modelling inground decay of wood poles for optimal...
TRANSCRIPT
Modelling Inground Decay of Wood Poles
for Optimal Maintenance Decisions by
Anisur Rahman M.Sc. Engineering (Engineering Management), PG Dip in Personnel
Management, B.Sc. Engineering (Mechanical)
A thesis submitted to the School of Mechanical, Manufacturing and Medical Engineering,
Queensland University of Technology For the degree of Master of Engineering (BN 72)
January 2003
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ABSTRACT
Wood poles are popular and widely used in the Power Supply Industries in all over the
world because of their high strength per unit weight, low installation and maintenance
costs and excellent durability. Reliability of these components depends on a complex
combination of age, usage, component durability, inspection, maintenance actions and
environmental factors influencing decay and failure of components. Breakdown or
failure of any one or more of these components can lead to outage and cause a huge
loss to any organisation. Therefore, it is extremely important to predict the next failure
to prevent it or reduce its effect by appropriate maintenance and contingency plans.
In Australia, more than 5.3 million wooden poles are in use. This represents an
investment of around AU$ 12 billion with a replacement cost varying between
AU$1500-2500 per pole. Well-planned inspection and maintenance strategies
considering the effect of environmental and human factors can extend the reliability
and safety of these components. Maintenance and sophisticated inspection is
worthwhile if the additional costs are less than the savings from the reduced cost of
failures.
Objectives of this research are to:
• Investigate decay patterns of timber components based on age and environmental
factors (e.g. clay composition) for power supply wood pole in the Queensland
region.
• Develop models for optimizing inspection schedules and Maintenance plans.
Deterioration of wood poles in Queensland is found mostly due to inground soil
condition. It is found that the moisture content, pH value (Acidity/ alkalinity), bulk
density, salinity and electrical conductivity have influence over the deterioration
process. Presence of Kaolin or Quartz has some indirect effect on the degradation
process. It allows more water to be trapped inside the soil that cause algae, moss and
mould to grow and attack the wood poles. On the other hand, by virtue of
permeability, soils with high quartz content allows more water to infiltrate, preventing
the growth of micro-organism.
This research has increased fundamental understanding of inground wood decay
process, developed testing methods for soil factors and proposed integrated models for
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performance improvement through optimal inspection, repair and replacement
strategies considering durability, environmental and human factors in maintenance
decisions. A computer program is also developed to analyse “what if” scenario for
managerial decisions.
This research has enhanced knowledge on the wood decay process in diverse
environmental conditions. The outcomes of this research are important, not only to
users of timber components with ingrond decay but also to the wood industry in
general (the housing sector, railways for wooden sleepers and other structural
applications such as timber bridges). Three refereed conference papers have already
come out of this research and two more papers for refereed journal publication are in
the process.
This research can be extended to develop models for:
• Qualitative as well as quantitative research database on lab/field wood decay
process;
• Assessment of the residual life of timber infrastructure;
• Optimal condition monitoring and maintenance plans for timber components
showing inground decay;
And
• Cost effective decisions for prevention of timber components and mitigation.
Findings of this research can be applied to other equipment or assets showing time
dependent failure rate and can be extended further to consider age/usage replacement
policies, downtime and liability costs.
Key Words: Reliability, maintenance models, inground decay, wood poles, environmental and human factors
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ACKNOWLEDGEMENT I wish to acknowledge the contribution of the following people to the completion of
this project:
• My supervisor, Dr. Gopinath Chattopadhyay for his sincere and constant support,
encouragement, and guidance throughout this project. He spent his valuable time
in discussing various solutions related to problems during this project.
• My associated supervisor, Dr. R.M Iyer for his assistance and direction in
developing models and preparation of refereed papers.
• Professor Joseph Mathew, Head of School of MMME, Associate Professor Doug
Hardgrave, Acting Head of School and Dr. Prasad Yarlagadda, Former Director
of Research Centre for providing financial support.
• Mr. Darren Lloyd, Mr. Ray Krosch, Mr. Mark Van Der Hurk and Mr.Peter
Coman for providing wood pole data from Energex database.
• Mr Less Dowes, Resource manager, Soil Laboratory, School of Civil
Engineering, for his assistance and guidance in sampling soils and testing.
• Associate Professor Ray Frost, and Mr Tony Raftery, Senior Technologist, X-
Ray Analysis facility, School of Physical and Chemical Science for their help in
chemical analysis of the soil samples.
• Mr. Venkatarami Reddy, Mr. David Ho and Dr. Netra Gurung for helping me in
collecting soil samples from different places and testing the samples in the
laboratory and analysis of data.
• Finally, to my wife Roushan and my children Afnan and Rakin for their love,
support and continuous encouragement throughout this project.
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STATEMENT OF ORIGINALITY,
I declare that to the best of my knowledge the work presented in this thesis is original
except as acknowledged in the text, and that the material has not been submitted,
either in whole or in part, for another degree at this or any other university.
Signed: ………………………….. Anisur Rahman
Date:
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LIST OF PUBLICATIONS
Publications Resulting from this Thesis
1. Chattopadhyay, G.N., Rahman, A. and Iyer, R.M (2002); “Modelling
Environmental and Human Factors in Maintenance of High Volume Infrastructure
Components”; 3rd Asia Pacific Conference on System Integration and
Maintenance, Cairns, Sept 2002 (based on Chapters 1, 2 and 4).
2. Rahman, A. and Chattopadhyay, G.N., (2003); “Identification and Analysis of Soil
factors for Predicting Inground Decay of Timber Poles in Deciding Maintenance
Policies”; Proceedings of the 16th International Congress and Exhibition on
Condition Monitoring and Diagnostic Engineering Management, COMADEM
2003, Vaxjo, Sweden, 27-29 August 2003 (based on Chapters 2, 4 and 5)
3. Rahman, A. and Chattopadhyay, G.N., (2003); “Estimation of Parameters for
Distribution of Timber Pole Failure due to In-ground Decay”; Proceedings of the
5th Operations Research Conference on Operation Research in the 21st Century,
the Australian Society of Operations Research, Sunshine coast, Australia, 9-10
May, 2003, (based on Chapters 4 and 5).
Papers under Preparation
1. Chattopadhyay, G.N., Rahman, A. “Modelling Inground Decay of Timber Poles
for Optimal Maintenance Decisions of Components of Complex Infrastructure”,
(in process based on Chapters 1, 3 and 5)
2. Rahman, A; Chattopadhyay, G.N. “Development of Testing Methods for Soil
Factors Influencing Inground Decay of Wood Poles”, (based on Chapters 1, 3 and
5)
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Contents
Abstract……………………………………………………………….........................2
Acknowledgement…………………………………………………………………....4
Statement of Source………………………………………………………………......5
List of Publications…………………………………………………………………...6
Contents ........................................................................................................................7
List of Figures.............................................................................................................11
List of Tables………………………………………………………………………..10
Nomenclatures............................................................................................................13
Chapter One Introduction and Scope of Work..............................................................................15 1.1 Introduction 15 1.2 Importance of Reliability Restoration 15 1.3 Reliability of Infrastructure Component 16 1.4 Objectives of this Research 17 1.5 Scope of the Research and the Thesis Outline 17 Chapter Two Overview of Reliability and Maintenance Models ..................................................20 2.1 Introduction 20 2.2 Reliability 20
2.2.1 Failure 20 2.2.2 Classification of failure 22 2.2.3 Failure Models 26 2.2.4 Distribution models in Reliability 29
2.3 Overview of Reliability and Maintenance models 37 2.3.1 Maintenance 37 2.3.2 Maintenance Actions 37 2.3.3 Optimum Maintenance Policy 40 2.3.4 Human Reliability 40 2.3.5 Periodic Inspection and Condition Monitoring 41 2.3.6 Maintenance Models and Policies 42 2.3.7 Maintenance and inspection models for Wood pole 53
2.4 Summary 54 Chapter Three Reliability of Wood Pole............................................................................................55 3.1 Introduction: 55 3.2 Deterioration or Degradation of Wood Pole 56
3.2.1 Factors Affecting the Reliability and Safety 56 3.2.2 Inground Decay of Wood Pole 60
3.3 Inspection and Maintenance of Wood Pole 62 3.3.1 Inspection Techniques 62 3.3.2 Maintenance Techniques for Wood Pole 66
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3.4 Human Factors 69 3.5 Summary 72 Chapter Four Modelling Maintenance Cost for Complex Infrastructure Component ...............73 4.1 Introduction 73 4.2 Development of Mathematical Cost Models 73
4.2.1 Assumptions 73 4.2.2 Notations 74 4.2.3 Modelling Expected Cost per Unit Time 74 4.2.4 Estimation of Quality of Maintenance 77 4.2.5 Modelling Human Factors and Inspection 78 4.2.6 Modelling Environmental Factors 79 4.2.7. Modelling Durability 80 4.2.8 Parameter Estimation 82 4.2.9 Numerical Example of Estimating Base Line Parameters 82 4.2.10 Estimation of Parameters considering Environmental & Human Factors 90 4.2.11 Numerical Example of the Developed Model 94
4.3 Sensitivity Analysis of the Developed Model 96 4.4 Summary 104 Chapter Five Collection and Analysis of Environmental Data ...................................................106 5.1 Introduction: 106 5.2 Identification of Influential Soil Factors 106
5.2.1 Selection of Sampling Sites106 5.2.2 Equipment and Instruments Used in Sampling and Testing 107 5.2.3 Sample Collection 109 5.2.4 Soil Testing 109 5.2.5 On Site Testing 110 5.2.6 Laboratory Testing 110 5.2.7 Preparation of Soil Sample for Chemical Analysis 111
5.3 Results of Soil Sample 112 5.3.1 Collection of Soil Data and Analysis 112
5.4 Data Analysis 117 5.4.1 Moisture Content 117
5.4.2 pH value 118 5.4.3 Bulk Density 119 5.4.4 Salinity 120 5.4.5 Effect of Electrical conductivity 121 5.4.6 Chemical Analysis Test Result 121
5.5 Summary 123 Chapter Six Conclusions and Recommendations for Future Works .......................................124 6.1 Contribution 124 6.2 Limitations 125 6.3 Scope for Future Works 126
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References.................................................................................................................129
Appendices................................................................................................................134
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List of Tables
Table 1.1: Cost analysis of maintaining wood poles ...................................................16
Table 2.1: Distinguishing features of bathtub curve ...................................................25
Table 3.1: Comparative study of inspection and maintenance ...................................55
Table 3.2: Inspection reliability ...................................................................................70
Table 4.1: In-ground decay degradation factor for round pole in Zone B...................81
Table 4.2: Mortality of fully impregnated wood pole..................................................83
Table 4.3: Pole life analysis .........................................................................................84
Table 4.4: Data analysis for failure distribution .........................................................85
Table 4.5: Regression analysis for mortality of fully impregnated wood pole............86
Table 4.6: Failure rate, reliability function and cumulative failure distribution..........88
Table 4.7: Calculation of log[r*(t)] and log t ...............................................................92
Table 4.8: Regression analysis of wood poles data considering environment ...........93
Table 4.9: Effect of variation of maintenance quality. ................................................96
Table 4.10: Effect of cost and quality of maintenance ................................................97
Table 4.11: Effect of environment on shape parameter (β) and characteristic life
parameter (η) ...............................................................................................................98
Table 4.12: Effect of environmental parameter………………………………………92 Table 4.13: Effect of durability..................................................................................101
Table 4.14: Effect of cost of replacement Cre ............................................................102
Table 4.15: Effect of variation of minimal repair ......................................................103
Table 5.1: Soil physical test result, Wynnum, Lota and Mansfield ...........................113
Table 5.2: Soil physical test result, Caboolture .........................................................114
Table 5.3: Soil physical test result, Holland Park......................................................114
Table 5.4: Soil physical test result, Chelmer .............................................................115
Table 5.5: Soil chemical test resul, Caboolture .........................................................115
Table 5.6: Soil chemical test result, Chelmer ............................................................115
Table 5.7: Soil chemical test result, Wynnum, Lota and Mansfield..........................116
Table 5.8: Soil chemical test result, Holland park .....................................................117
Table 5.9: Summary of soil test result .......................................................................117
Table 5.10: Summary of chemical analysis of soil sample........................................122
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List of Figure
Figure 2.1: Failure causes classification (Blischke & Murthy, 2000) .........................21
Figure 2.2: The Bathtub curve ....................................................................................24
Figure 2.3: Three basic reliability functions ................................................................27
Figure 2.4: Exponential distribution ............................................................................30
Figure 2.5: Failure rate of the Weibull Distribution. ...................................................32
Figure 2.6: The effect of standard deviation σ on the normal PDF.............................33
Figure 3.1: Reliability of complex infrastructure ........................................................56
Figure 3.2: Examples of assumed decay pattern for in-ground poles..........................58
Figure 3.3: Hazard zone for decay of wood in ground contact in Australia ................59
Figure 3.4: Brown rot...................................................................................................61
Figure 3.5: Soft rot .......................................................................................................62
Figure 3.6: A Shigometer............................................................................................63
Figure 3.7: Ultra-sonic machines like the IML............................................................64
Figure 3.8: Path of measured transit time between transducers, around the shaded
(decayed) area (Dolwin 2001). ....................................................................................65
Figure 3.9: Minimal repair of wood pole a. Bioguard bandage and b. Plucked for
insertion treatment .......................................................................................................67
Figure 3.10: Mechanical Reinstatement of Wood Pole ...............................................68
Figure 3.11: Quick Deuar Pole Reinstatement Method ...............................................69
Figure 4.1: Failure rate distribution Vs maintenance...................................................75
Figure 4.2: Effects of covariates on hazard rate. .........................................................80
Figure 4.3: Graphical representation of regression analysis ........................................86
Figure 4.4: Failure rate Plot .........................................................................................89
Figure 4.5: Reliability Plot...........................................................................................89
Figure 4.6: Cumulative distribution.............................................................................90
Figure 4.7: Regression analysis plot for wood pole data when Durability,
environmental and human factors are taken into consideration...................................93
Figure 4.8: Effect of variation of α ..............................................................................97
Figure 4.9: Effect cost and quality of Maintenance.....................................................98
Figure 4.10: Effect of environmental parameter........................................................100
Figure 4.11: Effect of durability modification. ..........................................................101
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Figure 4.12: Effect of cost of replacement.................................................................102
Figure 4.13: Effect of minimal repair. .......................................................................104
Figure 5.1: Equipment and instruments used for traditional soil sampling .............108
Figure 5.2: Procedures for sample collection and measuring the depth ....................109
Figure 5.3: WP 81 – pH, Salinity and conductivity meter.........................................110
Figure 5.4: S4 Explorer X-Ray Defractometer (Bruker AXS Inc.) ...........................111
Figure 5.5: Relationship of moisture content of soil with failure rate .......................118
Figure 5.6: Relationship of pH and failure rate .........................................................119
Figure 5.7: Relationship between Bulk density and failure rate ................................120
Figure 5.8: Effect of salinity of soil ...........................................................................120
Figure 5.9: Effect of electrical conductivity ..............................................................121
Figure 5.10: Effect of Quartz .....................................................................................122
Figure 5.11: Effect of Kaolin .....................................................................................123
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NOMENCLATURES
Unless otherwise specified, the symbols used in this thesis have the following
meaning.
C(N, x) Expected cost per unit time
C(N*,x*) Expected optimum unit cost
Cpm Cost of preventive maintenance
Cre Cost of replacement
Cmr Cost of minimal repair
τ Component/system life restoration due to maintenance actions
k Number of maintenance actions carried out till time t
t Time
x Interval between two successive preventive maintenance
x* Optimum maintenance interval
N Number of preventive maintenance actions to be carried out
before being replaced
N* Optimal number of preventive maintenance before being
replaced
Nx Length of service life
α Quality or effectiveness of maintenance.
β Weibull Shape parameter (for base line equation)
β* Weibull Shape parameter when durability, environmental, and
human factor are taken into consideration
ε Linear environmental parameter
η Characteristic life of Weibul distribution (for base line
equation)
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η* Weibull characteristic life parameter when durability,
environmental, and human factor are taken into consideration
θ Exponential environmental
ψ Environmental factor
ψt Time dependent environmental factor
F(t) Cumulative distribution function
f(t) Probability density function
R(t) Reliability function
r(t) Failure rate of equipment at time t
ri(t) Failure rate of equipment with effect of inspection reliability at
time t.
r*(t) Failure rate of equipment when environmental and human
factors are taken in to consideration.
rpm(t) Failure rate of equipment at time t undergoing maintenance
action.
σ Standard deviation of a normally distributed data set
µ Mean of a normally distributed data set
z Standardised normal variate
b Constant
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Chapter One
Introduction and Scope of Work
1.1 Introduction
In today’s increasingly competitive world, reliability of a system and its components is
extremely important. Production and maintenance managers are in tremendous
pressure to improve reliability of products/services to lower operating cost for
satisfactory customer services.
Beasley (1991) defines reliability as “the probability that a device or system will
operate for a given period of time under given operating conditions”.
The concept of reliability would be different for different persons even though the
functions might be the same. For example, a water tap might well be considered to be
operational or reliable if, in a domestic setting, the ‘off’ position allows one drop of
water to escape every 10 to 15 seconds. But if the same rate of dripping occurred
through a tap in a piece of chemical engineering equipment, then this might be
unacceptable and the tap would be considered failed or unreliable ((Beasley (1991)).
Failure of a system causes inconvenience, irritation and a severe disruption of service
and loss to the society and environment. These sufferings are worse when failure is
unexpected. Faulty design, bad workmanship, over stress on the component, age,
usage and environmental factors are the reasons for unexpected failures. Regular and
accurate inspection and proper maintenance actions are important to predict and
prevent unexpected failures of the system/component, thereby making it more reliable.
1.2 Importance of Reliability Restoration
Accurate inspection and data acquisition, effective maintenance and dependable
restoration of systems or components provide economic benefits. Bingel (1995) carried
out a study that gives the economic benefit of a comprehensive wood structure, when
the reliability of the wood pole component is extended through the maintenance
program. Summary of Bingel’s cost–benefit analysis of wood poles maintenance is
given bellow:
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Bingel examined, for every 100,000 poles owned by a utility around 2000 were
replaced annually with an average age of 30 years for replacements, 40 years for steel
truss and 45 years for fibre glass restoration.
Case I
When all the 2000 poles are replaced at an age of 30 years
Total cost of replacement = 2,000 × $1,500 = $ 3,000,000
Per year cost = $ 3,000,000/30 = $ 100,000;
Case II
Different maintenance decisions were taken as shown in the Table 1.1.
Table 1.1: Cost analysis of maintaining wood poles
Maintenance decision
No. of Poles
Installn. cost per pole $
Maintn. cosper pole $
Total cost =Installn. cost + Maintn. cost $
Average extended service life
Per year cost $
Steel Truss 1000 1,500 400 1,900,000 40 years 47500
Fibre Glass 600 1,500 800 1,380,000 45 years 30667
Replace 400 1,500 0 600,000 30 years 20000
Total per year cost of 2000 poles, when different maintenance actions are being taken
= $ (47500 + 30667 + 20000)
= $ 98167
Therefore, the savings is = $ (100,000 – 98167)) = $ 1883 for every 100,000 poles.
1.3 Reliability of Infrastructure Component Research on reliability of equipment including inspection plan and maintenance
strategies is of great interest to the reliability engineers and maintenance managers. In
the maintenance context, one of the most useful but ignored areas is modelling optimal
maintenance policies for high volume component of complex infrastructure. Review of
literatures shows that only a few papers are available in the area of modelling
maintenance policies for complex infrastructure and existing models so far have
concentrated only on operational factors and have not considered durability,
environmental and human factors. Modelling inground decay of wood poles due to soil
factors is important for predicting reliability and in deciding inspection and
maintenance decision.
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High frequencies of sophisticated inspection and maintenance can improve the system
reliability at an increased maintenance cost. However, it can reduce the cost of
downtime time. On the other hand, low frequency or conventional inspection and
replacement on failure can reduce the inspection and maintenance cost. However, it
can increase downtime cost and the risk associated with failure. Maintenance is
worthwhile if the additional cost of maintenance is less than savings from the reduced
failure costs. Therefore, a trade off is required between no inspection and
repair/replacement on failure vs. over inspection or unnecessary replacements.
1.4 Objectives of this Research
• To investigate inground decay patterns of timber components based on age and
environmental factors (e.g. clay composition) in the Queensland region.
• To develop testing procedures for soil factors.
• To understand age and effect of soil factors on failure/ degradation and to study
various types of maintenance and repair activities applied for pole restoration.
• To develop mathematical models for optimal maintenance and inspection policies
for infrastructure components.
• To analyse the models by using numerical examples.
• To integrate failure/ decay pattern of infrastructure components, the environmental
factors and human factors in inspection and maintenance decisions.
1.5 Scope of the Research and the Thesis Outline Scope of this research is as follows:
This thesis deals with identification of soil factors, development of testing
procedure for these factors and the development of mathematical models for
optimal maintenance plan for high volume components of complex infrastructure (
wood poles used in Power supply industries). More specifically, it provides new
theoretical models to study the following topics:
• Investigation of inground decay of timber materials due to influential soil
factors and based on this, development of decay pattern of such materials.
• Development of hazard map for soil factors for preventive measures.
• Understanding of human errors on inspection and maintenance decisions and
the effect of this on overall system reliability.
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• The use of various types of repair versus replacement strategies.
• Development of optimal inspection and maintenance policies for infrastructure
components.
Outline of this thesis is as follows
Chapter 2 provides an overview of reliability and various maintenance actions. This
chapter consists of two main sections. In the first section, reliability terms are defined
with citations and some important reliability models, useful for this research. The
second section of this chapter discusses the different categories of maintenance actions
and their effects on the reliability of systems. A clear understanding of these
theoretical aspects is useful for developing optimal maintenance policies. These are
discussed with critiques in the later part of this section.
Chapter 3 examines the problems of high volume components of complex
infrastructure. In this chapter, importance and various factors of reliability of
infrastructure component such as wood pole are discussed. This also provides us clear
pictures of the effects of different factors on the component reliability. This chapter
also describes of various actions, currently in use for inspection and maintenance of
wood poles.
Chapter 4 discusses the development of maintenance models and analysis. In this
chapter maintenance models are developed considering the durability of component,
environmental and human effects on the component reliability. This model is then
testified and analysed with real life data. Sensitivities of these models are analysed in
the later section.
Chapter 5 deals with modelling environmental and human factors in inspection and
maintenance decisions. This chapter focuses on identifying soil factors, influencing
inground decay of timber poles and on developing mechanism for measuring these
factors. Field data from Brisbane, Australia has been used for analysis. Various soil
factors are investigated and analysed with a view to develop functional relationship of
these factors and inground decay of wood pole. Results from this investigation can be
used in maintenance and replacement decisions of inground timber components used in
utility, construction, railway and transportation sectors.
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Finally the contribution and the limitations of this research and the scope for future
research are discussed in Chapter 6.
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Chapter Two
Overview of Reliability and Maintenance Models
2.1 Introduction All systems and their components degrade/fail over time. Failure may lead to a loss of
production, loss of properties or loss of life. A failed system may be replaced by an
identical new system or by a used system. Mathematical models form a base for an
objective measurement, and interpretation of results. Models are used for prediction of
failure, optimal design and control (Kovalenko et al (1997). Stochastic models
describe a process and the influence of a large variety of random phenomena on such a
process. Section 2.2 briefly defines failure and reliability. Section 2.3 provides an
overview of reliability and maintenance models.
2.2 Reliability Different authors defined reliability in different manners. Drummer and Winton (1990)
defined reliability as the probability of survival of all parts in a system, Davidson
(1994) defined reliability as a statement of probability (statistical concept) that a plant
or piece of equipment or component will not fail in a given time while working in a
given environment. Needless to say, the reliability of systems is becoming increasingly
important because of factors such as cost, competition, public demand, and usage of
untried technology. The only effective way to ensure reliability is to consider the
reliability factors seriously during their design and operation. Before introducing the
various reliability factors it is essential that the word “Failure” is fully defined and
understood.
2.2.1 Failure
The dictionary definition of failure is falling short of something expected, attempted or
desired or in some way deficient or lacking the desire. Blischke and Murthy (2000)
recommend a definition that failure is an event when a system is not available to
perform any function at a specified condition according to the determined schedule or
not capable to produce parts or to carry out schedule operation as per specification.
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Therefore, “Failure” of a component/system can be defined as the termination of a
component or system that carries out a specified function. A component or system is
designed to sustain certain nominal stresses through aging effect due to surrounding
environment and in operations. When these components or systems are operated,
failures or breakdowns may occur due to the increase of operational stresses above the
rated level.
Cause of failure
There are number of factors that lead to a component or system to fail. Among them,
wear and tear of the components due to usage, the frequency of usage, the downgraded
quality and strength of the component, environment or weather condition, load
imposed on the component structure and repair and rectification, quality or durability
of the component /system are most significant.
According to IEC 50(191), cause of failure is “the circumstances during design,
manufacturing or operation, which has led to a failure”. According to Blischke and
Murthy (2000) the causes of failure can be classified as following (Figure 2.1) in
relation to the life cycle of the system
[This figure is not available online. Please consult the hardcopy thesis available from the QUT library]
Figure 2.1: Failure causes classification (Blischke & Murthy, 2000)
• Design failure- inadequate and faulty design may cause accidental failure of the
system or the product and thereby increasing the failure rate this type of failures
are mostly occurred during early stage.
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• Weakness failure – Normally, it happens when the system is too weak to withstand
the normal stress designed for it. Such failure happen because of the
incompatibility of design with the operation.
• Manufacturing failure – Due to lack in proper quality control of the manufactured
product.
• Aging failure – Each system degrades over time only because of the age or usage.
• Misuse failure – Due to misuse of the system- operating beyond its design
specification.
• Mishandling failure – As a result of careless operation and lack of proper
maintenance.
2.2.2 Classification of failure
To understand the broad meaning of failure it is important to classify failure into
different categories on the basis of their occurrence and nature Villemeur (1992).
Classification of failure based on suddenness
• Gradual failure or drift failure –failure of an entity due to a gradual
degradation with the time or usage. These sorts of failures could be detected or
anticipated prior to the occurrence of failure by proper inspection. One
example of such type of failure could be failure of a ball bearing, operating for
a long time.
• Sudden failure – when failure of an entity occurs all of a sudden without any
prior notice and could not be predicted by examination or monitoring through
regular inspection. This type of failure does not result in a progressive loss of
the performance. It does not show any successive degradation or impairment
on the entity. In case of such failure, the failed entity would be thrown away
and replaced by a new one. Failure of an electric bulb is a very common
example of sudden failure.
• Classification of failure based on degree of failure – whether it is a complete
or partial failure
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• Complete failure – failure results in complete malfunction of the entity. Here,
failure results from a deviation in the entity characteristic(s) beyond specified
limits such as a situation that would lead to a complete loss of the capability of
the entity to perform its specified function.
• Partial failure – failure resulting from a deviation in characteristic(s) beyond
specified limits but unlike complete failure, it does not cause a complete
deficiency of the required function. Here, the failed entity still can be useable
for a short time, while waiting for repair/replacement. This partially failed item
could lead to a complete failure if not repaired/replaced early.
• Classification of failure combining suddenness and degree
• Catastrophic failure – failure which is a combination of both complete and
sudden breakdown. This type of failure is very dangerous and results in total
system loss and loss of lives.
• Degradation failure – this type of failure is a combination of partial and
gradual failure and ultimately turns to a complete failure. Item experiencing a
partial failure could end up with complete failure caused by the gradual
deterioration. One simple example of this type of failure is the deterioration of
an electricity wood pole due to decay or biological attack.
Classification of failure based on their date of occurrence in the system life time In this categorisation of failure, the failure types are explained (Ebeling (1997)) by a
curve called bath tub curve
Bath tub curve
In a real life situation, most components or systems, tend to have failure rate curves
with a similar kind of appearance. These curves may be obtained as a composite of
several failure distributions. Because of their shape, the failure rate function is
commonly referred to as the bathtub curve.
Burn-in failure or infant mortality
The first part of the curve is known as early failure period or infant mortality (as the
failures experience decreasing failure rates early in their life cycle). It exhibits defect
in design, manufacturing error, inappropriate construction or installation, faulty
operation or misuse. During this period, the weak parts or faults are weeded out.
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[This figure is not available online. Please consult the hardcopy thesis available from the QUT library]
Figure 2.2: The Bathtub curve (Ebeling 1997)
Stable failure period or intrinsic failure
The long nearly flat portion of the failure rate curve is called the stable failure period.
This is also known as the intrinsic failure period. Here, the failures of components or
systems occur in a random fashion at a uniform or constant rate. The failures occurring
in this phase are purely by chance or due to random factors and modeled by
exponential lifetime data.
Wear out failure
Finally, the ever increasing part of the curve is known as the wear out failure or fatigue
failure period, exhibits an increasing rate of failure, characterised mainly by complex
aging phenomena. Here, the component deteriorates due to accumulated fatigue and
outside shock (Modarres (1993)). The Table 2.1 summarises the distinguishing
features of the Bath tub curve.
25
Table 2.1: Distinguishing features of bathtub curve (Ebeling (1997))
[This figure is not available online. Please consult the hardcopy thesis available from the QUT library]
Classification of failure based on effects- failure could have different effects on the
entity. It could affect the system directly which merely requires a correction or easily
performable action or it could affect the system availability or safety. So, it is
important to assess the effect of failure. Here, failure is classified according to its
effect.
• Minor failure – The effect of this failure is not dangerous. The failure causes
minor or negligible damage to the entity or the environment.
• Significant failure – This type of failure causes a degradation of the system
performance without having any severe damage to the system or the
environment.
• Critical failure – Resulting in significant damage to the entity or the
environment but negligible loss of life.
• Catastrophic failure – Any failure, which could cause the complete damage to
the system and also could result in significant damage to the environment and
life.
26
2.2.3 Failure Models
This section introduces several quantitative measures for reliability of a unit. This unit
can be anything from a simple component to a complex system. The most important
measures for the reliability of a unit are as follows:
• The failure cumulative distribution function F(t)
• The reliability function R(t)
• The failure rate r(t)
• Mean time to failure MTTF
• Mean time between failure MTBF
Failure cumulative distribution function F(t)
The failure cumulative distribution function is given by
( )duuftFt
∫ ∞−=)( (2.1)
Where, f(t) is the probability density function and t is the time
The alternative equations to obtain the failure cumulative distribution function are as
follows:
( ) ( )tRtF −= 1 (2.2)
and
( ) ( )
−−= ∫
tduurtF
0exp1 (2.3)
Where, R(t), is the probability that the unit does not fail in the interval (0, t).
And, r(t) is the failure rate.
The reliability function R(t)
The reliability function of a component, a subsystem or system, R(t), is given by
( ) ( )∫∞
=t
duuftR (2.4)
Other equations also used to obtain the reliability function, R(t), are as follows
( ) ( )duuftRt
∫−=0
1 (2.5)
and
( ) ( )
−= ∫ duurtR
t
0exp (2.6)
27
Probability density function f(t)
Probability density function can be expressed by
( ) ( )
( )dt
tdRdt
tdFtf
−=
= (2.7)
In addition the following relationship can also be used to obtain the probability density
function:
( ) ( ) ( )tRtrtf = (2.8)
This PDF function describes the shape of the failure distribution. And the PDF has the
following two properties
( ) 0≥tf And ( ) 10
=∫∞
dttf
Given the PDF, f(t), then
( ) ( )∫=t
duuftF0
(2.9)
( ) ( )∫∞
=t
duuftR (210)
These three functions are illustrated in the Figure 2.3.
[This figure is not available online. Please consult the hardcopy thesis available from the QUT library]
Figure 2.3: Three basic reliability functions a. Probability density function, b. Cumulative distribution function, and c. the Reliability function
(Ebeling (1997)).
So, both the reliability function and the CDF function represent the area under the
curve defined by f(t). Since the area beneath the entire curve is equal to unity we can
define both the reliability and failure probability as bellow
28
( ) 10 ≥≥ tR And ( ) 10 ≤≤ tF
The failure rate function r(t)
Failure rate provides an instantaneous rate of failure. The failure rate function r(t) of a
component, subsystem or a system is defined as follows:
When, there is a large number of component in a system, and the time to failure of
each of the component is noted, an estimate of failure rate of a component at any point
of time can be made taking into account the ratio of number of items which fails in an
interval of time to the number of the original population which were operational at the
start of the interval. Failure rate has the property that if an item has not failed at time t,
then the probability that it fails in the interval t to t + ∆t is ( )∆tλ t and thus it is the
probability that an entity will fail in the next interval of time provided that it is good at
t, the start of the interval (Hokstad 1997). So, it is a conditional probability
From { } ( ) ( )ttRtRttTt ∆+−=∆+≤≤Pr we can have the following
equation.
{ } [ ( ) ( ) ] ( )tRttRtRtTttTt /Pr ∆+−=≥∆+≤≤
Then ( ) ( )( ) ttR
ttRtR∆
∆−− is the conditional probability of failure per
unit of time
Therefore,
( ) ( ) ( )[ ]
( )( )
( )( )( )tRtf
tRdttdR
tRttRttRtr
t
=⋅−
=
⋅∆
−∆+−=
→∆
1
.1lim0
(2.11)
The failure rate function r(t) provides an alternative way of describing a failure
distribution. Failure rate may be increasing, decreasing or constant when r(t) is an
increasing decreasing or constant function (Ebeling (1997)).
29
The failure rate function of an item can also be obtained from the following relationships:
( ) ( )( )
dttdR
tRtr 1
−=
and
( ) ( )( )tF
tftr−
=1
(2.12)
MTTF
Mean time to failure – applies to non repairable item.
This is the average time an item may be expected to function before failure. The
MTTF is defined as
( )∫∞
=0
dttRMTTF (2.13)
An alternative approach to obtain the same result is to be find the expected value of
the failure probability density function. Thus the expected value E(t) of a continuous
random variable is given by
( ) ( )∫∞
==0
dtttfTEMTTF (2.14)
MTBF
Mean time between failures- applies to repairable item. This is the average time
between two failures of same item. The MTBF tells us that how often, on the average,
we can expect experience an outage.
2.2.4 Distribution models in Reliability
The exact form of reliability depends on the underlying failure distribution of the
equipment/system. One can use either graphical or statistical methods to determine if a
given failure data set can be adequately modeled by one of the available failure
distributions. When reviewing reliability literature, reference is often made to the
failure distribution being Weibull, exponential, gamma, normal, hyper-exponential and
so on. Exponential Distribution
Exponential probability distribution is one of the important and easy to apply a
distribution function that defines a failure distribution with a constant failure rate.
30
Failure due to completely random or chance events will follow this distribution.
Parameter estimators and exact confidence interval can be easily computed under
different testing condition. This distribution is found to be typical for many electronic
components and pieces of industrial plant.
Development of CFR (constant failure rate) model begins by assuming that
( ) 0....0., >≥= λλλ tt
Where, λ is the exponential parameter
Then the reliability function, probability distribution function and cumulative
functions (Ebeling1997) are given by:
( ) tetF λ−−= 1 0≥t (2.15)
( ) tetf λλ −= for t > 0, λ > 0 (2.16)
= 0 otherwise
( ) ( ) ,t
teduuftR λ−∞
== ∫ for t > 0. (2.17)
And, ( ) ( )( ) λλ
λ
λ
===−
−
t
t
ee
tRtftr (2.18)
The Figure 2.4 gives the graphical representation of the exponential distribution
Figure 2.4: Exponential distribution
Weibull Distribution
Weibull distribution is widely used in component reliability analysis and in predicting
the failure. This simplifies the modeling of failure distribution while failure rate is
increasing or decreasing. Its flexibility (it can deal with increasing, decreasing and
constant hazards), mathematical simplicities, amenability to graphical analysis, and
λ
F(t)
f(t)
time0
31
ability to fit most of life time data make it attractive to the practicing reliability
engineer and also accounts for its wide use (Crowder (1991)).
In two parameter form the Weibull model the model can be expressed as follows:
Cumulative distribution function: ( )β
η
−
−=t
etF 1 (2.19)
PDF: ( )β
ηβ
ηηβ
−
−
=
t
ettf1
(2.20)
Reliability: ( )β
η
−
=t
etR (2.21)
The failure rate is
( ) ( )( )
1−
=
=
β
ηηβ t
tRtftr
0,;0 >∞<≤ ηβt (2.22)
Where, η is the scale parameter, known as characteristic life, which is value of t at
which there is an approximately 66.66% probability that the component will have
failed.
β is the shape parameter, which is related to behaviour of the hazard function
For β = 1, r(t) is constant (that is equivalent to the exponential distribution)
For β >1, r(t) is increasing.
For β<1 r(t) is decreasing, For β = 3.44 the Weibull distribution is very close to
Normal distribution.
The Weibull distribution has the best possibility of application (Carter (1986)).
The Figure 2.5 represents a two parameter Weibull distribution
32
[This table is not available online. Please consult the hardcopy thesis available
from the QUT library]
Figure 2.5: Failure rate of the Weibull Distribution, η =1 (Hoyland (1994)).
The three parameters (β =shape parameter, η=scale parameter and γ = shift or location
parameter (since it represents a simple time shift)) in the Weibull distribution:
The equations are
The Cumulative distribution function:
( ) [ ]{ }βηγ /)(1 −−−= tetF (2.23)
Corresponding density function:
( )[ ] [ ]ηηγβ β /1/)( 1−−= ttf ( )[ ]{ }βηγ /−− te (2.24)
The Reliability:
( ) [ ]{ }βηγ /)( −−= tetR (2.25)
And, the failure rate is given by
( )[ ] 1/)( −−= βηγηβ ttr (2.26)
Changes in the scale of t are dependent of the changes of η. While β is dimensionless
η and γ have the dimension of time.
The Extreme Value Distribution
This distribution is very similar to Weibull distribution and useful in cases where
hazard rate is initially constant but increases rapidly as the time increases. This plays
an important role in reliability analysis. This distribution is used to describe the failure
33
time of a component or product which is operating normally at specified condition and
suddenly breaks down because of a secondary causes such as overheating, fracture etc
when subjected to extreme condition (Elsayed (1996)).
The cumulative distribution function, probability distribution function and reliability
functions and failure rate function are expressed as:
( ) ( )11
−−−=
teb
etFα
α (2.27)
( ) ( )1−−=
tebtebetf
α
αα (2.28)
( ) ( )1−−=
teb
etRα
α (2.29)
( ) tbetr α= (2.30)
Where b is a constant and eα represents the increase in the failure rate per unit time.
[This graphs is not available online. Please consult the hardcopy thesis available from the QUT library] Figure 2.6: The effect of standard deviation σ on the normal probability density
function (Elsayed (1996))
Normal or Gaussian distribution is used successfully to model most of the mechanical
component/product that are subjected to repeated cyclic loads such as fatigue test. It is
also useful in analysing lognormal probabilities. Density function of the normal
provides bell shaped curve is shown in the Figure 2.6.
A random variable T is said to be normally distributed with mean µ and variance σ2,
( )2,~ σµΝT .
The cumulative distribution function is given by
34
( )( )
tdetFt
t′−=
−′−∞
∫2
2
2
211 σ
µ
πσ (2.31)
Where the probability density of T is
( )( )
2
2
2
21 σ
µ
πσ
−−
=t
etf for ∞<<∞− t . (2.32)
The reliability function for this distribution is determined from
( )( )
tdetRt
t′=
−′−∞
∫2
2
2
21 σ
µ
πσ. (2.33)
As there is no close form of this equation, it must be evaluated numerically. If the
transformation
σ
µ−=
Tz is made, then z will be normally distributed with a
mean of zero and a variance of one. The PDF of z is given by
( ) 2
2
21 z
ez−
=π
φ
and z is referred to as the standardised normal variate. Its cumulative distribution
function is
( ) ( )∫∞−
′′=Φz
zdzz φ .
Therefore,
( ) { }
−
Φ=
−
≤=
−
≤−
=≤=
σµ
σµ
σµ
σµ
ttz
tTtTtF
Pr
PrPr.
Therefore, reliability is
( )
−
Φ−=σ
µttR 1 .
And the failure rate function can not be written in closed form either. However,
( ) ( )( )
( )
−
Φ−==
σµt
tftRtftr
1 (2.34)
it can be shown to be an increasing function. Therefore, the normal distribution would
be used to model wear out phenomena only.
35
Log normal distribution
It is one of the most widely used distributions in describing the life data resulting from
a closely related failure mechanism. It is also used in predicting reliability from
accelerated life test data.
If the time to failure T has a lognormal distribution, the logarithm of T has a normal
distribution that is T is said to be normally distributed if ‘Y = ln T' is normally
distributed.
The cumulative distribution function of the lognormal is
( ) tdet
tFt
t′
′=
−′
−
∫2ln
21
0 21 σ
µ
πσ
or ( ) ( )
−
≤=≤=σ
µtztTtF lnPrPr . (2.35)
The probability density function of lognormal distribution is
( )2ln
21
21
−
−
= σµ
πσ
t
et
tf ∞<<∞− µ σ >0, t>0. (2.36)
Therefore the reliability function is
( ) [ ]
−
>=>=σ
µtztTtR lnPrPr . (2.37)
And the failure rate function is
( ) ( )tRt
t
trσ
σµφ
−
=
ln
. (2.38)
Gamma distribution
This distribution covers a wide range of failure rate function: increasing, decreasing or
constant failure rate and is suitable for describing the failure time of a component
whose failure takes place in n stages or the failure time of a system that fails when n
independent sub-failures have occurred.
36
This distribution is characterised by two parameters called shape parameter γ and scale
parameter θ . When 0<γ<1, the failure rate monotonically decreases from infinity to
1/θ as times increases from 0 to infinity. When γ>1, the failure rate monotonically
increases from 1/θ to infinity and when γ = 1 the failure rate is constant.
In this case the cumulative distribution function is written as
( )( ) tdettF
tt
t′
Γ′
=′
−
′
−
∫ θγ
γθ0
1
. (2.39)
The PDF of gamma distribution is
( ) ( )( )θ
γγ θθ
Γ−
=− tttf exp1
. (2.40)
Reliability function is then given by
( ) ( ) ( ) ( )( )
−++++=
−−
!1........
!21
12
θγγγ
θγ tttetR t (2.41)
and the failure rate is
( ) ( )( )
( )( )
( ) ( ) ( )( )
−++++
Γ−
==−
−
−
!1........
!21
exp
12
1
θγγγ
θγγ
θγ
θθ
ttte
tt
tRtftr
t
(2.42)
Power series model
In some cases in the real life situation, the hazard rate or failure data do not fit exactly
in any of the established failure distribution models such as exponential, Weibull and
so on. In such situations, a general power series model can be applicable to fit the
failure data. The number of terms in the power series model relates to the desired level
of fitness of the model to the empirical data. The failure rate function of the PSM is
given by
( ) nn tatataatr ..........2
310 +++= (2.43)
and the reliability function is:
( )
+
++−=+
1......
32exp
132
21
0 ntatata
tatRn
n (2.44)
37
2.3 Overview of Reliability and Maintenance models 2.3.1 Maintenance
Maintenance can be defined as the combination of all technical and associated
administrative actions intended to retain an item or system in or restore it to a state in
which it can perform its required function.
A system/equipment usually operates under severe condition of load, shocks,
corrosion, heat and humidity, and so on. This deteriorates slowly or rapidly unless
proper servicing and maintenance is carried out at appropriate time.
In case of failure, there are three options available: repair, overhaul and replacement
actions. Selection among these alternatives simply depends on the cost and the
resulting benefits from each option being taken. Therefore, the optimal selection of
such preventive or corrective actions and intervals are (which, balances the cost and
benefits) extremely important in the context of cost of operation and the risk to the
environment/community.
2.3.2 Maintenance Actions
Maintenance actions can be classified into two main categories, such as preventive
maintenance and corrective maintenance.
Preventive maintenance – preventive maintenance is a planned maintenance action to
be carried out during operation of the system to reduce the system deterioration and/or
risk of failure and if carried out properly it will retain the system in an operational and
available condition. Reliability of a system is greatly affected by the implementation
of an effective preventive maintenance program. Preventive actions can be divided
into three sub categories.
• Replacement actions
Planned replacement of component/system is carried out at constant intervals of time
or based on other criteria (e.g. Number of revolution etc). The components or the parts
could be replaced at predetermined age or usage. Replacement enables the system to
be “as good as new” condition (see curve a in the Figure 2.7). This means the failure
rate r(t) of the system is restored to zero.
38
b c a time
Figure 2.7: Failure rate with effect of a) Replacement, b) Minimal repair,
c) Overhauling. • Overhaul or Major repair action
An overhaul is a restorative maintenance action that is taken before equipment or
component has reached to a defined failed state (Jardine (1973)). An overhaul could
not make the system “as good as new” condition any more but can tune up the system
by replacing the worn out components. Overhauling/major repair is also termed as
imperfect repair. Imperfect repair of a component restores a substantial portion of wear
and the hazard rate falls in between “as good as new” and “as bad as old” (see curve c
in the Figure 2.7) (Coetzee (1997)), (Carter (1986)). It improves the reliability of the
system/component up to a certain level based on the scope and availability of the
upgrade.
• Minimal repair
A minimal repair makes insignificant improvement and the condition after
maintenance is “as bad as old” (see curve b in the Figure 2.7). It does not change the
total failure rate of the system since the aging of the other components is unchanged
(Elsayed (1996), (Barlow (1960)).
• Condition monitoring action
Action, like monitoring the condition of some equipment has an influence in
maintenance decision and helps in reducing the probability of failure based on the
accuracy of diagnosis and appropriate action.
r(t)
39
Corrective maintenance – These actions are carried out followed by an occurrence of
failure to return the system back to operation. This type of maintenance action does
not need any planned schedule to perform the activity. These actions can be sub
divided into two categories.
• Replacement at failure
In case of non-repairable system/component, the failed system/component is replaced
by a new one or used but good one.
• Repair actions
These types of maintenance actions are applicable to a system with repairable failed
components. Repairable actions can again be sub-divided into
a. Perfect repair or ‘As good as new’ repair
It is that type of maintenance action where the system would be brought back to “as
good as new” condition. The failure rate and the reliability of a system experiencing
perfect maintenance would be approximately the same as new system. Perfect repair is
assumed to be suitable for comparatively simple system with a very few number of
components.
b. Minimal repair
Under this repair policy when an item fails, it is repaired or restored minimally. It is
also known as “as bad as old”. Minimal repair is used for a large and complex system.
Repairing one or more components will not affect the total failure rate of the system
since the aging of the other components will ensure that the system failure rate will
remain unchanged (Elsayed (1996)), (Barlow (1960)).
c. Imperfect repair
When a system/component is repaired by replacing failed components and also other
aged components, then the failure rate of the repaired system becomes less than that of
failure before failure state but more than that of a new system. This type of repair is
assumed to be in between perfect and minimal repair.
40
2.3.3 Optimum Maintenance Policy
Maintenance improves reliability of systems by slowing its aging effect (Brown
(1983)). However, maintenance means costs - costs of inspection,
repair/replacement/overhaul and costs of downtime. But the high frequency of
maintenance actions increase the total cost of maintenance and reduce the cost of
failure and the cost of downtime time. Whereas low frequency of maintenance actions
decreases the cost of maintenance but increases the cost of failure and the cost of
downtime time as well as the risks associated with system failure. So, it is essential to
ensure that the total costs due to maintenance are less than the benefit from the actions
being taken. Therefore, there should exist an optimum maintenance policy depending
on the type of failure and it’s time distribution (Elsayed (1996)). The Figure 2.8
exhibits preventive maintenance action effort versus cost of failure that determines the
optimum maintenance policy.
[This graph is not available online. Please consult the hardcopy thesis available from the QUT library]
Figure 2.8: Optimum maintenance policy (Blischke & Murthy (2000))
2.3.4 Human Reliability
All engineering activities involve human participation and a significant amount of
failures occurs due to human errors. So, the human factor has a significant influence in
the system reliability. About 20-30 percent of system failures were directly or
indirectly caused by people working in the system (Meister (1966)). It may be in the
form of ignorance or negligence of responsible personnel, mishandling or
41
misinterpretation of instruments and inspection. Thorndike (1977) defined human
error as the failure to perform a prescribed task (or the commitment of a prohibited
action), which could result in the damage to the equipment and/or properties or
disruption of scheduled operations, regardless of how advanced the technology is or
the degree of automation inbuilt to the system. Therefore, less the errors committed by
personnel, higher is the system reliability. In earlier days, the human elements of
systems were totally ignored. William and Carter (1957) pointed out that realistic
system reliability analysis must take into consideration the role of humans in a system.
At present, there have been considerable advances in the study of human errors and
their effect on reliability of the system. Human reliability can be defined as the
probability that a job or task will be successfully completed by the personnel at any
required stage in the system operation within required time (Meister (1966)). A
formula for the system overall reliability can then be shown as follows:
Total system operational reliability
= reliability of the equipment or component × human reliability.
In the context of infrastructure reliability, most of the human errors arise from
inspection of the component. Inspection reliability will be used here to capture the
human factor.
Therefore, overall infrastructure component reliability
= reliability of the component × reliability of inspection
2.3.5 Periodic Inspection and Condition Monitoring
The primary purpose of condition monitoring is to determine the state of a
system/component with a view to control the condition of the system to ensure its
availability. The state or condition of the system is firstly identified and the
inspections are carried out to determine the values of these attributes. And finally,
depending on the state/condition, maintenance decision is taken. To improve the
reliability of any system, a preventive maintenance schedule is coupled with a periodic
inspection schedule. An alternative to the periodic inspection is the continuous
condition monitoring for high accuracy at higher cost.
42
2.3.6 Maintenance Models and Policies
Design of a highly reliable system demands an understanding of the failure
mechanism. Development of a maintenance models helps in achieving high reliability
of the system.
Jardine (1973) classified reliability based problems into two categories, such as
deterministic and probabilistic or stochastic. In deterministic problems timing and
outcome of the maintenance and inspection actions are assumed to be known with
certainty. On the other hand, in probabilistic problem, the timing and outcome are
dependent on chance of good or not good and may be described by the distribution of
time. In case of probabilistic models, inspection is essential to determine the state of
system or entity, whether it is good, not good or somewhere in between. Model on
reliability and maintenance are available in Jardine and Buzacott (1985) and Brown
and Proschan (1983). Makis and Jardine (1993) treat p, the probability of successful
repair, as a function of t (age of the item) and n (number of times the item has been
rectified until t). Malik (1979) introduced a “degree of improvement” in the failure
rate and called it the “the improvement factor”.
Constant replacement policy
The objective is to determine the optimal interval between preventive replacements of
equipment subject to failure.
Situation: - under this policy, both preventive and failure replacement actions are
performed. In preventive replacement action, components or parts are replaced at
predetermined times regardless of the age of the components or the parts being
replaced whereas, in failure replacement action, components or parts are replaced upon
failure.
Essential conditions of preventive replacement
• The total cost of replacement must be greater for maintenance after failure than
before, as preventive replacement action of equipment should be taken before
occurrence of failure.
• The failure rate of the equipment must be increasing
The total expected cost per unit time for preventive replacement at time tp denoted
E[C(tp)] is
43
( ) ( )ervaloflength
tervalintectedtotaltCE p
p int],0[intcosexp
][ =
Total expected cost in interval (0, tp)
= Expected cost of a preventive replacement + expected cost of failure replacement
( )][ pfp tNECC += (2.45)
Where, ( )][ ptNE is the expected no. of failure in interval (0, tp)
The length of interval = tp
Therefore,
( ) ( )p
pfpp t
tNECCtCE
][][
+= (2.46)
Where Cp is the cost of preventive replacement and Cf is the cost of failure
replacement
E[N(tp)] can be determined by either the renewal approach or by a discrete approach
Value of N(tp) using discrete approach is given by
( ) ( )[ ] ( )dttfitNtNEi
i
t
i ∫∑+−
=−−+=
11
011][ 1≥t (2.47)
Predetermined age/usage replacement policy
One disadvantage of the component replacement model is that the components or parts
are replaced at failure and constant interval of time since the last preventive
replacement. Under the replacement at a predetermined age/usage policy, the
component or part is replaced at failure or at a predetermined time/usage which comes
first.
Situation:- the time at which the preventive replacement occurs depends on the
age/usage of the equipment instead of making preventive replacements at fixed
intervals.
The objective of this model is to determine the optimal preventive replacement
age/usage for the equipment to minimise the total expected cost of replacement per
unit time.
44
Figure 2.9: Predetermined age replacement policy
Here, Cp is the cost of preventive replacement
Cf is the cost of failure replacement
f(t) is the probability density function of the failure times of the equipment
The replacement policy is to perform a preventive replacement once the equipment has
reached a specified age tp and this policy is illustrated in the Figure 2.9.
In this problem, there are two possible cycles of operation: one cycle being determined
by the equipment reaching its planned replacement age and the other being determined
by the equipment ceasing to operate due to failure occurring before the planned
replacement time which are shown in the Figure 2.10:
Or 0
Figure 2.10: Two cycles of operation: Cycle 1- Predetermined replacement, Cycle 2- Failure replacement.
The total expected replacement cost per unit time C(tp) is given by
( )lengthcycleected
percyclettreplacemenectedtotaltC p expcosexp
=
Failure replacement
Preventive replacement
Failure replacement
Preventive replacement
tp tp
time
operation
Cycle 1
operation
Cycle2
Preventive replacement
Failure replacement
0
45
Expected cycle length
= Length of a preventive cycle x Probability of preventive cycle
+ Expected length of a failure cycle x Probability of a failure cycle
( )pp tRt ×= + (expected length of failure cycle) ( )[ ]ptR−× 1 (2.48)
To determine the expected cycle length for consider the Figure 2.11. The mean time to
failure of the complete distribution is
( )dtttf∫∞
∞−
Figure 2.11: Normal distribution curve
Which the normal distribution equals the mode or peak of the distribution. It a
preventive replacement occurs at time tp then the mean time to failure is the mean of
the shaded portion of the figure 2.11 (Jardine (1973)). The mean of the shaded area is
( ) ( )[ ]p
ttRdtttfp
−∫ ∞−1/ , which is denoted by M(tp)
Therefore the expected length of the cycle
= ( ) ( ) ( )[ ]pppp tRtMtRt −×+× 1 (2.49)
Cost, ( ) ( ) ( )[ ]( ) ( ) ( )[ ]pppp
pfppp tRtMtRt
tRCtRCtC
−×+×
−×+×=
11
(2.50)
Modelling minimal repair
Repair and replacement of a failed component /s of a complex system of large number
of components usually restores function to the system but the proneness of the entire
system failure remains as it was just before failure. Barlow and Hunter (1960) first
introduced a minimal repair model. This was generalised and was modified by some
other researchers to fit more realistic situations. Ascher (1969) renamed this type of
f(t)
Mean t
tp
46
repair as ‘bad as old’ repair as the replacement or repair of the component/’s do not
affect the probability of failure of the system. Bassin (1969) called it ‘restoration
process. However, they emphasised probabilistic modelling under minimal repair and
this is equivalent to assuming a non-homogeneous Poisson process as model (Ascher
and Feingold (1984)).
The long run expected cost per unit time using a replacement age t the basic model is given by:
( ) ( )t
ctNctC rf +
= (2.51)
Where, cr and cf are the replacement cost and failure cost respectively and N(t)
represents the expected no. of failure.
The general assumptions of minimal repair model are- failure rate function of the
system. Failure rate of the system remains unaffected by the minimal repairs. cf < cr
and the system failures are immediately identifiable. Considering an adjustment cost
ca(ik) incurred at an age ik=1, 2, 3, …..And k >1, Tilquin and Cleroux (1975) modified
the basic model as follows:
( ) ( ) ( )( )t
tvcctNctC arf
*++= , (2.52)
Where, ( )( ) ( )( )∑ ==
tv
i aa ikctvc0
* and v(t) represents the number of adjustment in the
period (0, t). They assumed that ca (0) = 0 and ca(s) ≥ o for any k > s.
Considering minimal repair cost cf is a random variable, Cleroux (1979) modified the
above model and shows that if the modified minimal repair cost cf ≥ δcr ,the system
should be replaced otherwise minimal repair should be performed (here, δ is a
constant- percentage of replacement cost cr that decision makers select according to
experience). They give a simple algorithm for optimal replacement age. Similarly,
Boland and Proschan (1982) analysed a model where the minimal repair cost is not
fixed but rather, it depends on the number of minimal repair the system has suffered
since the last replacement epoch. The minimal repair cost of the Kth failure is given by
bkac f += (where, a > 0 and b ≥ 0, are the two constant) with variable minimal repair
cost fc (depends on the number of minimal repair). They find the t* that minimises the
total expected cost of minimal repair and replacement over a fixed horizon and the
long run expected cost per unit time. They used the calculus approach to find t* (the
optimal repair or replacement age) for a system with increasing failure rate
47
distribution. Again assuming the minimal repair cost as dependent on the system age,
Boland (1982) extends this to:
Long run expected cost per unit time ( )( ) ( )
t
cduuhuctC
t
rf∫ += 0 (2.53)
where h is the failure rate. He claims that for increasing failure rate, the optimal
replacement interval can be obtained by differentiating C (t) with respect to t and
equating it to zero. Hameed (1987) finds the optimal block replacement that minimises
the long run expected total cost per unit time.
Muth (1977) proposes a model in which, instead of fixed replacement interval, a
minimal repair is performed if the failure occurs before a fixed time t* or replacement
at the first failure after t* otherwise. Assuming a decreasing mean residual life function
after some age, he proves that his model yields the minimum long run total expected
cost per unit time.
The cost function to minimise Muth’s model is as follows:
( ) ( )( )trt
ctNctC rf
+
+= (2.54)
where ( )tr is the mean residual life function of the system at time t. He showed that
the optimum value of t can be found out analytically, but he did not present any
algorithm when it is not analytically traceable.
Some other maintenance models
Coetzee (1997) examines the present state of the analysis of the maintenance failure
data and then develops two formats of the Non homogeneous Poisson Process model
for practical use by maintenance analysts. This includes an identification framework,
goodness of fit test and optimisation modeling. This model is tested on two failure data
sets from literature.
Park et al (2000) consider a periodic preventive maintenance policy of a repairable
system with slow degradation that minimises the expected cost rate per unit time over
an infinite time span. They worked with a PM model proposed by Canfield (1986)
where the failure rates are monotonically increasing however the failure rate is
reduced after each PM. They also considered the case when the minimal repair cost
varies with time. Martorrel et al (1999) developed an age dependent reliability model,
taking into consideration the effect of maintenance and the working environment.
48
They considered accelerated life model (ATM) and proportional hazard model (PHM)
as tools to introduce the operational and environmental factors.
Pulcini (2000) modeled the failure data of repairable system whose failure intensity
shows a bathtub type non-monotonic behavior. He proposed an NHPP arising from the
superposition of two power law processes. The characteristics and mathematical
details of the model were illustrated and a graphical approach to obtain a crude but
easy estimate of the model parameter is presented. Here, the author cited two
numerical applications to illustrate the proposed model and the estimation procedures.
Sheu (1998) proposed a Bayesian approach to express and update the uncertain
parameters for determining optimal age replacement, when the failure density is
Weibull with uncertainty. Sheu (2000) also proposed a Bayesian approach to an
adoptive preventive model. They proposed the adoption of Bayesian approach to
estimate the Weibull parameters α and β. From Weibull distribution, they derived an
optimal policy so that the expected long run average cost is minimised.
Sarhan (1999) introduced a general hazard rate model
( ) 1−+= cbtath , (2.55) where a, b, c are the constant and greater than 0.
This model can be considered a mixture of an exponential and Weibull distribution
with shape parameter c. Estimation of independent random variables a and b were
based on Bayes’ estimation.
Yeh (1997) proposed state dependent maintenance policies for a system that
deteriorates continuously because of age or usages where each inspection identifies the
current status of the system. Yeh proposed the maximum likelihood procedure and
random likelihood procedure to approximate the optimal inspection and replacement
for multistate, semi Markovian deteriorating system, when the sojourn times
distribution each state is Earlong.
Instead of concentrating only on operational parameters, Martorell et al (1999)
presented an age dependent reliability model considering parameters related to
surveillance and maintenance effectiveness and working conditions of the equipment,
(both environmental and operational). They considered the accelerated life model
(ALM) and the proportional hazard model (PHM) to introduce the environmental as
49
well as operational factors. They cited an example of the model initially adopting the
Weibull distribution and then showed the result of the several sensitivity studies
Inspection models
One common assumption of inspection is that the state of the system remains
unknown, unless the inspection being performed and carried out inspections are
assumed to be perfect in the sense that they reveal the true status of the system, or the
information obtained through inspection are reliable. Valdez (1989) describes a two-
dimensional decision space of maintenance inspection problems, which are – a)
determination of the maintenance action and b) determination of the inspection
sequence.
Most of the inspection models are based on the basic model developed by Barlow et
al (1963) (pure inspection models for age replacement with no PM and replacement
only at failure) and are developed depending on the assumptions regarding time
horizon, available information, nature of the cost functions, objective and the system
constraints. The basic model developed by Barlow et al is:
Total cost per inspection cycle, ( ) ( )., 21 txcncxtC n −+=
Where, t is the time to failure, ( ).,........., 21 xxx = are the sequence of inspection, with
.......321 xxx << and n is such that nn xtx <<−1 . The optimal inspection policy x* is
the one that minimises ( )[ ]xTCE , , where, the system failure time, is a random
variable and assumed to be non-negative
Luss (1976) uses state dependent maintenance approach to determine the optimal
control limit policy and optimal inspection interval for states 0, 1, 2…α - 1. Luss
gives a very simple iterative procedure to find out optimal inspection interval for
different states. Luss (1977) addresses an optimal inspection scheme that minimises
the long run expected cost per unit time, considering the inspection times and down
times are not negligible and the inspection schedule is finite. He proposes two
algorithms to solve the problem. For more efficient algorithm, the system should have
increasing failure rate. Assuming a significant inspection cost, Anderson and
Friedman (1977) address a perfect optimal inspection plan to minimise the total cost
for a Brownian motion, except that instead of stopping the motion. The process is
assumed to be renewed by setting it to the origin. The optimal inspection times are
then found by reducing the stochastic problem to a free boundary problem, in analysis
50
and the solution is then made by iteration procedure. Assuming all inspections are
hazardous, Wattanapanom and Shaw (1979) propose some algorithms for finding
optimum inspection plan for system with uniform and exponential failure time
distribution. They present convergent algorithms for solving the optimisation
equations given to solve the basic model introduced by Barlow et al.
Menipaz(1979) considers inspection model of several cases where the cost of
inspection and downtime changes over time. He obtains the optimal inspection
policies for the cases – i). the system is inspected at different points of time, and
replaced at failure, ii). inspection is made up to a predetermined age and replaced, if it
is not yet failed iii). Inspection is performed discretely up to an age t and if still in
good condition, it is continuously inspected then on and replaced at failure. Similar to
Luss ‘s model Zuckerman (1986) presents a model where the status of the system is
monitored by inspection. Inspections and maintenance are instantaneous. At failure
detection, total cost would be the combined cost of periodic inspection, operating,
failure and preplanned replacement cost. He does not provide a general algorithm to
compute the optimal policy but notes the difficulties to find it. Using the basic cost
model of Barlow et al, Beichelt(1981) developed optimal inspection schedule when
the life time distribution of the system is either completely unknown or partially
known. It uses minimax approach for the partially unknown life distribution but does
not have any numerical procedure to obtain optimal inspection frequency. Nakagawa
et al (1984) proposes a modified inspection model considering a system that is check
periodically to see whether or not it needs to be replaced. He considers q=1-p as the
probability of the system as before the check and p is the probability of the system as
good as new. He also derives the total expected cost and the expected cost per unit
time up to failure. He suggests the use of numerical search procedure to find the
optimal inspection time rather than analytic solution. By using a ‘delay time analysis’
technique Chister and Waller (1984) propose models for optimal inspection and
replacement policies for both perfect and imperfect inspection (where the inspection
does not reveal the true status of the system). Delay time of a fault is the time lapse
from the first identification of the fault until its delayed replacement due to
unacceptable consequences. Because of the simplicity of the cost model, the optimal
inspection policy can be easily found out.
51
Abdel Hameed (1987) generalises compound Poisson Process used by Zuckerman
and models the deterioration by using pure jump Markov process. He finds the
optimal inspection plan that minimises the long run expected cost per unit time. When
the failure distribution is exponential, the periodic inspection is optimal. Nakagawa
and Yasui (1987) presents an algorithm to compute the near optimal inspection
policies when the failure distribution is not exponential by citing a numerical example
that shows that the approximation is good enough for Weibull distribution. The
inspection models discussed so far are intended to schedule optimal inspection time
for an operating system based on perfection of inspections.
Based on the condition based monitoring model developed by Turco (1984), Cheilbi
et al (1999) proposes a mathematical model and a numerical algorithm to solve the
problem of generating an optimal inspection strategy for randomly failing equipment.
Francis (2001) derives four optimum inspection policies for a single unit
stochastically failing or deteriorating system, using the inspection density n (x)
(number of checks per unit time) proposed by Keller, J.B. (1974), that are, 1). the
basic model, 2). the basic model with imperfect inspection (inspection can not reveal
the true state of the system without error), 3). the basic model with inspection time
(inspection takes a fixed amount of time), 4)the basic model with imperfect inspection
and inspection time. To overcome the disadvantage of the most commonly used
constant interval inspection policies for aging system, a number of policies have been
proposed by some researchers. Among them, he prefers the proposal of Keller, J. B.
(1974) - a smooth density n (x) to obtain a sequence of approximate inspection times
using the usual method of calculus of variation. For each model he obtains the
minimax inspection schedules minimising the total expected cost by applying the
calculus of variation.
Optimum Inspection Policies
This policy is one where the state of the component or equipment is determined in the
most economic way. The frequency of inspection should be influenced by the cost of
inspection and benefit from it. Obviously, more frequent inspection reduces the non-
detection cost but at the same time increases the cost of inspection. On the other hand,
a less frequent inspection or no inspection will reduce the cost of inspection but
increase the non-detection (failure) cost (loss) and increases the risks of failure.
52
Therefore, an optimal inspection strategy is required, to provide the correct balance
between the type and number of inspections and the resulting benefit. Depending on
the result of inspection, the maintenance action will be decided.
The expected cost of inspection per unit time (Jardine (1973)) is
( )l
c
EE
xxxc =....32,1 (2.58)
Where, Ec and El are the total expected cost of inspections per cycle and the expected
cycle length respectively.
The total expected cost of inspections per cycle can be expressed as
( ) ( ) ( )dttfcxckccE rkuk
x
x irck
k
]11[ 10
1+−+++= +
∞
=∑∫
+ (2.59)
Where, ci = inspection cost per inspection.
cr = cost of repair.
cu = cost per unit time of undetected failure or degradation
k = 1, 2, 3,
f(t) = the p.d.f of equipment time to failure.
and the expected cycle length is
( ) ( )dttfxTEk
x
x krlk
k∑ ∫
∞
=+
+−++=
01
1 1µ (2.60)
where, µ is the mean time to failure of the equipment and Tr is the time requested to
repair .
Therefore, expected cost of inspection per unit time (Jardine (1973)) is
( )( ) ( ) ( )
( ) ( )dtfxT
dtfcxckccxxxc
k
x
x kr
rkuk
x
x ir
k
k
k
k
∑∫
∑∫∞
=+
+
∞
=
+
+
−++
+−+++=
01
10
3211
1
1
]11[.....,
µ (2.61)
The optimum inspection schedule can be obtained by taking the derivative of the
above equation with respect to (x1, x2, x3, …) and equating it to zero and then solving
the resulting equation simultaneously.
According to Barlow and Proschan (1963) and Jardine (Jardine (1973)) and applying
the maximum likelihood estimation one can determine the optimum inspection
schedule, where the residual function is given by
53
( ) cl ELExxxLR −=....; 321 (2.62)
When L represents either an initial estimation of the minimum cost c(x1,x2,x3….) or a
value obtained from the previous cycle of iteration process. The schedule that
minimise R(x1,x2,x3….) is the same schedule that minimises c(x1,x2,x3….). The
following procedure determines the expected schedule
Step 1 choose a value for L
Step 2 choose a value for xi
Step 3 generate a schedule (x1,x2,x3….) using the following relationship
( ) ( )
( ) Lcc
xfxFxF
xu
i
i
iii −
−−−
=+1
1
step 4 compute R using ( ) cl ELExxxLR −=....; 321
Step 5 repeat step 2 through 4 with different values of xi until Rmax is obtained.
Step 6 repeat step 1 through 5 with different values of L until Rmax = 0.
The procedure for adjusting L until it is identical with the minimum cost can be
obtained from
( )lE
RLxxxc max
32 ,...,,1
−= (2.63)
2.3.7 Maintenance and inspection models for Wood pole
Krishnasamy et al(1985) proposes a reliability model (based on actual pole strength
values and other line details) for wood poles and also the overall line reliability. The
formation of the method requires detail information about the size and strength of each
pole, the span length, conductor details, and data on environmental loads i.e. extreme
wind and ice on each pole and any accessories such as transformer mounted on it.
The load due to wind alone is given by
12/2 DLCSVPw = (2.56)
And load due to wind on ice covered conductor is given by
( ) 2/221 LtDCSVPw += (2.57)
Where, Pw, Pwl, C, S, D, L, V and t are the load due to wind, load due to wind on ice
covered conductor, constant (.003), Span factor, Conductor diameter, Span length,
Annual extreme wind, Annual extreme ice condition respectfully.
54
Using design point method and Rackwitz Fiessler algorithm they developed pole
reliability model. The basic variables of the failure function were represented by
Exponential, Normal, Lognormal or Extreme Type 1 distribution. Considering the
probability of structural failure under wind load, ice load and the combined wind and
ice load, Krishnasamy et al (1987) proposed a probability based design method of
wood pole distribution lines that can provide quantitative estimates of the structural
reliability of individual pole component and the overall wood pole structures.
Bhuyan (1998) developed a procedure to asses the serviceability and reliability of
existing wood pole structures, above or bellow the ground level decay using an
instrumented drill and maps and a customised computer program. He cites an example
as well as a case study to discuss the uniqueness and the benefit of the method.
Gustavsen et al (2000) proposed a methodology for simulating the expected wood pole
replacement rate as a function of time, considering statistical nature of pole strength
and climatic loads. They outlined the dependency of replacement rate and costs on rate
of decay of pole strength, replacement strategies and reliability of assessing pole
strength. A probabilistic technique is used for maintenance purpose of overhead line,
which allows the user to investigate the influence of a change of pole management
program on expected line costs throughout its lifetime. This is achieved by simulating
the change in line strength and pole replacement as a function of time, taking into
account the uncertainties existing in the system.
2.4 Summary This chapter broadly focussed on reliability and maintenance models. Most of the
theories and models presented in this chapter were based on a number of assumptions,
which make it difficult to implement in some real life situations. These models forgo
important aspects such as the effect of environment, human factors in modelling
failure and maintenance decisions. Therefore, the next chapters aim at developing
more realistic maintenance and inspection models for infrastructure components which
considers those factors.
55
Chapter Three
Reliability of Wood Pole
3.1 Introduction
The function of wood poles used in Power Supply or Telecommunication Industries is
to support the overhead lines, the conductors and to resist the bending load from pole
top over a wide range of areas. Because of their high strength per unit weight, low
cost, excellent durability (service life of wood poles can vary from 25 to 50 years or
more (Bingel (1995)), wood poles are very popular throughout the world. In Australia,
more than 5.3 million wooden poles are in use (Greaves et al (1985)). This represents
an investment of around AU$ 12 billion with a replacement cost varying between
AU$1500-2500 per pole. Bingel (1995) conducted a comparative study to show the
cost of Electricity wood poles with and without maintenance. Expected savings over
40-year pole life was estimated to be around $400 per pole. Table 3.1 shows the
comparative study of wood pole with and without maintenance action.
Table 3.1: Comparative study of inspection and maintenance (Bingel (1995))
[This graph is not available online. Please consult the hardcopy thesis available from the QUT library]
Formatted: Font: 14 pt, Fontcolor: Auto
56
From the raw data of power supply companies in Australia it was observed that there
is a higher replacement rate of wooden poles in the clayey costal areas. Destructive
tests of discarded and failed poles showed a significant human error in inspection and
replacement decisions.
The outline of this chapter is as follows: In Section 3.2 fundamentals of wood pole
degradation process and factors causing decay of wood pole are discussed. Section 3.3
describes the different types of inspection and maintenance actions available for timber
component. Section 3.4 describes human errors involved in inspection decision for
wood pole. Finally, summary of this chapter is presented in section 3.5.
3.2 Deterioration or Degradation of Wood Pole
In deciding effective maintenance strategies for reliability and safety of power supply
wood pole, it is important to identify various decay conditions and factors behind
inground decay, measure these factors and develop models for correctly predicting
inground decay.
3.2.1 Factors Affecting the Reliability and Safety
The Figure 2.1 is a schematic diagram showing the major factors affecting the
reliability of infrastructure components (e.g. Wood pole).
Figure 3.1: Reliability of complex infrastructure
Reliability of Infrastructure
Human Factors Environmental Factors
Age and Usage Durability of Component
Maintenance
57
Age and usage
Age and usage have effects on the strength of wood pole. Hardwood species
characterised by a high strength and durability ratings are used for poles in
Queensland. In general, the rate of pole replacement increases with the age of pole
population.
Durability
In the context of infrastructure, durability can be defined, as the capacity of the
component or the system to perform a specified or intended function, for an
anticipated design period of time and can simply be defined as the resistance to decay.
The outer portion of wood is called Sapwood, which is normally lighter in colour, less
durable and of higher moisture content. The inner dry, durable section is called
Heartwood with no living cell. Bodig (1985) observed that pole strength is a function
of pole circumference.
Strength of wood pole is affected by the moisture content. The fibre saturation points
occur when the moisture contents fall to around 30%. The compressive strength of a
defect free pole is as much as 100% greater than that of the green timber when the
moisture content of the soil is around 12% (Pearson (1958)). The decay of wood
materials is extensive in the presence of Oxygen and Moisture, as this condition enable
metabolic activity and growth of aerobic microorganisms, such as bacteria, fungi etc.
Biological decay can extend where the moisture content is more than 20%. Three
stages are normally involved in degradation process. In stage 1- pole strength
decreases from 10 – 15% due to the formation of cracks, checks and the onset of
vibrational cycles; in stage 2- the treatment protects the pole from the Biological
attack; in stage 3- active biodegradation begins. Termites also affect wood poles by
using as shelter and food.
Environmental factors
The environmental factors in wood pole deterioration process are the soil condition
(clayey, sandy or acidic), climate and weather, cyclic wetting and drying, wind speed,
snowfall and temperature. Effect of environmental factors such as soil composition
and its properties (both physical and chemical properties) on in-ground decay has not
been studied well. In the first five to eight years of service life, the pole weakens under
58
the effect of applied loads. Then cracks that develop are due to environmental changes,
and wetting and drying cycles.
Most of the component failures are due to the decrease in peripheral dimensions at or
below ground level (Figure 2.2). High moisture content in the soil increases the chance
of biological attack. In clayey soils, the moisture and chemicals are trapped inside the
soil. These cause algae, moss, and mould to grow and attack the wood, thereby
causing rapid deterioration. On the other hand, by virtue of their permeability,
cohesion-less sandy soils allow drainage and protect from biological attack.
[This image is not available online. Please consult the hardcopy thesis available from the QUT library]
Figure 3.2: Examples of assumed decay pattern for in-ground poles
(Foliente et al (2001))
Foliente, et al (2001) have grouped Australian land into four in-ground-decay hazard
zones- A (lowest decay hazard rate), B, C and D (highest decay hazard rate) on the
basis of intensity of decay of wood due to ground condition. According to their
research eastern coastal areas have the highest decay hazard rate. On the other hand
sandy central areas have the lowest in-ground decay hazard rate. Figure 2.3 gives a
detail mapping of the in-ground hazard rate of wood structure in Australia:
59
[This image is not available online. Please consult the hardcopy thesis available from the QUT library]
Fig: 3.3 Hazard zone for decay of wood in ground contact in Australia
(Foliente et al (2001))
The following are the various soil variables that may influence the deterioration
process of inground portions of infrastructure components
Moisture content in the soil – moisture content in soil is an important feature for
reliability of infrastructure component. Raw data from industry shows that decay of
wood poles is more in black clayey areas in Queensland region compared to other
areas in Australia (Australian Clay Mineral Society, 1988: Wallace, 1988, Ward et al.,
1991; Middleton and Clem 1998). Clay minerals tend to form microscopic to sub-
microscopic crystals. They absorb or loose water with the humidity changes. When
mixed with limited water they form plastic. When water is absorbed, clays often
expand as the water fills the space between the stacked silicate layers and accelerate
the weathering process.
Moreover, decaying fungi, which need a source of water in contact with the wood can
cause severe structural damage to any wood member, even wood species such as
redwood and cedar. Decay will occur in untreated wood in direct contact with ground,
cement or concrete or exposed to a source of moisture such as rain. Some of the
damaged caused by decay fungi are as follows;
• Chemical composition– the presence of kaolin, quartz, chloride and other
chemicals may have effect on in-ground decay.
60
• Bulk density– weight of the soil itself exerts forces. It is determined by the bulk
density of the soil.
• pH value– presence of excessive acidity or alkalinity of groundwater in soils
could have detrimental effects on wooden, concrete or even on metal components
buried in the ground. The acidity or alkalinity can be diagnosed by the pH value
of the ground water. It provides information about the aggressive quality of soil
and special treatment for these components (Vickers (1983)).
• Salinity- the chloride (sometimes sulphate, carbonate or magnesium) content is
an indication of salinity. High salinity can cause decay of the wood or corrosion
of iron /steel including steel reinforcement in concrete (Head (1980)). A build up
of salts can also be threat to the infrastructure foundation.
• Electrical Conductivity. It can be used to determine the soluble salts in the
extract and hence soil salinity. The unit of Electricity conductivity is the siemens
/decisiemens or microsiemens.
3.2.2 Inground Decay of Wood Pole
Analysis of failure data shows that most of the failures of timber poles are due to the
decrease in timber strength and peripheral dimensions at or below ground level.
Therefore, the ground line condition of the pole is critical. The strength of wooden
fibres is particularly important in resisting bending load from the top. Due to
influential soil factors the diameter and the fibre strength of pole starts decreasing.
These factors cause rotting of fibres from centre to outward or outward to centre by
accelerating fungal and insect (termite) attack. Types of common inground decay of
wood due to fungal attack pole are:
Brown Rot (Cubic Rot)
Brown rot (Figure 3.4) fungi feed on the wood's cellulose, a component of the wood's
cell wall, leaving a brown residue of lignin, the substance which holds the cells
together. Infested wood may be greatly weakened, even before decay can be
seen. Advanced infestations of brown rot are evidenced by wood more brown in
colour than normal, tending to crack across the grain. When dried, wood previously
infested will turn to powder when crushed. Often, old infestations of brown rot which
have dried out are labelled as "dry rot."
61
Figure 3.4: Brown rot
White Rot
When white rot attacks wood, it breaks down both the lignin and cellulose causing the
wood to lose its colour and appear whiter than normal. Wood affected by white rot
normally does not crack across the grain and will only shrink and collapse when
severely degraded. Infested wood will gradually lose its strength and become spongy
to the touch.
Soft rot
Soft rot initially can be visible on the out side of the pole, which gradually progresses
towards the centre of the pole. This type of rot can therefore be found by scraping the
outside of the pole. Pearson et al (1958) stated that this rot in sap wood poles has been
Early Brown rot
Complete FailureAdvanced Brown rot
Intermediate Brown rot
62
observed with varying degrees of severity in Queensland and if left untreated, has
resulted in reduction of the service life by as much as 75%.
Figure 3.5: Soft rot
Figure 3.5 is a transmission electron micrograph of a group of wood cells in cross-
section undergoing biodegradation by 'soft-rot' fungal organisms. The fungal hyphae
penetrate inside the cell walls of the wood and produce large cavities in the
surrounding cell wall matrix.
3.3 Inspection and Maintenance of Wood Pole 3.3.1 Inspection Techniques
There are many methods of determining the strength characteristics of wood
components, which are categorised into two classes, traditional destructive testing that
can only be performed in the laboratory and non-destructive evaluation. Visual
inspection, sounding, excavation and boring techniques fall into the first category
while information from these type of techniques are inevitably limited and can
produce misleading results, because of their inability to quantify the defect. Their
ability to detect defects relies solely on the experience and judgement of the inspector.
Non-destructive evaluation or NDE devices are commercially successful technique
capable of predicting pole strength more accurately without disturbing it service. With
the introduction of NDE (Goodman et al, 1990), it is possible to locate problem areas
and fix them before they become hazardous.
63
Various devices for the detection of decay and defects are available, but they all have
limitations to some degrees. Some the popular and most widely used inspection
techniques are discussed bellow:
The Shigometer
The Shigometer is one of the modern examples of electrical resistance meter. Decay in
poles can now be detected almost accurately with a new device called Shigometer.
A hole of 3/32 inch (2.38 mm) in diameter is first drilled into the pole with a portable
battery-operated drill. This hole can be drilled to a depth of 12 inches (30.48 cm) in
less than a minute.
Results of experiments indicate that a pole can heal such a small wound very quickly.
A long, thin probe is next inserted into this hole to indicate the condition of the wood
at the tip of the probe, -especially whether decay is present or not.
Figure 3.6: A Shigometer
Although the Shigometer’s efficiency is influenced by a number of factors such as
moisture content, wood species, internal checks etc. This difficulty of interpretation
can be over come to a certain extent with experience.
The Condition-meter is another electrical resistance meter used to check wood
condition which is a self calibrating pulse resistance meter that records the electrical
resistance between two contact points on a rigid probe inserted into pre-drilled holes at
Formatted: Justified, Indent:Left: 0.4 cm, Right: -0.5 cm,Line spacing: 1.5 lines
Formatted: Justified, Indent:Left: 0.4 cm, Right: -0.5 cm,Line spacing: 1.5 lines
Formatted: Heading1,Heading 1 Char Char CharChar Char, Left, Indent: Left: 0.4 cm, Right: -0.5 cm, Linespacing: single
Deleted:
64
or bellow ground level. The pattern or reading indicates the condition of the wood. A
drop of 75% in reading indicates the presence of decay.
Sounding by impacting with hammer
Sounding the pole with a hammer is one of the oldest and cheapest techniques for
detecting decay or fault. An experienced inspector does this type of inspection. This
technique is reliable in detecting sound wood or wood decayed to the point of
replacement but is not reliable for detecting early decay of the component. Again, the
presence of seasonal cracks, internal cracks and pockets filled with moisture may
affect the produced sound, which is enough to misguide the inspectors. This technique
is highly subjective, relying on the knowledge and experience of the individual
inspector as the interpretation of produced sound may vary between individuals.
Sonic devices
These devices compare the time delay for a sound wave to travel through the
inspecting pole with the time delay through a section of sound wood. The variation of
velocity indicates the decay. The De-K Teckor, the Pole ultrasonic rot locator, the
pulsed ultrasonic non-destructive indicating tester are the examples of sonic devices
used in decay identification of wooden components.
Ultrasonic Devices
Figure 3.7 A : Ultra-sonic machines like the IML F500 reveal the relative density distribution of wood, which allows arborists to make recommendations to clients regarding decay
Figure 3.7 B: Several devices use stress waves for the inspection of cracks, cavities and rot. They work through the simultaneous measurement of the time of transmission of stress waves by sensors arranged around the trunk.
65
Ultrasonic devices employ pulses of sound with frequencies above the audible range
of 20,000 kHz. These devices measure the transit time of a sound wave across the
trunk or branch in a radial direction. The velocity of a sound wave in a specific
direction through healthy wood is more or less constant and is proportional to the
square root of elasticity over density:
nEV =
where V = Velocity, E = Modulus of Elasticity and n = Density.
The reason for the constant velocity is that although the gross density of wood can
vary considerably, the density of wood matter in which all air and water is excluded is
more or less constant. This velocity will vary depending on the direction of travel due
to changes in the elasticity. The velocity is greatest in the axial direction followed by
the radial and then the tangential directions. Where decay is present, the ratio of the
elasticity to density normally changes, giving an increase in the transit time (decrease
in the velocity). [This image is not available online. Please consult the
hardcopy thesis available from the QUT library].
Figure 3.8: Path of measured transit time between transducers, around the shaded (decayed) area (Dolwin 2001).
The ultrasonic signal is transmitted and received by transducers placed on either side
of the trunk. Where no decay is present the measured transit time will be a straight
path between the two transducers. Where decay or defects are present, the first pulse
received is likely to be the signal that passes around the decay. It is this increased
transit time that indicates the presence of decay or a defect.
The higher the frequency of the crystal in each transducer, however, the more quickly
the ultrasound will be attenuated Drilling /Boring
Drilling/Boring inspection is performed to confirm the type and severity of decay
when a wooden component is failed by a sounding test. Decayed wood shavings are
66
more brittle than a sound fibrous wood shaving. This method is intrusive and may
breach the preservative treated wood that may initialise interior decay.
None of the methods are 100% accurate because each of the methods is dependent on
operator’s experience and skills. So, in modelling overall reliability, it is important to
consider human reliability or reliability of inspection.
3.3.2 Maintenance Techniques for Wood Pole
Maintenance has a great impact on system reliability. A quality maintenance program
can play an important role in restoring or improving the reliability of infrastructure
component. A quality maintenance program includes several maintenance actions that
have to be carried out at regular interval to maintain the required system performance.
These actions are minimal repair, overhauling and replacement. Replacement enables
the system to be brought back in a state where the reliability of the system can be fully
restored. This means the failure rate of the system is restored to zero. A minimal repair
has an insignificant effect on reliability restoration and improvement of a system or
component. And the failure rate of the component remains almost the same as it was
before. In case of a wood pole, the most common example of minimal repairs actions
are insertion of the Pole saver rod into the decaying pole, Bioguard bandage,
application of highly toxic Chloropicrin, Vapam treatment. Pole saver rods are solid
rods of boron and fluoride as major ingredients that slowly release fungicide,
insecticides used in stopping fungus and termite attack in the heartwood of in-service
poles. The rods are very much effective in protecting wood poles from biodegradation
for up to 5 years.
The Bioguard bandage consists of discs or pills of solid wood preservative, which
are pre-formed into individual round recesses (8 mm deep and open on one side)
within an impervious flexible polyethylene outer cover. The bandage is wrapped
around the pole in the critical zone just bellow ground line.
67
Figure 3.9: Minimal repair of wood pole a. Bioguard bandage and b. Plucked for
insertion treatment
Insertion treatment- Application of highly toxic Chloropicrin treatment is done by
inserting Chloropicrin ampoules into a series of steeply sloping holes (400), drilled at
beginning at the ground line and moving upward 150 mm and around the pole 1200
apart.
The Vapam treatment utilises an agricultural fumigant, sodium N-
methyldithiocarbamate that effectively sterilises the affected area and prevents internal
decay by destroying insects.
Overhauling or major maintenance actions are the restorative maintenance actions that
are taken when treatment is ineffective or the decay is far beyond treatment. This type
of maintenance can have the capability to restore the component/system reliability to a
significant state. Quality of these maintenance actions falls in between that of
replacement and minimal repair. As for the example, a mechanical reinstatement can
extend the service life of a wooden pole by up to ten years. Mechanical reinstatement
and Quick Deuar Pole reinstatement are the examples of imperfect maintenance action
of wood poles in Australia.
Mechanical reinstatement
Most wood poles are condemned because of degradation of strength from decay at the
ground line. When this degradation of strength is far beyond treatment, mechanical
reinstatement or Quick Deuar Pole reinstatement is the last action to extend the life of
the pole. Reinstatement can provide additional ten-15 years of service life for a cost
around 20 – 30 percent of the pole (Bingel, 1995)).
a b
68
The Steel truss system, Fibre glass composite system, Steel encasement, Quick Deuar
pole reinstatement are all examples of mechanical reinstatement.
Steel truss method
Steel truss is the widely used restoration system in all over the world in which an open
or close section truss is driven to within
Figure 3.10: Mechanical Reinstatement of Wood Pole
6 inches of the butt to provide a solid foundation. 4 to 5 feet of the truss extend above
the ground level and banding secures it to the pole. A highly engineered c-shaped truss
restores wood poles to equivalent new pole bending strength even while assuming zero
wood bending strength at the ground line (Bingel, 1995).
Fibre Glass System
Because of inadequacy of sound wood at the banding location or some other
engineering criteria, fibreglass composite system is used for restoration instead of steel
truss system. The fibreglass “wrap” system designed with adequate strength for wood
utility poles have provided the most reliable field service. The steel encasement shows
an excellent strength performance, which consists of excavating around 3 feet deep,
applying wood preservatives to the pole and then bolting a steel shell to it. The
annulus is filled with acrylic modified mortar.
69
Quick Deuar pole reinstatement method
Figure 3.11: Quick Deuar Pole Reinstatement Method This technique was first introduced by Dr. Kris Deuar (Deuar, 2001) which is based on
the application of a variety of commercially available hot rolled steel profiles such as
universal columns, lean beam and so on. All profiles are modified by the addition of
rounded ends to provide for public safety and extra strength. Typically only one type
of hot-deep galvanized steel element is used, however, for large poles; two or three
steel elements can be used. Steel elements are driven into the ground.
3.4 Human Factors
In case of infrastructure components, especially power supply pole, railway sleepers or
components, human intervention can significantly affect maintenance decisions. We
consider human participation in woodpole maintenance program restricted to
inspection of pole only. Accurate assessment of pole strength should provide the
strength of the new pole and strength of the in- service pole, with a degradation rate of
pole strength over usage (Bodig 1985). Inspectors could make some mistakes in
evaluating the actual state of the system. For example, discarding a component that is
in perfect working order or failure to detect a degraded component, which may cause a
huge loss to the industry. These inspection errors are due to the inspector’s ignorance,
70
lack of adequate vigilance, misinterpretation of instruments and inaccuracy of testing
instruments.
Steps in estimating inspection reliability
In real life, inspection consists of numerous steps. The overall reliability of inspection
is dependent on reliability of human efforts in each step:
Figure 3.12: Inspection process.
The overall probability of success of an inspection
ni RRRRR ××××= ......3211 (3.1)
All the steps involved in inspection are in series and are independent and this
probability of success of inspection can be defined by human reliability and can be
presented by
Ri = 1-Fi, (3.2)
where, Fi is the probability of human error.
Again, in
eni E
EF = (3.3)
Where, Een = Total number of known error of given type.
Ein = Total number of inspections in any step.
Table 3.1 is an example of inspection reliability calculation procedure for a power
supply wooden pole performed by one person using an electrical resistance meter (eg.
Shigometer) device:
Table 3.2: Inspection Reliability
Steps Description Reliability (R) 1 Setting up the device 0.9990 2 Reading of gauges 0.9950 3 Actuating of switches 0.9989 4 Inspection consist functional test of the component 0.9999 5 Verify the component functionality surveillance 0.9999 6 Writing reports 0.9998 7 Data analysis and determining the type of maintenance
action 0.9990
Step 1 P1
Step 2 P2
Step 3 P3
Step n Pn
71
From the above example, we can construct the following event tree diagram (Figure 3.12):
Figure 3.12: Event tree diagram for a wood pole inspection process.
Therefore, the reliability of the above inspection is;
7654321 RRRRRRRRi ××××××= = 0.9898
Where, R1…..7 indicate reliability of each of the above seven steps.
The above analysis, gives us an idea that probability of success decreases with the
increase of the number of inspection steps and at the same time the probability of
failure decreases with the increase in the number of inspectors thus increases the
overall inspection reliability. But in real life, addition of each inspector will increase
the overall inspection cost. Therefore, it is important to optimise the steps in inspection
and the number of inspectors with appropriate tools and training.
Once again, the frequency of inspection should be influenced by the cost of inspection
and the expected benefit from it. Obviously, more frequent inspection may reduce the
non-detection cost but at the same time increase the cost of inspection. On the other
hand, a less frequent inspection or no inspection will reduce the cost of inspection but
will increase the non-detection cost and increase the risk of failure. Therefore, the
optimal inspection strategy discussed above provides the correct balance between the
number of inspections and the resulting benefit.
1
2
2
3
3
4
4
5
5
6
6
7
7
Success
Failure
Failure
Failure
Failure
Failure
72
3.5 Summary
In this chapter, reliability of wood poles, and various factors that affect the reliability
are discussed. Different types of decay or failure conditions of power supply wooden
poles, maintenance and inspection actions are also discussed. These will be helpful in
developing optimal maintenance and inspection models in subsequent chapters.
73
Chapter Four
Modelling Maintenance Cost for Complex Infrastructure Component
4.1 Introduction Maintenance keeps the system working and performing its required functions. It
restores or improves system reliability. In modeling a maintenance cost, it is essential
to determine the type of maintenance actions to be performed, the cost of doing or not
doing maintenance as well as the interval for performing maintenance actions.
The aim of this chapter is to develop mathematical models for infrastructure
components that consider the durability of components, environmental and human
factors that influence the deterioration process or failure of infrastructure components
and the effect of maintenance and inspection decisions on the failure rate of
infrastructure component. These models are expected to
• Develop optimum maintenance cost per unit time for different type of
maintenance actions
• Obtain optimal interval between maintenance actions
• Determine the number and type of maintenance actions.
The outline of this chapter is as follows: Section 4.2 deals with the development of
mathematical models and analysis of theses models with real life data. Sensitivities of
these models are analysed in section 4.3.
4.2 Development of Mathematical Cost Models In this section maintenance cost models are developed and some numerical examples
are cited for optimal maintenance policies for infrastructure component.
4.2.1 Assumptions
• Failure rate increases with time.
• Preventive maintenance restores a portion of life.
• The level of restoration is based on the quality of the maintenance performed
and cost of PM is dependent on the quality of PM.
74
4.2.2 Notations
C(x, N) : the expected cost per unit time
Cmr : the unit cost for minimal repair in $
Cpm : preventive maintenance cost in $
Cre : replacement cost in $.
rpm(t) : failure rate at time t , with maintenance.
r(t) : failure rate at time t when no maintenance is being performed.
N : number of times the maintenance action is performed before
being replaced by a new one.
x : time between two consecutive preventive maintenances.
K : number of maintenance actions carried out up to a certain point of
time t.
τ : time restoration of maintenance action.
Where, τ = αx
α : quality of the maintenance and it is dependent on the on the
type of maintenance actions and it lies between 0 and 1. Higher
the α, better the quality of maintenance.
4.2.3 Modelling Expected Cost per Unit Time The Figure 4.1 shows the effect of various maintenance actions hence based on this
graph; a conceptual mathematical model has been developed.
From Figure 4.1, the failure rate rpm(t) =r( t - kτ) (4.1)
75
x x x x x
r(t)
Ag
t
0
Legends
τ time of restoration r1(t) failure rate distribution after 1st maintenance r2(t) failure rate distribution after 2nd maintenance r3(t) failure rate distribution after 3rd maintenance
Figure 4.1: Graphical representation of failure rate distribution with maintenance
Expected Cost per unit time
= (Expected Cost of minimal repair
+ Expected Cost of preventive maintenance
+ Cost of replacements on failure)/Cycle time (4.2)
Expected Cost of minimal repair can be given by
= ( )( )
∑∫=
+N
k
xk
kx pmmr dttrC0
1 (4.3)
From the Figure 4.1, the failure rate rpm(t), at any point of time can be written by the
following equation:
( ) ( )τktrtrpm −=
76
Therefore, Expected cost of minimal repair
( )( )
−= ∑∫=
+N
k
xk
kxmr dtktrC0
1τ (4.4)
When failure rate follows the Weibull distribution, the failure rate expression is given
by:
( ) β
β
ηβ 1−
=ttr (4.5)
Therefore, by substituting equation 4.5 in equation 4.4, expected cost of minimal
repair is expressed as
= ( )
−∑ ∫=
−+N
k
xk
kxmr dtktC0
1)1( ββ τ
ηβ
= ( ) ( )xk
kx
N
k
kt1
0
+
=
−∑ βτ
ηβ
β
= ( ) ( )[ ]
−−+−∑=
N
kmr kkkkxC
0
1 βββ
β
ααη
(4.6)
Expected cost of preventive maintenance till time t
= ( ) pmCN 1− (4.7)
Expected Cost of replacement = Cre (4.8)
Therefore, the total expected cost per unit time C(x, N) can be found by substituting
equations 4.6, 4.7 and 4.8 in equation 4.2. The total expected cost per unit time C(x, N)
can therefore be expressed as
C(x, N) = Nx1 [ ( )( )
∑∫=
+N
k
xk
kx pmmr dttrC0
1+ ( ) pmCN 1− + Cre]
( ) ( ) ( )
+−+
−= ∑ ∫=
−+
repm
N
k
xk
kxmr CCNdtktCNx
NxC 11,0
1)1( ββ τ
ηβ
77
Finally:
( ) ( ) ( )[ ] ( )
+−+
−−+−= ∑=
repm
N
kmr CCNkkkkxC
NxNxC 111,
0
βββ
β
ααη
(4.9)
The optimum solutions for maintenance interval x, number of preventive maintenance
N, before a replacement action, and the cost per unit time can be obtained by
differentiating equation 4.9 with respect to time ‘t’ and then equating it to zero. In this
case, analytical expression is not traceable and MAPLE is used to solve the above
equation [see Appendix C].
4.2.4 Estimation of Quality of Maintenance
Quality of maintenance can be measured by its effectiveness on the reliability or
serviceability of the component that simply means how a maintenance action can
extend the service life. If a maintenance action has a capability to extend the service
life of a component with an expected service life of 30 years to 35 years, then the
quality of maintenance can be determined as follows:
Quality of maintenance
e
e
LLL 0−
=α = 35
3035 − = 0.15. (4.10)
Where, Le = Extended service life
L0 = Expected service life without this type of maintenance
Quality of maintenance ranges from 0 to 1.
The quality of minimal repair action can be considered as almost ‘0’, as this type of
maintenance action cannot make a significant improvement of reliability of the
component. As for example, a simple painting of a Power Supply wooden pole can
extend the service life of the pole in an average 1-3 years which gives a value of α that
falls in the range 0.02 to 0.08, if we consider the normal service life wooden pole to be
between 35 – 40 years. This range provides an almost zero restoration of reliability.
On the other hand, a mechanical reinstatement action has an ability to extend the
service life of wood pole ten to fifteen years that gives a value of α approximately
equal to 0.3 or more.
78
Once again the quality of maintenance is very much related to the cost of maintenance.
The greater the quality of maintenance, the higher the cost of maintenance. Certainly,
the cost of mechanical reinstatement is much higher than that of painting.
4.2.5 Modelling Human Factors and Inspection
For timber pole, we assume that the human participation is restricted only in
inspection. The over all inspection reliability as discussed previous chapter (when
inspections steps are assumed to be in series) is ;
ni RRRR ..............21 ×=
Where, Ri represents the inspection reliability and 1,2,3, n, indicate the inspectors
steps.
And the value of Ri can be obtained by the analysis of Human factors.
Overall System Reliability, R = System Reliability × Inspection Reliability.
= Rs(t) × Ri(t) (4.11)
Then the failure rate can be given by
( ) ( )( )tRtftr =
where, ( ) ( ) ( )tRtRtR si ×=
Therefore, when integrating human inspection reliability, the hazard rate model would
be
( ) ( )( ) ( )tRtR
tftrsi
i ×=
( ) ( )trtRi
1=
Assuming the failure rate follows the Weibull distribution, then the equation will be
given by
( ) ( )11 −= β
βηβ t
tRtr
ii (4.12)
79
4.2.6 Modelling Environmental Factors
As discussed earlier, there are number of environmental variables influencing the
decay of wood pole. Let ψ be the functional formate of all the environmental
parameters affecting the failure process. Then the expression for the failure rate
function can be given by:
( ) ( ) ( )( )tRtrtrtr
ii
ψψ ==∗ (4.13)
Where,
ψ : the functional form of environmental parameters
r(t) : base line failure rate ( when there is no environmental or human
hazard)
( )tr ∗ : failure rate, taken into consideration of the environmental and
human factors.
Operating environment is considered, the hazard rate of a system can be modelled as
the product of an arbitrary and unspecified base line failure rate r(t) and a positive
functional term ψ(z; β), basically independent of time, incorporating the effects of
environmental variables ( covariates) such as temperature, salinity moisture content etc
(Kumar et al, 1993).
When these covariates are taken into account, the new failure rate or hazard rate ( )tr ∗
can be modelled as
( ) ( ) ( )thztr ×=∗ βψ ; (4.14)
Where, ψ (z; β): incorporates the effect of the different factors through a row vector
z(covariates) and a column vector β of a regression parameter, which characterise the
effect of z. The covariates influence the failure rate in a manner such that the observed
hazard rate is either greater or smaller than the base line hazard rate depending on the
poor maintenance quality and the improved maintenance quality or new component
respectively (Kumar et al, 1993). Figure 4.2 shows the total failure rate under the
influence of covariates
80
Effect of covariates Observed failure rate Initial failure rate time
Figure 4.2: Effects of covariates on hazard rate
It is generally assumed that the functional form of ψ (z; β) is known, while baseline
hazard rate remains unspecified.
Different forms of ψ (z; β) may be used as shown bellow.
Exponential ( ) ( )ββψ zz exp; = (4.14)
Logarithmic ( ) { }ββψ zez += 1log; (4.15)
Inverse linear ( )β
βψz
z+
=1
1; (4.16)
Linear ( ) ββψ zz += 1; (4.17)
In reality, ψ might be time dependent. Therefore, the time dependent Proportional
hazard models are assumed (Kumar et al, 1993) to be
Exponential ( ) tet θθψ =; (4.18)
Logarithmic ( ) { }tet θθψ += 1log; (4.19)
Inverse linear ( )ε
εψt
t+
=1
1; (4.20)
Linear ( ) εεψ tt += 1; (4.21)
Where, θ and ε are shape parameters based on the co-relationship to the environmental
conditions.
4.2.7. Modelling Durability
Failure rates of the infrastructure component, especially, components made of organic
material such as wood or timber are largely affected by the durability of the
Failu
re ra
te
81
component. The higher the durability of the components, the lower is the failure rates.
In modelling durability, Foliente et al (2002) introduce an in-ground degradation
modification factor (kd) that can be applied for wood in ground contact, such as poles
and posts for a specified designed service life. Durability factors for other materials
can also be determined by using Foliente’s models for generation of the durability
factor. The following table shows an example of generating kd for Australian timber
poles, hard wood and soft wood with and without preservative treatment (AS 1604.1)
given in table 4.1.
Table 4.1: In-ground decay degradation factor for round pole in Zone B (Foliente et al (2002))
[This graph is not available online. Please consult the hardcopy thesis available from the QUT library]
82
Failure rate is inversely proportional to the degradation modification factor kd. The
durability degradation factors kd can be used in the design process and development of
maintenance decision models.
This is given by
( )( )
∑∫=
+N
k
xk
kxd
pmmr dtk
thC0
1 1}{
= ( )( )
∑∫=
+N
k
xk
kx pmd
mr dtthk
C0
11 (4.22)
Therefore, total cost would be given by
( ) ( ) ( )[ ] ( )
+−+
−−+−= ∑=
repm
N
kdmr CCNkkkkx
kC
NxNxC 1111,
0
βββ
β
ααη
(4.23) 4.2.8 Parameter Estimation
Estimation of base line parameters
Estimation of base line parameters β and η when effect of durability, environmental
factors is kept zero:
Assuming the failure rate follows the Weibull distribution. The reliability function
then can be presented as:
( )β
β
η
η
−
=−
=t
t
etF
etR
1
)(
( )( )β
η
=−−
ttF1ln
Now taking log of both sides
( )( )[ ] ηββ loglog1lnlog −=−− ttF (4.24)
4.2.9 Numerical Example of Estimating Base Line Parameters
The data in Table 4.2 shows life of poles in Launceston and Devonport District of
Tasmania. (Stillman (1994)) . The data are then analysed by using regression analysis
methods.
83
Table 4.2: Mortality of fully Impregnated Wood Pole
Year
Age
1 9 6 0
1 9 6 1
1 9 6 2
1 9 6 3
1 9 6 4
1 9 6 5
1 9 6 6
1 9 6 7
1 9 6 8
1 9 6 9
1 9 7 0
1 9 7 1
1 9 7 2
1 9 7 3
1 9 7 4
1 9 7 5
1 9 7 6
1 9 7 7
1 2 1 3 4 1 5 1 6 0 1 1 7 1 2 1 8 1 1 1 2 3 9 1 2 1 2 3 1 10 5 6 2 7 1 2 1 0 11 2 3 5 1 8 3 2 6 12 6 7 6 10 12 5 3 13 1 2 2 4 5 12 9 6 3 14 3 5 2 5 10 6 14 6 15 1 9 7 7 25 10 11 6 16 1 6 2 6 18 11 14 12 0 17 4 12 16 10 10 20 7 0 18 3 9 19 2 16 11 0 19 2 16 9 13 16 0 20 14 14 8 13 0 21 3 11 13 0 22 12 10 0 23 10 0 24 0 Fail 0 49 97 80 58 71 84 56 57 49 36 18 7 11 6 6 5 1
No. of ins. 97 551 924 996 1151 1307 1251 1153 1329 1196 1053 884 516 325 410 528 316 385
No. of Rem 97 502 845 916 1093 1236 1167 1097 1272 1147 1017 866 509 314 404 522 311 384
We analysed the Parameters of failure distribution, using the following notations and
assumptions
Notations:
t time interval
cht cohort of poles installed at t
rt number of poles that fail in each cohort
ct censored having reached age at t = cht - rt
Σct cumulative sum of ct commencing at the first cohort
dt poles that have failed at each age t
84
Σdt cumulative sum of dt commencing at the first cohort
nt number of poles in service at age t – the characteristic population
Table 4.3: Pole life analysis
t rt cht ct Sum of ct dt Sum of
dt nt 1 0 675 675 17579 0 682 18261 2 0 979 979 16904 1 682 17586 3 0 447 447 15925 0 681 16606 4 0 413 413 15478 1 681 16159 5 0 442 442 15065 4 680 15745 6 6 450 444 14623 5 676 15299 7 3 483 480 14179 4 671 14850 8 1 385 384 13699 6 667 14366 9 5 316 311 13315 6 661 13976
10 6 528 522 13004 9 655 13659 11 6 410 404 12482 24 646 13128 12 11 325 314 12078 30 622 12700 13 7 516 509 11764 49 592 12356 14 18 884 866 11255 44 543 11798 15 36 1053 1017 10389 50 499 10888 16 49 1196 1147 9372 76 449 9821 17 57 1329 1272 8225 70 373 8598 18 56 1153 1097 6953 79 303 7256 19 84 1251 1167 5856 60 224 6080 20 71 1307 1236 4689 56 164 4853 21 58 1151 1093 3453 49 108 3561 22 80 996 916 2360 27 59 2419 23 79 924 845 1444 22 32 1476 24 49 551 502 599 10 10 609 25 0 97 97 97 0 0 97
Assumptions:
Failure time is a continuous variable
The variables from the table 4.2 are expressed in logarithmic form in the following
table 4.3. This tabulation is based on the assumption that failures occur continuously
And plotted as log(t) vs log[-ln(1-F(t))] :
85
Table 4.4: Data analysis for failure distribution
1 2 3 4 5 6 7
Time t dt nt pt Log t F(t) Log[-ln(1-F(t))]
1 0 18261 0.00000 0.0000 0.0000 0.0000 2 1 17586 0.00006 0.30103 0.0001 -4.2452 3 0 16606 0.00000 0.47712 0.0001 -4.2452 4 1 16159 0.00006 0.60206 0.0001 -3.9253 5 4 15745 0.00025 0.69897 0.0004 -3.4284 6 5 15299 0.00033 0.77815 0.0007 -3.1550 7 4 14850 0.00027 0.84510 0.0010 -3.0135 8 6 14366 0.00042 0.90309 0.0014 -2.8577 9 6 13976 0.00043 0.95424 0.0018 -2.7405 10 9 13659 0.00066 1.00000 0.0025 -2.6059 11 24 13128 0.00183 1.04139 0.0043 -2.3653 12 30 12700 0.00236 1.07918 0.0067 -2.1747 13 49 12356 0.00397 1.11394 0.0106 -1.9711 14 44 11798 0.00373 1.14613 0.0144 -1.8397 15 50 10888 0.00459 1.17609 0.0190 -1.7182 16 76 9821 0.00774 1.20412 0.0267 -1.5678 17 70 8598 0.00814 1.23045 0.0348 -1.4503 18 79 7256 0.01089 1.25527 0.0457 -1.3298 19 60 6080 0.00987 1.27875 0.0556 -1.2427 20 56 4853 0.01154 1.30103 0.0671 -1.1581 21 49 3561 0.01376 1.32222 0.0809 -1.0739 22 27 2419 0.01116 1.34242 0.0920 -1.0152 23 22 1476 0.01491 1.36173 0.1070 -0.9465 24 10 609 0.01642 1.38021 0.1234 -0.8805 25 0 97 0.00000 1.38021 0.1234 -0.8805 Where,
p(t) = dt/nt
F(t) = Cumulative sum of p(t)
And
R(t) = 1- F(t)
SUMMARY OUTPUT Regression Statistics
Multiple R 0.987281
R Square 0.974724
Adjusted R Square 0.973575
86
Standard Error 0.176659
Observations 24
Table 4.5: Regression analysis for Mortality of fully Impregnated Wood Pole
df SS MS F Significance
F Regression 1 26.47741 26.47741 848.4018 4.58E-19 Residual 22 0.686589 0.031209 Total 23 27.164
Coefficients Standard
Error t Stat P-value Lower 95% Upper 95%
Intercept -5.90099 0.133414 -44.2306 5.44E-23 -6.17767 -5.6243 X Variable 1 3.567039 0.122464 29.12734 4.58E-19 3.313064 3.821013
Figure 4.3: Graphical representation of regression analysis
Very high R Square (0.973575) of this regression analysis indicates a very good linear
fit which gives gradient of the slope and the y intersections 3.56 and –5.89 respectably.
Therefore,
β = 3.56
and
log[- ln(1-F(t))] is the y interception which can be computed by setting t = 1,
Regression Analysis
-6
-5
-4
-3
-2
-1
00 0.5 1 1.5
log t
log[
-ln{1
-F(t)
}]
87
Therefore,
From log[ln(1-F(t))] = -βlogη
Or, -5.89=-3.56logη
η = 45.133. Therefore,
The estimate of base line shape parameter of Weibull distribution of the above data β
is estimated as 3.56
And characteristic life η is 45.133
We can also estimate the above parameters by using Weibull Probability paper by
plotting cumulative failure distribution against age at failure. In this case study, the
largest F(t) value is very small and it is not compatible to scale when using the
Weibull Probability paper. However, when we plotted these data on the Weibull graph
paper it also gives us the same result ( see Appendix E). It gives base line parameter
values, β = 3.6 and η = 45.5. The method of maximum likelihood Estimation can also
be used to estimate the parameter by using the following equation
The β and η value can be determined by using FORTRAN NAG files and other
softwares.
By using this base line parameter values, β = 3.56 and η = 45.133 we can calculate
the base line failure rate r(t), reliability R(t) and the cumulative failure distribution
F(t) by using the reliability formulae. This is presented in the table 4.6:
( )β
ηβ
ηηβηβ
−−
=
= ∏
ixi
n
in exxxL
1
11 ,;.....,
88
Table 4.6: Base line failure rate, reliability function and cumulative failure distribution of wood pole data .
For β = 3.96, and η = 45.133 and considering Weibull distribution
1 2 3 4 Time t h(t) R(t) F(t)
1 1.11147E-06 1 0
2 8.64862E-06 0.9999 0.0001 3 2.87195E-05 0.9999 0.0001
4 6.7297E-05 0.9999 0.0001
5 0.000130271 0.9996 0.0004
6 0.000223473 0.9993 0.0007
7 0.000352686 0.999 0.001
8 0.000523654 0.9986 0.0014
9 0.000742089 0.9982 0.0018
10 0.001013674 0.9975 0.0025
11 0.001344066 0.9957 0.0043
12 0.0017389 0.9933 0.0067
13 0.002203791 0.9894 0.0106
14 0.002744335 0.9856 0.0144
15 0.00336611 0.981 0.019
16 0.004074678 0.9733 0.0267
17 0.004875587 0.9652 0.0348 18 0.005774372 0.9543 0.0457
19 0.006776552 0.9444 0.0556
20 0.007887637 0.9329 0.0671 21 0.009113124 0.9191 0.0809
22 0.010458497 0.908 0.092
23 0.011929234 0.893 0.107
24 0.013530798 0.8766 0.1234
25 0.015268648 0.8766 0.1234
89
These functions can be represented graphically by the following manner:
Failure rate function r(t)
Figure 4.4: Failure rate Plot
Reliability function R(t)
Figure 4.5: Reliability Plot
Base line Failure rate
0.0000E+00
5.0000E-03
1.0000E-02
1.5000E-02
2.0000E-02
1 3 5 7 9 11 13 15 17 19 21 23 25 Years
r(t)
Reliability function
0.8
0.820.840.860.88
0.9 0.920.940.960.98
11.02
1 3 5 7 9 11 13 15 17 19 21 23 25
Time in years
R(t)
90
Cumulative failure distribution
Figure 4.6: Cumulative distribution
4.2.10 Estimation of parameters when Durability, Environmental and Human factors are taken into consideration: In real life situations, the infrastructure components are used in different environments
and their operations are interfered with by the environment itself, activities of human
being or they may have different durability status. The Environmental factors would
obviously affect the reliability characteristics of these components. It is therefore,
desirable to isolate these factors in order to estimate their influence. This requires that
these factors should be identified and quantified by numeric variables generally called
covariates.
For a better estimate of reliability characteristics, the use of regression models is
suggested because of the possibility of including covariates. Well known regression
models most often used for reliability analysis fall into two categories. The first
category includes parametric models such as linear, exponential, Weibull, lognormal
and extreme value distribution. In this research work, the parametric models are taken
into consideration.
Cumulative failure distribution
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Time in years
F(t)
91
As the hazard rate distribution of the component changes with the durability,
environmental and the human factors, the parameter of the Weibull distribution needs
to be re-estimated.
Assuming the failure distribution follows the Weibull distribution, the hazard rate
when durability, environmental and human factors are taken into consideration can be
expressed by
( ) 1*
*
−
∗
∗∗ = β
βη
β ttr
By taking logarithm on both sides of the above equation:
( )[ ] ( )LogtLogtrLog 1−+= ∗
∗
∗∗
∗ βη
ββ
… (4.25)
Stallman’s data for wood pole can again be used to estimate the revised Weibull
parameters and this can be done by formatting the following table:
Let us assume ψt be linear function of ε.
Therefore,
ψt = 1+ tε, where, ε is the linear shape parameter assumed to be 0.1.
Inspection Reliability is assumed 0.9.
The value of ε depends on the environmental variables and their quantitative influence
in the deterioration process. Inspection reliability depends on the accuracy of
instrument used in inspection, inspector’s experience and judgement capabilities
Calculation of r*
( ) ( )di kRthtr ψ
=∗
= β
β
ηβε 1)1( −
×+ tR
t
i
= 56.3
156.3
133.4556.3
9.0)1.01( −
×+ tt
Base line shape parameter β = 3.56 (obtained from the regression analysis of base line
equation). And base line characteristic life parameter η = 45.133 (obtained from the
regression analysis of base line equation)
From the above expression, log[r*(t)] and log t is calculated in table 4.7
92
Table 4.7: Calculation of log[r*(t)] and log t 1 2 3 4 5 6 7 8 9 10 11 Time t
No. of poles nt
Failed dt
f(t) F(t) R(t) r(t) Ri ψt log[r*(t)] log t
1 18261 0 0.000 0.0000 1.000 0.0000 0.9 1.1 -5.25489 0.000 2 17586 1 0.0001 0.0001 0.9999 0.00003 0.9 1.2 -4.44737 0.301 3 16606 0 0.0000 0.0001 0.9999 0.00008 0.9 1.3 -3.96234 0.477 4 16159 1 0.0001 0.0001 0.9999 0.00016 0.9 1.4 -3.61089 0.602 5 15745 4 0.0003 0.0004 0.9999 0.00028 0.9 1.5 -3.33293 0.698 6 15299 5 0.0003 0.0007 0.9996 0.00044 0.9 1.6 -3.10243 0.778 7 14850 4 0.0003 0.0010 0.9990 0.00066 0.9 1.7 -2.90492 0.845 8 14366 6 0.0004 0.0014 0.9986 0.00093 0.9 1.8 -2.73181 0.903 9 13976 6 0.0004 0.0018 0.9982 0.00125 0.9 1.9 -2.57753 0.954 10 13659 9 0.0007 0.0025 0.9975 0.00164 0.9 2.0 -2.43826 1.000 11 13128 24 0.0018 0.0043 0.9957 0.00209 0.9 2.1 -2.31122 1.041 12 12700 30 0.0024 0.0067 0.9933 0.00261 0.9 2.2 -2.19440 1.079 13 12356 49 0.0040 0.0106 0.9894 0.00321 0.9 2.3 -2.08620 1.113 14 11798 44 0.0037 0.0144 0.9856 0.00388 0.9 2.4 -1.98452 1.146 15 10888 50 0.0046 0.0190 0.9810 0.00463 0.9 2.5 -1.89108 1.176 16 9821 76 0.0077 0.0267 0.9733 0.00546 0.9 2.6 -1.80238 1.204 17 8598 70 0.0081 0.0348 0.9652 0.00637 0.9 2.7 -1.71866 1.230 18 7256 79 0.0109 0.0457 0.9543 0.00737 0.9 2.8 -1.63940 1.255 19 6080 60 0.0099 0.0556 0.9444 0.00847 0.9 2.9 -1.56411 1.278 20 4853 56 0.0115 0.0671 0.9329 0.00965 0.9 3.0 -1.49243 1.301 21 3561 49 0.0138 0.0809 0.9191 0.01094 0.9 3.1 -1.42401 1.322 22 2419 27 0.0112 0.0920 0.9080 0.01232 0.9 3.2 -1.35856 1.342 23 1476 22 0.0149 0.1070 0.8930 0.01380 0.9 3.3 -1.29583 1.361 24 609 10 0.0164 0.1234 0.8766 0.01539 0.9 3.4 -1.23561 1.380 25 97 0 0.0000 0.1234 0.8766 0.01539 0.9 3.5 -1.17768 1.380
SUMMARY OUTPUT
Regression Statistics
Multiple R 0.9992642
R Square 0.998529
Adjusted R Square 0.998465
Standard Error 0.0420235
Observations 25
93
Table 4.8: Regression Analysis of Wood poles data when Durability, environmental and human factors are taken into consideration
ANOVA
df SS MS F Significance
F Regression 1 27.5707135 27.5707 15612.22 4.41E-34 Residual 23 0.04061732 0.00177 Total 24 27.6113308
Coefficients Standard
Error t Stat P-value Lower 95% Upper 95%
Lower 95.0%
Intercept -5.366654 0.02532566 -211.91 2.36E-39 -5.419044 -5.31426 -5.4190445 X Variable 1 2.964573 0.02372629 124.949 4.41E-34 2.915492 3.013655 2.91549154
-6
-5
-4
-3
-2
-1
00 0.5 1 1.5
log[
h*(t)
]
log t
Figure 4.7: Regression analysis plot for wood pole data when Durability, environmental and human factors are taken into consideration
Regression analysis of column 10 & 11 of the table 3 gives an equation
( )[ ] LogttrLog 96.236.5 +−=∗
When comparing this equation with the equation 425, it gives
Slope of the equation m = (β* - 1)
2.96 = (β∗ - 1)
β∗ = 3.96
y axis interception c is given by
∗∗
∗
=βη
βLogc
94
( ) 96.396.336.5
∗=−
ηLog
5.32=∗η
Where, β* and η* are the Weibull shape parameter and characteristic life parameter
respectively when environmental and human factors are taken into account.
If environmental parameter is not taken into consideration, the regression analysis
gives value of β and η as 3.56 and 45.133 respectably (Base line condition).
4.2.11 Numerical Example of the Developed Model
Here, we need to find out the following variables for this model:
The expected cost per unit time C*(N, x )
The optimal interval between maintenance x*
Optimal number of maintenance N*
Assumption:
ψt is time dependent and follows a linear function of ε
Accuracy of inspection is almost 100%. We assume inspection reliability is 0.9.
Full length preservative- treated softwood diameter Diameter 250≤ d≤ 400mm. for
which With durability factor kd = 0.95
Environmental parameter, ε = 0.1
Inspection reliability, RI =0.9
Weibull shape parameter, β* = 3.96
Weibull characteristic life, η* = 32.51
The following cost values were only assumed which is approximate to the actual
value.
Cost of minimal repair, Cmr = $100
Cost of preventive maintenance, Cm = $500
Cost of replacement, Cre = $2500
Maintenance effectiveness, α = 0.8
Durability factor kd = 0.95 (assuming)
Substituting all the values in the equation 4.23 and solve for the optimum values of
cost rate, maintenance interval and number of maintenance.
95
( ) ( ) ( )[ ] ( )
+−+
−−+−= ∑−
=repm
N
kdmr CCNkkkkx
kC
NxNxC 1111,
1
0 *
**
*
*
ββ
β
β
ααη
Solving 4.23 in Maple gives the following results
• The expected cost rate per unit time C*(N, x)= $ 59.96 per year
• The optimal interval between maintenance, x*= 30
• The number of maintenance action before replacement, N*= 2
Quality of maintenance is often associated with cost, for example, an improved
reliability of inspection and maintenance needs additional money to be spent on repair
action, more sophisticated instruments and or frequent inspection.
96
4.3 Sensitivity Analysis of the Developed Models
This section discusses the effects of various parameters on the developed model, such
as the effect of Weibull parameters (β and η), environmental parameters (ε), functional
form of environmental parameters, effect of durability factor and also the effect of
quality (α) and cost of various maintenance actions such as Cre, Cmr, and Cpm.
Effect of Quality of Maintenance Action (α)
The reliability of the component/ system improves with the quality of maintenanceα.
The Table 4.9 and the Figure 4.8 shows the effect of variation of α, while keeping all
other parameters constant.
Table 4.9: Effect of variation of maintenance quality.
α 0.05 0.1 0.2 0.3 0.4 0.5 0.8
N* 2 2 2 3 3 3 2
x* 20.17 20.76 22.09 18.92 20 20 30
C(N*,x*) 102.80 99.85 93.88 87.19 80.32 75.6 60.93
N* x* 40.34 41.52 44.18 56.76 60 60 60
Replacement cost Cre = $2500,
Preventive maintenance cost Cpm= $600,
Minimal repair cost Cmr = $100,
Durability factor kd = 0.95
Environmental parameter ε = 0.1
Inspection reliability Ri =0 .95
Shape parameter β = 3.96
Characteristic life η = 32.51
97
.
Figure 4.8: Effect of variation of α
From the above table and plot it is seen that the service life (N*x*) is increasing but the
maintenance cost per unit time (year) C(N*,x*) is decreasing with the quality of
maintenance. This indicates that higher quality of maintenance yields a greater service
life (Reliability).
The cost of maintenance increases with the increase of quality as the cost of preventive
maintenance varies with the quality of maintenance and higher quality maintenance
involves higher costs. The Table 4.10 and the Figure 4.9 shows this effect
Table 4.10: Effect of cost and quality of Maintenance
α 0.05 0.1 0.2 0.3 0.4 0.5 0.8
Cpm $100 $200 $400 $600 $900 $120 $180
N* 6 4 3 3 3 2 2
x* 8.62 12.37 17.05 18.80 20 28.09 30
C(N*,x*) 77.50 82.35 86.31 87.19 88.89 91.41 94.00
98
Figure 4.9: Effect cost and quality of Maintenance It is observed that with the increase of quality and PM cost, the number of
maintenance decreases while interval between two successive maintenance increases.
Here cost per unit time also increases but at the same time the service life or the
reliability of component also increases with the quality of maintenance.
Effect of Environmental Parameters
Shape parameter β and the characteristic life parameter η are largely affected by the
variation of the environment. This can be shown by the following Table 4.11 for wood
pole.
Table 4.11: Effect of environment on shape parameter (β) and characteristic life
parameter (η)
ε 0.1 0.2 0.4 0.6 0.8 1.0
β* 3.96 4.1 4.24 4.31 4.36 4.39
η* 32.51 28.95 25.03 22.94 21.56 20.54
99
Now, by substituting these β*, and η* values for various environmental shape
parameter values in the model equation, the optimal number of maintenance (N*) ,
interval between two successive maintenances (x*) and the maintenance cost per unit
time (C(N*, x*)) can estimated and that is showen in the following table. This will
provide the effect of environmental factors on the decision parameters N*, x*, and
C(N*, x*).
Here we assume
Replacement cost Cre = $2500,
Preventive maintenance cost Cpm= $600,
Minimal repair cost Cmr = $100,
Durability factor kd = 0.95
Inspection reliability Ri =0 .95
Quality of maintenance α = 0.3
Table 4.12: Effect of environmental parameter
ε 0.1 0.2 0.4 0.6 0.8 1.0
N* 3 3 3 3 3 3
x* 18.65 16.14 13.58 12.28 11.48 10.84
C(N*,x*) 88.49 101.04 118.14 130.70 139.82 147.23
N* x* 55.47 48.42 40.47 36.84 34.32 32.52
100
Figure 4.10: Effect of environmental parameter The Table 4.12 and the Figure 4.10 show that for a particular quality of maintenance
as optimal number of maintenances remain the same with the variations of the
environmental parameter but the interval between two successive maintenances
decreases whereas, the cost of maintenance per unit time increases with the increase
of environmental parameter value ε. Therefore, it is clear that the service life or the
reliability of the component is largely affected by the environmental conditions. The
higher the environmental hazards, the higher the maintaining cost and the lower the
reliability.
Effects of Durability Modification Factor of Component:
The effect of variation of durability of components on expected cost rate, number of
preventive maintenance and the interval between two successive maintenances in
optimal situation can be represented by the following table and plot. Durability
degradation factors were considered as the representation of the durability of
components.
Here, we assume all other parameters remain constant
Here we assume
101
Replacement cost Cre = $2500,
Preventive maintenance cost Cpm= $600,
Minimal repair cost Cmr = $100,
Environmental parameter ε = 0.1
Inspection reliability Ri =0 .95
Quality of maintenance α = 0.3
Table 4.13: Effect of Durability
kd 0.95 0.85 0.75 0.65 0.55 0.45
N* 3 3 3 3 3 3
x* 18.65 18.03 17.56 16.7 16.00 15.05
C(N*,x*) 88.49 91.31 93.11 97.59 98.89 108.23
N* x* 55.47 54.09 52.68 50.1 48 45.15
Figure 4.11: Effect of durability modification.
102
From the Table (4.13) and the Figure (4.11), it is observed that as durability
modification factor kd decreases, the number of optimal maintenance remains constant
for all other parameters remaining constant but the expected cost rate per unit time
C(N*,x*) increases and the interval between two successive maintenance and the total
service life (N*x*) or reliability decreases. Therefore, it can be concluded that a
component with higher durability provides less maintenance cost and higher service
life in any environmental condition.
Effect of Cost of Replacement Cre
The following table (4.14) shows the scenario of the optimal number of preventive
maintenances, interval between two successive PM and the expected cost rate per unit
time as the replacement cost increases while keeping all other parameters as constant.
Table 4.14: Effect of Cost of Replacement Cre
Cre $2000 $2500 $3000 $3500 $4000 $4500
N* 2 3 3 3 3 3
x* 22.25 18.64 19.25 19.86 20.00 20.00
C(N*,x*) 78.16 88.50 97.28 105.82 114.17 122.30
Figure 4.12: Effect of cost of replacement
103
The table and the plot above show that there is a trend towards increasing of the
number of optimal PM and the interval between two successive PM as the cost of
replacement increases. The reason behind this trend is that when the replacement costs
increases it is better to perform more PM rather than replace it early to achieve an
optimum solution. The table also shows that the expected cost rate per unit time also
increases with the increase of replacement cost. Therefore, we will be expecting an
increase of number of preventive maintenance, when the cost of replacement is
increasing and decreasing the chance of replacement.
Effect of Variation of Minimal Repair
Here, we assume the cost of minimal repairs are constant throughout the life cycle of
the component (our assumption is to carryout the same type of minimal repair), for the
simplification of the model, although these costs may vary with type of minimal repair
decision and also with the time. The effect of variation of minimal can be observed by
simulating the model with different costs of minimal repair keeping all other
parameters constant. This can be represented by the following table:
Here we assume
Replacement cost Cre = $2500,
Preventive maintenance cost Cpm= $600,
Durability modification factor kd = 0.95,
Environmental parameter ε = 0.1
Inspection reliability Ri =0 .95
Quality of maintenance α = 0.3
Table 4.15: Effect of Variation of Minimal Repair
Cmr 50 100 150 200 300 500
N* 2 3 3 3 3 3
x* 27.75 18.65 16.85 15.65 14.10 12.40
C(N*,x*) 74.82 88.50 98.03 105.82 116.80 132.90
104
Figure 4.13: Effect of minimal repair.
The Table (4.15) and the Figure (4.13) shows that like the cost of replacement, the cost
of minimal repair has some effects on the model. As Cmr increases, the number of PM
increases, the interval between two successive PM decreases and finally, expected cost
per unit time increases. In the case of less minimal repair costs this data suggested
performing more minimal repair in between two successive PM so that it provides
longer interval between PM. On the other hand, there will be an increased need to
maintain and replace at an increased Cmr.
4.4 Summary
Reliability of an infrastructure component is affected by different kinds of
environmental and operational mechanisms. The deterioration process is more rapid
when such components are exposed to more adverse environmental conditions than
when it is exposed to a less adverse environment. The adverse environment alters the
parameters, therefore the expected cost per unit time, time between maintenances and
the number of maintenances vary with the variations of environmental condition.
Similarly, parameters are also affected by the durability of the component and the pre-
installation treatment procedures. The numerical example of the model shows
105
significant variations in the maintenance related decision variables for different
environmental conditions.
For simplification, the models developed in this chapter assumes the linear format of
environmental parameters but in real life situation, the functional form could be
exponential, logarithmic, inverse linear or even empirical and each of the formats may
provide different results. To develop the functional form of environmental parameters,
it is essential firstly to identify the affecting environmental factors and then, secondly
to determine the numerical effect of these factors on the decay or failure process of
infrastructure component. The next chapter will discuss the identification of influential
environmental factors and will develop a relationship model of these factors with the
inground decaying process of Power supply wood poles.
106
Chapter Five
Collection and Analysis of Environmental Data
5.1 Introduction
Failures or decay rate of infrastructure components are influenced by different
environmental factors. In timber or wood pole some of these factors are soil condition
and composition, wind and ice load, cyclic wetting and drying, temperature, humidity,
attack of insects, and other micro-organisms. These factors could affect the reliability
characteristics of components under study. Reports from the industries confirm this.
This chapter is on identifying influential variables and mechanism for measuring those
variables. The relationship of the decay or failure rates of infrastructure component
and these variables is analysed using soil data. Data are collected from south eastern
coastal areas in Australia, with high replacement rate and inland areas with less
replacement. These are used to model the effect of soil factors for inground timber
decay.
5.2 Identification of Influential Soil Factors
In order to ensure reliability of infrastructure component, it is important to identify the
influential environmental factors and their effects on the failure or deterioration
process. Few studies have so far focused on this. This study analyses following soil
factors and their influence over the reliability of wooden poles:
• Chemical composition
• Moisture content
• Bulk density
• pH value
• Salinity.
• Electrical Conductivity
5.2.1 Selection of Sampling Sites
As high volume component of infrastructure, such as power supply wooden poles, are
spread over a wide range of area and soil condition and compositions vary from place
to place, Selection of sampling location depends on the availability of failure data of
107
the component from the related industries. In this study data on service life of wood
poles were supplied by ENERGEX (Appendix A). From this data, some of the suburbs
in Brisbane with high replacement rate such as Wynnum, Lota and Manly and
Caboolture were selected and at the same time some of the suburbs in Brisbane with
less replacement rate such as Chelmer and Holland park were also chosen to collect
soil sample.
Australian Standards (AS 1289.1.4.1) specifies the procedure for selection of sites
within the selected locations. It states that selection should be is as follows
• At random by using random number table;
• Determined and recorded the boundaries of test area ;
• Established the length(X) and width(Y) of the area and number of sites to be
sampled.
• For each site, select a random number and multiply it with the length of the area
to obtain a longitudinal distance from the start point.
• Measure the width of the area this area and multiply it with another random
number to obtain a lateral distance from the datum edge
• And the intersection of the results of the above two steps will define the location
of the site. Another three random numbers are used to select three sample points
for each site.
Undisturbed samples needs to be obtained by employing some type of sampler, usually
incorporating the use of sample tube and immediately covering it with layer of wax in
order to protect it from the external interference (Vickers (1983)), (Lambe (1951)). As
the satisfactory storage and maintaining natural moisture content and other properties
of soil sample is very difficult, it is always recommended to inspect and test the
samples at the sites or immediately after their arrival at the laboratory. But when the
sample sites are far away from the testing laboratory or further studies are to be
undertaken, some storage is essential. Each collected sample should have a label or tag
that gives an identification number, location of sampling and sampling date and type
of soil.
5.2.2 Equipment and Instruments Used in Sampling and Testing
• 50 / 76 mm hand auger with an extension rod, Sample bags, and 30m tape for soil
sample collection.
108
• Sample extruder, weigh balance, scale and ruler (see the Figure 5.1).
Figure 5.1: Equipment and instruments used for traditional soil sampling and testing, a. Hand Augar and sample extractor with accessories, b. Sample mould,
c. Sample extruder, d. Nata weigh balance, e. Drying oven.
Hand Augar
Extractor Sample tubes
a
c
b
d e
109
• Electronic tester for measuring PH value, salinity and conductivity – A TPS WP-
81 electronic Ph-Cod-Salinity can be used.
• Compact Gauge for Soil Compactness
• Moisture can (tin or aluminium), Oven with temperature control to determine
moisture content.
• Volumetric flask, 250ml or 500ml, Vacuum pump or aspirators for supplying
vacuum, Mortar and pestle, Balance weighting to 0.1g, Supply of de-aired,
temperature-stabilised water for determination of specific gravity of Soil
• X-Ray spectrum for chemical composition Analysis.
5.2.3 Sample Collection
i) Location of sample is marked in the label. (Such as pole 53; sample 1 collection
date October/ 2002 will be label as 53/1/Oct02). Samples are to be taken from
0.6m below the ground level and within 1- 1.5m range from the pole.
Figure 5.2: Procedures for sample collection and measuring the depth of drill. a.
Drilling with the Augar, b. Sample extraction, c. Measuring the depth.
ii) Three samples are to be collected locally for each location for specific soil
composition.
5.2.4 Soil Testing
Physical and chemical testing of the samples are done both on the site and off the site
(in the laboratories) as expressed bellow. On site testing are those that can be done
immediately after collecting the samples from the sites whereas, all other testing that
a b c
110
can be performed in the soil testing laboratory and chemical laboratory are termed as
off the site testing
5.2.5 On Site Testing
• pH value, Conductivity and salinity of the sample Testing: A WP 81 – pH,
Salinity and conductivity meter made by TPS ( Figure 5.3 ) is used in this study. A
solution is made by mixing one part by vol. of soil and five parts by vol. of water
and the pH of soil is measured by dipping the probe of the electronic pH tester to
determine the pH of that soil sample. Similarly conductivity and salinity can be
determined by using specified probe.
Figure 5.3: WP 81 – pH, Salinity and conductivity meter
5.2.6 Laboratory Testing
• Moisture Content: It is calculated by subtracting the final weight after drying the
sample from the initial weight taken dived by the initial wt as follows (Bowels
1970):
% of moisture content = 100×−
i
fi
mmm
where, mi and mf are the initial and final wt. of the sample.
• Soil Compactness : it is estimated by determining the bulk density:
111
Bulk density:
4
,2hdV
whereVm
π
ρ
=
=
where m, V, d and h are the mass, volume of the sample , diameter and height of the
cylindrical sample respectively.
• Chemical Composition: Soil samples are analysed by X-Ray Powder
Diffractometry (Figure 5.4) ) at the chemical Laboratory.
Figure 5.4: S4 Explorer X-Ray Defractometer (Bruker AXS Inc.)
5.2.7 Preparation of Soil Sample for Chemical Analysis
The collected samples are grounded to power in a swing machine for chemical
analysis. The following steps are carried out for sample preparation.
There are two types of analysis
112
Soil Analysis
• Soils samples are put in a container and distilled water are then added.
• The mixture is then fractionised by using an ultrasonic device (Branson Sonics). This
procedure breaks up the clay in the soil sample for analysis.
• The mixture is then left for settlement for ten to fifteen minutes.
• This analyse the clay content of the sample.
Chemical Analysis
• The samples are powderised to less than 100 micron using a swing mill.
• 2.7 gms of soil are used to mix with 0.3 gms. of corundum. The corundum is used to
determine the percentages of mineral compositions for the test.
• The mixture is then further breakdown to 5 micron in the crushing lab.
• The final samples are then pour into a beaker and placed in the oven. The dried
powdered sample from the beaker is then used for analysis.
• When dried up, it is put on a X-Ray Defractomer (Figure 5.4) for chemical
composition analysis.
5.3 Results of Soil Sample
5.3.1 Collection of Soil Data and Analysis
The soil samples are collected from suburbs with high and less failure rate in Brisbane.
The soil samples were collected from the ground within a meter radius from from the
centre of each pole. The soil samples are tested for pH value, moisture content,
salinity, conductivity and bulk density. Chemical composition tests are carried out in
the chemistry laboratory in QUT. The sample test results are presented in Tables 5.1 –
5.8 and details are presented in the Appendix B1 and B2. A summary of the test result
are presented in the Table 5.9 (for physical properties) and Table 5.10(analysis).
113
Table 5.1: Soil Physical Test Result Wynnum, Lota and Mansfield
Sample No.
Soil Description
Initial wt
After Oven wt Height
Diam-eter
Moisture Content
Bulk Density
PH value
Cond-uctvty
Salin-ity
gms gms mm mm % gm/cc �s ppt
12933 clay 92.72 73.68 41 38 20.53 1.99 4.87 56.9 24.4 6154
Earnest St
Light brown
124.16 98.46 57.5 38 20.69 1.90 5.56 39 16.7
6332 Pelmagel
St Sandy 157 148.2 79 38 5.598 1.75 5.37 224 82.4
389159 Sand +
Black clay 125.4
8 110.5 55 38 11.93 2.01 5.58 20.85 8.85 62731
Beltana St Brown Clay 75.1 66.49 40 38 11.46 1.65 6.06 130.4 58.4
16504 Sandy 73 62.81 13.95 #DIV/0
! 6.01 47.7 20.3
6452 Brown Clay 148.5 120.3 68 38 19.025 1.92 6.73 127 56.9 6375
Orallo St Clay 84.3 66.96 41 38 20.569 1.81 6.71 99 43.7 5857
Haylock Light
Brown Clay 134.2 109.4 58 38 18.48 2.04 6.93 38.7 16.5 11784
Tulia St Black Clay
+Sandy 173 158.7 75 38 8.27 2.03 5.87 29.3 12.1 6174
Belgamba St
Dark brown Clay 89.9 78.47 38 38 12.71 2.087 6.85 30.8 12.9
5928/2 106.1 96.92 50 38 8.72 1.87 6.3 26.59 11.1
5968/2 106.6 99.71 58 38 6.52 1.62 6.28 28.3 11.9
5971/1 122.7 98.48 60 38 19.78 1.80 7.04 24.3 13.2
6096/1 63.5 50.95 29 38 19.76 1.93 5.04 35.1 14.7
6207/1 152 139.4 75 38 8.28 1.79 3.86 351 164
10845/1 101.3
5 96.1 38 5.18 #DIV/0 7.28 20 804
11554/1 177.8
7 145.7 75 38 18.04 2.09 5.02 32 13.6
14795/1 89 86.66 50 38 2.63 1.57 6.32 14.76 6.02
17929/2 134.7 116.6 69 38 13.46 1.72 5.52 60 25.8
31991/1 172.4 141.7 75 38 17.81 2.02 4.33 833 404
114
Table 5.2: Soil Physical Test Result Caboolture
Sample No.
Soil Descript.
Initial wt
After Oven wt Height
Diamet.
Moisture Content
Bulk Density
PH value
Condti-vity.
Salin-ity
gms gms mm mm % gm/cc �s ppt
96019 clayey 81.9 67.43 47 38 17.67 1.54 5.3 48.7 20.9
310484 Black clay 72.65 62.27 36 38 14.29 1.78 5.31 200.7 91.5
80441 Sand +clay 143.66 120.67 71 38 16.00 1.78 4.94 134 60.2
97632 Sand +clay 154.2 141.28 80 38 8.38 1.70 5.8 98 49
102795 Yellowis
h clay 118.26 90.67 58 38 23.33 1.80 4.87 218.5 100
105689
Yellow Brown Clay 177.47 140.75 79 38 20.69 1.98 4.64 43.6 18.4
Mean 16.72 1.76 5.14 123.9 56.67
Table 5.3: Soil Physical Test Result Holland Park
Sample No.
Soil Descrpt
Initial Weight
After Oven Weight Height Diamet
Moisture Content
Bulk Density
PH value
Conduc-tivty Salinity
gms gms mm mm % gm/cc �s ppM Roscoe St HP 1 181.2 161.4 61 38 10.95 2.62 5.8 29.4 14 Greenmount St 106.18 96.92 50 38 8.72 1.873 6.3 26.59 11.1 Seville Rd HP 2 157.89 136.04 49 38 13.84 2.84 5.42 50.3 24.6 Mar St HP 3 161.97 146.2 58 38 9.74 2.46 6.95 110 56.5
Mean 10.81 2.45 6.12 54.07 26.55
115
Table 5.4: Soil Physical Test Result Chelmer
Sample No. Initial wt
After Oven wt Height Diametr
Moisture Content
Bulk Density
PH value Conductiv Salinity
gms gms mm mm % gm/cc � ppM 15483 Acacia St
157.36
138.16 48 38 12.20 2.89 5.26 91.7 46.1
25360 Roseberry
105.58 98.44 40 38 6.76 2.33 6.95 45.1 21.9
25647 Glenwood St
166.03
146.03 53 38 12.046 2.76 6.61 46.1 22.5
6654 179.6
9 153.4
1 55 38 14.62 2.88 4.99 251 132
6599 161.9
7 146.2 58 38 9.736 2.467 6.95 110 56.5
7689 194.8 175.4 66 38 9.935 2.603 5.75 67.6 32.6
Mean 10.88 2.65 6.09 101.916 51.93
Table 5.5: Soil Chemical Test Result Caboolture
Sample Number
Chemical Composition (Weight %)
Quartz Kaolin Muscovite Sanidine Goethite Hematit Albite Amorph
Other
310484 61.13 29.26 Very low
105689 39.11 46.48 6.2 .01 80441 44.33 41.14 9.93 0.48 102795 32.4 47.5 20.02 0.95 96019 52.96 30.88 9.94 0.21 97632 60.01 20.73 12.12 0.18
Table 5.6: Soil Chemical Test Result Chelmer
Sample Number
Chemical Composition (Weight %)
Quartz Kaolin Muscovite Sanidine Goethite Hemati Albite Amorph
Other 25360 31.64 28.7 13.67 33.56 10.2
7689 66.50 7.78 8.54 14.46 1 25647 30.65 30.30 10.8 30.64 2.98
6599 37.10 10.42 16.37 41.9 7.3 15483 43.43 51.02 11.70 12.4
6654 41.58 52.70 9.78 5.7 Average 41.82 30.15 23.67 6.60
116
Table 5.7: Soil Chemical Test Result Wynnum, Lota and Mansfield
Sample No.
Soil Descrptn
Chemical Composition (Weight %)
Quartz Kaolin Amo-rphus
Mixed layer illitesmectit Goethit Hematite
Albi-te low
Magntite Other
12933 Wynn-um
Dark brown clay 25.10 31.90 14.70 1.20 10.10 0.30 5.5
6154 Wynn-um
Yellow clay 36.90 39.3 11.40 0.00
6332 Wynn-um Sandy 88.60 1.10
389157 Wynn-um
Sandy (Black clay) 85.30 4.20 0.20
6375 Wynn-um Clay 51.70 36.70 0.20
5857 Lota Red clay 40.80 40.20 4.00 2.70 1.40 0.50 11784 Lota
Black clay 87.50 1.80 0.20
6174 Lota Red clay 60.20 28.40 0.50 62731 Lota
Brown clay 74.30 6.80 1.70 2.00 0.10 2.20
16509 Lota Sandy 24.50 34.80 0.20 21.40 6.00 6452 Lota
Brown clay 36.40 40.0 1.40 6.80 4.20
31991 Lota Sandy 77.8 2.2 0.49 6.8 6096 Wynn-um Clay 31.5 64 0.18 8.5 10845 Lota Sandy 98.4 1.5 0.01 0.2 5971 Manly
Yellow mud 48.0 32.8 3.36 6.7
11554 Manly
Pure sand 0
117
Table 5.8: Soil Chemical Test Result Holland park
5.4 Data Analysis
The Table 5.9 shows the mean value of each of the physical properties of soil. And
these soil properties are analysed with respect to the failure data of power supply
timber poles,
Table 5.9: Summary of Soil Test Result
Suburbs Moisture Content
Bulk Density pH valu Conductivity Salinity Failure rate
Chelmer 10.88 2.66 6.01 101.91 51.93 0.027
Holland Park 11.58 2.45 6.12 54.07 26.55 0.031
Wynnum 13.5 1.86 5.88 108.03 86.73 0.037
Caboolture 16.73 1.764 5.143 123.91 56.67 0.046
5.4.1 Moisture Content
The percentage of moisture content in the soil depends on the rain falls, humidity, and
temperature of the air and draught. Metrological conditions of the sampling site are
recorded during sample collection. As the climate changes over seasons, it is required
to do soil sampling of same areas in different seasons. The table 5.9 shows that the
moisture content of the soil around the poles in Wynnum, Lota and manly suburbs
ranged from 5 to 20 percent with mean moisture content 13.5%. Similarly, the average
moisture contents of Caboolture, Holland Park and Chelmer were found 16.73%,
11.58% and 10.88% respectively. The average service life of wood in Caboolture,
Wynnum, Holland Park, and Chelmer are 22, 27, 32, and 37 years respectively (
Sample Number
Chemical Composition (Weight %)
Quartz Kaolin Muscovite Sanidine Goethite Hemtit Albite Amorp Other
Roscoe St HP 1 45 48.3 Greenmount St 56.21 21.02 9.38 Seville Rd HP 2 65.1 29.34 Mar St HP 3 53.1 25.03 12.1
118
Details in Appendix A). This estimates the failure rate of these suburbs 0.045, 0.037,
0.031, and 0.027 respectively. A relation of failure rate with soil moisture content is
presented in the Figure 5.5.
Moisture content vs Failure rate
Moisture content of soil
10 11 12 13 14 15 16 17 18
Failu
re ra
te
0.026
0.028
0.030
0.032
0.034
0.036
0.038
0.040
0.042
0.044
0.046
Figure 5.5: Relationship of moisture content of soil with failure rate
The moisture content of the surrounding soil has an influence over the wood pole
failure or decay process. The failure rate increases with the moisture content of the
soil.
5.4.2 pH Value
The pH values of collected samples were found between 4 and 7 with a mean of 5.88,
in Wynnum, Lota and Mansfield suburbs which indicates a slightly acidic soil. The
average pH value of soil samples collected from Caboolture, Holland Park and
Chelmer were 5.143, 6.12 and 6.01 respectively which is also slightly acidic. The
effect of acidity or alkalinity of soil on the failure rate of wood pole is analysed and
presented in the Figure 5.6.
119
pH of Soil vs Wood Pole Failre rate
pH value
5.0 5.2 5.4 5.6 5.8 6.0 6.2
Failu
re ra
te
0.026
0.028
0.030
0.032
0.034
0.036
0.038
0.040
0.042
0.044
0.046
Figure 5.6: Relationship of pH and failure rate
The decay or failure rate of wood pole is slightly related to the acidity of the
surrounding soil. The failure rate increases slightly with the increase of soil acidity.
This is consistent with the findings of Charman (2000) which tells that the soil acidity
problem appears to have occurred when pH value falls bellow 5.5 and at pH value less
than 5.5, the soil can be very much toxic to the wooden component and can severely
affect the buried portion of wood material (Charman (2000)).
5.4.3 Bulk Density
Bulk density of the samples collected from Wynnum, Lota and Mansfield were ranged
from 1.5 to 2.1 gm/cm2 with a mean of 1.86 which consistent with Charman’s (2000)
findings. This means that the decay of component of that area are not affected by the
bulk density of soil. Similarly, the mean of bulk density of soil in other selected
suburbs Caboolture, Holland Park, and Chelmer are recorded 1.764, 2.54, and 2.66
gm/cm2 respectively. These values indicate a clear trend of increasing of failure rate
with the increase of bulk density, although the bulk densities of the collected samples
were found well bellow the aggressive condition. The trend is shown by in Figure 5.7.
the bulk density may have effect on drainage and can be related to moisture trapped
during heat season.
120
Bulk density vs Failure rate Plot
Bulk density gm/cc
1.6 1.8 2.0 2.2 2.4 2.6 2.8
Failu
re ra
te
0.025
0.030
0.035
0.040
0.045
0.050
Figure 5.7: Relationship between Bulk density and failure rate
5.4.4 Salinity
The mean salinity of coastal suburbs Wynnum, Lota and Mansfield are found 86.73
ppM and the mean salinity of another coastal suburb Caboolture is recorded as 56.67
ppM. The mean salinity of Chelmer and Holland Park are as 51.93 and 26.55 ppM
respectively. Salinity of Soil vs Wood pole failure rate
Salinity of Soil ppM
20 30 40 50 60 70 80 90
Woo
d po
le fa
ilure
rate
0.025
0.030
0.035
0.040
0.045
0.050
Figure 5.8: Effect of salinity of soil
The Figure 5.8 shows a relationship of salinity of soil with failure rate of wood pole.
Tthe salinity of the selected suburbs has a very slight influence on the failure rate of
wood poles
121
5.4.5 Effect of Electrical Conductivity
The mean electrical conductivities of the collected samples are 101.91 µS in Chelmer,
54.07 µS in Holland park, 108.03µS in Wynnum and 123.91µS in Caboolture. This is
shown in Figure 5.9 Electrical Conductivity VS Failure rate
Electrical conductivity µS
50 60 70 80 90 100 110 120 130
Failu
re ra
te
0.025
0.030
0.035
0.040
0.045
0.050
Figure 5.9: Effect of electrical conductivity
The failure rate of wooden pole increases with the increase of electrical conductivity.
Therefore, it is observed that like moisture content, pH value, salinity, the electrical
conductivity has a relation with the in ground decay process of timber component.
5.4.6 Chemical Analysis Test Result
The Table 5.11 exhibits the summary of chemical analysis test result of the samples
collected from various suburbs in Brisbane (for details see Appendix B2) which shows
that the sampled soils are composed of a number of different ingredients and most of
them are uncommon to all the samples. The mean value major chemical components
and failure rate of each of the suburbs are shown in the Table5.11. The uncommon or
ingredients with very low percentages were not taken into consideration.
122
Table 5.10: Summary of Chemical Analysis Test Result
Suburbs Failure rate Quartz Kaolin Albite (low)
Amorphus
Caboolture 0.046 48.32 35.998 11.64 0.366
Wynnum, Lota and Manly
0.037 57.14
24.38
Holland Park
0.031 50.85 30.92
Chelmer 0.027 41.82 30.15 23.67 6.60
Quartz (SiO2) and Kaolin (Al2Si2O5(OH)4 are the two common ingredients present in
collected samples. The average percentage by weight of quartz against failure rate as
in analysed in the Figure 5.10. This exhibits no significant relationship between failure
rate of wood pole and the amount of quartz content. From chapter three, we know that
the sandy soil (because of high percentages of quartz content), allows the soil less
moist. This helps in preventing the Biological attack. This in turn, allows a better
decay condition of timber materials.
Effect of Quartz SiO 2 on the Failure rate
Percentage of Quartz SiO2
40 42 44 46 48 50 52 54 56 58
Failu
re ra
te
0.025
0.030
0.035
0.040
0.045
0.050
Figure 5.10: Effect of Quartz
The main factor of clay is the presence of kaolin and Table 5.9 and the Figure 5.11,
show that kaolin is directly inactive to the in-ground decay process. But because of its
non-permeable property, kaolin allows more water to trap inside the soil that causes
123
algae, moss, and mould to grow and attack the in-ground timber component, and
results in rapid deterioration. Effect of Kaolin on the Failure rate
Kaolin Al2Si2O5(OH)4
22 24 26 28 30 32 34 36 38
Failu
re ra
te
0.025
0.030
0.035
0.040
0.045
0.050
Figure 5.11: Effect of Kaolin
The effect of Albite ((Na,Ca)Al(SiAl)308) and Amorphus on the decay/failure rate is
not clear . But it needs more sample testing to come into a conclusion about the effect
of Albite and Amorphus. To ensure the effect of other chemical compound we need
more detail analysis of the samples.
5.5 Chapter Summary
This chapter investigate and analyse the soil samples collected from different areas
(Appendix A) in Brisbane, Australia. It shows that Moisture content, pH value, Bulk
Density and Conductivity of soil have direct relation to timber pole failure rate,
whereas Quartz and Kaolin have indirect effect. Soil with high Quartz content means
sandy soil which allows water to drain and prevents moisture to trap inside.
124
Chapter Six
Conclusions and Recommendations for Future Works
This chapter provides summary of this thesis and discusses contribution, limitations
and finally suggests directions for future research.
6.1 Contribution
In Chapter 1, reliability of infrastructure components was discussed. Scope for this
research along with outline of this thesis was provided.
In Chapter 2 overview of reliability models was given along with discussion. Various
inspection and maintenance models are available in literature. The existing models
related to maintenance decisions have so far considered operational factors and are
based on number of simplified assumptions. Most of the papers have not considered
the durability of the component linked to environmental factors. This, as a result, leads
to bias on decisions.
Factors affecting inground decay of wood poles were discussed in Chapter 3. High
volume components such as, power supply poles, distribution line crossbars, railway
sleepers, water and sewerage pipes are widely used in all over the world. Reliability of
these components depends on a complex combination of age, usage, durability,
inspection, maintenance actions and environmental factors influencing decay and
failure of components.
In Australia, more than 5.3 million wooden poles are in use. This represents an
investment of around AU$ 12 billion with a replacement cost varying between
AU$1500-2500 per pole. Well-planned inspection and maintenance strategies
considering the effect of environmental and human factors can extend the reliability
and safety of these components. Maintenance and inspection is worthwhile if the
additional costs are less than the savings from the reduced cost of failures. This
chapter analyses some of the plans and their effectiveness.
125
Models for optimal maintenance decision are developed in Chapter 4. Stillman’s
(1994) data on the pole population situated in the Launceston and Devonport District
of Tasmania and Energex data for Brisbane, Australia are analysed.
This research investigates decay patterns of timber components based on age and
environmental factors (e.g. soil factors) for infrastructure components in the
Queensland region. It also develops models for optimizing inspection schedules and
Maintenance plans.
Identification of environmental factors and modelling based on these factors were
carried out in Chapter 5. Soil data were collected from selected suburbs in Brisbane. It
is found that the moisture content, pH value (Acidity/ alkalinity), bulk density, salinity
and electrical conductivity have influence over the deterioration process. It is also
found out that presence of Kaolin or Quartz has some indirect effect on the
degradation process. Kaolin allows more water to be trapped inside the soil that causes
algae, moss and mould to grow and attack the wood poles. On the other hand, by
virtue of permeability, soils with high quartz content allows more water to infiltrate,
preventing the growth of micro-organism.
This research has increased fundamental understanding of in-ground wood decay
process, developed testing methods for soil factors and proposed integrated models for
optimal inspection, repair and replacement strategies considering durability,
environmental and human factors. A computer program is developed to analyse “what
if” scenario for managerial decisions. Three refereed conference papers have already
come out of this research and two more papers for refereed journal publication are in
the process.
6.2 Limitations
This research has enhanced knowledge on the wood decay process in diverse
environmental conditions. The outcomes of this research are invaluable, not only to
users but also to the wood industry in general (the housing sector, railways for wooden
sleepers and other structural applications such as wooden bridges). In spite of the
contribution mentioned in above this research has following limitations:
126
• Human factors and quality of maintenance model developed in Chapter 3 is
based on the event tree analysis which is a probabilistic model and validation
needs to be carried out using field and laboratory data. In developing this model,
inspection failure rate is assumed to be independent of time. This assumption is
limited to some extent because human reliability depends on human
concentration on a particular job, education, skills, motivation and training. It is
an area of interest and could be investigated in the future.
• In this thesis we assume environmental factors to be linear and the value of
environmental shape parameter ε as 0.1. In practice, identification of
environmental factors and estimation of environmental parameters are more
complicated. It needs information on:
° Failure or decay pattern of components when exposed to different
environmental conditions.
° The correlation between the failure or decay rate and the environmental
variables
° The functional form of environmental parameters– whether it is Linear,
Exponential, Logarithmic, Weibull or empirical.
° Whether ψ is time dependent or not.
° Estimation of environmental parameters from environmental samples.
• Soil condition varies over seasons. In wet season, presence of moisture content
may be higher than that in the dry season. Due to limited time this was not
covered well.
• Higher quality of maintenance is often associated with higher costs. It is
therefore necessary to find the tradeoffs between cost of maintenance and the
desired level of maintenance quality. Due to the lack of cost vs quality of
maintenance data, this important aspect is not analysed in detail.
6.3 Scope for Future Work
There is enormous scope for further research in many areas related to this research.
Some topics are:
• Models can be developed for:
127
Qualitative as well as quantitative research database on lab/field wood
decay process.
Assessment of the residual life of timber infrastructure.
Optimal condition monitoring and maintenance plans for timber
components and
Cost effective decisions for prevention of inground decay and mitigation.
Optimal inspection plan and improvement of inspection accuracy.
• The effect of time value of money is ignored in our models. Since the service
life of component is long, the effect of discounting value will change the cost
per unit time. This can be considered to modify the model by considering
inflation and discounted cash flow method.
• Models developed in this thesis assume only one-dimensional (age) constant
interval preventive maintenance policy up to final replacement. Considering
Age/usage policy one can develop two-dimensional models.
• The use of both field and laboratory failure data from infrastructure industries
can be used for estimating the parameters of the models.
• Soil-data can be collected from various location in the hazard zone A, B, C, and
D in Australia. At the same time failure or decay data of infrastructure
component from selected zones can be obtained from electricity, bridge and
building industries. Samples need be collected in different times of the year to
take into account seasonal variations and build that into the model.
• Cost effective methods can be developed for data collection and quick but at the
same time accurate analysis of environmental factors in the field.
• Quality of maintenance and related costs could be included to these models.
• Quality of maintenance is modelled as deterministic and can be modelled as
stochastic based on variability of the process.
128
Findings of this research can also be applied to other equipment or assets showing
time dependent failure rate and can be further extended by considering cost of
downtime and liability.
129
References 1. Australian Standard AS 1289.1.4.1-1998. “Methods for testing soils for
engineering purpose”. 2. Ascher, H.E. and Fingold, H., (1969), “ Bad as old analysis of system failure
data”, In Annals of Assurance Sciences. Gordon and Breach, New York. 3. Ascher, H.E. and Fingold, H., (1984), “Repairable systems reliability:
Modelling, Inference, Misconceptions and their Causes”, Marcel Dekker, New York.
4. Anderson, R., Friedman, (1977). “Optimal inspection in a stochastic control problem with costly observations.” Mathematics of Operations Research Vol. 2: p 155-190.
5. Basin, W.M., (1969), “Increasing hazard functions and overhaul policy”, ARMS, IEEE-69, C8-R, pp 173-180.
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134
Appendices
135
Appendix A
Mean Service life of Wood Pole in Different Suburbs in Brisbane (Courtesy from ENERGEX)
Sample Data from 01/01/2001 - 31/12/2001 (Reinstated Poles) Pole Suburb Avg Life (Yrs) GALLANANI 37 BEERBURRUM 37 CHELMER 36 CALAMVALE 36 CAINBABLE 35 COAL CREEK 34 ALEXANDRA HEADLAND 34 BILINGA 34 CARINA 34 EAST DEEP CREEK 33 FERNY HILLS 33 GEEBUNG 32 COORPAROO 32 HOLLAND PARK 32 CANNON HILL 32 CHERMSIDE 32 BELLBIRD PARK 32 BEAUDESERT 32 CROWLEY VALE 32 DUNDAS 31 CARINDALE 31 BEACHMERE 31 ALBANY CREEK 31 CLAYFIELD 30 BELMONT 30 FIG TREE POCKET 30 BRENDALE 30 COES CREEK 30 CAMP HILL 30 CALOUNDRA 30 CHAPEL HILL 30 FOREST HILL 30 BALMORAL 29 EAGLEBY 29 AMAMOOR 29 COALFALLS 28 BUDDINA 28 COOROY 28 BURLEIGH WATERS 28 CURRUMBIN WATERS 28 BELLI PARK 28
136
Sample Data from 01/01/2001 - 31/12/2001 (Reinstated Poles) Pole Suburb Avg Life (Yrs) EATONS HILL 28 GAILES 28 CAPALABA WEST 28 BONGAREE 27 CLEVELAND 27 WYNNUM 27 BUDERIM 27 ANNERLEY 27 CAROLE PARK 27 BIRKDALE 27 GILSTON 27 DUNWICH 26 COOLANA 26 EAST IPSWICH 26 ALEXANDRA HILLS 26 ARANA HILLS 26 GATTON 26 CORINDA 26 BURNSIDE 26 BEENLEIGH 25 GLENORE GROVE 25 BULIMBA 25 CAPALABA 25 BOLLIER 25 BELLARA 25 BORALLON 25 BRASSALL 25 BASIN POCKET 24 COOLANGATTA 24 CEDAR VALE 24 CAMP MOUNTAIN 23 BUARABA 23 CURRUMBIN 23 EVERTON HILLS 23 FERNVALE 23 GHEERULLA 22 BLACKSOIL 22 BRAY PARK 22 CABOOLTURE 22 BUNDAMBA 22 CAMIRA 21 ANDREWS 21 BIDDADDABA 21 ELANORA 21 BRIGHTVIEW 20 CASHMERE 20 AUSTINVILLE 20
137
Sample Data from 01/01/2001 - 31/12/2001 (Reinstated Poles) Pole Suburb Avg Life (Yrs) FLINDERS VIEW 20 CHANDLER 20 DAISY HILL 19 CHUWAR 19 CURRUMBIN VALLEY 19 EUMUNDI 19 EDENS LANDING 18 COOROIBAH 16 ESK 16 BONOGIN 14
138
Appendix B1
Soil Physical test result
Wynnum, Lota and Mansfield
Sample No.
Soil Description
Initial Weight
Weight After Oven Height Diameter
Moisture Content
Bulk Density
PH value Conductivity Salinity
gms gms mm mm % gm/cc �s ppt 12933 clay 92.72 73.68 41 38 20.53 1.99 4.87 56.9 24.4 6154
Earnest St Light brown 124.16 98.46 57.5 38 20.69 1.90 5.56 39 16.7
6332 Pelmagel
St Sandy 157 148.21 79 38 5.59 1.75 5.37 224 82.4
389159 Sand +
Black clay 125.48 110.51 55 38 11.93 2.01 5.58 20.85 8.85 62731
Beltana St Brown Clay 75.1 66.49 40 38 11.46 1.65 6.06 130.4 58.4
16504 Sandy 73 62.81 13.958 #DIV/0! 6.01 47.7 20.3
6452 Brown Clay 148.54 120.28 68 38 19.025 1.92 6.73 127 56.9 6375
Orallo St Clay 84.3 66.96 41 38 20.569 1.81 6.71 99 43.7
5857 Haylock
Light Brown Clay 134.2 109.4 58 38 18.479 2.04 6.93 38.7 16.5
11784 Tulia St
Black Clay +Sandy 173 158.69 75 38 8.27 2.03 5.87 29.3 12.1
6174 Belgamba
St Dark brown
Clay 89.9 78.47 38 38 12.71 2.08 6.85 30.8 12.9 5928/2 106.18 96.92 50 38 8.72 1.87 6.3 26.59 11.1 5968/2 106.67 99.71 58 38 6.52 1.62 6.28 28.3 11.9 5971/1 122.76 98.48 60 38 19.78 1.80 7.04 24.3 13.2 6096/1 63.5 50.95 29 38 19.76 1.93 5.04 35.1 14.7 6207/1 152 139.41 75 38 8.28 1.78 3.86 351 164
10845/1 101.35 96.1 38 5.18 #DIV/0! 7.28 20 804 11554/1 177.87 145.78 75 38 18.04 2.09 5.02 32 13.6 14795/1 89 86.66 50 38 2.63 1.57 6.32 14.76 6.02 17929/2 134.76 116.62 69 38 13.46 1.72 5.52 60 25.8 31991/1 172.4 141.7 75 38 17.81 2.02 4.33 833 404
139
Caboolture
Sample No.
Soil Descriptn
Initial Weigh
Weight After Oven Height Diameter
Moisture Content
Bulk Density
PH value Conductivity Salinity
gms gms mm mm % gm/cc �s ppt
96019 clayey 81.9 67.43 47 38 17.67 1.537 5.3 48.7 20.9
310484 Black clay 72.65 62.27 36 38 14.29 1.780 5.31 200.7 91.5
80441 Sand +clay 143.66 120.67 71 38 16.00 1.78 4.94 134 60.2
97632 Sand +clay 154.2 141.28 80 38 8.38 1.70 5.8 98 49
102795 Yellowish clay 118.26 90.67 58 38 23.33 1.798 4.87 218.5 100
105689
Yellowish Brown Clay 177.47 140.75 79 38 20.69 1.98 4.64 43.6 18.4
Mean 16.72636 1.76 5.143 123.917 56.67
Holland Park
Sample No.
Soil Descriptn
Initial Weight
Weight After Oven Height Diameter
Moisture Content
Bulk Density
PH value Conductivity Salinity
gms gms mm mm % gm/cc �s ppM Rosco St HP 1 x 181.2 161.36 61 38 10.95 2.62 5.8 29.4 14 Greenmount St 106.18 96.92 50 38 8.72 1.87 6.3 26.59 11.1 Seville Rd HP 2 157.89 136.04 49 38 13.84 2.84 5.42 50.3 24.6 Mar St HP 3 161.97 146.2 58 38 9.74 2.46 6.95 110 56.5
Mean 10.81 2.45 6.12 54.07 26.55
140
Chelmer
Sample No.
Soil Description
Initial Weight
Weight After Oven Height Diameter
Moisture Content
Bulk Density
PH value Conductivity Salinity
gms gms mm mm % gm/cc � ppM 15483 Acacia St 157.36 138.16 48 38 12.20 2.8912 5.26 91.7 46.1 25360 Rosebery 105.58 98.44 40 38 6.76 2.32 6.95 45.1 21.9 25647 Glenwod St 166.03 146.03 53 38 12.04 2.76 6.61 46.1 22.5
6654 179.69 153.41 55 38 14.62 2.88 4.99 251 132 6599 161.97 146.2 58 38 9.73 2.46 6.95 110 56.5 7689 194.75 175.4 66 38 9.93 2.60 5.75 67.6 32.6
Mean 10.88 2.65 6.085 101.9167 51.933
141
Appendix B2
Soil Chemical Analysis Result SIROQUANT RESULTS Scan File : C:\D\DATA-XRD\3500\3546-3.cps 2-12-2002 Sample No. 25647 (Calibration Corrected) (Background Removed) Task File : C:\D\DATA-XRD\3500\3546-3.tsk Global Chi^2 : 4.05 Contrast Corrected Weight % PHASE WEIGHT% ERROR Quartz 26.6 % 0.35 Albite (high) 12.0 % 0.45 Albite (low) accurate study 13.8 % 0.44 Muscovite 9.4 % 0.47 Kaolin 26.3 % 0.65 Corundum 12.0 % 0.35 Spiked sample Amorphous Content 16.4 2.4 Original sample Amorphous Content 18.2 2.7 Quartz 24.7 0.3 Albite (high) 11.2 0.4 Albite (low) accurate study 12.8 0.4 Muscovite 8.7 0.4 Kaolin,1Md disordered,obs. (hkl) fi 24.4 0.6 SIROQUANT RESULTS Scan File : C:\D\DATA-XRD\3500\3546-4.cps 2-12-2002 Sample No 6599 (Calibration Corrected) (Background Removed) Task File : C:\D\DATA-XRD\3500\3546-4.tsk Global Chi^2 : 5.68 Contrast Corrected Weight % PHASE WEIGHT% ERROR Quartz 29.8 % 0.49 Albite (high) 25.8 % 0.85 Albite (low) accurate study 8.8 % 0.66 Muscovite 13.5 % 0.65 Kaolin 8.6 % 0.38 Corundum 13.5 % 0.45 Spiked sample Amorphous Content 25.7 2.5 Original sample Amorphous Content 28.5 2.7 Quartz 24.6 0.4 Albite (high) 21.3 0.7 Albite (low) accurate study 7.3 0.5 Muscovite 11.2 0.5 Kaolin,1Md disordered,obs. (hkl) fi 7.1 0.3
142
SIROQUANT RESULTS Scan File : C:\D\DATA-XRD\3500\3546-5.cps 2-12-2002 Sample No HP-1 (Calibration Corrected) (Background Removed) Task File : C:\D\DATA-XRD\3500\3546-5.tsk Global Chi^2 : 7.25 Contrast Corrected Weight % PHASE WEIGHT% ERROR Quartz 45.0 % 0.58 Kaolin 48.3 % 0.66 Corundum 6.7 % 0.45 SIROQUANT RESULTS Scan File : C:\D\DATA-XRD\3500\3546-6.cps 2-12-2002 (Calibration Corrected) (Background Removed) Task File : C:\D\DATA-XRD\3500\3546-6.tsk Global Chi^2 : 6.29 Contrast Corrected Weight % PHASE WEIGHT% ERROR Quartz 61.7 % 0.98 Kaolin 27.8 % 1.07 Corundum 10.6 % 0.55 Spiked sample Amorphous Content 5.3 4.9 Original sample Amorphous Content 5.9 5.5 Quartz 64.9 1.0 Kaolin,1Md disordered,obs. (hkl) fi 29.2 1.1 SIROQUANT RESULTS Scan File : C:\D\DATA-XRD\3500\3546-7.cps 2-12-2002 (Background Removed) Task File : C:\D\DATA-XRD\3500\3546-7.tsk Global Chi^2 : 4.96 Contrast Corrected Weight % PHASE WEIGHT% ERROR Quartz 34.8 % 0.41 Kaolin 40.9 % 0.64 Albite (low) accurate study 9.4 % 0.41 Corundum 15.0 % 0.37 Spiked sample Amorphous Content 33.4 1.6 Original sample Amorphous Content 37.1 1.8 Quartz 25.7 0.3 Kaolin,1Md disordered,obs. (hkl) fi 30.3 0.5 Albite (low) accurate study 6.9 0.3 SIROQUANT RESULTS Scan File : C:\D\DATA-XRD\3500\3546-8.cps 2-12-2002
143
(Calibration Corrected) (Background Removed) Task File : C:\D\DATA-XRD\3500\3546-8.tsk Global Chi^2 : 5.30 Contrast Corrected Weight % PHASE WEIGHT% ERROR Quartz 34.8 % 0.46 Kaolin 44.1 % 0.70 Albite (low) accurate study 8.2 % 0.41 Corundum 12.9 % 0.37 Spiked sample Amorphous Content 22.7 2.2 Original sample Amorphous Content 25.2 2.5 Quartz 29.9 0.4 Kaolin,1Md disordered,obs. (hkl) fi 37.9 0.6 Albite (low) accurate study 7.0 0.3 SIROQUANT RESULTS Scan File : C:\D\DATA-XRD\3500\3546-9.cps 2-12-2002 (Calibration Corrected) (Background Removed) Task File : C:\D\DATA-XRD\3500\3546-9.tsk Global Chi^2 : 6.70 Contrast Corrected Weight % PHASE WEIGHT% ERROR Quartz 60.8 % 0.93 Kaolin 29.1 % 1.02 Corundum 10.1 % 0.53 Spiked sample Amorphous Content 0.5 5.3 Original sample Amorphous Content 0.5 5.8 Quartz 67.3 1.0 Kaolin,1Md disordered,obs. (hkl) fi 32.2 1.1 SIROQUANT RESULTS Scan File : C:\D\DATA-XRD\3500\3546-10.cps 2-12-2002 Sample No 105689 (Calibration Corrected) (Background Removed) Task File : C:\D\DATA-XRD\3500\3546-10.tsk Global Chi^2 : 5.55 Contrast Corrected Weight % PHASE WEIGHT% ERROR Quartz 38.2 % 0.48 Kaolin 45.4 % 0.64 Albite (low) accurate study 6.1 % 0.44 Corundum 10.2 % 0.39 Spiked sample Amorphous Content 2.2 3.7 Original sample Amorphous Content 2.4 4.2 Quartz 41.5 0.5 Kaolin,1Md disordered,obs. (hkl) fi 49.4 0.7 Albite (low) accurate study 6.7 0.5
144
SIROQUANT RESULTS Scan File : C:\D\DATA-XRD\3500\3546-11.cps 2-12-2002 Sample No 80441 (Calibration Corrected) (Background Removed) Task File : C:\D\DATA-XRD\3500\3546-11.tsk Global Chi^2 : 6.02 Contrast Corrected Weight % PHASE WEIGHT% ERROR Quartz 41.5 % 0.53 Kaolin 38.5 % 0.72 Albite (low) accurate study 9.3 % 0.44 Corundum 10.7 % 0.41 Spiked sample Amorphous Content 6.6 3.5 Original sample Amorphous Content 7.3 4.0 Quartz 43.0 0.6 Kaolin,1Md disordered,obs. (hkl) fi 40.0 0.7 Albite (low) accurate study 9.6 0.5 SIROQUANT RESULTS Scan File : C:\D\DATA-XRD\3500\3546-12.cps 2-12-2002 Sample No 102795 (Calibration Corrected) (Background Removed) Task File : C:\D\DATA-XRD\3500\3546-12.tsk Global Chi^2 : 5.20 Contrast Corrected Weight % PHASE WEIGHT% ERROR Quartz 26.9 % 0.29 Kaolin 43.5 % 0.54 Albite (low) accurate study 18.6 % 0.35 Corundum 11.0 % 0.32 Spiked sample Amorphous Content 9.3 2.6 Original sample Amorphous Content 10.3 2.9 Quartz 27.1 0.3 Kaolin,1Md disordered,obs. (hkl) fi 43.8 0.5 Albite (low) accurate study 18.8 0.3 SIROQUANT RESULTS Scan File : C:\D\DATA-XRD\3500\3546-13.cps 2-12-2002 Sample No 96019 (Calibration Corrected) (Background Removed) Task File : C:\D\DATA-XRD\3500\3546-13.tsk Global Chi^2 : 6.68 Contrast Corrected Weight % PHASE WEIGHT% ERROR Quartz 50.6 % 0.78 Kaolin 29.5 % 0.96 Albite (low) accurate study 9.5 % 0.52
145
Corundum 10.5 % 0.49 Spiked sample Amorphous Content 4.4 4.4 Original sample Amorphous Content 4.9 4.9 Quartz 53.8 0.8 Kaolin,1Md disordered,obs. (hkl) fi 31.3 1.0 Albite (low) accurate study 10.1 0.6 SIROQUANT RESULTS Scan File : C:\D\DATA-XRD\3500\3546-14.cps 2-12-2002 Sample No. 97632 (Calibration Corrected) (Background Removed) Task File : C:\D\DATA-XRD\3500\3546-14.tsk Global Chi^2 : 6.57 Contrast Corrected Weight % PHASE WEIGHT% ERROR Quartz 57.9 % 0.93 Kaolin 20.0 % 1.12 Albite (low) accurate study 11.7 % 0.55 Corundum 10.4 % 0.51 Spiked sample Amorphous Content 3.4 4.8 Original sample Amorphous Content 3.8 5.3 Quartz 62.2 1.0 Kaolin,1Md disordered,obs. (hkl) fi 21.5 1.2 Albite (low) accurate study 12.6 0.6
146
APPENDIX C
MAPLE Programming for the Solution of the Model -1 Quality of maintenance Alpha = 0.1 Environmental parameter epsilon = 1 Cost of replacement, Cre = $2500; Cost of PM, Cpm= $200 and Cost of minimal repair Cmr=$100 with(linalg): > x:='x':RealN:='RealN':RealXmin:='RealXmin':m:='m': > RealCmin:='RealCmin':a:='a':C:='C':N:='N': > cmr:=100.0: al:=0.1:cpm:=200:cre:=2500:be:=4.39: nu:=20.54: ep:=1.0: > a:=(sum((k-k*al+1)^be-(k-k*al)^be,k=0..N));
> m:=60:xmin:=array(1..m):Cmin=array(1..m): > for N from 1 to m do > C:=(1/N/x)*(cmr*(x^be/nu^be)*a+(N-1)*cpm+cre); > y1:=diff(C,x);xmin[N]:=fsolve(y1=0,x,0.1..m); > Cmin[N]:=subs( x=xmin[N], C ); > if xmin[N]<m/N then > Cmin[N]:=subs( x=xmin[N], C ); > else > xmin[N]:=m/N;Cmin[N]:=subs( x=xmin[N], C );fi: > od: > > RealCmin:=Cmin[1]:RealXmin:=xmin[1]: > for N from 1 to m do > if Cmin[N]<RealCmin then > RealCmin:=Cmin[N]: > RealXmin:=xmin[N]: > RealN:=N: > fi: > od: > [RealN,RealXmin,RealCmin]; [4, 7.287000811, 137.7265620] > >
:= a ∑ = k 0
N
− ( ) + .9000000000 k 1.
439100
.6296868392 k
439100
147
APPENDIX C
MAPLE Programming for the Solution of the Model-2 Quality of maintenance Alpha = 0.8 Environmental parameter epsilon = 0.1 Cost of replacement, Cre = $2500; Cost of PM, Cpm= $600 and Cost of minimal repair Cmr=$100 > with(linalg): Warning, new definition for norm Warning, new definition for trace > x:='x':RealN:='RealN':RealXmin:='RealXmin':m:='m': > RealCmin:='RealCmin':a:='a':C:='C':N:='N': > cmr:=100.0: al:=0.8:cpm:=600:cre:=2500:be:=3.96: nu:=32.51: ep:=0.1: > a:=(sum((k-k*al+1)^be-(k-k*al)^be,k=0..N));
> m:=60:xmin:=array(1..m):Cmin=array(1..m): > for N from 1 to m do > C:=(1/N/x)*(cmr*(x^be/nu^be)*a+(N-1)*cpm+cre); > y1:=diff(C,x);xmin[N]:=fsolve(y1=0,x,0.1..m); > Cmin[N]:=subs( x=xmin[N], C ); > if xmin[N]<m/N then > Cmin[N]:=subs( x=xmin[N], C ); > else > xmin[N]:=m/N;Cmin[N]:=subs( x=xmin[N], C );fi: > od: > > RealCmin:=Cmin[1]:RealXmin:=xmin[1]: > for N from 1 to m do > if Cmin[N]<RealCmin then > RealCmin:=Cmin[N]: > RealXmin:=xmin[N]: > RealN:=N: > fi: > od: > [RealN,RealXmin,RealCmin]; [2, 30, 59.93615750] >
:= a ∑ = k 0
N
− ( ) + .2000000000 k 1.
9925
.001706391908 k
9925
148
APPENDIX C
MAPLE Programming for the Solution of the Model -3 Quality of maintenance Alpha = 0.8 Environmental parameter epsilon = 0.2 Durability modification factor kd = 0.95 Cost of replacement, Cre = $2500; Cost of PM, Cpm= $200 and Cost of minimal repair Cmr=$100 > with(linalg): Warning, new definition for norm Warning, new definition for trace > x:='x':RealN:='RealN':RealXmin:='RealXmin':m:='m': > RealCmin:='RealCmin':a:='a':C:='C':N:='N': > cmr:=100.0: al:=0.8:cpm:=600:cre:=2500:be:=4.1: nu:=28.94: s:=0.95: > a:=(1/kd)*(sum((k-k*al+1)^be-(k-k*al)^be,k=0..N));
> m:=60:xmin:=array(1..m):Cmin=array(1..m): > for N from 1 to m do > C:=(1/N/x)*(cmr*(x^be/nu^be)*a+(N-1)*cpm+cre); > y1:=diff(C,x);xmin[N]:=fsolve(y1=0,x,0.1..m); > Cmin[N]:=subs( x=xmin[N], C ); > if xmin[N]<m/N then > Cmin[N]:=subs( x=xmin[N], C ); > else > xmin[N]:=m/N;Cmin[N]:=subs( x=xmin[N], C );fi: > od: > > RealCmin:=Cmin[1]:RealXmin:=xmin[1]: > for N from 1 to m do > if Cmin[N]<RealCmin then > RealCmin:=Cmin[N]: > RealXmin:=xmin[N]: > RealN:=N: > fi: > od: > [RealN,RealXmin,RealCmin]; [2, 30, 66.02118435] >
:= a 1.052631579
∑ = k 0
N
− ( ) + .2000000000 k 1.
4110
.001362143876 k
4110
149
Appendix D
Result of various scenarios by changing the parameters
Cre = $2500, Cpm= $500, Cmr = $100, α = 0.6 and kd = 0.95
ε 0.1 0.2 0.4 0.6 0.8 1.0
β* 3.96 4.1 4.24 4.31 4.36 4.39
η* 32.51 28.95 25.03 22.94 21.56 20.54
N* 3 3 4 4 4 4
x 20.31 20.10 15.83 15.50 15.30 14.31
C*(N, x) 66.90 72.67 84.37 85.52 86.09 90.45
Cre = $2500, Cpm= $300, Cmr = $100, α = 0.3 and kd = 0.95
ε 0.1 0.2 0.4 0.6 0.8 1.0
β* 3.96 4.1 4.24 4.31 4.36 4.39
η* 32.51 28.95 25.03 22.94 21.56 20.54
N* 4 4 4 4 4 4
x 15.00 12.99 10.92 9.88 9.20 8.73
C*(N, x) 75.77 86.63 101.86 112.00 119.92 126.15
Cre = $2500, Cpm= $200, Cmr = $100, α = 0.1 and kd = 0.95
ε 0.1 0.2 0.4 0.6 0.8 1.0
β* 3.96 4.1 4.24 4.31 4.36 4.39
η* 32.51 28.95 25.03 22.94 21.56 20.54
N* 4 4 4 4 4 4
x 12.56 10.86 9.13 8.26 7.68 7.28
C*(N, x) 82.44 94.37 111.11 122.77 130.77 137.72
150