introduction to engineering reliability engineering reliability concepts
TRANSCRIPT
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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
Point Estimate Method
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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Point Estimate Method
Figure from Baecher and Christian (2003)
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Point Estimate Method
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
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Response Surface Methodology (RSM)
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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Response Surface Methodology (RSM)
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