include uncertainties arising from fabrication and modeling errors. The uncertainties

in structural loads generally depend on the source of the loadings (e.g., dead, live,

earthquake, wind). Typical COVs can be assigned to each of loading type. The COVs

are approximately 30%, with the exception of the nearly deterministic dead loads

(COV = 10%) and the highly variable earthquake loads (COV = 138%). The

uncertainty in a design soil property is a function of inherent soil variability,

measurement error, and transformation uncertainty. (Phoon et al. 1995). However, in

this research study the COV of the inherent variability COV for N value were

considered between 25% and 50%. As noted above, most of the COVs in structural

design are fairly small but the COVs in geotechnical design are large.

4.2.4 Limit State Function

The loads �� and resistance �� are treated as random variables. A random variable is a

parameter that can take different values that are not predictable. However, the

distribution of the frequency of occurrence of those random values can be

characterized using a distribution function (e.g., normal, lognormal, etc.) and

statistical parameters such as the mean and standard deviation. The probability of

failure, Pf, is represented in the reliability analysis by the reliability index β. The

reliability index β represents the distance measured in standard deviations between the

mean safety margin and the failure limit. The load and resistance factors are set such

that the probability of failure (i.e., failure occurs when R-Q is less than zero) as

determined from the reliability analysis, is acceptably small.

42

The limit-state functions ��(X) for the various failure modes are formulated as ��(v) = ��(v) − ��(�) where �� and �� denote the modal capacities and demands,

respectively, and are given below:

Bearing Capacity:

The performance function or limit state of interest for bearing capacity of soil can be

defined as following equation

� = (@A − H�) − �FGD… … … … … … … … … … … … … … … … … … …(4.7)

Where, (@A − H�) is the net ultimate bearing capacity of soil.

�FGD is the upward soil pressure below the base.

If g < 0, the footing fails. When, g ≥ 0 the footing is safe.

The equations that are commonly used to evaluate the allowable bearing capacity of

soil proposed by Meryerhof (1956, 1974) based on 25mm of foundation settlement.

Bowles revised Meyerhof’s equations because he believed that Meryerhof’s equation

might be conservative.

qall = )33.01(1

06.0

2

70

B

D

B

BN+

+′… … … … … … … … … … … … … …(4.8)

when (1+0.33B

D) ≤1.33 and B > 1.20 m

Where: qall = Allowable bearing pressure in kPa, for ∆H = 25 mm settlement.

D = Depth of foundation (m),

B = Width of footing (m).

According to BNBC, 2006, the factor of safety for shallow foundation is 2.50. Hence

the ultimate bearing capacity of soil is obtained, @A = @_��2.50.

Mean of resistance is taken as the mean of net ultimate bearing pressure, which is

obtained from SPT value, and COV of resistance is the same as COV of net ultimate

bearing pressure obtained from SPT value. Applied pressure of the footing Q is

assumed as a deterministic parameter, which is obtained by applying an appropriate

factor of safety.

43

Bending Moment:

Flexural design of a footing slab is just like a design of beam or one way slab. When a

reinforced concrete isolated column footing slab is loaded up to failure, three distinct

flexural failure modes are possible. The particular failure mode that occurs is dictated

by the percentage of reinforcement steel located in the tension zone. If the one way

slab is lightly reinforced, the slab will fail due to sudden yielding of the steel which

cannot carry the stress redistribution caused by the cracking of concrete; such a failure

is of the brittle type, characterized by a rapid crack development. If the slab is over

reinforced, the slab will fail by crushing of the concrete, also in a brittle fashion. The

following two limit-state functions define analytically the conditions of light and

heavy reinforcement:

�� = ?< − �.6m�l ^W… … … … … … … … … … … … … … … … … … … … .(4.9)

�4 = ?< − J.mKY{�Z′�l �JJ�JJ&�l ^W… … … … … … … … … … … … … … … … ...(4.10)

The condition g1 < 0 corresponds to a lightly reinforced member, whereas the

condition g2 > 0 indicates an over-reinforced member, since in the latter case the

tension reinforcement area As is larger than the balanced one. The beam is moderately

reinforced otherwise, namely when the condition {(g1>0) ∩ (g2<0)} holds. The

conditional probabilities of flexural failure given that the beam is lightly, moderately,

or over-reinforced are determined respectively by using the following limit-state

functions:

�6 = =� X1.25^ℎ4\78′[ − h… … … … … … … … … … … … … … … … .(4.11)

�B = =�?<7; XW − p`�l�.E�Z′a[ −h… … … … … … … … … … … … … … … …(4.12)

�K = =� X�6^W478′[ − h… … … … … … … … … … … … … … … … … … (4.13)

Where, M is the external bending moment produced by the upward soil pressure

beneath the footing. The model factors for g3, g4, and g5 should be treated as having

different means and COV. However, due to the scarcity of experimental data for

lightly and over-reinforced beams, the distribution parameters for the moderately

reinforced case have been adopted uniformly for the three cases. This approximation

is further justified by the negligible contribution of g3 and g5 to the failure probability.

Based on the results of the statistical studies reported by (MacGregor et al., 1983) on

44

the resistance of reinforced concrete members, a mean of 1.10 and COV, of 0.12 have

been chosen for Bf. These statistics have been adopted by (Israel et al., 1987).

One way Shear:

The performance function or limit state function of one way shear is defined as the

following equation: � = =��8 − �8P�Q Where, �8 = (0.17\78′^W), the shear strength provided by concrete. =� = Shear model uncertainty (ACI) factor, �8P�Q = the critical shear force developed at a distance d from the column face. �8P�Q can be obtained using the following equation

�8P�Q = @A= eXD�f4 [ − Wg… … …. … … … … … … … … … … … …… … .(4.14)

Punching Shear:

The performance function or limit state function of punching shear is defined as

following equations:

�� = =�0.17 X1 + 4YZ[\78ʹ^JW − �8P�Q … …. … … … … … … … … … … … (4.15)

�4 = =�0.17 X1 + _`.Mab [\78ʹ^JW − �8P�Q … …. … … … … … … … … … … (4.16)

�6 = =�0.33\78ʹ^JW − �8P�Q … …. … … … … … … … … … … … …. … (4.17)

Where, �8P�Q = the critical punching shear force developed at a distance d/2 from the

column face.

=�=punching shear model uncertainty

d< = 40 for interior column

d< = 30 for edge column

d< = 20 for corner column ^J = Perimeter of critical section of footing in mm.

�8 = Ratio of long side to short side of concentrated load or reaction area.

789 = Uniaxial cylinder (compressive) Strength of concrete in MPa.

d = Effective depth in mm.

c1 and c2 are the dimensions of column. Hence, the �8P�Q can be obtained using the

following equation �A,8P�Q = (IS + SS + :�) − @A(V� + W)(V4 + W)… … … … … … … … .. ... (4.18)

45

4.3 RESULT AND DISCUSSION

4.3.1 Failure Probability of Footings considering Gravity Load

Based on the basic random variables and limit-state functions which is defined in this

study is consistent with traditional mechanical models of reinforced concrete

behavior, the structural failure probability and geotechnical failure probability of

footings are calculated according to Monte Carlo’s Simulation (MCS) method using

spread sheet. The failure probabilities of shallow footing for the flexural moment,

flexural shear, punching shear, and bearing capacity of soil considering individual

failure modes for gravity loading only are shown in Table 4.2, Table 4.3 and Table

4.4, respectively.

Table 4.2: Failure probabilities of footings of model building-1 under gravity loads

Footing

ID

Gravity Loads Width of

footing, B

Probabilities of failure for

DL LL Flexural

Moment

Flexural

Shear

Punching

Shear

Bearing

Capacity

(kN) (kN) (m) ��

F1 810 354 2.02 2.5x10-5 2.5x10-5 4.5x10-3 5.6x10-3

F2 1261 635 2.58 5.0x10-5 2.5x10-5 5.5x10-3 5.8x10-3

F3 1505 1150 3.08 2.5x10-5 5.0x10-5 3.8x10-3 6.4x10-3


Footing

ID


footing, B


DL LL Flexural

Moment

Flexural

Shear

Punching

Shear

Bearing

Capacity

(kN) (kN) (m) ��

F1 1146 506 2.41 5.0x10-5 2.5x10-5 5.8x10-3 6.4x10-3

F2 1725 990 3.09 2.5x10-5 2.5x10-5 5.2x10-3 5.3x10-3

F3 2005 1820 3.68 5.0x10-5 2.5x10-5 4.9x10-3 6.3x10-3

46


Footing

ID


footing, B


DL LL Flexural

Moment

Flexural

Shear

Punching

Shear

Bearing

Capacity

(kN) (kN) (m) ��

F1 1491 657 2.74 5.0x10-5 2.5x10-5 5.5x10-3 6.2x10-3

F2 2192 1271 3.48 5.0x10-5 2.5x10-5 5.1x10-3 6.0x10-3

F3 2537 2271 4.10 2.5x10-5 2.5x10-5 5.1x10-3 6.9x10-3

Fig. 4.2 Live load to dead load ratio on the failure probabilities of footings under

gravity loads only


gravity loads only

0.0E+00

1.0E-05

2.0E-05

3.0E-05

4.0E-05

5.0E-05

6.0E-05

7.0E-05

0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

Pro

babi

litie

s of

failu

re

LL to DL ratio

Effect of LL to DL ratio on the failure probability of footing under gravity loads

Flexural failure

One way shear failure

0.0E+001.0E-032.0E-033.0E-034.0E-035.0E-036.0E-037.0E-038.0E-03

0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

Fa

ilure

pro

babi

litie

s

LL to DL ratio

Effect of LL to DL ratio on the failure probabilities of footing under gravity loads

Punching failure

soil bearing failure

47

From the analytical investigation of this research presented in the Table 4.2, Table 4.3

and Table 4.4, it is seen that the failure probability of shallow footings against bearing

capacity of soil is higher than any other types of failure modes of footings. It is also

observed from Fig. 4.2 and Fig. 4.3 that the failure probability of shallow footings

against punching shear is higher than any other structural failure modes under the

gravity loads. The failure probability of shallow footings against one way shear and

flexural moment is not critical designed following BNBC, 2006.

4.3.2 Failure Probability of Footings considering Gravity plus earthquake Load

The structural failure probability and geotechnical failure probability of RC shallow

footings are calculated according to Monte Carlo’s Simulation (MCS) method using

spread sheet. The failure probabilities of shallow footing for the flexural moment,


failure modes for combined gravity load plus earthquake load are presented in Table

4.5, Table 4.6 and Table 4.7, respectively.

Table 4.5: Failure probabilities of footings of model building-1 considering gravity

loads plus lateral loads

Footing

ID

Gravity

Loads

EQ

Load

Width

of

footing,

B


DL LL Flexural

Moment

Flexural

Shear

Punching

Shear

Bearing

Capacity

(kN) (kN) (kN) (m) ��

F1 810 354 64 2.04 1.7x10-4 2.5x10-5 5.5x10-4 9.6x10-3

F2 1261 635 84 2.60 1.0x10-4 2.5x10-5 8.0x10-4 8.6x10-3

F3 1505 1150 1 3.08 2.5x10-5 5.0x10-5 9.1x10-4 6.4x10-3

48



Footing

ID

Gravity

Loads

EQ

Load

Width

of

footing,

B


DL LL Flexural

Moment

Flexural

Shear

Punching

Shear

Bearing

Capacity

(kN) (kN) (kN) (m) ��

F1 1146 506 94 2.41 1.7x10-4 2.5x10-5 2.3x10-3 1.0x10-2

F2 1725 990 127 3.09 7.5x10-5 2.5x10-5 1.3x10-3 9.0x10-3

F3 2005 1820 5 3.68 5.0x10-5 2.5x10-5 4.9x10-3 6.3x10-3



Footing

ID

Gravity

Loads

EQ

Load

Width

of

footing,

B


DL LL Flexural

Moment

Flexural

Shear

Punching

Shear

Bearing

Capacity

(kN) (kN) (kN) (m) ��

F1 1491 657 137 2.74 3.0x10-4 4.5x10-5 2.5x10-3 1.1x10-2

F2 2192 1271 180 3.48 7.5x10-5 2.5x10-5 1.4x10-3 9.4x10-3

F3 2537 2271 10 4.10 2.5x10-5 2.5x10-5 7.5x10-4 6.9x10-3


earthquake load

0.0E+00

1.0E-04

2.0E-04

3.0E-04

4.0E-04

5.0E-04

6.0E-04

0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

Pro

babi

litie

s of

failu

re

Ratio of LL to DL

Effect of LL to DL ratio on the reliability of footing under earthquake loads

Flexural failure

Beam shear failure

49


earthquake loads

From the results of the analytical investigation of this study presented in Table 4.5,

Table 4.6 and Table 4.7, it is seen that the failure probability of shallow footings

against bearing capacity of soil increases with the increase of seismic load. The

earthquake load highly influence the failure probability of shallow footings against

bearing capacity of soil. However, from the Fig. 4.5 it is seen that the earthquake load

has no influence on the failure probability of shallow footings against punching shear

failure of footing. The failure probability of shallow footings against one way shear

and flexural moment is not critical considering earthquake load. But, the failure

probability of shallow footings against bearing capacity of soil is very critical

considering earthquake load.

4.3.3 Reliability of Footings considering gravity loads

The structural reliability and geotechnical reliability of RC shallow footings are

calculated from the inverse of the standard normal cumulative distribution function of

failure probability. The reliability indices of shallow footing for the flexural moment,


failure modes for gravity loads only are presented in Table 4.8, Table 4.9 and Table

4.10, respectively.

0.0E+00

2.0E-03

4.0E-03

6.0E-03

8.0E-03

1.0E-02

1.2E-02

1.4E-02

1.6E-02

0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

Fai

lure

pro

babi

lity

Ratio of LL to DL

Effect of LL to DL ratio on the failure probability of footing under earthquake loads

Punching shear failure

Bearing failure of soil

50

Table 4.8: Reliability indices of footings of model building-1under gravity loads

Footing

ID


footing, B

Reliability Indices for

DL LL Flexural

Moment

Flexural

Shear

Punching

Shear

Bearing

Capacity

(kN) (kN) (m) β β β β

F1 810 354 2.02 4.06 4.06 2.61 2.53

F2 1261 635 2.58 3.79 4.06 2.54 2.52

F3 1505 1150 3.08 4.06 4.06 2.66 2.49

Table 4.9: Reliability indices of footings of model building-2 under gravity loads

Footing

ID


footing, B


DL LL Flexural

Moment

Flexural

Shear

Punching

Shear

Bearing

Capacity


F1 1146 506 2.41 3.89 4.06 2.52 2.49

F2 1725 990 3.09 4.06 4.06 2.56 2.55

F3 2005 1820 3.68 3.89 4.06 2.58 2.50

Table 4.10: Reliability indices of footings of model building-3 under gravity loads

Footing

ID


footing, B


DL LL Flexural

Moment

Flexural

Shear

Punching

Shear

Bearing

Capacity


F1 1491 657 2.74 3.89 4.06 2.53 2.50

F2 2192 1271 3.48 3.79 4.06 2.54 2.50

F3 2537 2271 4.10 3.89 3.89 2.57 2.46

From the results of the probabilistic analysis of footings, presented in the Table 4.8,

Table 4.9 and Table 4.10, it is seen that the reliability index of all footings for bearing

capacity of soil is lower than any other modes of footing’s failures. And the reliability

of shallow footings against punching shear failure is second critical considering

gravity loads only. Therefore, the structural reliability of shallow footings highly

depends on the punching shear capacity of footing. In other word, in selecting the

structural reliability of footing the punching shear is critical. From Table 4.8, Table

51

4.9 and Table 4.10, one can also understand that the overall reliability of shallow

footing depends greatly on the reliability index obtained against bearing failure of soil

designed following BNBC, 2006.

4.3.4 Reliability of Footings considering gravity plus earthquake loads

The structural reliability and geotechnical reliability of RC shallow footings are

calculated from the inverse of the standard normal cumulative distribution function of

failure probability. The reliability indices of shallow footing for the flexural moment,


failure modes for gravity loads plus the effect of seismic loads are presented in Table

4.11, Table 4.12 and Table 4.13, respectively.

Table 4.11: Reliability indices of footings of model building-1 considering gravity

loads plus earthquake loads

Footing

ID

Gravity

Loads

EQ

Load

Width

of

footing,

B


DL LL Flexural

Moment

Flexural

Shear

Punching

Shear

Bearing

Capacity

(kN) (kN) (kN) (m) β β β β

F1 810 354 68 2.02 3.58 4.06 3.26 2.34

F2 1261 635 84 2.58 3.72 4.06 3.16 2.38

F3 1505 1150 1 3.08 4.06 4.06 3.20 2.48

Table 4.12: Reliability indices of footings of model building-2 considering gravity


Footing

ID

Gravity

Loads

EQ

Load

Width

of

footing,

B


DL LL Flexural

Moment

Flexural

Shear

Punching

Shear

Bearing

Capacity


F1 1146 506 94 2.41 3.58 4.06 2.84 2.32

F2 1725 990 127 3.09 3.79 4.06 2.60 2.36

F3 2005 1820 05 3.68 3.89 4.06 2.58 2.50

52

Table 4.13 Reliability indices of footings of model building-3 considering gravity


Footing

ID

Gravity

Loads

EQ

Load

Width

of

footing,

B


DL LL Flexural

Moment

Flexural

Shear

Punching

Shear

Bearing

Capacity


F1 1491 657 137 2.74 3.43 3.89 2.81 2.29

F2 2192 1271 180 3.48 3.79 4.06 2.97 2.35

F3 2537 2271 10 4.10 3.89 3.89 3.16 2.46

On the basis of the analytical results of this research presented in Table 4.11, Table

4.12 and Table 4.13 respectively, it is seen that the magnitude of earthquake load has

large influence on the reliability of footings against bearing failure of soil. The

reliability index of all footings against bearing capacity of soil decreases with the

increase of earthquake load. Only in the interior footings where the seismic load effect

is negligible, the bearing capacity reliability index remaining unchanged after seismic

loading is taken into account. The reliability indices against bearing capacity of soil

are deceased largely at exterior footings due to effect of earthquake loads. However,

the reliability of shallow footings against one way shear, flexural moment and

punching shear is not critical after the seismic load is taken into account.

4.3.5 Live load to dead load ratio

The effect of the load ratio that is mean live load to mean dead load ratio was

evaluated for shallow footings of each building. The influence of that load ratio on the

reliability of footing is presented in Fig 4.6, Fig.4.7 and Fig. 4.8, respectively. The

reliability indices decrease as the load ratio increase, due to larger variability of live

load, i.e., COV = 25% as compared to COV=10% for the dead load. It is observed

that the punching reliability and bearing capacity reliability decreases with the

increase of live load to dead load ratio. Whereas flexural reliability and one way shear

reliability has minor effect on the live load to dead load ratio.

53

Fig. 4.6 Live load to dead load ratio on the reliability of footings of model building-1



1.00

1.50

2.00

2.50

3.00

3.50

4.00

4.50

0.2 0.3 0.4 0.5 0.6 0.7 0.8Rel

iabi

lity

Inde

ces,

β

Live load to Dead load ratio

LL to DL ratio on the reliability of footings under gravity loads of model building-1

Flexural moment

Flexural Shear

Punching Shear

Bearing Capacity

1.00

1.50

2.00

2.50

3.00

3.50

4.00

4.50

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Rel

iabi

lity

Inde

x β

Dead Load to Live Load ratio


Flexural moment

Flexural Shear

Punching Shear

Bearing Capacity

1.00

1.50

2.00

2.50

3.00

3.50

4.00

4.50

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Re

liabi

lity

Inde

ces,

β

Live load to Dead load ratio


Flexural moment

Flexural Shear

Punching Shear

Bearing Capacity

54

4.3.6 Effect of COV of soil on the reliability of footings under gravity loads

Effect of coefficient of variation (COV) of soil on the bearing capacity reliability of

shallow footings of model building-1, model building-2 and model building-3 is

presented in Table 4.14, Table 4.15 and Table 4.16, respectively. It is observed that

the COV of bearing capacity of soil is higher than the COV of any other variables.

Therefore, the effects of COV of soil on the reliability of footings under gravity loads

are presented in Table 4.14, Table 4.15 and Table 4.16, respectively.

Table 4.14 Influence of COV of bearing capacity of soil on the reliability of footings

of model building-1

Footing ID

Gravity Loads Width of footing,

B

Factor of

Safety

Coefficient of

Variation of soil

Reliability Indices for Bearing Capacity

DL LL

(kN) (kN) (m) FS COV% β

F1 810 354 2.02 2.5

30 3.43

35 2.90

40 2.52

45 2.26

50 2.02

F2 1261 635 2.58 2.5

30 3.37

35 2.88

40 2.53

45 2.23

50 2.00

F3 1505 1150 3.08 2.5

30 3.20

35 2.77

40 2.49

45 2.21

50 1.98

55


of model building-2

Footing ID


B

Factor of

Safety

Coefficient of

Variation of soil


DL LL


F1 1146 506 2.41 2.5

30 3.40

35 2.89

40 2.51

45 2.20

50 2.08

F2 1725 990 3.09 2.5

30 3.33

35 2.82

40 2.54

45 2.21

50 2.01

F3 2005 1820 3.68 2.5

30 3.17

35 2.70

40 2.41

45 2.17

50 1.92



Footing ID


B

Factor of

Safety

Coefficient of

Variation of soil


DL LL


F1 1491 657 2.74 2.5

30 3.41

35 2.85

40 2.53

45 2.16

50 1.98

F2 2192 1271 3.48 2.5

30 3.31 35 2.80

40 2.50

45 2.17

50 1.97

56


of model building-03(continued)

Footing ID


B

Factor of

Safety

Coefficient of

Variation of soil


DL LL


F3 2537 2271 4.10 2.5

30 3.14

35 2.68

40 2.40

45 2.12

50 1.95

Fig. 4.9 Influence of COV of soil on the reliability against soil bearing of footings

From the results of probabilistic analysis presented in Table 4.14, Table 4.15, and

Table 4.16 and in the Fig. 4.9, it is seen that reliability of footing increases with the

decrease of COV. When the COV of bearing capacity of soil ≤ 30%, the reliability of

shallow footing is above average to good designed following BNBC, 2006, that is

considering a factor of safety (FS) = 2.50. On the other hand, it is also observed from

the analysis of this study that if the COV of bearing capacity of soil ≥45%, the

reliability of shallow footing is unsatisfactory to poor designed following BNBC.

1.00

1.50

2.00

2.50

3.00

3.50

20 25 30 35 40 45 50 55

Rel

iabi

lity

inde

x, β

COV of bearing capacity of soil

Effect of COV of soil on the reliability of footing under gravity loads using FS=2.50

57

4.3.7 Effect of COV of soil on the reliability of footings under earthquake load

Effect of coefficient of variation (COV) of soil on the bearing capacity reliability of

shallow footings considering earthquake loads of model building-1, model building-2

and model building-3 are presented in Table 4.17, Table 4.18 and Table 4.19,

respectively.

Table 4.17 Effect of COV of live load on the reliability of footings considering

earthquake load for model building-1

Footing ID

Loads Width of footing, B

Factor of

Safety

Coefficient of Variation of soil


Bearing Capacity

DL LL EQ

(kN) (kN) (kN) (m) FS COV% β

F1 810 354 68 2.02 2.5

30 3.10

35 2.69

40 2.35

45 2.07

50 1.90

F2 1261 635 84 2.58 2.5

30 3.08

35 2.70

40 2.38 45 2.11

50 1.91

F3 1505 1150 01 3.08 2.5

30 3.15

35 2.77

40 2.46 45 2.24

50 2.02



Footing ID


Factor of

Safety

Coefficient of Variation of

soil


Bearing Capacity DL LL EQ


F1 1146 506 94 2.41 2.5

30 3.10

35 2.69

40 2.35

45 2.07

50 1.90

58


earthquake load for model building-2 (continued)

Footing ID


Factor of

Safety


soil


Bearing Capacity

DL LL EQ


F2 1725 990 127 3.09 2.5

30 3.08

35 2.70

40 2.38

45 2.11

50 1.91

F3 2005 1820 05 3.68 2.5

30 3.15

35 2.77

40 2.46

45 2.22

50 1.98



Footing ID


Factor of

Safety


soil


Bearing Capacity DL LL EQ


F1 1491 657 137 2.74 2.5

30 3.07

35 2.71

40 2.38

45 2.01

50 1.88

F2 2192 1271 180 3.48 2.5

30 3.03

35 2.73

40 2.33 45 2.10

50 1.86

F3 2537 2271 10 4.10 2.5

30 3.11

35 2.71

40 2.42 45 2.18

50 1.95

59

Fig. 4.10 Influence of COV of soil on the reliability against soil bearing of footings

under earthquake loads

From the results of probabilistic analysis presented in Table 4.17, Table 4.18, and

Table 4.19 and in the Fig. 4.10, it is seen that reliability of footing decreases with the

increase of COV. When the COV of bearing capacity of soil ≤ 30%, the reliability of

shallow footing is above average under the earthquake loads designed following

BNBC, 2006. On the other hand, it is also observed from the analysis of this study

that if the COV of bearing capacity of soil ≥40%, the reliability of shallow footing is

unsatisfactory to poor designed following BNBC.

4.3.8 Effect of Factor of Safety on the reliability under gravity loads

The factor of safety (FS) in calculating the allowable soil bearing capacity has greater

influence on the reliability of footing. The effect of factor of safety in calculating the

allowable soil bearing capacity on the reliability of shallow footings of model

building-1, model building-2 and model building-3 are presented in Table 4.20,

Table 4.21 and Table 4.22, respectively.

1.00

1.50

2.00

2.50

3.00

3.50

25 30 35 40 45 50 55

Re

liabi

lity

inde

x

COV of soil

Effect of COV of soil on the reliability of footing under earthquake load using FS=2.50

mean soil bearing=292kPa

COV of EQ load=138%

60

Table 4.20 Effect of Factor of Safety on the reliability against bearing failure of soil

of model building-1

Footing ID

Gravity Loads Factor of

Safety

Width of

footing, B

Coefficient of Variation

of soil


DL LL

(kN) (kN) FS (m) COV% β

F1 810 354

2.00 1.81

40

1.92 2.50 2.02 2.52 3.00 2.22 3.06 3.50 2.40 3.48 4.00 2.56 3.80

F2 1261 635

2.00 2.30

40

1.89 2.50 2.58 2.50 3.00 2.82 3.04 3.50 3.05 3.51 4.00 3.26 3.89

F3 1505 1150

2.00 2.73

40

1.87 2.50 3.08 2.49 3.00 3.34 3.02 3.50 3.61 3.48 4.00 3.86 3.72


of model building-2

Footing ID


Safety

Width of

footing, B


of soil


DL LL


F1 1146 506

2.00 2.15

40

1.90 2.50 2.41 2.51 3.00 2.64 3.03 3.50 2.85 3.44 4.00 3.04 3.78

F2 1725 990

2.00 2.76

40

1.89 2.50 3.09 2.50 3.00 3.38 3.06 3.50 3.65 3.50 4.00 3.91 3.86

61


of model building-2 (continued)

Footing ID


Safety

Width of

footing, B


of soil


DL LL


F3 2005 1820

2.00 3.30

40

1.85 2.50 3.68 2.48 3.00 4.03 3.03 3.50 4.35 3.43 4.00 4.65 3.70


of model building-3

Footing ID


Safety

Width of

footing, B


of soil


DL LL


F1 1491 657

2.00 2.45

40

1.92 2.50 2.74 2.52 3.00 3.00 3.06 3.50 3.24 3.48 4.00 3.46 3.80

F2 2192 1271

2.00 3.11

40

1.89 2.50 3.48 2.50 3.00 3.81 3.04 3.50 4.12 3.51 4.00 4.40 3.89

F3 2537 2271

2.00 3.66

40

1.87 2.50 4.10 2.49 3.00 4.50 3.02 3.50 4.85 3.48 4.00 5.18 3.72

62

Fig. 4.11 Effect of factor of safety on the reliability against soil bearing of footings

From the analytical investigation of the study carried out in this research presented in

Table 4.20, Table 4.21 and Table 4.22, and in Fig. 4.11, it is seen that the reliability

index β against soil bearing capacity is 2.49 to 2.52 considering a factor of safety (FS)

= 2.5. The reliability index β = 2.49 to 2.52 indicates the performance of footing is

poor to below average (US Army Corps of Engineers 1997). If one can use a factor of

safety (FS) = 3.00 in footing design, the performance of footing will be above

average.

4.3.9 Effect of Factor of Safety on the reliability under earthquake loads

The factor of safety (FS) in calculating the allowable soil bearing capacity has greater

influence on the reliability of footing. The effect of factor of safety in calculating the

allowable soil bearing capacity on the reliability of shallow footings of model

building-1, model building-2 and model building-3 under the earthquake load is

presented in Table 4.23, Table 4.24 and Table 4.25, respectively.

1.00

1.50

2.00

2.50

3.00

3.50

4.00

1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50

Re

liabi

lity

indi

ces

Factor of Safety in soil bearing capacity

Effect of factor of safety on the reliability of footing against soil bearing capacity under gravity loads

Mean bearing capacity=292kPa

COV=40%

63


of model building-1under earthquake load

Footing ID


Safety

Width of

footing, B

Coefficient of

Variation of soil


Bearing Capacity

DL LL EQ

(kN) (kN) (kN) FS (m) COV% β

F1 810 354 68

2.00 1.81

40

1.74 2.50 2.02 2.34 3.00 2.22 2.88 3.50 2.40 3.40 4.00 2.56 3.77

F2 1261 635 84

2.00 2.30

40

1.72 2.50 2.58 2.40 3.00 2.82 2.85 3.50 3.05 3.37 4.00 3.26 3.73

F3 1505 1150 01

2.00 2.73

40

1.75 2.50 3.08 2.41 3.00 3.34 2.98 3.50 3.61 3.40 4.00 3.86 3.69


of model building-2 under earthquake load

Footing ID


Safety

Width of

footing, B

Coefficient of

Variation of soil


Bearing Capacity

DL LL EQ


F1 1146 506 94

2.00 2.15

40

1.73 2.50 2.41 2.37 3.00 2.64 2.85 3.50 2.85 3.28 4.00 3.04 3.72

F2 1725 990 127

2.00 2.76

40

1.75 2.50 3.09 2.40 3.00 3.38 2.83 3.50 3.65 3.30 4.00 3.91 3.72

64


of model building-2under earthquake load (continued)

Footing ID


Safety

Width of

footing, B

Coefficient of

Variation of soil


Bearing Capacity

DL LL EQ


F3 2005 1820 05

2.00 3.30

40

1.76 2.50 3.68 2.41 3.00 4.03 3.00 3.50 4.35 3.41 4.00 4.65 3.68


of model building-3 under earthquake load

Footing ID


Safety

Width of

footing, B

Coefficient of

Variation of soil


Bearing Capacity

DL LL EQ


F1 1491 657 137

2.00 2.45

40

1.70 2.50 2.74 2.33 3.00 3.00 2.84 3.50 3.24 3.24 4.00 3.46 3.71

F2 2192 1271 180

2.00 3.11

40

1.72 2.50 3.48 2.39 3.00 3.81 2.80 3.50 4.12 3.27 4.00 4.40 3.71

F3 2537 2271 10

2.00 3.66

40

1.73 2.50 4.10 2.41 3.00 4.50 2.89 3.50 4.85 3.28 4.00 5.18 3.67

65

Fig. 4.12 Effect of factor of safety on the reliability against soil bearing of footings

From the analytical investigation of this study carried out in the research and

presented in Table 4.23, Table 4.24 and Table 4.25, and in Fig. 4.12, it is seen that

the reliability index β against soil bearing capacity varies from 2.33 to 2.40

considering a factor of safety (FS) = 2.50 for constant COV=40% under the

earthquake loads. In this case the performance of footing is poor (US Army Corps of

Engineers 1997). If one can use a factor of safety (FS) = 3.00 in footing design, the

performance of footing will be below average. To get the performance of shallow

footing above average considering earthquake load, a factor of safety at least 3.50

should be used.

4.3.10 Effect of COV of live load on the reliability of footings under gravity load

In this analytical research, the COV of live load is considered as 25% to determine the

reliability of shallow footings. However, the COV of live load for lightly loaded

industrial building is unknown in context of Bangladesh. Therefore the author

considered a wide range of COV of live load to evaluate the effect of COV of live

load on the reliability of shallow footing for industrial building in Bangladesh.

Therefore, the influences of COV of live load on the reliability of footings of model

building-1, model building-2 and model building-3 are presented in Table 4.26, Table

4.27, Table 4.28, respectively.

1.0

1.5

2.0

2.5

3.0

3.5

4.0

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

Rel

iabi

lity

indi

ces

Factor of safety (FS)

Effect of factor of safety on the reliability of footing under earthquale loads

COV=40%

Mean soil capacity=292kPa

66

Table 4.26 Influence of COV of live load on the reliability of footings of model

building-01

Footing ID


footing, B

Factor of

Safety

Coefficient of

Variation of Live Load


Bearing Capacity

Reliability Indices for Punching shear

DL LL

(kN) (kN) (m) FS COV% β β

F1 810 354 2.02 2.5

20 2.55 2.65

25 2.53 2.63

30 2.50 2.60

35 2.47 2.58

40 2.44 2.57

F2 1261 635 2.58 2.5

20 2.55 2.57

25 2.54 2.55

30 2.51 2.54

35 2.45 2.53

40 2.39 2.52

F3 1505 1150 3.08 2.5

20 2.53 2.65

25 2.50 2.64

30 2.42 2.63

35 2.37 2.59

40 2.34 2.58


building-02

Footing ID


footing, B

Factor of

Safety

Coefficient of

Variation of Live Load


Bearing Capacity


DL LL


F1 1146 506 2.41 2.5

20 2.54 2.64

25 2.52 2.62

30 2.51 2.60

35 2.48 2.58

40 2.43 2.57

67


building-02 (continued)

Footing ID


footing, B

Factor of

Safety

COV of Live Load


Bearing Capacity

Reliability Indices for Punching shear DL LL


F2 1725 990 3.09 2.5

20 2.55 2.57

25 2.54 2.55

30 2.52 2.54

35 2.46 2.53

40 2.40 2.52

F3 2005 1820 3.68 2.5

20 2.53 2.64

25 2.51 2.63

30 2.41 2.62

35 2.36 2.59

40 2.33 2.58


building-03

Footing ID


footing, B

Factor of

Safety

COV of Live Load


Bearing Capacity

Reliability Indices for Punching

shear DL LL


F1 1491 657 2.74 2.5

20 2.53 2.64

25 2.52 2.63

30 2.50 2.61

35 2.48 2.59

40 2.43 2.58

F2 2192 1271 3.48 2.5

20 2.54 2.56

25 2.53 2.55

30 2.51 2.54

35 2.46 2.51

40 2.39 2.50

F3 2537 2271 4.10 2.5

20 2.53 2.64

25 2.50 2.63

30 2.42 2.62

35 2.36 2.59

40 2.32 2.57

68

Fig. 4.13 Effect of COV of live load on the reliability against soil bearing of footings

Fig. 4.14 Effect of COV of live load on the reliability against punching shear of

footings

From Table 4.26, Table 4.27 and Table 4.28 and Fig 4.13 and Fig. 4.14, it is

understood that the reliability index β for both soil bearing capacity and punching

shear is decreasing with increasing of COV of live load. On the basis of analytical

study carried out in the research, the reliability index against punching shear failure

1.0

1.5

2.0

2.5

3.0

10 15 20 25 30 35 40 45

Re

liabi

lity

inde

x

COV of live load

Effect of COV of live loads on the reliability against soil capacity for FS=2.50 under gravity load

Footing, F1

Footing, F2

Footing, F3

1.0

1.5

2.0

2.5

3.0

10 15 20 25 30 35 40 45

Re

liabi

lity

indi

ces

COV of live load

Effect of COV of live load on the reliability of footing against punching shear under gravity loads

Footing, F1

Footing, F2

Footing, F3

69

under gravity loads varies from 2.65 to 2.52 depends on the COV of live load. It is

also observed that the reliability index against bearing failure of soil under gravity

loads varies from 2.55 to 2.34 depends on the COV of live load. In the case of interior

footing where the intensity of live load is large, the reliability against punching shear

and also the reliability against soil bearing decreases highly with the increase of COV

of live load.

4.3.11 Effect of COV of live load on the reliability under earthquake load

The COV of live load for lightly loaded industrial building is unknown in context of

Bangladesh. Therefore, a wide range of COV of live load is considered to determine

the effect of COV of live load on the reliability of shallow footing under the

earthquake load. Therefore, the influences of COV of live load on the reliability of

footings under the earthquake loads are presented in Table 4.29 for model building-1,

in Table 4.30 for model building-2, in Table 4.31 for model building-3, respectively.



Footing ID

Loads Width of

footing, B

Factor of

Safety

COV of Live Load


Bearing Capacity


DL LL EQ

(kN) (kN) (kN) (m) FS COV% β β

F1 810 354 68 2.02 2.5

20 2.38 3.37

25 2.35 3.25

30 2.31 3.18

35 2.27 3.05

40 2.26 2.98

F2 1261 635 84 2.58 2.5

20 2.40 3.13

25 2.39 3.11

30 2.34 3.00

35 2.32 2.98

40 2.30 2.82

F3 1505 1150 01 3.08 2.5

20 2.55 3.03

25 2.48 2.92

30 2.43 2.83

35 2.41 2.64

40 2.35 2.54

70



Footing ID

Loads Width of

footing, B

Factor of

Safety

COV of Live Load


Bearing Capacity


DL LL EQ


F1 1146 506 94 2.41 2.5

20 2.36 2.81

25 2.33 2.79 30 2.32 2.78

35 2.30 2.73

40 2.28 2.64

F2 1725 990 127 3.09 2.5

20 2.39 3.17

25 2.36 3.06 30 2.33 3.04

35 2.32 2.94

40 2.31 2.75

F3 2005 1820 05 3.68 2.5

20 2.55 2.96

25 2.53 2.92

30 2.43 2.75

35 2.41 2.65

40 2.34 2.47



Footing ID

Loads Width of

footing, B

Factor of

Safety

COV of Live Load


Bearing Capacity


shear

DL LL EQ


F1 1491 657 137 2.74 2.5

20 2.29 2.91

25 2.31 2.84

30 2.28 2.81

35 2.24 2.73

40 2.25 2.65

F2 2192 1271 180 3.48 2.5

20 2.37 3.08

25 2.32 3.00

30 2.30 2.94

35 2.29 2.79

40 2.25 2.70

71


earthquake load for model building-3 (continued)

Footing ID

Loads Width of

footing, B

Factor of

Safety

COV of Live Load


Bearing Capacity


shear

DL LL EQ


F3 2537 2271 10 4.10 2.5

20 2.50 3.35

25 2.48 3.17

30 2.40 3.07

35 2.35 2.86

40 2.31 2.78

Fig. 4.15 Effect of COV of live load on the reliability of footings against soil bearing

capacity under earthquake loads.

1.0

1.5

2.0

2.5

3.0

10 15 20 25 30 35 40 45

Rel

iabi

lity

inde

x, β

COV of live load

Effect of COV of live load on the reliability against bearing failure of soil under earthquake loads

72

Fig. 4.16 Effect of COV of live load on the reliability of footings against punching

shear under earthquake loads.

From the results of the analytical investigation presented in Table 4.29, Table 4.30

and Table 4.31 and in Fig 4.15 and Fig. 4.16, it is seen that the reliability index β for

both soil bearing capacity and punching shear is decreasing with increasing of COV

of live load. On the basis of analytical study carried out in the research, the reliability

index against punching shear failure under gravity loads varies from 2.65 to 2.52

depends on the COV of live load. It is also observed that the reliability index against

bearing failure of soil under gravity loads varies from 2.55 to 2.34 depends on the

COV of live load. In the case of interior footing where the intensity of live load is

large, the reliability against punching shear and also the reliability against soil bearing

decreases highly with the increase of COV of live load.

1.0

1.5

2.0

2.5

3.0

3.5

10 15 20 25 30 35 40 45

Re

liabi

lity

inde

x, β

COV of live load

Effect of COV of live load on the reliability against punching shear failure of footings

73

4.3.12 Effect of COV of earthquake load on the reliability against soil bearing

capacity of footings

The COV of highly varied earthquake load is also unknown in context of Bangladesh.

Therefore, a wide range of COV of earthquake load is considered to evaluate the

effect of COV of earthquake load on the reliability of shallow footings under the

constant COV of soil capacity. However, the reliability of footings under the

earthquake loads having different COV values are presented in Table 4.32 for model

building-1, in Table 4.33 for model building-2, in Table 4.34 for model building-3,

respectively.

Table 4.32 Effect of COV of earthquake load on the reliability of footings for model

building-1

Footing ID

Loads Width of footing,

B

Factor of

Safety

COV of Earthquake

Load

Reliability against Bearing Capacity

DL LL EQ

(kN) (kN) (kN) (m) FS COV β

F1 810 354 68 2.02 2.5

0.60 2.38

0.80 2.36

1.00 2.33

1.20 2.32

1.40 2.31

F2 1261 635 84 2.58 2.5

0.60 2.44

0.80 2.39

1.00 2.37

1.20 2.36

1.40 2.35

F3 1505 1150 01 3.08 2.5

0.60 2.52

0.80 2.52

1.00 2.50

1.20 2.49 1.40 2.48

74

Fig. 4.17 Effect of COV of earthquake load on the reliability of footings against soil

bearing capacity for Model building-1.


building-2

Footing ID

Loads Width of footing,

B

Factor of

Safety

COV of Earthquake

Load

Reliability against Bearing Capacity

DL LL EQ


F1 1146 506 94 2.41 2.5

0.60 2.39

0.80 2.37

1.00 2.36 1.20 2.35

1.40 2.30

F2 1725 990 127 3.09 2.5

0.60 2.40

0.80 2.39

1.00 2.37

1.20 2.36

1.40 2.35

F3 2005 1820 05 3.68 2.5

0.60 2.52

0.80 2.51

1.00 2.50

1.20 2.50

1.40 2.49

1.50

1.70

1.90

2.10

2.30

2.50

0.40 0.60 0.80 1.00 1.20 1.40 1.60

Rel

iabi

lity

Inde

x

COV of earthquake loads

Effect of COV of earthquake load on the reliability of footings using a FS=2.50 for constant COV of soil of 40%

Footing, F1

Footing, F2

Footing, F3

75




building-3

Footing ID


Factor of

Safety

COV of Earthquake

Load

Reliability against Bearing

Capacity DL LL EQ


F1 1491 657 137 2.74 2.5

0.60 2.36

0.80 2.34

1.00 2.32 1.20 2.30

1.40 2.29

F2 2192 1271 180 3.48 2.5

0.60 2.40

0.80 2.35

1.00 2.34

1.20 2.33

1.40 2.32

F3 2537 2271 10 4.10 2.5

0.60 2.48

0.80 2.48

1.00 2.47

1.20 2.47

1.40 2.46

2.00

2.10

2.20

2.30

2.40

2.50

2.60

0.4 0.6 0.8 1 1.2 1.4 1.6

Rel

iabi

lity

Inde

x

COV of earthquake loads


Footing, F1

Footing, F2

Footing, F3

76



It is seen from Table 4.32, Table 4.33 and Table 4.34 and in Fig 4.17, Fig.4.18 and

Fig. 4.19, that the reliability index β decreasing with increasing of COV of earthquake

load. On the basis of analytical study carried out in the research, the reliability index

of corner footings against bearing failure of soil under earthquake load are lower than

other footings of same building. In the case of interior footings where the intensity of

earthquake load is lower, the reliability of footings against soil bearing capacity is

approximately similar in both cases of gravity loads and under earthquake loads.

4.4 CONCLUSIONS

The structural failure probability and bearing failure probability of shallow footings

are calculated according to Monte Carlo’s Simulation (MCS) method using spread

sheet. The following conclusion can be drawn from the reliability analysis of RC

shallow footing:

(i) The failure probability of shallow footings against bearing capacity of soil

is higher than any other types of failure modes of footings.

(ii) The failure probability of shallow footings against punching shear is

higher than any other structural failure modes.

2.00

2.10

2.20

2.30

2.40

2.50

2.60

0.40 0.60 0.80 1.00 1.20 1.40 1.60

Rel

iabi

lity

Inde

x

COV of earthquake load


Footing, F1

Footing, F2

Footing, F3

77

(iii) The failure probability of shallow footings against one way shear and

flexural moment is not quite lower for gravity loads and the combined

gravity plus earthquake loads.

(iv) The failure probability of shallow footings against bearing capacity of soil

increases with the increase of earthquake load. However, the earthquake

load has no influence on the failure probability of shallow footings against

punching shear failure of footing.

(v) The reliability index of all footings for bearing capacity of soil is lower

than other modes of footing’s failures. And the reliability of shallow

footings against punching shear failure is second critical considering

gravity loads only. Therefore, the structural reliability of shallow footings

highly depends on the punching shear capacity of footing. In other word,

in selecting the structural reliability of footing the punching shear is

critical.

(vi) The reliability indices decrease as the load ratio increase, due to larger

variability of live load, i.e., COV = 25% as compared to COV=10% for the

dead load. It is observed that the punching reliability and bearing capacity

reliability decreases with the increase of live load to dead load ratio.

Whereas flexural reliability and one way shear reliability has minor effect

on the live load to dead load ratio.

(vii) The reliability increases with the decrease of COV of bearing capacity of

soil. When the COV of bearing capacity of soil is ≤ 30%, the reliability of

shallow footing is above average to good designed following BNBC.

(viii) The reliability index β for both soil bearing capacity and punching shear

are decreasing with increasing of COV of live load.

78

Chapter 5

CONCLUSIONS and RECOMMENDATIONS

5.1 INTRODUCTION

The main objective of the study was to evaluate the reliability of RC shallow footing

designed using Bangladesh National Building Code (BNBC), 2006. To evaluate the

reliability and corresponding failure probabilities of RC shallow footing, three model

buildings were analyzed and designed following BNBC, 2006. To achieve the

objectives of this research after the deterministic design of model buildings, the

reliability of footings of the model buildings was calculated using Monte Carlo

Simulation (MCS). The specific conclusions that are derived from the analytical

investigation carried out in the research is stated in the following sub-section.

5.2 SPECIFIC CONCLUSION

Based on the results obtained from the analytical investigation of the present study on

reliability analysis of reinforced concrete isolated shallow footings, the following

specific conclusions can be made:

(i) The reliability of footings for different failure modes is different. The

reliability against soil bearing capacity is lower than that of other failure

modes of footing designed following BNBC, 2006. The failure probability

of footings due to punching shear is larger than any other modes of

structural failure.

(ii) The failure probability of footings due to beam shear and flexural moment

is not quite low under combined gravity plus earthquake load.

(iii) The reliability decrease as the live load to dead load ratio increase, due to

larger variability of live load, that is COV = 25% as compared to

COV=10% for the dead load.

79

(iv) The reliability index β against soil bearing capacity varies from 2.49 to

2.55 using a factor of safety (FS = 2.50) under the gravity loads, while the

reliability index for that varies from 2.29 to 2.46 under earthquake load.

(v) The performance of RC shallow footings designed using BNBC, 2006 is

below average under the gravity loads, on the other hand, the performance

of footing designed using BNBC, 2006 is poor under the earthquake load.

(vi) From the results of analytical investigation of this study it is recommended

that a factor of safety (FS) = 3.50 may be used to increase the reliability of

footings under the earthquake load. Because the performance of shallow

footing designed considering FS=3.50 is above average under earthquake

load.

(vii) The reliability index for punching shear varies from 2.52 to 3.26

considering combined gravity plus earthquake loads.

(viii) The reliability of corner footings under earthquake load is lower than other

footings of same building. In the case of interior footings where

earthquake load is lower, the reliability of footings against soil bearing

capacity is approximately similar in both cases of gravity loads and under

earthquake loads.

5.3 RECOMMENDATION FOR FURTHER STUDY

The following recommendations are suggested for further study regarding the topic:

(i) The reliability of RC shallow footings can be calculated considering the effect

of wind loads.

(ii) The failure probability and corresponding reliability of RC shallow footing

may be evaluated taking the settlement of soil into account.

80

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86

SYMBOLS and NOTATIONS

fy Yield strength of steel

fcʹ Cylinder strength of concrete

\78′ A measure of concrete splitting strength

DL Dead Load

LL Live Load

EQ Earthquake Load

γ Unit weight of soil

Bf Flexural model uncertainty

Bv Shear model uncertainty (ACI)

Bv Punching shear model uncertainty for seismic loads

Bv Punching shear model uncertainty for gravity loads

qu Bearing capacity of soil based on N value

�� Inverse of the standard normal cumulative distribution function

CDF Cumulative distribution function

COV Coefficient of variation

pdf Probability density function

FORM First Order Reliability Method

SORM Second Order Reliability Method

MCS Monte Carlo Simulation �� Load Variable �� Resistance variable � Reliability index �� Probability of failure

g Performance function �� Mean value of resistance �� Mean value of Loads

BNBC Bangladesh National Building Code

SPT Standard Penetration Test

D Depth of foundation

B Width of footing