Now we consider the case where some information about the in-situ conditions is known that leads to a probability distribution on the shear force P. Clearly, for such a column, the probability distribution for the force P will, in general, depend on the structure and the site it is located in.
Figure 7.3 Failure region and safe region in the Ω space.
Pm
P DS = “shear failure”
DS = “none”
Ωn
[Ωn]C
Suppose that P is a lognormally distributed random variable: P~ LN (µP, σP). All the remaining properties of the column are exactly the same as in the first case. We want to find a fragility function for the column using this information about P: P(DS = “shear failure”| IDR = z).
First, we have to find a failure region corresponding to the event DS = “shear failure” (Ωn) in the space of the uncertain parameters: Ω = P×Pm = [0, ∞)×[0, ∞). Since
“shear failure” is defined as a reduction of the shear spring stiffness to αK1, it is easy to see that DS = “shear failure” is equivalent to P ≥ Pm. This failure region is shown as the shaded area in the Figure 7.3.
Second, given that event IDR = z has happened, parameters P and Pm can not take arbitrary values in the space Ω, because equality IDR = z imposes restrictions on the possible values of the external force P and maximum shear strength Pm. Therefore, we have to identify the region Ωz that corresponds to the event IDR = z. For the safe region, the external force P is found, as before, by (7.1). It is easy to see that P does not depend on Pm in this region. Therefore, within the safe region, Ωz is defined by
1 2 2
2 1
K L K
K P zLK
= + , Pm∈ [
1 2 2
2 1
K L K
K zLK
+ , ∞) (7.3)
In the failure region, the forces and deformations are related as follows
2 2
1
1 K
PL K
P P K
Pm m
+
−
+
=
∆ α
Therefore, the relation between P and Pm is easily derived as
( )
2 1 2
2 1 2
1 2
1 2
L K K
K LK P z
L K K
P K m
α α α
α
+ + +
= − , Pm∈ [0,
1 2 2
2 1
K L K
K zLK
+ ] (7.4)
Equations (7.3) and (7.4) define the region Ωz, which is a 1-D surface in the space Ω. Equation (7.4) defines the section of surface Ωz that lies within the failure zone.
Figure 7.4 Surface Ωz for different values of IDR
Figure 7.5 Surface Ωz for different values of α
Ωz3(IDR = z3)
Pm
P DS = “shear failure”
Ωz2(IDR = z2) Ωz1(IDR = z1)
(Ωn)
z1 < z2 < z3
DS = “none”
([Ωn]C)
Ωz(IDR = z)
Pm
P
DS = “shear failure”
(Ωn)
DS = “none”
([Ωn]C)
α= 1
α= 0 0<α< 1
Figure 7.4 shows surface Ωz for several values of IDR. Since α ranges within the interval [0, 1], the tangent of the line (7.4) also changes within the interval [0, 1]. Figure 7.5 shows surface Ωz for different values of α and the same value of IDR.
If the surface Ωz is a straight line (α = 1), then the fragility function will coincide with the CDF of capacity (because P and Pm are independent). Therefore, it should be true that the more Ωz deviates from a straight line, the more the site-specific fragility function should differ from general fragility function (CDF of capacity). Thus, for better demonstration it is desirable to take α = 0. However, as shown latter, taking α = 0 causes mathematical difficulties. Therefore, we pick α by observing the following conditions: α
≠ 0 and α << 1.
The procedure for estimating the fragility function is outlined in Equations 3.18 - 3.20. In order to be able to perform this procedure, we need to know the joint probability function of the vector X* that for the present case is equal to [Pm, P]. Since P and Pm are taken as independent random variables with lognormal probability distributions, the joint probability density function is just the product of their probability density functions
( )
y f(
p p)
f( )
p f( )
pfY = Pm,P m, = PmLN m PLN (7.5)
We define a new coordinate system as follows: Y = [Pm, IDR], then transformation of variables ψ -1(Y ) takes the form
( )
( )
ϕα α α
α ϕ ϕ ψ
IDR IDR P
L K K
K P LK
L K K P K
P P
P IDR IDR
P P Y P
X
m m
m m
m m
m
<
<
+ + +
= −
=
<
=
= − =
∗
0 1
:
2 1 2
2 1 2
1 2
2 1
(7.6)
Therefore, the Jacobian (3.20) of the transformation ψ -1(Y ) is equal to
( )
ϕα α α
α α
α
ϕ ϕ ϕ
IDR L P
K K
K LK L
K K
K LK L
K K J K
P IDR J
m m
<
+ <
= +
+
= −
<
=
=
0 1 1 0 ,
0 , 0 1
2 1 2
2 1 2
1 2
2 1 2
1 2
2
(7.7)
Now we can see that in the case α = 0, the Jacobian becomes zero, making the PDF for a certain area to be equal to zero too. Although there is nothing non-physical about this situation, we choose to avoid such an extreme case in this example.
The fragility function is given by (3.18). The safe region for the present case is expressed in terms of Pm and, given that IDR = z, is defined as before: zϕ <Pm. Then, the fragility function is calculated as
P( DS = “shear failure” | IDR = z) = z
∫
ϕfPmIDR(
pm z)
dpm0
| | (7.8)
where the conditional PDF of Pm is found as fPm|IDR
(
pm |z)
= fPm,IDR(
pm,z)
fIDR( )
z . The PDF of IDR is obtained according to the second equation of (3.19) that for the present case takes the particular form
( )
=∫
∞( )
0
,IDR m, m
P
IDR z f p z dp
f m (7.9)
where the joint PDF of Pm and IDR is found according to (3.19) and in the present case takes the form
( )
( )
( )
<
+ <
+ + +
−
<
= ϕ
α α α
α α
α
ϕ ϕ
ϕ
z L p
K K
K z LK
L K K
K p LK
L K K p K f
p z z
p f z p f
m m
m P P
m m
P P
m IDR P
m m
m 1 0
, , ,
2 1 2
2 1 2
1 2
2 1 2
1 2
2 ,
,
, (7.10)
where the joint PDF fPm,P
( )
o,o is given by (7.5).Equations 7.8 – 7.10 provide the algorithm for estimating the fragility function of the in-situ column. We use this algorithm to calculate the fragility function for a sample column. The parameters of the sample column are chosen based on the parameters of the columns of a seven-story hotel in Van Nuys, California. The general stiffness parameter ϕ = 10000 kips, length of the column L = 100 in, the translational (in shear) stiffness K1 = 6000 kips/in, the rotational stiffness is derived as K2=L2K1
(
ϕ L) (
K1−ϕ L)
≅ 1000000 kips⋅in, and the expected shear strength E[Pm] = 60 kips. The coefficient of variation of the shear strength is assumed to be δvPm = 0.15. The parameters of the probability distribution of the force P are chosen to provide a probability of failure (unconditional on IDR, given only that a seismic event has happened) to be 2% with coefficient of variation δvP = 0.6. The estimate of the coefficient of variation is based on the assumption that the force P is produced by natural hazard and has higher uncertainty than material properties.The estimate of the probability of failure is based on the result of two earthquakes that happened at the site. The building has 150 columns in the lateral force resisting frame;
none of them failed during the first earthquake and 6 of them failed in shear during the second one, giving an estimate of the probability of failure of 6/(2*150) = 0.02.
Assuming that the Van Nuys building is a typical example of the structural design of its time, we call the assembly with 2% probability of failure the “normal strength” design.
The stiffness parameter ϕ is not used for estimating parameters of the probability
distributions. Using E[Pm], δvPm, δvP, and the probability of failure, the four parameters of the probability distributions are estimated as: µPm = 10.991, σPm = 0.1492, µP = 9.8116, σP = 0.5545, where the units of both Pm and P are pounds (lb).