Estimating Structural Reliability Under Hurricane Wind Hazard : Applications to

Wood Structures

Balaji Rajagopalan, Edward Ou, Ross Corotis and Dan Frangopol

Department of Civil, Environmental and Architectural Engg.

University of Colorado

Boulder, CO

COALESCE – Spring 2004

Acknowledgements

Funding for this work is provided by NSF grant SGER (CMS-0335530)

Results from this work are being written up as a paper for Probabilistic Mechanics Conference – 04 (Albuquerque, NM, July 2004)

Motivation• Insured losses in the US from “natural hazards”

reached $22 billion in 1999• Second largest loss during 1990’s - $26 billion in

1992 due to Hurricane Andrew (in Florida and Louisiana) Topics (2000 - Munich)

• The U.S. House of Representatives, is working on bill H.R. 2020 - Hurricane, Tornado and Related Hazards Research Act, to promote :

inter-disciplinary research in understanding and mitigating windstorm related hazard impacts

new methodologies for improved loss estimation and risk assessment

Property Loss due to Hurricanes in the US

Hurricane Tracks - 2000

ENSO as a “free” mode of the coupled ocean-atmosphere dynamics in the Tropical Pacific Ocean

Significant Differences in Atlantic Hurricane attributes relative to NINO3 phases

Rajagopalan et al., 2000

Motivation(i) Often, structural reliability is estimated in isolation of

realistic likelihood estimates of hurricane frequencies and magnitudes.

(ii) Knowledge of year-to-year variability in occurrence and steering of hurricanes in the Atlantic basin is not incorporated in structural reliability estimation.

(iii) The estimation of losses is purely empirical, based on the wind speed and no consideration of structural information. (For example, a new structure and a 25 year old structure are assumed to have the same probability of failure for a given wind speed.)

(iv) The life cycle cost of structures is also not considered substantial misrepresentation of losses and consequently sub-optimal decision making.

Proposed Framework

Structural Reliability Estimation

Steps:1. Generate scenarios of maximum wind speeds

conditioned on large-scale climate information. - i.e. simulate from conditional PDF

f(wind speed | climate)“Load Scenarios”

2. Scenarios generated for different large-scale climate states (El Nino, La Nina)

3. Convert the maximum wind speed to 3-second gust (gust correction factor, Simiu, 1996)

4. “convolute” with fragility curves to estimate the failure probability – consequently the reliability

5. Considered 25 year time horizon, wooden structures

Walls - W

Roof Cover - T

Openings - O

Roof Sheathing - S

Roof to Wall Connections - C

Data for wind scenario

1. Historical Hurricane track data from http://www.nhc.noaa.gov

2. Get the historical track for the region of interest

(2deg X 2deg box over N. Carolina)

3. Estimate the annual maximum hurricane wind speed for the grid box (wind speed)

4. Climate information (e.g., El Nino index) is obtained from http://www.cdc.noaa.gov (climate index)

5. Simulate scenarios from the conditional PDF f(wind speed | climate)

Nonparametric Methods

• Kernel Estimators

(properties well studied)• Splines• Multivariate Adaptive Regression Splines (MARS)

• K-Nearest Neighbor Bootstrap estimators• Locally Weighted Polynomials

• http://civil.colorado.edu/~balajir

0

0.25

0.5

0.75

1

xt

0 25 50 75 100 125

time

•

••

•••

S

DiD2D1D3•

•

1

3

2

Values of xt

A time series from the model

xt+1 = 1 - 4(xt - 0.5)2

Logistic Map Example

State

0

0.25

0.5

0.75

1

xt+1

0 0.25 0.5 0.75 1

xt

A B

1

1

2 3 4

2

3

4

State

x*A x*B

k-nearest neighborhoods A and B for xt=x*A and x*B respectively

4-state Markov Chain discretization

K-NN Local Polynomial

Nonparametric Methods

• A functional (probability density, regression etc.) estimator is nonparametric if:

It is “local” – estimate at a point depends only on a few neighbors around it.

(effect of outliers is removed)

No prior assumption of the underlying functional form – data driven

Classical Bootstrap (Efron):

Given x1, x2, …... xn are i.i.d. random variables with a cdf F(x)

Construct the empirical cdf

Draw a random sample with replacement of size n from

n

iniIF

1/)()(ˆ xxx

)(ˆ xF

Moving Block Bootstrap (Kunsch, Hall, Liu & Singh) :

Resample independent blocks of length b<n, and paste them together to form a series of length n

k-Nearest Neighbor Conditional Bootstrap (Lall and Sharma)

Construct the Conditional Empirical Distribution Function:

Draw a random sample with replacement from

n

ikiK

krBiIiIF

1/)(*))(()(*)|(ˆ DDxxDx

*)|(ˆ DxF

Define the composition of the "feature vector" Dt of dimension d.

(1) Dependence on two prior values of the same time series.Dt : (xt-1, xt-2) ; d=2

(2) Dependence on multiple time scales (e.g., monthly+annual)Dt: (xt-1, xt-21, .... xt-M11; xt-2, xt-22, ..... xt-M22) ; d=M1+M2

(3) Dependence on multiple variables and time scales Dt: (x1t-1, .... x1t-M11; x2t, x2t-2, .... x2t-M22); d=M1+M2+1

Identify the k nearest neighbors of Dt in the data D1 ... Dn

Define the kernel function ( derived by taking expected values of distances to each of k nearest neighbors, assuming the number of observations of D in a neighborhood Br(D*) of D*; r0, as n , is locally Poisson, with rate (D*))

for the jth nearest neighbor

Selection of k: GCV, FPE, Mutual Information, or rule of thumb (k=n0.5)

1...k =j 1/j

1/jK(j)k

1i

Applications to date….

• Monthly Streamflow Simulation

• Multivariate, Daily Weather Simulation

• Space and time disaggregation of monthly to daily streamflow

• Monte Carlo Sampling of Spatial Random Fields

• Probabilistic Sampling of Soil Stratigraphy from Cores

• Ensemble Forecasting of Hydroclimatic Time Series

• Downscaling of Climate Models

• Biological and Economic Time Series

• Exploration of Properties of Dynamical Systems

• Extension to Nearest Neighbor Block Bootstrapping -Yao and Tong

ENSO index

Joint PDF of Max. Wind Speed and ENSO index

La Nina Years

El Nino YearsAll Years

Neutral Years

Histogram of #of Hurricane Occurrences over N. Carolina –With Respect to Large-scale Climate

ENSO index

WindSpeed

Joint PDF of Max. Wind Speed and ENSO index

WindSpeed

Joint PDF of Max. Wind Speed and ENSO indexAll Year Simulations

ENSO index

WindSpeed

Joint PDF of Max. Wind Speed and ENSO index

Historical CDF

El Nino YearSimulations

Failure Due to Panel Uplift

Failure due to Roof-to-wall Separation

Gust Effect - Failure due to Panel Uplift

Summary

• Integrated (Interdisciplinary) framework to estimate infrastructure risk due to hurricane hazard is presented

• Nonparametric method is used to generate hurricane wind scenarios conditioned on large-scale climate state (El Nino, La Nina etc.)

• Large-scale climate state appears to impact the number of hurricanes, maximum wind speed and consequently, infrastructure risk (over N. Carolina)

Further Extensions– Extension to other types of structures

(concrete, bridges etc.)

– Investigate gust correction factors for hurricane winds

– Study the impact of time-varying infrastructure risk estimation on the loss estimates

– Incorporate other relevant climate information for Hurricane occurrence and steering (such as, North Atlantic Ocean and Atmospheric conditions)

– Integrating life-cycle cost for optimal decision making on maintenance and replacement