Fragility 2.0

Fragility analysis Any mechanical model describing the behavior of a structure is intended to define mappings from the seismic loads ug(t) to the corresponding responses Rt such as: ug(t) ↦R(t)

(1.1)

From the mechanical model of the structure, r(t) depends on a vector X of uncertain parameters. Rt=gX=g(X1,…,Xq)

(1.2)

Let Rt;X be the response of a structure at time t and Rmax(X) the maximum response of the structure. Suppose n nominally identical structures have been subjected to a seismic load ug(t) and that (Rmax,1,…, Rmax,n) be the largest responses of these specimens. To build a fragility model that is consistent with the experimental or experience data, we need to find a value of X of X such that the mean square error between the observed and calculated responses i.e. i=1nRmax,1-RmaxX2 is minimized.

Damage states It is necessary to define the limit states or damage states of a structure for development of fragility curves. Damage state typically refers to certain serviceability criteria, and a structure is said to have reached a damage state d if the structural response Rt exceeds Rd*, the critical response value of d. For a structure, multiple damage states can be considered and a fragility curve corresponding to each damage state must be developed. For example, it is possible to define three damage states of a structure corresponding to (i) slight (repairable) (ii) moderate and (iii) major damage (unrepairable). Data-based fragility Suppose n identical specimens of a structure are subjected to the same seismic load ug(t). Denote the maximum responses of these by Rmax,1,…, Rmax,n. If pd denotes the probability that a structure enters the damage state d when subjected to a given ug(t), we can write pd=PRmax, i>Rd*

(1.3)

Bayesian framework for fragility evaluation using experimental data in conjunction with analyses The failure of each specimen subjected to the seismic load can be considered as a Bernoulli trial. Let Yd be the total number of specimens entering the damage state d out of n specimens. We can write MeanYd= npd VarYd=npd(1-pd)

(1.4)

COV= 1-pdnpd Pd=Ydn is an unbiased estimator of pd. The variance of Pd is VarPd= pd(1-pd)n

(1.5)

If it is required that the coefficient of variation does not exceed υd,0, the minimum number of experiments to be conducted is given by n≥1-pdυd,02 pd

(1.6)

Since pd is not known, one can still obtain the approximate number of specimens needed to be tested for a specified accuracy using Equation (1.5) by estimating the mean of Pd. Assume that pd∈[0,1] can be initially represented by a Beta distribution. Within the Bayesian framework, the prior density (Ang and Tang, 2007; Benjamin and Cornell, 1970) is f'pd∝pdr'-11-pdt'-1 , pd∈[0,1]

(1.7)

where r', t'>0 are parameters depending on the prior information based on expert opinions or experiments on similar structural systems. The normalizing constant is missing in the Equation (1.7). Now, if n specimens of the structure have been tested and nd enter the damage state d, the likelihood function (Benjamin and Cornell, 1970) of pd given this experimental data can be written as: lpddata=Pdatapd ∝pdnd1-pdn-nd

The posterior density function of pd is given by:

(1.8)

f''pd=lpddata×f'pd (1.9) f''pd∝pdnd+r'-11-pdn-nd+t'-1

The posterior density function is also a Beta distribution with parameters (nd+r',n-nd+t'). This posterior cumulative distribution function F''pd in Equation (1.9) can be used to compute the confidence limits on pd. F''p=0pf''udu

(1.10)

Multiple damage states The Bayesian framework can be extended to the case where several damage states are of interest. Suppose m damage states exist and let pdkξ, k=1,…,m, denote the unknown probabilities that a structure enters the damage state dk when subjected to a seismic action of intensity ξ. The structure is considered to enter any one of the m damage states under loading. The prior density of the unknown probabilities pd1ξ,…,pdmξ can be modeled as a Dirichlet distribution which is a multinomial extension of the Beta distribution: f'pd1ξ,…,pdmξ∝k=1mpdkξαk'-1, pdkξ≥0 (1.11) k=1mpdkξ=1.0

If n specimens of a structure are tested with ndkξ≥0 enter damage state dk, k=1,…,m. k=1mndkξ=n

(1.12)

The likelihood function given the experimental data can be written as l∝pd1ξnd1(ξ)∙pd2ξnd2(ξ)…pdmξndm(ξ) (1.13) l∝k=1mpdkξndk(ξ)

The posterior density function from Equations (1.10) & (1.12) is f''pd1ξ,…,pdmξ∝k=1mpdkξαk'+ndk(ξ)-1, pdkξ≥0 (1.14) k=1mpdkξ=1.0

The posterior density function is also of Dirichlet type with parameters αk'+nd1ξ, k=1, …, m. Example: Two damage states

Assume that two damage states d1 and d2 can be identified in the structure. pd1ξ and pd2ξ denote the probabilities of the structure entering d1 and d2 respectively. The probability pd3ξ that there is no damage in the structure is given by pd3ξ=1-pd1ξ+pd2ξ

(1.15)

The prior density of the unknown probabilities is f'pd1ξ,pd2ξ,pd3ξ∝pd1ξα1'-1.pd2ξα2'-1.pd3ξα3'-1

(1.16)

If a total of which n specimens are tested with nd1ξ entering d1 , nd2ξ entering d2 and nd3ξ=n-nd1ξ+nd2ξ suffering no damage, the likelihood function is given by l∝pd1ξnd1(ξ)∙pd2ξnd2(ξ)∙pd3ξnd3(ξ)

(1.17)

The posterior density function from Equations (1.16) & (1.17) is f''pd1ξ,…,pdmξ∝pd1ξα1'+nd1(ξ)-1∙pd1ξα2'+nd2(ξ)1∙pd1ξα3'+nd3(ξ)-1

(1.18)

Fragility Curves The fragility curves corresponding to a non-exceedence probability of ε can be constructed with points ξ,F''-1(ε) where F'' denotes the posterior cumulative distribution function of pd(ξ).

Bayesian framework for updating fragility models with availability of experimental data Bayesian analysis can be used to update and improve the existing fragility models when new experimental or experience data becomes available. The process consists of the following three steps: •

Step 1. Mechanical model of the structure should be developed to relate the response of the structure relating the response of the structure r(t) to the vector of random parameters X.

•

Step 2. Model Calibration of the mechanical model can be done using the available experimental data. The posterior densities of various random parameters are calculated corresponding to the available information.

•

Step 3. Fragility Models can be constructed from calibrated models

The following examples illustrate this approach. Example 1 (one damage state) Consider a steel beam of length L and rectangular cross-section subjected to a 4-point loading as shown in the figure below:

P

P a

a

b h

P

L  M max = P ⋅  − a  2  M h σ max = max ⋅ I 2

I=

bh 3 12

P

Figure 1: The 4-point load beam

Mechanical Model The ultimate strength σu of the steel is assumed to follow a triangular distribution. The density function fΣσ of the ultimate strength is shown in Figure 2 where η∈0,1 is an uncertain parameter.

f Σ (σ )

h=

µσ ( 1 − η )

1 ηµσ

µσ = 400MPa

µσ ( 1 + η )

σ

Figure 2: The density function of the ultimate strength of steel

Only one damage state is being considered - the beam is assumed to ‘fail’ if the maximum bending stress due to applied loads exceeds its ultimate strength i.e. if σmax>σu. Alternatively, one might assign various damage states corresponding to yielding, plastic behavior and finally, the ultimate strength. Suppose the prior information on η can be quantified by a uniform distribution as shown in Figure 3.

f Η (η )

η 0.10

0.15

0.20

Figure 3: Prior distribution of η (uniformly distributed between 0.1 and 0.2)

The prior density of η is given by fΗ'η=10.2-0.1

(1.1)

Model Calibration Suppose n nominally identical beams are tested in a laboratory by slowly increasing the load P on each beam from zero to a specified maximum value of ξ. The following have been observed: •

n1ξ specimens out of n specimens failed at P =P1,P2,…Pn1respectively i.e. σmaxPi>σui,i=1,2…n1

•

The remaining n-n1ξ specimens could sustain the load ξ without failing i.e. σmaxPi<σui,i=n1+1,…n

The likelihood function of η corresponding to the observed data is given by lηdata∝i=1n1(ξ)fΣσmaxPi×1-FΣξn-n1(ξ)

(1.1)

where fΣσ and FΣσ respectively denote the probability density function and the cumulative distribution function of the ultimate strength of the beam. The posterior density function of η, fΗ''ηcorresponding to the observed data is given by fΗ''η∝fΗ'η∙lηdata ∝10.2-0.1∙i=1n1(ξ)fΣσmaxPi×1-FΣξn-n1(ξ)

(1.2)

The constant of proportionality is missing in Equations (1.20) and (1.21). It can be computed by normalizing the area under the curve to be unity. Taking L=10m ; a = 1m ; b = 0.1m ; h = 0.3m , the maximum bending stress, σmax in the beam subjected to a seismic load of intensity P is given by: σmaxP=6P∙L2-abh2=2666.7P

(1.3)

If n=10 nominally identical beams are tested by loading each specimen from 0 to ξ=140kN. Table 1 lists the observations recorded.

Table 1: Laboratory test results for a 4-point loading of 10 identical beams Maximum applied

Failure

load (kN)

(kN)

1

160

127.1

2

160

148.8

3

160

151.3

4

160

152.6

5

160

152.7

6

160

158.3

7

160

158.7

8

160

158.8

9

160

Did not fail

10

160

Did not fail

Beam #

•

Load

From Table 1, it can be seen that n1ξ=4 out of 10 beams failed before the maximum stress level could be reached.

•

The actual strengths of the 6 beams that did not fail cannot be determined though it is clear that for these beams that σu>σmaxξ. This is called censored data.

Using this data, we can find the posterior density function of η, fΗ''η using Equations (1.20) and (1.21). The prior and posterior density functions of η are plotted in Figure 4.

Figure 4: Prior distribution and posterior distributions of η

Fragility Curves Having known the posterior density, fΗ''(η) , it is possible to revise the fragility curves. The fragility curve is given by

Pσu≤σmaxξ=0∞Pσu≤σmaxξ|η∙fΗηdη

(1.1)

Figure 5 presents the prior and posterior (updated) fragility curves. Figure 5: Prior and Posterior (updated) fragility curves

Example 2 (Multiple damage states) This example illustrates the Bayesian approach for updating the fragility model when multiple damage states are relevant. Consider the beam subjected to a 4-point loading as shown in Figure 1. Suppose the following damage states are defined: 1. Yield strength damage state (d1) - The structure enters this damage state if the applied

bending stress exceeds the yield strength of the material of the beam. 2.

Serviceability damage state (d2) – The structure enters this damage state if the maximum deflection of the beam exceeds 0.02 m.

Mechanical Model The yield strength σy of the steel is assumed to follow a triangular distribution. The density function fΣσ of the yield strength is shown in Figure 6 where η∈0,1 is an uncertain parameter. The prior information on η is quantified by a uniform distribution between 0.1 and 0.2 as shown in Figure 7.

f Σ (σ )

h=

µσ (1 −η )

1

ηµσ

µσ (1 +η )

µσ = 250 MPa

σ

Figure 6: The density function of the yield strength of steel

f Η (η )

η 0.10

0.15

0.20

Figure 7: Prior distribution of η (uniformly distributed between 0.1 and 0.2)

The maximum deflection of the beam under the 4 point loading is given by δmaxP=Pa24EI3L2-4a2

(1.1)

The variation of maximum deflection is also modeled as a triangular distribution with the mean deflection given by Equation (1.24). The distribution is shown in Figure 8 where θ∈0,1 is an uncertain parameter. The prior information on θ is quantified by a uniform distribution between 0.05 and 0.15 as shown in Figure 9. It is assumed that the random variables representing deflection and strength are independent of each other.

f∆ (σ )

h=

µ δ (1 − θ )

1 θµ ∆

µδ = δ max ( P)

µδ (1 + θ )

σ

Figure 8: The density function of deflection of the beam for a load intensity of P

fΘ (θ )

0.05

0.10

0.15

θ

Figure 9: Prior distribution of θ (uniformly distributed between 0.05 and 0.15)

Model Calibration Suppose n=10 nominally identical beams with dimensions - L=10m ; a = 1m ; b = 0.1m ; h = 0.3m are tested in the laboratory. The following are determined for each beam: (a) Maximum deflection under a load of ξ=70000 N (b) The yield stress of the material of beam

The results of the testing are given in Table 2. Table 2: Laboratory test results for a 4-point loading of 10 identical beams Maximum applied

Maximum Deflection

load (kN)

(m)

1

70

2

Serviceability Damage State

Observed Yield Stress (Nm-2)

0.0186

No

2.14E+08

70

0.0187

No

2.22E+08

3

70

0.0187

No

2.25E+08

4

70

0.0190

No

2.34E+08

5

70

0.0198

No

2.37E+08

6

70

0.0199

No

2.40E+08

7

70

0.0199

No

2.43E+08

8

70

0.0202

Yes

2.43E+08

9

70

0.0204

Yes

2.55E+08

10

70

0.0211

Yes

2.59E+08

Beam #

Following a similar procedure as detailed in Example 1, it is possible to evaluate the posterior density functions of η and θ using Equations (1.20) and (1.21). These posterior densities are shown in Figure 10.

Figure 10(a): Prior distribution and posterior distributions of η

Figure 10(b): Prior distribution and posterior distributions of θ

Fragility Curves The structure can enter any one of the following states: i.

Only d1 - where the fragility is defined as Pfd1=Pσy≤σmaxξ

ii. Only d2 - where fragility is defined as Pfd2=Pδmaxξ≤0.02 iii. Both d1 & d2 - where fragility is defined as Pσy≤σmaxξ & δmaxξ≤0.02

iv. No damage The fragility curves for the above conditions can be constructed based on Equation (1.23). Figures 11 to 13 present the prior and updated fragility curves for the above states.

Figure 11: Prior and posterior (updated) fragility curves for damage state d1

Figure 12: Prior and posterior (updated) fragility curves for damage state d2

Figure 13: Prior and posterior (updated) fragility curves for both damage states d1 & d2

Factors affecting fragility Prior densities of uncertain parameters In the previous example, the probability density function of the strength of the beams fΣσ depends on an uncertain parameter η∈0,1. The prior information on η is assumed to be a uniform distribution between 0.1 and 0.2 (denoted as U(0.1,0.2) ). Uncertainty in these prior distributions can have a significant effect on fragility calculations. If it is assumed that η is a uniform distribution between 0.1 and 0.4, that is, fΗ(η)= U(0.1,0.4), the resulting fragility curves are considerably different from the fragility curves of

the previous example. Figure 14 compares the fragility curves when η is a uniform distribution between 0.1 and 0.4.

Figure 14: Comparison between fragility curves for yield strength damage state for different prior densities on η

Density function of the strength The density function assumed for the yield strength has a direct effect in the calculation of fragility. Figure 15 compares the fragility curves obtained from modeling the yield strength as a triangular and normal distributions.

Figure 14: Comparison between fragility curves for yield strength damage state assuming different density functions for yield strength

Fragility of SDOF system The fragility of a SDOF system subjected to harmonic loading as shown in Figure 15 is examined. It is considered that the structure enters the ‘damage state’ if the amplitude of the steady state vibration of the structure exceeds a limiting amplitude of u*.

P0sinΩt

Figure 15: A general SDOF system

Mechanical Model The maximum steady-state displacement umax of the structure in Figure 15 is given by umax= P0k-mΩ22+(cΩ)212

(1.1)

It can be seen that the value of umax is dependent on the values of P0,k,m,Ω and c. In this case, we consider that the stiffness k of the structure is uncertain, which in turn results in an uncertainty in umax. Figure 16 plots umax with stiffness value k of the structure. Assume that the density function of stiffness variations is given by fKk;θ where θ=(θ1,θ2…θn) is a set of uncertain parameters. Let the prior joint density function of θ be represented by fΘ1,Θ2… Θn'θ1,θ2…θn.

Figure 16: Maximum steady state displacement Vs stiffness of a SDOF system

u k1(u)

k2(u)

Using Equation (1.25), an expression for the density function of Umax can be derived in terms of stiffness variations fKk;θ and uncertain parameters θ. From Figure 16, it can be seen that any given amplitude of steady state motion, u corresponds to two values of stiffness k1(u) and k2(u). We can write the cumulative distribution function (CDF) of Umax as Pumax≤u =PK≤k1(u)+Pk≥k2(u) (1.2) FUmaxu;θ=FKk1u+1-FKk2u

where FKk is the CDF of stiffness variations and can be evaluated by FKk=…FKk;θ fΘ1,Θ2…Θn'θ1,θ2…θn dθ1 dθ2…dθn

(1.3)

Differentiating Equation (1.26) and using the relation between umax and k from Equation (1.25), the density function, fUmaxuobs,i;θ in terms of fKk;θ can be obtained as fUmaxu;θ=4P0u3×fKk1u;θ+fKk2u;θΔ(u) k1(u)=mΩ2-Δu; k2(u)=mΩ2+Δu

(1.4)

Δu=P0u2-CΩ2

Lastly, the fragility can be defined as the probability that the maximum steady state displacement of the system, umax exceeds the limit state displacement u* at a given loading amplitude P0 i.e. Pumax≥u* | P0. From Equation (1.26), we can write Pumax≥u* | P0=Pk≤k2*-Pk≤k1* =FKk2*-FKk1*

(1.5)

The FKk are determined from Equation (1.27).

Model Calibration Suppose m identical specimens of the structure are tested to measure the maximum steady state displacement for the loading condition above. We can employ the experimental data to update the fragility curves in a Bayesian framework. Let the set of observed amplitudes be uobs,1,… uobs,i…uobs,m. The likelihood function of uncertain parameters, l(θ|data) corresponding to

the m observations can be written as

lθdata∝i=1mfUmaxuobs,i;θ

(1.6)

Using Equations (1.28) and (1.30), the likelihood function, l(θ|data) is simplified as lθdata∝i=1nfKk1uobs,i;θ+fKk2uobs,i;θ

(1.7)

where k1uobs,i and k2uobs,i are the two values of stiffness that can yield the ith observation, uobs,i. The posterior joint density of the uncertain parameters fΘ1,Θ2…Θn''θ1,θ2…θn can be

now be written using the prior information and the likelihood function: fΘ1,Θ2…Θn''θ1,θ2…θn=lθdata× fΘ1,Θ2…Θn'θ1,θ2…θn

(1.8)

Updated Fragility Curves Using the posterior densities of the random parameters, we can update the fragility curve by Pumax≥u* | P0=Pk≤k2*-Pk≤k1* =FK''k2*-FK''k1* FK''k=…FKk;θ fΘ1,Θ2…Θn''θ1,θ2…θn dθ1 dθ2…dθn

(1.9) (1.10)

Seismic Fragility of Real-Life Systems In the last section, the simple case of a SDOF system subjected to harmonic loading was considered to estimate the fragility of steady-state amplitude due to an uncertainty in stiffness. To extend the same approach to a MDOF system, the following considerations need to be addressed: •

Typically in MDOF systems, uncertainties in multiple factors like stiffness, damping ratios, ground motion input can influence the response Rt of the structure. Hence, the definition of the mechanical model should probabilistically characterize (using prior information) all required kinds of uncertainties.

•

Several configurations of random parameters can yield the same response as observed in an experiment. It is a complex process to determine all the configurations of random parameters and their probabilities of occurrence that correspond to an observed value of response so as to obtain the likelihood function of uncertain parameters.

•

In MDOF systems, a closed-form relationship between the response Rt and the physical uncertainties in the mechanical model of the structure is not available. This makes the determining posterior distributions of uncertain parameters of the model difficult using likelihood functions very difficult;

Monte-Carlo Simulation based Fragility In the previous section, the limitations associated with determining the seismic fragility of a MDOF system by direct consideration of uncertain parameters and their statistics are outlined. One way to overcome this problem is to estimate the fragility empirically by conducting multiple analyses in a Monte Carlo simulation. The set of uncertain parameters θ=(θ1,θ2…θn) can be modeled and simulated as random variables to perform multiple analyses with each instance of simulation to obtain multiple responses. The fragility can be estimated empirically from the set of multiple responses. The procedure is illustrated using a primary-secondary type of coupled system. Fragility of Primary-Secondary Systems –Example Monte-Carlo based Fragility The primary-secondary system as shown in Figure 18 is considered for this example. The fragility of the secondary system is defined as the probability that the relative displacement of the secondary system exceeds 0.2 m given a particular peak ground acceleration (PGA) i.e. Pf=P[us≥0.2|ugmax]. The seismic response of the secondary system can be obtained by a

coupled analysis using modal synthesis approach. The secondary system response is sensitive to various uncertainties in structural parameters as well as ground motion input.

Figure 18: A Primary-Secondary system

USNRC (1978) and ASME (2007) require that an uncertainty of ±15% to be considered in primary system’s frequencies. Accordingly, the frequencies of primary and secondary system are modeled as uniform random variables within the ±15% range, represented by Ωp and Ωs respectively. 100 different models were simulated by random sampling of frequency sets (Ωp,Ωs). The uncertainty in ground motion is accounted by considering 75 different time

histories all normalized to a unit PGA. Each of the 100 systems was analyzed with 75 earthquake histories producing 7500 secondary system responses. It must be noted that all of these responses correspond to a peak ground acceleration of unit PGA. To obtain the responses at a different PGA, the responses can be scaled by the value of the PGA with the assumption of linear elastic structural dynamics. The responses of the simulation may be arranged in the form of a matrix as Us= u1,1⋯u1,100⋮⋱⋮u75,1⋯u75,100

(1.1)

where ui,j represents the maximum secondary system relative displacement of jth model subjected to ith earthquake time history. The fragility curve can be directly evaluated over the entire set of responses using Pf=P[k.ui,j≥0.2|PGA=k]. Figure 19 plots the empirical fragility curve for the system.

Bayesian Updating of Fragility Curve Suppose m=10 nominally identical models of the P-S system are tested by subjecting them to base excitation at a level of unit PGA. Table 3 lists the observations recorded. Table 3: Maximum secondary system relative displacements as recorded Model #

Maximum Secondary System Displacement (m)

PGA(g)

1

0.208

1

2

0.224

1

3

0.092

1

4

0.225

1

5

0.177

1

6

0.087

1

7

0.117

1

8

0.163

1

9

0.233

1

10

0.234

1

To update the fragility curve within the Bayesian framework, it is necessary to characterize the variation of the secondary system response by a probability density function with random parameters. The experimental data can be used to obtain the posterior densities of the random parameters of such a probabilistic model. In this example, the response of the secondary system is modeled by a Generalized Extreme Value (GEV) distribution. The cumulative density function of a variable following generalized extreme value distribution is given by FUu;μ,σ,ξ=exp-1+ξu-μσ-1ξ

(1.2)

The model has three random parameters- θ =(μ,σ,ξ). Each column of Us in Equation (1.35) contains 75 responses for a single model. Fitting a GEV distribution for each column of the 100 columns yields a set of parameter values for μ,σ,ξi. The 100 sets of parameter values,

μ,σ,ξi=1…100 can be used to estimate the prior distributions for the parameters μ,σ and ξ. The

prior distributions of the random parameters are estimated as given in Equation (1.37). fΝ'μ=U0.7,2.4 fΣ'σ=U0.1,0.3

(1.3)

fΞ'ξ=U0.15,0.25

where Ua,b indicates a uniform distribution between a & b and zero elsewhere. Figure 19 compares the empirical fragility with the fragility computed from the probabilistic model. The Bayesian updating process and the fragility curve is similar to the earlier examples. The following Equations (1.38) – (1.39) outline the process: fΝ,Σ,Ξ'μ,σ,ξ=fΝ'μ.fΣ'σ.fΞ'ξ lμ,σ,ξ|data: uobs,1,…uobs,m∝i=1mfUuobs,i|μ,σ,ξ

(1.4)

fΝ,Σ,Ξ''μ,σ,ξ=fΝ,Σ,Ξ'μ,σ,ξ×lμ,σ,ξ|data Pus≥0.2ugmax=Pus≥0.2ugmax,μ,σ,ξ×fΝ,Σ,Ξ''μ,σ,ξdμ dσ dξ

(1.5)

Figure 20 gives the updated fragility using the set of 10 experimental results.

Figure 19: Empirical and probabilistic model based fragility estimates for secondary system displacement

Figure 20: Prior and posterior fragility estimates for the secondary system displacement