Transcript IVW Cursus 7 Oktober 2003

IVW Cursus
7 Oktober 2003
Betrouwbaarheid van
systemen en elementen
Pieter van Gelder
TU Delft
Inhoud
• Tijdsafhankelijke faalkansen
• Faalkansberekening op systeem niveau
– Parallel en serieel
– Afhankelijkheden
• Faalkansberekening op element niveau
– Niveau III, II, en I
1
P(X x)
0.6
X
F (x)
0.8
0.4
0.2
0
x
0.5
fX(x)
0.4
P(x < X  x+d x)
0.3
0.2
0.1
0
x x+d x
Random Variable T:
Time to Failure
J.K. Vrijling and P.H.A.J.M. van Gelder, The effect of inherent uncertainty in time and space on the reliability of
flood protection, ESREL'98: European Safety and Reliability Conference 1998, pp.451-456, 16 - 19 June 1998,
Trondheim, Norway.
T = time to failure
• The Hazard Rate, or instantaneous failure rate is
defined as:
• h(t) = f(t) / [1 - F(t) ] = f(t) / R(t)
• f(t) probability density function of time to failure,
• F(t) is the Cumulative Distribution Function (CDF) of
time to failure,
• R(t) is the Reliability function (CCDF of time to failure).
• From: f(t) = d F(t)/dt , it follows that:
• h(t) dt = d F(t) / [1 - F(t) ] = - d R(t) / R(t) = - d ln R(t)
Integrating this expression
between 0 and T yields an
expression relating the
Reliability function R(t) and the
Hazard Rate h(t):
Bathtub Curve
Constant Hazard Rate
• The most simple Hazard Rate model is to
assume that: h(t) = λ , a constant. This
implies that the Hazard or failure rate is not
significantly increasing with component age.
Such a model is perfectly suitable for
modeling component hazard during its useful
lifetime.
• Substituting the assumption of constant
failure rate into the expression for the
Reliability yields:
• R(t) = 1 - F(t) = exp (- λt)
• This results in the simple exponential
probability law for the Reliability function.
Non-Constant Hazard Rate
• One of the more common non-constant
Hazard Rate models used for evaluation of
component aging phenomenon, is to assume
a Weibull distribution for the time to failure:
• Using the definition of the Hazard function
and substituting in appropriate Weibull
distribution terms yields:
• h(t) = f(t) / [1 - F(t) ] = β t β -1 / t β
• For the specific case of: β = 1.0 , the
Hazard Rate h(t) reverts back to the
constant failure rate model described
above, with: t = 1/ λ . The specific
value of the β parameter determines
whether the hazard is increasing or
decreasing.
β values, 0.5, 1.0, and 1.5.
β values, 0.5, 1.0, and 1.5.
Effect of inherent uncertainty on
decrease in hazard rate
• Example: PAC dike and sea dike
Reliability
Maintainability
Availability
Reliability
Reliability is the probability that a
process or a system will operate without
failure for a given period and under
given operating conditions.
R(t) = e-lt
This equation is the exponential reliability function, it applies
only for cases of “constant failure rate”, where l is failure rate.
Maintainability
Maintainability is the probability that a
process or a system that has failed will
be restored to operation effectiveness
within a given time.
M(t) = 1 - e-mt
where m is repair (restoration) rate
The Concept of Availability
Reliability
Maintainability
Availability
Availability
Availability is the proportion of the
process or system “Up-Time” to the total
time (Up + Down) over a long period.
Availability =
Up-Time
Up-Time + Down-Time
Process
Requirements
Process Effectiveness
Capability
Performance
Availability
Reliability
Maintainability
Dependability
Capability: A measure of the ability of a process to
satisfy given requirements (A measure of Quality - no
time dependency)
Availability: A measure of the ability of a process to
complete a mission without excessive down time
(Depends on Reliability and Maintainability)
Dependability: A measure of the ability of a process
to commence and complete a mission without failure
(Depends on Reliability and Maintainability)
System Operational States
B1
Up
Down
A1
B2
A2
Up: System up and running
Down: System under repair
B3
A3
t
Mean Time To Fail (MTTF)
MTTF is defined as the mean time of the occurrence of the
first failure after entering service.
MTTF =
B1
Up
Down
B1 + B2 + B3
3
A1
B2
A2
B3
A3
t
Mean Time Between Failure
(MTBF)
MTBF is defined as the mean time between successive
failures.
MTBF =
B1
Up
Down
(A1 + B1) + (A2 + B2) + (A3 + B3)
3
A1
B2
A2
B3
A3
t
Mean Time To Repair (MTTR)
MTTR is defined as the mean time of restoring a process or
system to operation condition.
MTTR =
B1
Up
Down
A1 + A2 + A3
3
A1
B2
A2
B3
A3
t
Availability
Availability is defined as:
A=
Up-Time
Up-Time + Down-Time
Availability is normally expressed in terms of MTBF and
MTTR as:
A=
MTBF
MTBF + MTTR
Reliability/Maintainability Measures
Reliability R(t)
(Failure Rate) l = 1 / MTBF
R(t) = e-lt
Maintainability M(t)
(Maintenance Rate) m = 1 / MTTR
M(t) = 1 - e-mt
Types of Redundancy
• Active Redundancy
• Standby Redundancy
Active Redundancy
A
Input
Output
Div
B
Divider
Both A and B subsystems are operative at all times
Note: the dividing device is a Series Element
Standby Redundancy
A
Input
Output
SW
B
Switch
Standby
The standby unit is not operative until a failure-sensing device
senses a failure in subsystem A and switches operation to
subsystem B, either automatically or through manual selection.
Series System
Input
A1
A2
An
Output
ps = p1 + p2 +……. + pn - (-1)n joint probabilities
For identical and independent elements:
ps ~ 1 - (1-p)n < np (>p)
ps :
pi :
Probability of system failure
Probability of component failure
Parallel System
A
Input
Output
B
Multiplicative Rule
ps = p1.p2 … pn
ps :
Probability of system failure
M
o
d
e
l
l
e
r
i
n
g
•Haringvliet outlet sluices
•
Lifetime distribution
for one component
t
t
start
t
t
Time
Replacement strategies of large numbers of similar components in hydraulic structures
Series / Parallel System
A1
Input
A2
Output
C
B1
B2
System with Repairs
Let MTBF = q
and system MTBF = qs
A
Input
Output
B
For Active Redundancy (Parallel or duplicated system)
qs = ( 3l + m )/ ( 2l2 )
qs = m / 2l2 = MTBF2 / 2 MTTR
l << m
A
Input
Output
SW
B
Switch
Note: The switch is a
series element, neglect
for now.
Standby
Note: The standby
system is normally
inactive.
For Standby Redundancy
qs = ( 2l + m )/ (l2 )
qs = m / l2
= MTBF2 / MTTR
System without Repairs
For systems without repairs, m = 0
For Active Redundancy
qs = ( 3l + m )/ ( 2l2 )
qs = 3l / ( 2l2 ) = 3 / ( 2l )
qs = (3/2) q where q = 1/l
qs = 1.5 MTBF
For Standby Redundancy
qs = ( 2l + m )/ (l2 )
qs = 2l/ l2 = 2/ l
qs = 2q
where q = 1/l
qs = 2 MTBF
Summary
Type
With Repairs
Without Repairs
Active
MTBF2 / 2 MTTR
1.5 MTBF
Standby
MTBF2 / MTTR
2 MTBF
Redundancy techniques are used to increase the system MTBF
Example: wire
• Limit state function:
•
Z=R-S
• with:
variable
R
distribution
mean
standard deviation
R
normal
60 kN
5 kN
S
normal
40 kN
10 kN
S
Probability densities
0.08
Probability density (1/N)
0.07
R
0.06
0.05
S
0.04
0.03
0.02
0.01
0
0
20
40
R,S (N)
60
80
Joint probability density
80
70
60
R (kN)
50
40
Z>0
failure: Z<0
30
20
10
0
0
20
40
S (kN)
60
80
Analytical

dr  fR,S r, s) ds
r
80
• Failure probability70
PZ  0 ) =
=

 dr  f r, s) ds
R,S

r
r
dr
50
R (kN)

40
Z>0
30
• Independent
• R and S:

=

 fR r ) dr  fS s) ds

r
falen: Z<0
20
10
0
0
20
40
S (kN)
r
80
Analytical
• Elaborate:
PZ  0 ) =


 f r ) dr  f s) ds
R

S
r1
• R and S normally distributed:
2




1
1
r

m
R
fR r ) =
exp  
 
 2  R  
 R 2


2




r

m
1
1
S
 
fS r ) =
exp  
 2 S  
 S 2


• Fill in and calculate
Analytical
• In this case simple approach possible:
– variables normally distributed
– Z is linear in variables
• Then Z also normally distributed.
– Mean
m Z = mR  mS = 60  40 = 20 kN
– Standard deviation
 Z2 =  R2 +  S2 = 52 + 102 = 125   Z = 11.2 kN
Probability density of Z
0.04
Probability density (1/kN)
0.035
0.03
0.025
0.02
0.015
0.01
0.005
0
-20
0
20
Z (kN)
40
60
Failure probability
Probability density (1/kN)
0.04
0.035
0.03
0.025
0.02
0.015
0.01
0.005
0
-20
0
20
Z (kN)
40
60
Limit State
Z < 0 failure
Z=0
failure
boundary
Z > 0 no failure
R ( X1 )
Het aantal ongevallen met druktankwagens in Nederland in de
periode 1983 - 1992 bedraagt slechts één. Als beste schatter voor de
gemiddelde uitstroomfrequentie volgt met dit gegeven 0.1/ jaar.
P(Xt = x) = (nt)x/x! e-nt (x=0, 1, 2, ...);
n = 0.1 jr-1.
Simulatie van het aantal ongevallen met druktankwagens in de periode 1983-1992
3
Aantal ongevallen
2.5
2
1.5
1
0.5
0
0
20
40
60
Simulatie i (i=1,...,100)
80
100
MCS of the failure probability
•
•
•
•
•
Draw R and S
Z= R - S
if Z<0, n = n+1
Repeat
p = n / total
Failure probability
200 samples for S,R
80
70
60
R (kN)
50
40
Z>0
30
failure: Z<0
20
10
0
0
20
40
S (kN)
60
80
Estimator failure probability
n
p
n
= Z 0
n
=
 Iz )
i
i=1
=0.015
n
with
1 z i  0
Izi ) = 
0 zi  0
failure
no failure
n
1
2


)
)
 p2 =
I
z

p

nn  1) i=1
 p = 0.01
Uncertainty in failure probability
Accuracy independent of number of variables!
Importance sampling
200 samples for S,R
80
70
60
R (kN)
50
40
actual distribution
30
sampling distribution
20
10
0
0
failure: Z<0
20
40
S (kN)
60
80
Importance sampling
– Sampling from different distribution:
sampling distribution
– Correction in statistical analysis afterwards:
 fR,S ri , si ) 
Izi ) 



)
h
r
,
s
i=1
 R,S i i 
=
n
n
p
– Prior knowledge required
actual density
sampling density
Importance sampling
• Increased variance sampling:
Standard deviations increased by factor 1.5
80
70
60
R (kN)
50
40
Z>0
30
20
failure: Z<0
10
0
0
20
40
S (kN)
60
80
Importance sampling
– Possible reduction of number of samples w.r.t.
crude Monte Carlo: order 10-100.
– For increased variance sampling little prior
knowledge required
– Variation: adaptive importance sampling
iterative adaptation of sampling distribution.
FORM
• Example:
 d2 f
Z=
4
• with:
• f
• d
S
R=
tensile strength
diameter of wire
• S = 100 kN
• f = 300 N/mm2
• d = N (30 mm, 3 mm)
normal
m

S = 100kN
 d2 f
4
400
Z (kN)
300
Example
200
100
0
density (1/mm)
-100
15
20
25
30
d (mm)
35
40
45
20
25
30
35
40
45
0.2
0.15
0.1
0.05
0
15
400
Z (kN)
300
Linearisation in m
Example
d
200
100
0
density (1/mm)
-100
15
20
25
30
d (mm)
35
40
45
20
25
30
35
40
45
0.2
0.15
0.1
0.05
0
15
FORM
• Linearization in mean:
– Error in failure probability
– Error larger as non-linearity increases
• Solution:
– Better choice linearization point
400
Z (kN)
300
Linearisation in Z=0
Example
200
100
0
density (1/mm)
-100
15
20
25
30
d (mm)
35
40
45
20
25
30
35
40
45
0.2
0.15
0.1
0.05
0
15
FORM
• For Z-function with 1 stoch. variable:
– Z = 0 corresponds to 1 punt
– Linearization in Z=0 trivial and unnecessary
• More stoch. variables:
– Z = 0 corresponds to line, surface, …
– Linearize where on Z=0 line of surface?
FORM
• Elaborated example:
 d2 f
Z=
• with:4
• f
• d
S
R=
tensile strength
diameter of wire
• S = 100 kN
• f = N(290 N/mm2, 25 N/mm2)
• d = N (30 mm, 3 mm)
S = 100kN
 d2 f
4
Contour lines Z-function
f
FORM-design point
f
Linearisation
Z’ = a + b*f + c*d
Design point
• Design point:
– Point on line or surface Z=0 with the highest
probability density
– Assessment is ‘constrained optimization’
– Z-function is linearized in design point
– Directly gives estimate of failure probability
FORM
• Linearized Z-surface:
Z' = a + b f + c d
• with given a, b and c
• So Z’ normally distributed with:
m Z' = a + b m f + c md
 Z2' = b2  f2 + c 2  d2
FORM: sensitivities
• Formula for variance of Z:
 Z2' = b2  f2 + c 2  d2
•
indicates which influence variables have on
uncertainty in Z en hence on failure probability.
Expressed in a’s:
a = b f
f Z
ad =
c d
Z
Example, Frechet -> Normal
Frechet en Vervangende Normale Verdeling
1
10
Vergelijking tussen Gumbel en Normale verdeling in ontwerppunt
0
Gumbel
Normaal
0.9
10
0.8
0.7
10
-1
-2
1-CDF
CDF
0.6
0.5
Vervangend Normaal
10
-3
0.4
10
0.3
-4
Frechet
0.2
10
-5
0.1
0
10
0
5
10
H
15
20
-6
0
0.5
1
1.5
x
2
2.5
Resume
• Tijdsafhankelijk falen
• Systeembetrouwbaarheid
• Elementbetrouwbaarheid