30079_20.pdf

CHAPTER 20

RELIABILITY IN MECHANICAL DESIGN

B. S. Dhillon

Department of Mechanical Engineering

University of Ottawa

Ottawa, Ontario, Canada

20.1 INTRODUCTION

487

20.5.3 Failure Rate Modeling and

Parts Count Method

496

20.2 BASICRELIABILITY

NETWORKS

20.5.4 Stress-Strength Interference

Theory Approach 497

20.5.5 Network Reduction Method 498

20.5.6 Markov Modeling

488

20.2.1 Series Network

488

20.2.2 Parallel Network

488

498

20.5.7 Safety Factors

500

20.2.3 k-out-of-n Unit Network

489

20.2.4 Standby System

490

20.6 DESIGNLIFE-CYCLE

COSTING

20.3 MECHANICALFAILURE

MODES AND CAUSES

501

491

20.7 RISKASSESSMENT 501

20.7.1 Risk-Analysis Process and Its

Application Benefits

20.4 RELIABILITY-BASED DESIGN

491

502

20.7.2 Risk Analysis Techniques

502

20.5 DESIGN-RELIABILITYTOOLS 492

20.5.1 Failure Modes and Effects

Analysis (FMEA)

492

20.8 FAILUREDATA

504

20.5.2 Fault Tree

494

20.1 INTRODUCTION

The history of the application of probability concepts to electric power systems goes back to the

1930s. 1 " 6 However, the beginning of the reliability field is generally regarded as World War II, when

Germans applied basic reliability concept to improve reliability of their Vl and V2 rockets.

During the period from 1945-1950 the U.S. Army, Navy, and Air Force conducted various studies

that revealed a definite need to improve equipment reliability. As a result of this effort, the Department

of Defense, in 1950, established an ad hoc committee on reliability. In 1952, this committee was

transformed to a group called the Advisory Group on the Reliability of Electronic Equipment

(AGREE). In 1957, this group's report, known as the AGREE Report, was published, and it subse-

quently led to a specification on the reliability of military electronic equipment.

The first issue of a journal on reliability appeared in 1952, published by the Institute of Electrical

and Electronic Engineers (IEEE). The first symposium on reliability and quality control was held in

1954. Since those days, the field of reliability has developed into many specialized areas: mechanical

reliability, software reliability, power system reliability, and so on. Most of the published literature

on the field is listed in Refs. 7, 8.

The history of mechanical reliability in particular goes back to 1951, when W. Weibull 9 developed

a statistical distribution, now known as the Weibull distribution, for material strength and life length.

The work of A. M. Freudenthal 10 ' 1 1 in the 1950s is also regarded as an important milestone in the

history of mechanical reliability.

The efforts of the National Aeronautics and Space Administration (NASA) in the early 1960s

also played a pivotal role in the development of the mechanical reliability field, 1 2 due primarily to

two factors: the loss of Syncom I in space in 1963, due to a bursting high-pressure gas tank, and the

loss of Mariner III in 1964, due to mechanical failure. Many projects concerning mechanical relia-

Mechanical Engineers' Handbook, 2nd ed., Edited by Myer Kutz.

bility were initiated and completed by NASA. A comprehensive list of publications on mechanical

reliability is given in Ref. 13.

20.2 BASIC RELIABILITY NETWORKS

A system component may form various different configurations: series, parallel, fc-out-of-n, standby,

and so on. In the published reliability literature, these configurations are known as the standard

configurations. During the mechanical design process, it might be desirable to evaluate the reliability

or the values of other related parameters of systems forming such configurations. These networks are

described in the following pages.

20.2.1 Series Network

The block diagram of an "n" unit series network is shown in Fig. 20.1. Each block represents a

system unit or component. If any one of the components fails, the system fails; thus, all of the series

units must work successfully for the system to succeed.

For independent units, the reliability of the network shown in Fig. 20.1 is

R s = R 1 R 2 R 3 --R n

(20.1)

where R s = the series system reliability

n = the number of units

Ri = the reliability of unit i; for i = 1, 2, 3, • • • , n

For units' constant failure rates, Eq. (20.1) becomes 1 4

R,(t) = e~^ . e~^ . e~^ - - - e~^

(20.2)

g-jS

A,/

where R s (t) = the series system reliability at time t

A 1 = the unit i constant failure rate, for / = 1, 2, 3, • • • , n

The system hazard rate or the total failure rate is given by 1 4

**>-<jr3M*

where A 5 (O = the series system total failure rate or the hazard rate

Note that the series system failure rate is the sum of the unit failure rates. In mechanical or in other

design analysis, when the failure rates are added, it is automatically assumed that the units are acting

in series. This is the worst-case design assumption—if any one unit fails, the system fails. In engi-

neering design specifications, the adding up of all system component failure rates is often specified.

The system mean time to failure is expressed by 1 3

MTTF 3 = lim R s (s) = -^-

(20.4)

E A,

1=1

where MTTF 5 = the series system mean time to failure

s (in brackets) = the Laplace transform variable

R s (s) = the Laplace transform of the series system reliability

20.2.2 Parallel Network

The block diagram of an "n" unit parallel network is shown in Fig. 20.2. As in the case of the series

network, each block represents a system unit or component. All of the system units are assumed to

Fig. 20.1 Block diagram representing a series system.

Fig. 20.2 Parallel network block diagram.

be active and at least one unit must function normally for the system to succeed, meaning that this

type of configuration may be used to improve a mechanical system's reliability during the design

phase.

For independent units, the reliability of the parallel network shown in Fig. 20.2 is given by 1 3

R p =l-(l- R 1 )(I - R 2 ) - - • (1 - R n )

(20.5)

where R p = the parallel network reliability

For constant failure rates of the units, Eq. (20.5) becomes

R p (t) = 1 - (1 - <T A ")(1 - e~^} •••(!- <T A «0

(20.6)

where R p (i) = the parallel network reliability at time t

Obviously, Eqs. (20.5) and (20.6) indicate that system reliability increases with the increasing values

of n.

For identical units, the system mean time to failure is given by 1 4

MTTF^ - lim R p (s) = 7 S T

(20.7)

5-0

A /=i i

where MTTF p = the parallel network mean time to failure

R p (s) = the Laplace transform of the parallel network reliability

A = the constant failure rate of a unit

20.2.3 fr-out-of-n Unit Network

This arrangement is basically a parallel network with a condition that at least k units out of the total

of n units must function normally for the system to succeed. This network is sometimes referred to

as partially redundant network. An example might be a Jumbo 747. If a condition is imposed that at

least three out of four of its engines must operate normally for the aircraft to fly successfully, then

this system becomes a special case of the k-out-of-n unit network. Thus, in this case, k = 3 and

n = 4.

For independent and identical units, the k-out-of-n unit network reliability is 1 4

R** = 2 m #(i - Rr- 1

(20.8)

i=* w

where

M = _o!_

\ij

i!(/i-i)!

R = the unit reliability

R Un = the k-out-of-n unit network reliability

Note that at k — 1, the k-out-of-n unit network reduces to a parallel network and at k = n, it becomes

a series system.

For constant unit failure rates, Eq. (20.8) is rewritten to the following form: 1 3

RvM = S ( n } e~ ixt (1 - e-*T-'

(20.9)

«•=* Vv

where R^M = is the k-out-of-n unit network reliability at time t

The system mean time to failure is given by 1 3

MTTF^ = Hm R^(S) = 7 Z T

(20.10)

5-»o

A i=k I

Rk/ n ( s ) = m e Laplace transform of the k-out-of-n unit network reliability.

20.2.4 Standby System

The block diagram of an (n + 1) unit standby system is shown in Fig. 20.3. Each block represents

a unit or a component of the system. In the standby system case, as shown in Fig. 20.3, one unit

operates and n units are kept on standby.

During the mechanical design process, this type of redundancy is sometimes adopted to improve

system reliability.

If we assume independent and identical units, perfect switching, and standby units as good as

new, then the standby system reliability is given by 1 4

* (ct V /

RM = E 1 A(f)<fry e-&*o*/n

where MTTF^ n = the mean time to failure of the k-out-of-n unit network

(20.11)

^o |>

where R ss (t) = the standby system reliability at time t

n = the number of standbys

A(O = the unit hazard rate or time-dependent failure rate

For two non-identical units (i.e., one operating, the other on standby), the system reliability is

expressed by 1 5

RJt) = RM + \* fodiWJit - t,) Jt 1

(20.12)

where R 0 (t) = the operating unit reliability at time t

R 5 M = the standby unit reliability at time t

/ 0 (*i) = m e operating unit failure density function

For known reliability of the switching mechanism, Eq. (20.12) is modified to

R u (t) = RM + R^ P/ 0 ('i)*»(f - *i) ^i

(20.13)

where R sw = the reliability of the switching mechanism

Fig. 20.3 An (n + 1) unit standby system block diagram.

For identical units and constant unit failure rates, Eq. (20.13) simplifies to

R ss (t) = e~ xt (l + R sw Xt)

(20.14)

where A = the unit constant failure rate

20.3 MECHANICAL FAILURE MODES AND CAUSES

There are certain failure modes and causes associated with mechanical products. The proper identi-

fication of relevant failure modes and their causes during the design process would certainly help to

improve the reliability of design under consideration.

Mechanical and structural parts function adequately within specific useful lives. Beyond those

lives, they cannot be used for effective mission, safe mission, and so on. A mechanical failure may

be defined as any change in the shape, size, or material properties of a structure, piece of equipment,

or equipment part that renders it unfit to perform its specified mission satisfactorily. 1 3 One of the

factors for the failure of a mechanical part is the specified magnitude and type of load. The basic

types of loads are dynamic, cyclic, and static. There are many types of failures that result from

different types of loads: tearing, spalling, buckling, abrading, wear, crushing, fracture, and creep. 1 6

In fact, there are many different modes of mechanical failures. 1 7

• Brinelling

• Thermal shock

• Ductile rupture

• Fatigue

• Creep

• Corrosion

• Fretting

• Stress rupture

• Brittle fracture

• Radiation damage

• Galling and seizure

• Thermal relaxation

• Temperature-induced elastic deformation

• Force-induced elastic deformation

• Impact

Field experience has shown that there are various causes of mechanical failures, including 1 8 de-

fective design, wear-out, manufacturing defects, incorrect installation, gradual deterioration in per-

formance, and failure of other parts.

Some of the important failure modes and their associated characteristics are presented below. 1 9

• Creep. This may be described as the steady flow of metal under a sustained load. The cause

of a failure is the continuing creep deformation in situations when either a rupture occurs or

a limiting acceptable level of distortion is exceeded.

• Corrosion. This may be described as the degradation of metal surfaces under service or storage

conditions because of direct chemical or electrochemical reaction with its environment. Usu-

ally, stress accelerates the corrosion damage. In hydrogen embrittlement, the metal ductility

increases due to hydrogen absorption, leading either to fracture or to brittle failure under

impact loads at high-strain rates or under static loads at low-strain rates, respectively.

• Static failure. Many of the materials fail by fracture due to the application of static loads

beyond the ultimate strength.

• Wear. This occurs in contacts such as sliding, rolling, or impact, due to gradual destruction

of a metal surface through contact with another metal or non-metal surface.

• Fatigue failure. In the presence of cyclic loads, materials can fail by fracture even when the

maximum cyclic stress magnitude is well below the yield strength.

20.4 RELIABILITY-BASED DESIGN

It would be unwise to expect a system to perform to a desired level of reliability unless it is specif-

ically designed for that reliability. The specification of desired system/equipment/part reliability in

the design specification due to factors such as well-publicized failures (e.g., the space shuttle Chal-

lenger disaster and the Chernobyl nuclear accident) has increased the importance of reliability-based

design. The starting point for the reliability-based design is during the writing of the design

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