Seismic fragility curves for structures using non-parametric representations

doi:10.1007/s11709-017-0385-y

Front. Struct. Civ. Eng.

2017, Vol. 11

Issue (2) : 169-186 https://doi.org/10.1007/s11709-017-0385-y

RESEARCH ARTICLE

Seismic fragility curves for structures using non-parametric representations

Chu MAI, Katerina KONAKLI, Bruno SUDRET(

)

Risk, Safety & Uncertainty Quantiﬁcation, Institute of Structural Engineering, ETH Zürich, Zürich CH-8093, Switzerland

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Abstract

Fragility curves are commonly used in civil engineering to assess the vulnerability of structures to earthquakes. The probability of failure associated with a prescribed criterion (e.g., the maximal inter-storey drift of a building exceeding a certain threshold) is represented as a function of the intensity of the earthquake ground motion (e.g., peak ground acceleration or spectral acceleration). The classical approach relies on assuming a lognormal shape of the fragility curves; it is thus parametric. In this paper, we introduce two non-parametric approaches to establish the fragility curves without employing the above assumption, namely binned Monte Carlo simulation and kernel density estimation. As an illustration, we compute the fragility curves for a three-storey steel frame using a large number of synthetic ground motions. The curves obtained with the non-parametric approaches are compared with respective curves based on the lognormal assumption. A similar comparison is presented for a case when a limited number of recorded ground motions is available. It is found that the accuracy of the lognormal curves depends on the ground motion intensity measure, the failure criterion and most importantly, on the employed method for estimating the parameters of the lognormal shape.

Keywords earthquake engineering fragility curves lognormal assumption non-parametric approach kernel density estimation epistemic uncertainty

Corresponding Author(s): Bruno SUDRET

Online First Date: 17 April 2017 Issue Date: 19 May 2017

Cite this article:

Chu MAI,Katerina KONAKLI,Bruno SUDRET. Seismic fragility curves for structures using non-parametric representations[J]. Front. Struct. Civ. Eng., 2017, 11(2): 169-186.

URL:

https://academic.hep.com.cn/fsce/EN/10.1007/s11709-017-0385-y
https://academic.hep.com.cn/fsce/EN/Y2017/V11/I2/169

Fig.1 (a) Maximal drifts versus IM before scaling; (b) maximal drifts versus IM after scaling within the bin marked with red color. (Note: a large bin is considered in the figure only to facilitate visualization of the method)

Fig.2 (a) Steel frame structure; (b) hysteretic behavior of steel material at section 1-1 for an example ground motion.

parameter	distribution	support	$μ X$	$σ X$
I_a (s× g)	lognormal	(0, + ∞)	0.0468	0.164
D_5-95 (s)	beta	[,]	17.3	9.31
t_mid (s)	beta	[0.5, 40]	12.4	7.44
$ω m i d / 2 π (H z)$	gamma	(0, + ∞)	5.87	3.11
$ω' / 2 π (H z)$	two-sided exponential	[-2, 0.5]	-0.089	0.185
$ς f$	beta	[0.02, 1]	0.213	0.143

Tab.1 Statistics of synthetic ground motion parameters according to Ref. [].

Fig.3 Examples of synthetic ground motions.

		PGA	Sa
$δ o$	approach	median	log-std	median	log-std
0.7%	MLE	0.35 g	0.70	0.49 g	0.36
	LR	0.37 g	0.64	0.44 g	0.13
	KDE	0.36 g		0.45 g
1.5%	MLE	1.10 g	0.56	1.66 g	0.31
	LR	0.87 g	0.64	1.47 g	0.24
	KDE	1.08 g		1.53 g
2.5%	MLE	1.76 g	0.56	2.82 g	0.37
	LR	1.55 g	0.64	3.29 g	0.24
	KDE	1.82 g		3.04 g

Tab.2 Steel frame structure−parameters of the obtained fragility curves

Fig.4 Paired data {(IM_i, D_i), i = 1,…,N} and fitted models in log-scale (the units of the variables in the fitted models are the same as in the axes of the graphs)

Fig.5 Fragility curves with parametric and non-parametric approaches using PGA and Sa as intensity measures (LR: linear regression; MLE: maximum likelihood estimation; bMCS: binned Monte Carlo simulation; KDE: kernel density estimation)

Fig.6 Histograms and fitted normal distributions for lnD at two levels of PGA and Sa. (a) PGA = 0.5 g; (b) PGA = 1.5 g; (c) Sa = 0.5 g; (d)Sa = 1.5 g

$δ o$	approach	PGA	Sa
0.7%	bMCS	0.0003 g	0.005 g
0.7%	KDE	0.0005 g	0.005 g
1.5%	bMCS	0.037 g	0.054 g
1.5%	KDE	0.037 g	0.050 g
2.5%	bMCS	0.114 g	0.090 g
2.5%	KDE	0.120 g	0.080 g

Tab.3 Log-standard deviation of median IM

Fig.7 Estimated and mean bootstrap fragility curves and 95% confidence intervals for the binned Monte Carlo simulation and the kernel density estimation approaches. (a) Binned Monte Carlo simulation (PGA); (b) Kernel density estimation (PGA); (c) Binned Monte Carlo simulation (Sa); (d) Kernel density estimation (Sa)

Fig.8 Bridge configuration [] and hysteretic behavior.

reference	level	description	damage	drift ratio $δ o$
[]	II	Operationa	Minor	0.01
[]	III	Life safety	Moderate	0.03
[]	II	Operationa	Minor	0.005
[]	III	Life safety	Moderate	0.015

Tab.4 Bridge performance and respective drift-ratio threshold

Fig.9 Paired data {(IM_i, D_i), i = 1,…,N} and fitted models in log-scale (the units of the variables in the fitted models are the same as in the axes of the graphs).

Fig.10 Fragility curves with parametric and non-parametric approaches using PGA and P_sa as intensity measures (LR: linear regression; MLE: maximum likelihood estimation; KDE: kernel density estimation.

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