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Background Documents on Risk Assessment in Engineering
Document #5
Optimization with a LQI Acceptance Criterion
JCSS Joint Committee of Structural Safety
Date: November 2008 JCSS author:
Rüdiger Rackwitz, Technical University Munich
Optimization with a LQI Acceptance Criterion
H. Streicher and R. RackwitzDraft, February 2006
1. Introduction
This example collection covers some typical applications in structural reliability and discussesseveral important aspects. It is meant to be the basis by exampleapplications for practicableaccep-tance criteria for structural codes. The first three examples show the influence of the interest ratesin an optimization for the public or the owner, the optimal and acceptable solutions for differentcost values and the importance of the coefficient of variation of the resistance and load variables.Especially in the first two examples a number of parameter studies are performed. The next exam-ple deals with the realistic design for the buckling of a reinforced concrete column. Two examplesfor the optimal design under reliability constraints for different load combinations of intermittentload processes will be evaluated. Finally, an example from earthquake engineering is included.The obective functions are based on a systematic recontruction policy and a constant benefit rate.Optimization is carried out by the methods described in [9] but other methods are also possible.All LQI considerations are based on [6] . Usually, the LQI-criterion is added to the cost-benefitoptimization task as a constraint. In [6] it is shown that the LQI-criterion requires
dC
Y
(
p
)
≥−
G
x
kN
PE
dh
(
p
)
(1)with
p
a vector of design parameters. Using
dF
(
x
) =
f
(
x
)
d
x
one again solves this equation as anoptimization task
Minimize:
S
(
p
) =
C
(
p
) +
G
x
kN
F
h
(
p
)
(2)because eq. (1) is the first-order optimality condition of eq. (2) if no cost-benefit optimization isperformed.From [6] one concludes that the ’’societal value of a statistical life’’ is
G
x
=
gq
C
x
where ’’x’’stands for the particular mortality reduction scheme. For constant mortality reductions
∆
,
a Eu-ropean population with
GDP
≈
25000
PPPUS$(year 2000), the part available for risk reduction
g
≈
1
7500
PPPUS$ and
q
≈
0
.
1
5
, C
≈
45
and
G
≈
5
Mill
US$ if no discounting and no age-averaging. If discounting and age-averaging is performed it is
C
d
∆
≈
1
6
and
G
d
∆
≈
1
.
9
Mill
US$if discounting and age-averaging is performed and constant mortality reductions are envisaged bya safety measure. These values then enable to compute the societal willingness-to-pay to save astatistical life by
SWTP
=
gq
C
x
dm
.The examples have taken from several sources but many of them are contained in a collectionof examples in [2] . Therefore, some variations in important parameters, especially for the LQI-acceptability criterion, remain.1
2. Example 1: Resistance-demand problem
Theexamplehasalready beengiven in[5] and in [7] insomewhat differentform andwith differ-ent parameters. A single-modesystem is considered where failureis defined if arandom resistanceor capacity is exceeded by a random demand. The demand is modelled as a one-dimensional,stationary marked Poissonian pulse process of disturbances (earthquakes, wind storms, explo-sions, etc.) with stationary renewal rate
λ
and random, independent sizes of the disturbances
S
i
,i
=
1
,
2
,....
. The disturbances are assumed to be short as compared to their mean interarrivaltimes. We study normally distributed and log-normally resistance and disturbances. The resistancehas mean
p
and a coefficient of variation
V
R
.
The disturbances are independent and have meanequal to unity and coefficient of variation
V
S
so that
p
can be interpreted as central safety factor
. p
is taken as the only optimization parameter. For failure once a disturbance occurs with normallydistributed variables we have:
P
f
(
p
) =
Φ
Ã
−
p
−
1
p
(
pV
R
)
2
+
V
2
S
!
(3)In this model both resistances and disturbances can have negative values which usually is incor-rect from a physical point of view. Therefore, we truncate the distributions at zero and a morecomplicated formula for the failure probability results:
P
f
(
p
) =
1Φ
(
1
V
S
)
1Φ
(
1
V
R
)
Z
∞
0
1
√
2
π
V
S
ϕ
(
s
−
1
V
S
)
Z
s
0
1
√
2
π
V
R
p
ϕ
(
r
−
pV
R
p
)
drds
(4)For the log-normal case we have:
P
f
(
p
) =
Φ
−
ln
n
p
q
1
+
V
2
S
1
+
V
2
R
op
ln((
1
+
V
2
R
)(
1
+
V
2
S
))
(5)An appropriate objective function for systematic reconstruction which will be maximized, is thengiven by
Z
(
p
) =
b
γ
−
C
(
p
)
−
(
C
(
p
) +
H
M
+
H
F
)
λ
P
f
(
p
)
γ
(6)where
C
(
p
) =
C
0
+
C
1
p
a
and
H
M
the initial material cost.The acceptability criterion can be written as
ddpC
(
p
)
≥−
G
∆
kN
PE
ddp
λ
P
f
(
p
)
(7)The parameter assumptions are:
C
0
=
1
0
6
, C
1
=
1
0
4
, a
=
1
.
25
, H
M
= 3
C
0
, V
R
= 0
.
2
, V
S
= 0
.
3
and
λ
=
1
[
1
/year
]
. The LQI-data are
e
= 77
, g
= 25000
, C
∆
= 25
, q
= 0
.
1
6
, kN
PE
=
1
0
sothat
H
F
≈
7
·
1
0
6
and
G
∆
(
ρ
,
δ
)
≈
5
·
1
0
6
. Monetary values are in US$. Optimization will first beperformed for the public. Therefore, compensation cost are included in the objective function. Thebenefit rate is
b
= 0
.
02
C
0
and the interest rate is
γ
= 0
.
0
1
85
.
For the failure probability model eq.(3) one finds the optimum for
p
∗
= 5
.
54
and a limiting value of
p
lim
= 3
.
86
. If one uses the modelin eq. (4) the optimum is at
p
∗
= 3
.
00
and the limiting criterion eq. (7) results in
p
lim
= 2
.
49
.
Finally, for the model in eq. (5) the optimum is at
p
∗
= 4
.
40
and has a limit of
p
lim
= 3
.
45
. Thethree objective functions are shown in figure 1. One concludes from this study that the stochasticmodels for the failure function must be as realistic as possible. All subsequent investigations are2
based on log-normal distributions.
0 2 4 6 8 1010.500.51p
Z ( p )
plim(nt)plim(ln)plim(n)
Figure 1: Objective functions and limiting criteria for the failure models in eq. (1) (dashed line),(2) (solid line) and (3) (dotted line)The influence of the cost
C
1
on the optimal and acceptable solution is shown in figure 2, where
C
1
varies from
1
000
.
0
to
1
00000
.
0
and the corresponding optimal and acceptable failure rates aregiven. For
C
1
>
1
2000
the objective function
Z
(
p
∗
)
is negative.
1
.
10
3
1
.
10
4
1
.
10
5
1
.
10
6
1
.
10
5
1
.
10
4
1
.
10
3
0.01
as
:=
L o g ( F a i l u r e R a t e )
Log (C
1
)
1
.
10
3
1
.
10
4
1
.
10
5
1
.
10
6
1
.
10
5
1
.
10
4
1
.
10
3
0.01
as
:=
L o g ( F a i l u r e R a t e )
Log (C
1
)
Figure 2: Failure rates for various cost
C
1
. Dashed lines correspond to acceptable, solid lines tooptimal results.The optimal solution is acceptable for all cost values
C
1
. It is most important to ensure that allnecessary investments into life saving in order to guarantee the safety for human life and limb aredone and the resulting failure rate of a technical facility is acceptable . This is already fulfilled if the acceptability criterion derived from the LQI is fulfilled.The owner uses some typical value
b
= 0
.
07
·
C
0
. Compensation cost (Life saving cost) arenot included in the optimization. The objective of the owner is in general to maximize the benefit3

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