Handbook of Reliability, Availability, Maintainability and Safety in Engineering Design - Part 8 doc

3.2 Theoretical Overview of Reliability and Performance in Engineering Design 53
Fig. 3.6 Power train system reliability of a haul truck (Komatsu Corp., Japan)
Fig. 3.7 Power train system diagram of a haul truck
54 3 Reliability and Performance in Engineering Design
Table 3.2 Power train system reliability of a haul truck
Output shaft assembly Transmission sub-system Power train system
No. of components 5 50 100
Group reliability 0.99995 0.99950 0.99900
Output shaft assembly reliability =(0.99999)
5
= 0.99995
Transmission sub-system reliability =(0.99999)
50
= 0.99950
Power train system reliability =(0.99999)
100
= 0.99900
components are considered to have the same reliability of 0.99999. The reliability
calculations are given in Table 3.2.
The series formula of reliability implies that the reliability of a group of series
components is the product of the reliabilities of the individual components. If the
output shaft assembly had five components in series, then the output shaft assem-
bly reliability would be five times the product of 0.99999 = 0.99995. If the torque
converter and transmission assemblies had a total of 50 different components, be-
longing to both assemblies all in series, then this sub-system reliability would be
50 times the product of 0.99999 = 0.99950. If the power train system had a total of
100 different components, belonging to different assemblies, some of which belong
to different sub-systems all in series, then the power train system’s reliability would
be a 100 times the product of 0.99999 = 0.99900.
The value of a component reliability of 0.99999 implies that out of 100,000
events, 99,999 successes can be expected. This is somewhat cumbersome to en-

visage and, therefore, it is actually more convenient to illustrate reliability through
its converse, unreliability. This unreliability is basically defined as
Unreliability = 1−Reliability .
Thus, if component reliability is 0.99999, the unreliability is 0.00001. Th is implies
that only one failure out of a total of 100,000 events can be expected. In the case of
the haul truck, an event is when the component is used under gearshift load stress
every haul cycle. I f a haul cycle was an average o f 15 min, then this would imply
that a power train component would fail about every 25,000 operational hours. The
output shaft assembly reliability of 0.99995 implies that only five failures out of
a total of 100,000 events can be expected, or one failure every 20,000 events (i.e.
haul cycles). (This means one assembly failure every 20,000 haul cycles, or every
5,000 operational hours.) A sub-system (power converter and transmission) relia-
bility of 0.99950 implies that 50 failures can be expected out of a total of 100,000
events (i.e. haul cycles). (This means one sub-system failure every 2 ,000 haul cy-
cles, or every 500 operational hours.) Finally, the power train system reliability of
0.99900 implies that 100 failures can be expected out of a total of 100,000 events
(i.e. haul shifts). (This means one system failure every 1,000 haul cycles, or every
250 operational hours!) Note how the reliability decreases from a component reli-
ability of only one failure in 100,000 events, or every 25,000 operational hours, to
the eventual system reliability, which has 100 components in series, with 100 fail-
3.2 Theoretical Overview of Reliability and Performance in Engineering Design 55
Single component reliability
1.00
0
0.2
0.4
0.6
0.8
1
1.2

0.98 0.96 0.94 0.92 0.9 0.88 0.86
N = 10
N = 20
N = 50
N = 100
N = 300
Reliability of N series components
Fig. 3.8 Reliability of groups of series components
ures occurring in a total of 100,000 events, or an average of one failure every 1,000
events, or every 250 operational hours.
This decrease in system reliability is even more pronouncedfor lower component
reliabilities. For example, with identical component relia bilities of 0.90 (in other
words, one expected failure out of ten events), the reliability of the power train
system with 100 components in series would be practically zero!
R
System
=(0.90)
100
≈ 0 .
The following Fig. 3.8 is a graphical portrayal of how the reliability of groups of
series componentschangesfor different valuesof individualcomponentreliabilities,
where the reliability of each component is identical. This graph illustrates how close
to the reliability value of 1 (almost 0 failures) a component’s reliability would have
to be in order to achieve high group reliability, when there are increasingly more
components in the group.
The effect o f redundancy in system reliability When very high system reliabili-
ties are required, the designer or m anufacturer must often duplicate components or
assemblies, and sometimes even whole sub-systems, to meet the overall system or
equipment reliability goals. I n systems or equipment such as these, the components
are said to be redundant, or in parallel.

Just as the reliability of a group of series components decreases as the number of
components increases, so the opposite is true for redundant or parallel components.
Redundant components can dramatically increase the reliability of a system. How-
ever, this increase in reliability is at the expense o f factors such as weight, space,
and manufacturing and maintenance costs. When redundant components are being
analysed, the term unreliability is preferably used. This is because the calculations
56 3 Reliability and Performance in Engineering Design
Component No.1
Reliability R1 = 0.90
Component No.2
Reliability R2 = 0.85
Fig. 3.9 Example of two parallel components
are easier to perform using the unreliability of a component. As a specific example,
consider the two parallel components illustrated below in Fig. 3.9, with reliabilities
of 0.9 and 0.85 respectively
Unreliability: U =(1−R1)×(1−R2)
=(0.1) ×(0.15)
= 0.015
Reliability of group: R = 1 −Unreliability
= 1−0.015
= 0.985.
With the individual component reliabilities of only 0.9 (i.e. ten failures out of
100 events), and of 0.85 (i.e. 15 failures out of 100 events), the overall system re-
liability of these two components in parallel is increased to 0.985 (or 1 5 failures
in 1,000 events). The improvement in reliability achieved by components in paral-
lel can be further illustrated by referring to the graphic por trayal below (Fig. 3.10 ).
These curves show how the reliability of groups of parallel components changes for
different values of individual component reliabilities.
From these graphs it is obvious that a significant increase in system reliability is
obtained from redundancy.

To cite a few examples from these graphs, if the reliability of one component
is 0.9, then the reliability of two such components in parallel is 0.99. The reliability
of three such components in parallel is 0.999. This means that, on average, only one
system failure can be expected to occur out of a total of 1,000 events. Put in more
correct terms, only one time out of a thousand will all three components fail in their
function, and thus result in system functional failure.
Consider now an example of series and parallel assemblies in an engineered in-
stallation, such as the slurry mill illustra ted below in Fig. 3.11. The system is shown
with some major sub-systems. Table 3.3 gives reliability values for some of the
critical assemblies and components. Consider the overall reliability of these sub-
3.2 Theoretical Overview of Reliability and Performance in Engineering Design 57
0
0.2
0.4
0.6
0.8
1
10.90.80.70.60.50.40.30.20.10.00
1.2
N = 5
N = 3
N = 2
Single component reliability
Reliability of N parallel components
Fig. 3.10 Reliability of groups of parallel components
Fig. 3.11 Slurry mill engineered installation
58 3 Reliability and Performance in Engineering Design
Table 3.3 Component and assembly reliabilities and system reliability
of slurry mill engineered installation
Components Reliability

Mill trunnion
Slurrying mill trunnion shell 0.980
Trunnion dri ve gears 0.975
Trunnion dri ve gears lube (×2 units) 0.975
Mill drive
Drive motor 0.980
Drive gearbox 0.980
Drive gearbox lube 0.975
Driv e gearbox heat exchanger (×2 units) 0.980
Slurry feed and screen
Classification feed hopper 0.975
Feed hopper feeder 0.980
Feed hopper feeder motor 0.980
Classification screen 0.950
Distribution pumps
Classification underflow pumps (×2 units) 0.980
Underflow pumps motors 0.980
Rejects handling
Rejects con veyor feed chute 0. 975
Rejects conve yor 0.950
Rejects conveyor drive 0.980
Sub-systems/assemblies
Slurry mill trunnion 0.955
Slurry mill drive 0.935
Classification 0.890
Slurry distri bution 0.979
Rejects handling 0.908
Slurry mill system
Slurry mill 0.706
systems once all of the parallel assemblies and components have been reduced to

a series configuration, similar to Figs. 3.4 and 3.5.
Some of the major sub-systems, together with their major components, are the
slurry mill trunnion, the slurry mill drive, classification, slurry distribution, and re-
jects handling.
The systems hierarchy of the slurry mill first needs to be identified in a top-level
systems–assembly configuration, and acco rdingly is simply structured for illustra-
tion purposes:
3.2 Theoretical Overview of Reliability and Performance in Engineering Design 59
Systems Assemblies
Milling Slurry mill trunnion
Slurry mill drive
Classification Slurry feed
Slurry screen
Distribution Slurry distribution pumps
Rejects handling
Slurry mill trunnion:
Trunnion shell×Trunnion drive gears×Gears lube (2 units)
=(0.980×0.975) ×[(0.975+ 0.975) −(0.975 ×0.975)]
=(0.980×0.975 ×0.999)
= 0.955 ,
Slurry mill drive:
Motor×Gearbox×Gearbox lube×Heat exchangers (2 units)
=(0.980×0.980 ×0.975) ×[(0.980+ 0.980) −(0.980×0.980)]
=(0.980×0.980 ×0.975×0.999)
= 0.935 ,
Classification:
Feed hopper×Feeder×Feeder motor×Classification screen
=(0.975×0.980 ×0.980×0.950)
= 0.890 ,
Slurry distribution:

Underflow pumps (2 units)×Underflow pumps motors
=[(0.980 + 0.980) −(0.980×0.980)] ×0.980
=(0.999×0.980)
= 0.979 ,
Rejects handling:
Feed chute×Rejects conveyor×Rejects conveyor drive
=(0.975×0.950 ×0.980)
= 0.908 ,
Slurry mill system:
=(0.955×0.935 ×0.890×0.979×0.908)
= 0.706 .
60 3 Reliability and Performance in Engineering Design
The slurry mill system reliability of 0.706 implies that 294 failures out of a total
of 1,000 events (i.e. mill charges) can be expected. If a mill charge is estimated to
last for 3 .5 h, this would mean one system failure every 3.4 charges, or about every
12 operational hours!
The staggering frequency of one expected failure every operational shift of 12 h,
irrespective of the relatively high reliabilities of the system’s components, has a sig-
nificant impact on the approachto systems design for integrity (reliability, availabil-
ity and maintainability), as well as on a proposed maintenance strategy.
3.2.1 Theoretical Overview of Reliability and Performance
Prediction in Conceptual Design
Reliability and performance p rediction attempts to estimate the probability of suc-
cessful performance of systems. Reliability and perfor mance prediction in this con-
text is considered in the conceptual design phase of the engineering design process.
The most applicable methodology for reliability and performance prediction in the
conceptual desig n phase in cludes basic concepts of mathematical modelling such
as:
• Total cost models for design reliability.
• Interference theory and reliability modelling.

• System reliability modelling based on system per formance.
3.2.1.1 Total Cost Models for Design Reliability
In a paper titled ‘Safety and risk’ (Wolfram 1993), reliability and risk prediction is
considered in determining the total potential cost of an engineeringproject. With in-
creased design reliability (including strength and safety), project costs can incr ease
exponentially to some cut-off point. The tendency would thus be to achieve an ‘ac-
ceptable’ design at the least cost possible.
a) Risk Cost Estimation
The total potential cost of an engineering project compared to its design reliability,
whereby a minimumcost point designatedthe economic optimum reliability is deter-
mined, is illustrated in Fig. 3.12. Curve ACB is the normal ‘first cost curve’, which
includes capital costs plus operating and maintenance costs. With the inclusion of
the ‘risk cost curve’ (CD), the effect on total project cost is reflected as a concave or
parabolic curve. Thus, designs of low reliab ility are not worth consideration because
the risk cost is too high.
3.2 Theoretical Overview of Reliability and Performance in Engineering Design 61
C
B
First cost curve
Apparent economic
optimum reliability
Risk cost curve
(Capital costs plus operating and maintenance costs)
Increased risk of failure Strength, safety and reliability
First
cost
Risk
cost
D
A

C
O
S
T
DESIGN RELIABILITY
*
Fig. 3.12 Total cost versus design reliability
The difference between the ‘risk cost curve’ and the ‘first cost curve’ in Fig. 3.12
designates this risk cost, which is a function o f the pr obability and consequences of
systems failure on the project.
Thus, the risk cost can be formulated as
Risk cost = Probability of failure×Consequence of failure.
This probability and consequence of systems failure is related to process reliability
and criticality at the higher systems levels (i.e. process and system level) that is
established in the design’s systems hierarchy,or systems b reakdown structure (SBS).
According to Wolfram, there would thus appear to be an economically optimum
level of process r eliability (and safety). However, this is misleading, as the predic-
tion of processreliability and the inherentprobabilityof failure do not reflect reality
precisely, and the extent of the error involved is uncertain. In the face of this un-
certainty, there is the tendency either to be conservative and move towards higher
predicted levels of design reliability, or to rely on previous designs where the in-
dividual process systems on their own were adequately designed and constructed.
In th e first case, this is the same as selecting larger safety factors when there is
ignorance about how a system or structure will behave. In the latter case, the combi-
nation and integration of many previously designed systems inevitably give rise to
design complexity and consequent frequent failure, where high risks of the integrity
of the design are encountered.
Consequently, there is a need to develop good design models that can reflect re-
ality as closely a s possible. Furthermore, Prof. Wolfram contends that these design
models need not attempt to explain wide-ranging phenomena, just the criteria rele-

vant to the design. However, the fact that engineering design should be more precise
62 3 Reliability and Performance in Engineering Design
close to those areas where failure is more likely to occur is overlooked by most de-
sign engineers in the early stages of the design process. The questions to be asked
then are: which areas are more likely to incur failure, and what would the probabil-
ity of that likelihood be? The penalty for this uncertainty is a substantial increase in
first costs if the project economics are feasible, or a high risk in the consequential
risk costs.
b) Project Cost Estimation
Nearly every engineering design project will include some form of first cost estimat-
ing. This initial cost estimating may be performed by specific engineeringpersonnel
or by separate cost estimators. Occasionally, other resources, such as vendors, will
be required to assist in first cost estimating. The engineering d esign project manager
determines the need for cost estimatin g services and making arrangements for the
appropriate services at the ap propriate times. Ordinarily, cost estimating services
should be obtained from cost estimators employed by the design engineer. First cost
estimating is normally done as early as possible, when planning and scheduling the
project, as well as finalising the estimating approach and nature of engineering input
to be used as the basis for the cost estimate.
Typesoffirstcostestimates First cost estimates consist basically of investment or
capital costs, operating costs, and maintenance costs. These types of estimates can
be evaluated in a number of ways to suit the needs of the project:
• Discounted cash flow (DCF)
• Return on investment (ROI)
• Internal rate of return (IRR)
• Sensitivity evaluations
Levels of cost estimates The most important consideration in planning cost esti-
mating tasks is the establishment of a clear understanding as to the required level or
accuracy of the cost estimate.
Basically, each level of the engineering design process has a corresponding level

of cost estimating, whereby first cost estimations are usually performed during the
conceptual and preliminary design phases. The following cost estimate accuracies
for each engineering design phase are considered typical:
• Conceptual design phase: plus or minus 30%
• Preliminary design phase: plus or minus 20%
• Final detail design phase: plus or minus 10%
The percentages imply that the estimate will be above or below the final construc-
tion costs of the engineered installation, by that amount. Conceptual or first cost
estimates are generally used for project feasibility, initial cash flow, and funding
purposes by the client. Preliminary estimates that inclu de risk costs are used for
‘go-no-go’ decisions by the client. Final estimates are used for control purposes
during procurement and construction of the final design.