Introduction to Engineering Reliability engineering Reliability Concepts

8/12/2019 Introduction to Engineering Reliability engineering Reliability Concepts

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Delivering Integrated, Sustainable,

Water Resources Solutions

Introduction to Engineering

Robert C. Patev

North Atlantic Division Re ional Technical

Specialist

(978) 318-8394


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Topics

Reliability

Non-Probabilistic Methods

Probabilistic Methods First Order Second Moment

Point Estimate Method

Response Surface Modeling


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intended function for a specific period of time

under a given set of conditions

R = 1 - Pf

Reliability is the probability that unsatisfactory


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ReliabilityReliability

Reliabili ty and Probabality of Failu re

0.6

0.8

1

(t)

0.2

0.4R(t),

Probabality of Failure

1 2 3 4 5 6

Tim e, t (years )


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Probability of Failure, Pf

Easily defined for recurring events andreplicate components (e.g., mechanical andmechanical parts)

Probability of Unsatisfactory Performance,P(u) Pup

Nearly impossible to define for non-recurring. .,

gravity structures)


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Reliabilit

pdf

value

f or up


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Reliability

Demand Capacitypdf pdf

Demand Capacity

d c d c

value value

f up


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Reliability

Safety Margins

If limit at SM = C-D = 0

If limit at SM = C/D = 1

SM

SM = Safety Margin

Pup

0 or 1


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Basic Principles of Reliability Analysis

Identify critical components Use available data from previous design and analysis

Define performance modes in terms of past levels of

unsatisfactory performance Calibrate models to experience

Model reasonable maintenance and repair scenarios andalternatives


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Non-Probabilistic Reliability Methods

Historical Frequency of Occurrence

Expert Opinion Elicitation


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Reliability Index Methods

First Order Second Moment (Taylor Series)

Advanced Second Moment (Hasofer-Lind)

Point Estimate Method-

Monte Carlo Simulation


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Use of known historical information for records at site to estimate

For example, if you had 5 hydraulic pumps in standby mode andeac ran or ours n s an y an a e ur ng s an y.The failure rate during standby mode is:

ota stan y ours = ours = , oursFailure rate in standby mode = 3 / 10,000

= 0.0003 failures per hour


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Curves are available from manufacturers for differentmotors, pumps, electrical components, etc...

Curves are developed from field data and failed

components Failure data may have to be censored

However, usually this data id not readily available for

equipment at Corps projects except mainlyhydropower facilities

curve as well as many textbooks on the subject


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Solicitation of experts to assist in determiningpro a es o unsa s ac ory per ormance or ra es ooccurrence.

Need ro er uidance and assistance to solicit andtrain the experts properly to remove all bias anddominance.

ou e ocumen e we or Some recent projects that used EOE


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Probabilistic Methods

Reliability models are: defined by random variables and their underlying

based on limit states (analytical equations) similar

to those use in the design of engineeringcomponents

based on capacity/demand or factor of safetyrelationshi s

One method is the Reliability Index or method


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Reliability

Method - Normal Distribution

E(SM) / (SM)

if limit at SM = C-D = 0 SM)

=

if limit at SM = C/D = 1

SM = Safety Margin

Pup

0 or 1


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Taylor Series Finite Differenceorne , an osen ue ,

First-order expansion about mean value

Linearapproximation of second moment , , ,

Easy to implement in spreadsheets

Requires 2n+1 sampling (n = number of variables)

Results in a Reliability Index value ( )

Based on E(SM) and (SM) Problem: caution should be exercised on non-linear limit states


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ay or er es n e erence

Variable C

C C

C

FS = 0B B

C CFai

Safe

Variable BB


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Reliability Example

Determine the reliability of a tension bar

Ft A P

Limit State = Ft A / P


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Reliability Example

Random Variables

Ultimate tensile strength, Ft

mean, = 40 ksi; standard deviation, = 4 ksi assume norma s r u on

Load, P

mean, = 15 kips; standard deviation, = 3 kips,

assume normal distribution Area, A

= , .in2


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Reliability Example

Mean FS FS = 40(0.50)/15 = 1.333

Standard Deviation FS Ft FS+ = 44 (0.5)/15 = 1.467 FS- = 36 (0.5)/15 = 1.20

P FS+ = 40 (0.5)/18 = 1.111 FS- = 40 (0.5)/12 = 1.667

= 1.467 - 1.200 / 2 2 + 1.111 - 1.667 / 2 2 1/2

FS = (0.1342 + 0.2782)1/2

FS = 0.309


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Reliability Example

= E SM -1/ SM = 0.333/ 0.309

= 1.06

= .

R = 1 P(u) = 0.86


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y x


Based on analytical equations like TSFD

Quadrature Method n s e c ange n per ormance unc on or a com na ons o

random variable, either plus or minus one standard deviation For 2 random variables - ++, +-, -+, -- (+ or is a standard deviation)

Results in a Reliability Index value () Based on E(SM) and (SM)


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Figure from Baecher and Christian (2003)


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Figure from Baecher and Christian (2003)


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Advanced Second Moment

(Hasofer-Lind 1974, Haldar and Ayyub 1984) Based on analytical equations like PEM

Uses directional cosines to determine shortest distance tomulti-dimensional failure surface

Accurate for non-linear limit states


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Reliability

RandomVariable, X2

C-D >0 Safe

Random

C-D < 0 Fail

Variable, X1


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Shortcomings Instantaneous - capture a certain point in time

Index methods cannot be used for time-epen ent re a ty or to est mate azarfunctions even if fit to Weibull or similar

Incorrect assumptions are sometimes madeon underlying distributions to use toestimate the probability of failure


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Monte Carlo is the method (code name) forsimulations relating to development of atomic bombduring WWII

Traditional static not dynamic (not involve time), U(0,1)

Non-Traditional multi-inte ral roblems d namic time

Applied to wide variety of complex problems involvingrandom behavior

Procedure that generates values of a random variablebased on one or more probability distributions


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Usage in USACE Develo ment of numerous state-of-the-art USACE

reliability models (structural, geotechnical, etc..)

Used with analytical equations and other advanced

Determines Pf directly using output distribution

Convergence must be monitored

Variance recommend


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Reliability

equation:

R = 1 - P(u)

w ere, u = pu Npu = Number of unsatisfactoryperformances at limit state < 1.0

N = number of iterations


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Hazard Functions

Background Previousl used reliabilit index methods

Good estimate of relative reliability

Easy to implement Problem: Instantaneous - snapshot in time


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Hazard Functions/Rates

Started with insurance actuaries in England in late 1800s

They used the term mortality rate or force of mortality Brought into engineering by the Aerospace industry in 1950s

Accounts for the knowledge of the past history of the component

Basically it is the rate of change at which the probability of failure

changes over a time step Hazard function analysis is not snapshot a time (truly

cumulative) Utilizes Monte Carlo Simulation to calculate the true probability of

Easy to develop time-dependent and non-time dependentmodels from deterministic engineering design problems


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-

PhaseConstant failure

rate phase

-

phase

h(t)Electrical

Mechanical

Lifetime, t


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Ellingwood and Mori (1993)

L(t) = exp [-t[1-1/t FS(g(t)r) dt]]fR(r)dr

0

t

0

FS = CDF of load

g(t)r = time-dependent degradation

R = mean rate of occurrence of loading

ose - orm so u ons are no ava a e excep or afew cases

Solution: Utilize monte carlo simulations to examinethe life cycle for a component or structure


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Degradation of Structures Relationship of strength (R) (capacity) vs. load (S) (demand)

f r R

fS(r) S

t me, t


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Initial Random Variables for

Strength and Load

Reinitialize with new set of

random variablesPropagate variables for life

cycle (e.g. 50 years)

Run life cycle and document

year in which unsatisfactoryFor I = 1 to N

where N = number of MCS

Develo L t h t


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Developed for the ORMSSSeconomists/planners to assist in per ormingtheir economic simulation analysis for

h(t) = P[fail in (t,t+dt)| survived (0,t)]

h(t) = f (t) / L(t)

.

No. of survivors up to t

D li i I t t d S t i bl


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Response Surface Methodology (RSM)

Reliability is expressed as a limit statefunction, Z which can be a function of, n,

Z = g (X1,X2, X3,)

Z = C - D > 0

where D is demand and C is capacity

D li i I t t d S t i bl


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Reliability (in 2 variable space)

Safe C > D

Limit State

1

Surface

Fail C < D

X2

Delivering Integrated Sustainable


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Utilizes non-linear finite element analysis todefine to the res onse surface

Not closed form solution but close

a roximation Constitutive models generally not readily

available for performance limit states

Typical design equations generally are notadequate to represent limit state for



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Response Surface Methodology (RSM) Accounts for variations of random variables on

response surface , .

found in navigation structures

Calibrated to field observations/measurements eve op response sur ace equa ons an useMonte Carlo Simulation to perform the reliabilitycalculations

Recent USACE Applications Miter Gates (welded and riveted)

Tainter Gates

Tainter Valves (horizontally and vertically framed)

Alkali-Aggregate Reaction



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Reliability (in 2 variable space)

Safe C > D Limit State

1

esponse

Surface

Res onse

Fail C < DX2

Points



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2.0

Concrete Strain in Anchorage Region vs Compressive Strength

1.5

)

ps

1.0

Strain(

0.5

6400 psi

.

0 20 40 60 80 100 120

Time (years)



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Proposed Methodology for I-Wall Reliability

Poisson ratio constant

Random variables E, Su (G, K) Limit state based on deflection ( at ground

surface

, cr .



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Response Surface Modeling Concept (under development)

E max, Su min, (E max, Su max,

(EBE , SuBE ,

Range

of

Su

u

(E min, Su max,

cr

, ,



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Reliability

Preferred Methods For non-time dependent reliability problems Linear Taylor Series Finite Difference, Point

Estimate or Monte Carlo Simulation

Assume any distributions for MCS

Non-Linear Advanced Second Moment or Monte

For time-dependent reliability problems Hazard Function/Rates usin Monte Carlo

Simulation

Documents

Introduction to Engineering Reliability engineering Reliability Concepts