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ABSTRACT

The provision of a safe structural system is an object of structural design. Therefore the way
safety is prescribed in design is concern for Engineers. It is generally agreed in codified designs
that decisions are made under uncertainties and as such safety consideration is best done in a
probabilistic format. It is only in this way that consistent and uniform safety level can be ensured.
The limit state method assumes a probabilistic concept and it is on this basis that BS. 8110 (1985)
for reinforced concrete design was developed. However the implicit reliability in designing to the
code specification is not obvious in practical designs.
The BS. 8110 (1985) is assessed using a two span reinforced concrete beam to ascertain the
consistency and uniformity of reliability level of designs to the code specifications. It is found out
that reliability indices depend on the design input especially the load ratio, meaning that under
variable load the reliability will be inconsistent. The results also show that the conventional
method will produce designs with reliability index taking on a value among many possible ones.
This implies that the designer does not have control over the choice of a required reliability level.
Consequently, the work provides reliability plots that can be used as design charts to specify
design variables for known reliability levels.

 

 

TABLE OF CONTENTS

Title Page i
Declaration ii
Certification iii
Dedication iv
Acknowledgement v
Abstract vii
Chapter One: Introduction 1
1.1 Background to the study 1
1.2 The Need For a Rational Approach to Safe Structural Design 3
1.2.1 Limit State Design Method 6
1.3 Research Goal 7
1.4 Specific Objectives 8
1.5 Methodology 8
Chapter Two: Literature Review 10
2.1 Introduction 10
2.2 Review of Developments in Structural Engineering 10
2.30 Theoretical basis for Structural reliability 12
2.3.1 First Order Second Moment Reliability Indices 15
2.3.2 Hasofar and Lind Safety Index 16
2.3.3 Generalised First Order Reliability Indices 17
2.4. Classification of Reliability Methods 19
2.5 Review of Limit State Design Code 20
2.5.1 Loading and Safety Factors 22
2.5.2 Method of Partial Factor of Safety 23
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Chapter Three: Determination of Implied Reliability
3.1 Introduction 27
3.2 Idealisation of Functions of loads and load effect 27
3.3 Limit State Equations For Flexural Criteria 29
3.4 Singly Reinforced Sections 29
3.4.1 Interior Support 29
3.4.2 Doubly Reinforced Section 31
3.5 Limit State Equations for Shear Criteria 33
3.6 Parameters Identification for Limit State Function 34
3.7 Evaluation of Limit State Functions 36
Chapter Four: Discussion of Results 37
4.0 Evaluation of Limit State Functions 37
4.1 Singly Reinforced section in Flexure 37
4.1.1 Tensile Requirement at Support 37
4.1.2 Compression Requirement at interior Support 42
4.1.3 Tensile Requirement at Mid Span 45
4.1.4 Compression Requirement for Mid Span 49
4.2 Doubly Reinforced Section in Flexure 49
4.2.1 Compression Requirement for Interior support 50
4.2.2 Compression Requirement Mid Span 59
4.3 Shear Criteria 61
4.4 Reliability Design Charts 64
4.4.1 Uniform Reliability Level (ISOREL) 64
Chapter Five: Summary, Conclusion and Recommendation
5.1. Summary and Conclusion 66
5.2 Recommendations 67
10
References 68
Appendix I: Program Code Listing 75
Appendix II: Data 94
Appendix III: Graphs 110
Appendix IV: Design Example 128
11
LIST OF TABLES
Table 2.1: Load Combination and Values of partial Factors of Safety for Loads 23
Table 2.2: Values of partial Safety Factors Strength of Materials 23
Table 2.3: Classification of Failure Types arid Consequences 26
Table 3.1: Parameters For Flexural Requirement 35
Table 3.2: Parameters For Shear Requirement 35
Table 4.1: Load and Reinforcement Ratios for Specified Reliability index of a Singly
Reinforced section at interior support 40
Table 4.2 Reliability Indices of Mid span and support of 4m beam 49

CHAPTER ONE

INTRODUCTION
1.1 BACKGROUND TO THE STUDY
The field of Structural Engineering basically emanated from the need to provide safe and
functional structures within the dictates of system and environmental constraints. These
objectives require an intuitive understanding of the interaction of design variables and
resultants necessary to keep the system in the safe mode. This is necessary so as to
prevent an event of failure and its associated social economic problems. The effort of
Structural Engineers thus primarily centres on providing rational analyses and discourses
on theories and that in the final analysis border directly or indirectly on precision and
safety consideration of structures.
The various theories on mechanical properties of structures and materials, of forms and
shapes, and structural techniques, are tools employed to provide knowledge on the
possible states of a structure with which the Engineer to makes design decision regarding
safety and economy. It can be recognised that consideration for structural safety is a
fundamental requirement in making provision of engineering facilities.
The way safety is prescribed is however dependent on the level of sophistication of
prevalent engineering knowledge. Conwan (1971), observed that before the eighteenth
century, structures were constructed based on empirical rules, established by experience
and changing circumstances.
This was obviously due to the fact that at that time little was known about the mechanical
properties of structures and materials. It was also obvious judging from massiveness of
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the structural legacy of that period, that specification of large structural sections and sizes;
and relative non—complexity of structural forms made adherence to empirical rules
feasible.
Developments in modern structural engineering which started with the use of scientific
methods of experiment prediction to analyse structural response in relation to actions led
to the formulation of deterministic models to predict the state of the structure. The
expressions assumed an exact state of the structure which is not possible considering
random nature of input variables. Even though uncertainty in the state of the structure is
acknowledged by the prescription of safety factors, these factors are themselves
deterministic in the sense that it is perceived that once a factor of safety is applied to a
deterministic value, safety is guaranteed. Deterministic safety factors derivation is not
adequately rationalised. Freudenthal (1963)stated that:
“The most careful and rigorous structural analysis is largely deprived of
its merits if the accuracy of its results is limited by the use of empirical
multipliers, so—called safety factors varying between 1.0 and 5.0 and
selected rather arbitrarily on the basis of consideration that are not always
relevant or even rational”
There is a consensus among researchers in structural safety that deterministic approach to
matter concerning safety can best be described as inadequate considering the lack of
categorical statement that can be made on the practical values of load, strength and
response. According to Ang (1973), absolute safety of a structure cannot be guaranteed
due to uncertainties. Surhman et al (1983), state that “recognising that load and resistance
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are random variables, the rational approach to design would be through the use of
probabilistic method.”. Ang and et al (1974), also noted that the scatter exhibited by the
design variables introduces the need for probabilistic approach to reliability level
determination.
Structural reliability methods are developed in response to the realisation of the need for
probabilistic approach. The methods provide bases for a rational discuss of structural
safety problems.
1.2 THE NEED FOR A RATIONAL APPROACH TO SAFE
STRUCTURAL DESIGN
Structural Engineering is concerned with solving structural problems which are
characterised by randomness and their solutions must therefore take into consideration
inherent uncertainties. There exist uncertainties in the response of structures due to the
randomness of load borne by the structures as well as variable material properties. Some
of the uncertainties have been identified as (Ang (1973)):
i) unpredictability of the future loading conditions,
ii) inability to obtain and express the in—place material properties accurately,
iii) the use of simplified assumptions in predicting the behaviour of the structure due
to the loading under consideration,
iv) the limitations in the numerical methods used, and human factor e.g. errors,
omission e.t.c.
These are broadly classified as phenomenological, decision, modeling, physical and
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statistical uncertainties as well as human factor error (Afolayan, 1992).
In design, it is assumed that state of the structure can either be in the safe or failure
modes. Design input specification can therefore take on values in the safe set failure
domain. Usually in practice, we cannot be sure that values in the failure will not occur.
This brings about reliability problems. The safety requirement arises from the random
nature of the design parameters. Design therefore must seek to achieve and acceptable
probability that structures will perform satisfactorily or will not fail, within set criteria.
Failure in this instance may be defined as a condition of total collapse of the structure or
an element thereof, or show functional deficiency.
We need to have a measure of reliability according to which we design the system to have
any specified degree of safety. In designing for safety, the principle involve assessing
load effects that lead to the worst conditions in the life time of the structure and the
capacity of materials to resist the loads. A substantial margin is then provided to take care
of the inherent uncertainties. The way the margin is specified will vary widely from one
Engineer to another if left to the judgement of individual Engineers. To standardise
decision making with respect to analysis and design for safety, design codes are
developed. Code specifications are assumed to have taken into consideration all matter
relating to safety and of course economy of the designed structures.
Until recently code parameters were determined by judgement rather than by rational
analysis (Madsen et al, (1986)) as in the case of allowable stress method where an
empirical factor of safety is applied to ultimate stress to obtain allowable stress. A code
parameter becomes acceptable if the code, having being in use for some period of years,
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proved to have performed poorly. The need for precision and optimisation has brought
about continual review of codes of practice and it is discovered that it can be done most
rationally when probabilistic approach is considered (Ang, 1973; Surhman et al, 1983;
Ang et al, 1974). According to Brown (1960), it is the obligation of a Structural Engineer
to back up his safety consideration with as much logic and rational justification for safety
margins as necessary. In pursuance of rational approach there is an international focus of
researchers in applying probability concepts in analysing safety and uncertainties and it
has been shown that in developing a consistent code, design criteria are most rationally
formulated within a probabilistic framework (Ellingwood et al, 1974). Saul et al (1986),
also remarked that implementation of probabilistic based codes procedures would provide
harmonious philosophy to design and construction of structures.
Structural reliability method which evolved as a result of probabilistic consideration of
safety provides a rational basis for analysis of safety. It is an analytical tool that can be
utilised to arrive at either a point or an interval estimate of the probability that the
structural system will perform its specified function without failure when subjected to
loads and other environmental factors. The object of reliability analysis is to emphasise
the systematic and consistent use of analytical tools with a view to providing bases for
reducing the uncertainty involved in the reliability estimate (Afolayan, 1992). The
performance of reliability analysis will among other things indicate
i) the effects of increasing or decreasing safety factors on the structural
reliability
ii) the sensitivity of the structural reliability to other design parameters
iii) the additional information needed to improve accuracy of the reliability
estimate
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From this understanding consistent and systematic ways of making design decisions will
emerge.
1.2.1 Limit State Design Method
A method of design is defined by the way the margin for safety is prescribed through a
factor of safety either applied to the strength, the loading or combination of both. Mosley
et al (1989), explained that the permissible stress method in which designs are obtained
by applying a safety factor to the ultimate strength of the materials does posses some
inconsistencies resulting from the arbitrary way the permissible stress is defined and that
it is unsuitable for semi—plastic materials, non-linear structures and stability of structures
subjected to overturning forces. Also the load factor method is deficient because it does
not take variability of strength into consideration.
As earlier observed, the permissible stress and load factor methods though admit
uncertain nature of structural design through the application of safety factors, do not
consider rational reliability concept in their derivation. The introduction of probability
based safety analyses allows for developments of formal methods to obtain probability of
failure through systematic analysis of the uncertainties in all variables, although Ayyub et
al (1984), remarked that this development does not eliminate the need for one’s discretion,
rather the ways in which judgement is exploited is presumably more effective.
The limit state design concept resulted from the probabilistic considerations and it is
assumed to be more logical in its presentation of safety margin. The concept aims at
achieving a consistent and acceptable probability that structures being designed will
perform satisfactorily during their intended life (BS. 8110, Part 1, 1985 ). The philosophy
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involves setting limits for various design criteria and minimising the chances of the limits
being exceeded. Common among possible criteria are ultimate limit and serviceability
limit states.
Although the limit state concept is probability based, the existing codes on limit state
design especially with regard to reinforced concrete design, do not reveal the extent of
reliability of a particular structure or its components, consequent upon which the proof of
consistency or uniformity in reliability levels of structures designed to the existing limit
state code specification are not apparent.
1.3 RESEARCH GOAL
The current thinking is that a rational safety analysis should provide a uniform and
consistent reliability for effective optimisation of safety and economy. This should be the
object of any design code. In normal situation, designs carried out using design codes to
predict expected performance of structures subjected to varying design inputs given the
same criteria should produce uniform reliability.
In their calibration, the existing limit state design codes on reinforced concrete provide
partial safety factors which presumably cater for uncertainties inherent in design
variables. Examples are British Standard code of practice (CP 110) of 1972 and the more
recent British Standard Structural Use of Concrete (BS 8110) of 1985.
The method of partial factors of safety can best be described as ‘modified deterministic’
in the sense that a quantity displaying uncertainty is considered in terms of means and
standard deviation and are subsequently employed in deterministic operations using a
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constant characteristic value in place of the uncertain quantity. This raises some questions
as to the consistency and uniformity in reliability of resultant designs. To make matter
worse, adherence to design codes does not show the reliability to which designs are
carried out leaving the Engineer to accept that his design is safe without information on
the extent of reliability.
This research aims at investigating the implicit safety level provided by designing to
B.S.8110 (1985) specifications using structural reliability methods.
1.4 SPECIFIC OBJECTIVES
The specific objectives of the research are:
i) To appraise the implicit reliability of BS. 8110 (1985) for uniformity and
consistency in design of reinforced concrete beams
ii) To propose a basis for consistent design of rein forced concrete beams through
the use of reliability charts.
1.5 METHODOLOGY
This study will be basically analytical in approach. The first task will be to form safety
functions by obtaining the margin between strength functions and actions for the limit
being considered. The basic variables in the limit state functions are considered as
random variables. The means, standard deviations coefficients of variation and
distributions of the basic variables will be obtained from literature on works done in the
relevant areas of statistical test on loading and material strength.
Reliability calculation will then be performed using first order reliability method A
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computer program, FORM5 (1988) will be used for the approximate computation of
probability integrals. The program is capable of:
i) transforming non—guassian variables into independent standard normal variables
locating the most likely failure point through optimisation procedure
ii) linearising the limit state function in that point, and
iii) estimating probability of failure using the standard normal integral.
The necessity for these will be realised in the course of the study.
Having determined the reliability indices for various conditions from the above
procedures, the indices will be used to produce plots that will indicate implicit reliability
levels. The plots will be used to produce design charts from which a known reliability
can be prescribed for specific design in order to preserve uniformity in reliability level.

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