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Lin2008-WindborneDebrisRA.pdf
Probabilistic Engineering Mechanics 23 (2008) 523–530
www.elsevier.com/locate/probengmech
Windborne debris risk assessment
Ning Lin ∗ , Erik Vanmarcke
Department of Civil and Environmental Engineering, Princeton University, Princeton, NJ, USA
Received 31 July 2007; accepted 28 January 2008
Available online 4 March 2008
Abstract
A probabilistic model, based on Poisson random measure theory, is developed for predicting windborne debris damage in residential areas. It
is intended for incorporation into improved methodology to estimate economic losses due to hurricanes to individual houses or entire residential
developments. A sample application to a coastal housing development is provided.
c 2008 Elsevier Ltd. All rights reserved.
Keywords: Windborne debris; Hurricane risk; Poisson; Probability; Statistics; Wind tunnel
1. Introduction
Windborne debris is known to be a major source of damage
in strong winds such as during hurricanes. Windborne debris
modeling is logically a key component in evaluating hurricane
risk to buildings (and, especially, clusters of buildings).
However, there are few published articles on windborne debris
compared to the literature on loads caused by wind pressure,
and especially on the inherent probabilistic aspects of the
debris-damage problem. Twisdale et al. [1] developed an
integrated methodology to analyze debris risk in residential
areas, combining numerical models of the hurricane wind
field and debris generation, trajectory, and impact. They also
produced reliability curves for typical residential environments,
which lie at the basis of recommendations in ASTM [2] for
debris impact risk analysis. However, their probabilistic model
assumes that impact parameters (e.g., number of impacts,
momentum at impact) are identically distributed for all the
houses in the study area, and hence it is mainly suitable for
estimating mean debris risk in an area. This paper proposes a
methodology to predict the debris risk to individual buildings
in a development, as well as the aggregate debris risk. First,
a debris risk assessment methodology is established based on
Poisson random measure theory (see, e.g., Cinlar [3]). Second,
the approach is shown, with reasonable approximation, to be
amenable to matrix formulation. Next, probability distributions
∗ Corresponding author.
E-mail address: [email protected] (N. Lin).
c 2008 Elsevier Ltd. All rights reserved.
0266-8920/$ - see front matter doi:10.1016/j.probengmech.2008.01.010
characterizing the stochastic debris trajectories are proposed
and their parameters are obtained based on the experimental
debris trajectory models developed by Lin et al. [4,5]. Finally,
the way to incorporate the debris-damage model into hurricane
risk analysis is discussed.
2. Debris risk model
The debris risk model relies on Poisson random measure
theory to predict the impact damage to a residential area
under hurricane wind conditions, due to debris generated
from building sources. Other debris sources (e.g., street signs,
trash cans) are, at this stage of model development, ignored,
as they are mainly susceptible to vertical updrafts occurring
during tornadoes. We first define a relatively isolated region,
containing the structures of interest, such that the region is
not likely to interact with the outside in terms of debris
damage. Denote the residences by integers 1, 2, . . . , I , where
I is their total number within the region. Every house can
generate debris and can be hit by debris generated from any
house (itself included) in the region, depending on the wind
conditions (e.g., mean wind speed and direction, and degree
of turbulence). We use the subscript i (i = 1, 2, . . . , I ) to
identify the properties of a house when it is regarded as a debris
source house, and j ( j = 1, 2, . . . , I ) when it is regarded
as an impact target house. Let A j denote the area occupied
by house j and E the union of all A j . Suppose there are
Si types of debris generated from house i, and denote S =
max{Si, i = 1, 2, . . . , I }. In a residential area, for instance, the
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N. Lin, E. Vanmarcke / Probabilistic Engineering Mechanics 23 (2008) 523–530
roof structure of some houses may become partially damaged
and roofing materials may become windborne as debris. The
common debris types that have been observed in hurricane
damage surveys are roof covers, roof sheathings, and 2 × 4
timbers.
We focus first on the impacts due to one type of debris
generated from one arbitrary house, under a prescribed
wind condition. Let the number of items of type s debris
(s ≤ Si ) generated from house i be a random variable
L s,i , and assume it has the Poisson distribution, with mean
value λs,i . Define the landing positions of these debris
items as independent identically distributed random variables
1 , . . . , X ls,i , . . . , X Ls,i , with µ (dx) as their common
X s,i
s,i
s,i
s,i
probability distribution. Define Hs,i (A j ) as the number of
debris items that will land on the area occupied by house j.
Hence, the number of items of debris hitting house j may be
expressed as:
Hs,i (A j ) =
L s,i
X
l
s,i
δ(X s,i
, A j ),
where δ is the identity kernel, so that δ(x, A) = 1 if x belongs
to A, and δ(x, A) = 0 otherwise.
Under these conditions, it can be shown that Hs,i is a
Poisson random measure on E, with mean value υs,i (dx) =
λs,i µs,i (dx). This means that for every A j (A j ⊂ E, j =
1, 2, . . . , I ), the random variable Hs,i (A j ) has the Poisson
distribution with the following mean value υs,i (A j ) (i.e., the
mean number of hits on house j):
Z
µs,i (dx).
(2)
υs,i (A j ) = λs,i
Aj
Specifically, the probability that house j suffers exactly N
impacts is:
e−υs,i (A j ) [υs,i (A j )] N
.
N!
(3)
Define the horizontal impact momentum of debris item ls,i
(hitting on the ground or on a house) as a random variable
ls,i
Ys,i
(As noted earlier, vertical momentum is of secondary
importance in the hurricane wind condition; see Lin et al. [4]).
ls,i
ls,i
We may write Ys,i
= m s,i u ls,i
s,i , where u s,i is the horizontal
impact speed of the debris item and m s,i is its mass (assumed
to be a known constant for a given debris type). It has
been found, from experiments and numerical simulations, that
the horizontal impact speed of a debris item is primarily a
function of the distance traveled (e.g., Lin et al. [4], Holmes
et al. [6], and Lin et al. [5]). Hence, we define φs,i (dy|x) as the
common conditional distribution, for every item of debris ls,i =
ls,i
ls,i
1, 2, . . . , L s,i , of Ys,i
given X s,i
, the landing position of the
1 , . . . , X ls,i , . . . , X Ls,i } and
debris item. Denoting X s,i = {X s,i
s,i
s,i
1 , . . . , Y ls,i , . . . , Y Ls,i }, it is reasonable to assume
Ys,i = {Ys,i
s,i
s,i
ls,i
that the different elements Ys,i
are conditionally independent
given X s,i . Then let Ms,i denote the random measure formed
by X s,i and Ys,i . It can be shown that Ms,i is a Poisson random
ζj
Aj
= λs,i
Z
µs,i (dx)Φs,i (y > ζ j |x),
(4)
Aj
where Φs,i (y > ζ j | j) = 1 − Φs,i (y ≤ ζ j | j) is the
complementary cumulative distribution function evaluated at
the threshold ζ j . Then the probability that house j suffers N
over-threshold impacts is:
(1)
ls,i =1
P{Hs,i (A j ) = N } =
measure on E × R + (where R + stands for the positive real line),
with mean measure νs,i (dx, dy) = υs,i (dx)φs,i (dy|x). Suppose
the impact resistance of a particular (type of) vulnerable area
(e.g., windows) of house j is ζ j , a threshold expressed in
terms of the horizontal impact momentum. Let Ms,i (A j, y >
ζ j ) denote the number of debris items that landed within the
area occupied by house j and that have horizontal momentum
exceeding the threshold ζ j . Let αs,i ( j) denote the mean number
of these over-threshold impacts on house j,
Z ∞
Z
µs,i (dx)
φs,i (dy|x)
αs,i ( j) = νs,i (A j , y > ζ j ) = λs,i
P{Ms,i (A j , y > ζ j ) = N } =
e−αs,i ( j) [αs,i ( j)] N
.
N!
(5)
Consider now the behavior of arbitrary types of debris
generated from all houses, under a prescribed wind condition,
in the region. It can be assumed, to the first approximation,
that debris items fly independently of each other, as debris
items typically fly close to the mean wind direction so that the
probability that debris items interact with each other in the air is
relatively small. (Since the generation of an item of debris may
depend on the generation of other debris items from the same
house, a useful extension of the model might be to consider
randomly-sized clusters of debris items, and interdependence
in the generation of different types of debris.)
We construct Ms,i for s = 1, 2, . . . , S and i = 1, 2, . . . , I
as described above. (Ms,i = 0, in case s > Si .) Then Ms,i are
independent Poisson random measures on E × R + , with mean
measures νs,i , respectively. Let M be the sum of Ms,i . Again,
M is a Poisson random measure on E × R + , with mean measure
ν given by:
ν(dx, dy) =
S X
I
X
νs,i (dx, dy)
s=1 i=1
=
S X
I
X
λs,i µs,i (dx)φs,i (dy|x).
(6)
s=1 i=1
Denoting by M(A j, y > ζ j ) the total number of overthreshold impacts on house j, we have
M(A j , y > ζ j ) =
S X
I
X
Ms,i (A j , y > ζ j ).
(7)
s=1 i=1
Similarly, υ(A j ), the mean total number of impacts on house j,
is
Z
S X
I
S X
I
X
X
υ(A j ) =
υs,i (A j ) =
λs,i
µs,i (dx),
(8)
s=1 i=1
s=1 i=1
Aj
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N. Lin, E. Vanmarcke / Probabilistic Engineering Mechanics 23 (2008) 523–530
and α( j), the mean total number of over-threshold impacts on
house j, is
α( j) =
S X
I
X
αs,i ( j) =
s=1 i=1
=
S X
I
X
s=1 i=1
S X
I
X
νs,i (A j , y > ζ j )
s=1 i=1
λs,i
Z
µs,i (dx)Φs,i (y > ζ j |x).
(9)
Aj
The probability P( j, N ) that house j suffers, in total, N
over-threshold impacts is:
P( j, N ) = P{M(A j , y > ζ j ) = N } =
e−α( j) [α( j)] N
.
N!
(10)
Furthermore, denoting by Pζ ( j, n) the probability of n overthreshold impacts on the vulnerable area of house j, we may
write:
Pζ ( j, n) =
∞
X
Pζ ( j, n|N )P( j, N ),
(11)
N =n
wherePζ ( j, n|N ) is the conditional probability of n impacts
on the “vulnerable area” (e.g., the windows) of house j, given
that there are N impacts on house j. Here, we define “debris
damage” to a house as meaning that its vulnerable area suffers
at least one over-threshold impact. Then the probability of
debris damage to house j, denoted by PD ( j), is:
PD ( j) =
∞
X
Pζ ( j, n) =
n=1
∞ X
∞
X
Pζ ( j, n|N )
n=1 N =n
e−α( j) [α( j)] N
.
N!
(12)
If we assume that the hits on the building envelope are
uniformly distributed (as in [1]), and denoting by q j the
vulnerable fraction (the ratio of the vulnerable area to the
area of the building envelope), then Pζ ( j, n|N ) can be
approximated by the binomial distribution,
N
Pζ ( j, n|N ) =
q nj (1 − q j ) N −n .
(13)
n
Substituting Eq. (13) into Eq. (12), we obtain the following
expression for the probability of debris damage to a house:
PD ( j) = 1 − e−q j α( j) ,
the buildings (so that it is, by definition, isolated from the
outside in terms of debris effect), and (2) the distribution of
debris landing locations is likely to be widely spread out in
space, due to turbulence effects and other complex factors,
e.g., imperfect debris-item shapes owing to highly variable
pressure damage. Consequently, the calculation of α( j), the
only quantity required to estimate the probability of debris
damage using Eq. (14), can usually be simplified.
Denoting by µs,i ( j) the value of the distribution µs,i at
the center point (defined in terms of longitude and latitude) of
house j and by υs,i ( j) the mean number of hits, the integral in
Eq. (8) may be approximated as a summation:
(14)
which depends on its vulnerable fraction and mean number of
over-threshold impacts.
3. Discretization and matrix formulation
In practice, if the study region is relatively large compared
to an individual building’s plan area and the continuous
probability density of debris landing positions varies negligibly
within the typical area size of the building, it can be assumed
that the value of µs,i is constant within the area occupied
by each house (A j ). This assumption generally holds because
(1) the debris problem often involves a dense residential
development, with the study region defined to include all
υ( j) =
S X
I
X
υs,i ( j) =
s=1 i=1
S X
I
X
λs,i µs,i ( j)A j .
(15)
s=1 i=1
Similarly, house locations may also be approximated by their
center points so that the horizontal impact momentum is only
conditioned on the distances between the center points of
the source house and the target house. Let φs,i (dy| j) be the
conditional distribution of the horizontal momentum of type s
debris generated from house i, when hitting house j. Eq. (9) for
the mean total number of over-threshold impacts becomes:
α( j) =
S X
I
X
αs,i ( j)
s=1 i=1
=
S X
I
X
λs,i µs,i ( j)A j Φs,i (y > ζ j | j).
(16)
s=1 i=1
It is worth mentioning that the scale of debris flight distances
in the relatively large study region is different in concept
from the scale of debris impact on the vulnerable area of a
particular building. A house may be approximated as a point
when evaluating the probability of debris hits and the possible
impact intensities, neglecting the effect of house dimensions in
order to simplify the calculation. However, the specific debris
impact location on the building envelope matters, since the
debris can only cause damage when it hits the vulnerable area
with an over-threshold momentum, as is specifically considered
in Eq. (11).
Adopting the above approximations, a matrix presentation
can be formulated. In particular, the square matrix (Os ) I ×I of
the mean number of impacts by type s debris (i.e., the matrix
element (Os )i, j = υs,i ( j) is the mean number of hits on house
j by type s debris generated from house i) may be expressed by
matrix multiplication in terms of the diagonal matrix (Λs ) I ×I
with diagonal elements (Λs )i,i = λs,i , the square matrix
(Θs ) I ×I with elements (Θs )i, j = µs,i ( j), and the diagonal
matrix (A) I ×I with diagonal elements (A) j, j = A j , as follows:
Os = Λs Θs A.
(17)
Also, the square matrix (∆s ) I ×I of the mean number of
over-threshold impacts by type s debris, with elements
(∆s )i, j = αs,i ( j), may be obtained through element-byelement multiplication of (Os ) I ×I and (Φs ) I ×I , where (Φs ) I ×I
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N. Lin, E. Vanmarcke / Probabilistic Engineering Mechanics 23 (2008) 523–530
0
is the square matrix with elements (Φs )i, j = Φs,i (y > ζ j | j):
x
We may assume that the values of the standard deviations, σs,i
∆s = Os · Φs .
z
and σs,i
, equal some fraction of ds,i . For example, in case
0
(18)
0
Summing up the matrices for all types of debris, we obtain
the matrix of the mean total number of impacts, denoted by
(O) I ×I , and the matrix of the total number of over-threshold
impacts, denoted as (∆) I ×I , respectively:
O=
S
X
Os ,
(19)
∆s .
(20)
s=1
and
∆=
S
X
s=1
Finally, the value of υ( j) is the sum of the elements in the jth
column of the matrix O, and the value of α( j) equals the sum
of the elements in the jth column of the matrix ∆.
4. Probabilistic modeling of debris trajectories
The aim of the stochastic debris trajectory model is to obtain
the probability distribution of debris landing positions, µs,i ,
and the conditional distribution of the horizontal momentum,
φs,i , for each type of debris generated from each house,
under prescribed wind conditions (including mean wind speed
and direction). Although investigations on building damage
have been carried out after each significant storm, historical
information on debris flight behavior is generally not available.
Therefore, the probability distributions of debris trajectory
parameters may have to be established based on experiments
and/or numerical simulations, and with some help from
intuition and common sense, considering the complexity of the
problem in real situations.
We model the probabilistic distribution of the landing
position µs,i as 2D Gaussian. This is motivated by the
Tachikawa’s [7] observation in wind-tunnel experiments on
debris trajectories. He placed a catch-net perpendicular to the
direction of the wind at various distances in front of the debris
original position and found that debris impact locations were
almost always (approximately) uniformly distributed within
circles in the center of the net, suggesting the Gaussian
distribution to be a good candidate, at least for an initial
model. Given a mean wind speed and direction, we first
rotate the geographical x–z coordinate system (x j -longitude
and z j -latitude of house j, the effect of earth’s curvature being
neglected) to a new x 0 –z 0 system such that the wind direction
is in the positive x 0 direction. Then assume that debris will
most likely fly in (or close to) the mean wind direction and thus
the mode (and mean) of the 2D Gaussian distribution of debris
landing is in the positive x 0 direction at the most likely debris
flight distance (at landing), denoted as ds,i , from its original
position (house i). Thus we express the Gaussian distribution
of debris landing as
"
µs,i ( j) =
−
1
0
0
x σz
2π σs,i
s,i
e
(x 0j −xi0 −ds,i )2 (z 0j −z i0 )2
+
x 0 )2
z 0 )2
2(σs,i
2(σs,i
#
.
(21)
0
z
x = (1/3)d
σs,i
s,i and σs,i = (1/12)ds,i , the debris item is
estimated to have the probability 0.9919 of landing within a
rectangular area centered at its most likely landing point, and
with the length of the longer side equal to 2ds,i and the length
of the shorter side equal to (1/2) ds,i . Note that it is possible that
a debris item will hit its source building or fly in the direction
opposite to the mean wind, due to local turbulence effects.
Although intuitively negligible, this possibility is allowed for
in the Gaussian model (and has a probability less than 0.002
in this example). Note that, in reality, the target house that is
located slightly closer than the distance ds,i to the source house
may have the largest probability of its envelope being hit, due
to the effects of house dimensions. In this context, we suggest
replacing ds,i in Eq. (21) with (ds,i −h/2), where h is the typical
height of house eaves in the region under study, so that µs,i
will more realistically represent the probability distribution of
houses being hit.
The Gaussian distribution could also be adopted, mainly
for analytical convenience, for the conditional probability
distribution φs,i of debris horizontal impact momentum,
although the preferred stochastic model is the lognormal
distribution (as lognormal random variables are non-negative).
The random variable Ys,i| j = m s,i u s,i| j is the horizontal
momentum of type s debris generated from house i when
hitting house j; its parameters are the mean Ȳs,i| j = m s,i ū s,i| j
u
Y
and standard deviation σs,i|
j = m s,i σs,i| j , where ū s,i| j and
u
σs,i| j are the respective mean and standard deviation of the
conditional distribution of the impact speed u s,i| j . (Owing to
u
current lack of data, we may again assume the value of σs,i|
j to
be some fraction of ū s,i| j , like (1/6) ū s,i| j .) Other distributions,
e.g., Weibull and truncated Gaussian, will be considered in
further study.
Although statistical data on debris trajectories in real
storms are not available at present, experiments and numerical
simulations serve to estimate the parameters of the distributions
µs,i and φs,i . Lin et al. [4], Holmes et al. [6] and Lin et al. [5]
conducted extensive wind-tunnel experiments and numerical
simulations to study debris flight behavior in straight-line
winds (as approximations to the winds in traditional boundary
layers as well as in hurricanes). In particular, an empirical
model of the debris horizontal trajectory was proposed by Lin
et al. [4,5] for the three generic debris types classified by Wills
et al. [8], namely: ‘compact-like’ (e.g., roof gravel), ‘plate-like’
(e.g., roof covers and sheathings), and ‘rod-like’ (e.g., 2 × 4
timbers). Simple empirical expressions were derived to estimate
the most likely flight distance and horizontal impact speed
for each debris type. These expressions are used in our study
to rapidly estimate ds,i and ū s,i| j in the study region, for a
given uniform mean wind velocity. We assume, in the current
study, that the same types of debris generated from different
houses have similar flight trajectories, so that ds,i has the same
statistical properties for different source houses generating the
same types of debris. The effects of geometric characteristics
of the source house (i.e., height and roof configuration) on
N. Lin, E. Vanmarcke / Probabilistic Engineering Mechanics 23 (2008) 523–530
the debris trajectory were neglected. Modifications might be
necessary if the model is used to study debris risk in a region
with buildings of very different shapes. Key aspects of the
empirical model for flight trajectories are summarized in the
Appendix.
Further experimental and numerical studies on debris trajectories may, of course, yield finer classifications and improved
empirical expressions. The above-mentioned probability distributions (Gaussian and lognormal) of the debris landing position and impact strength also need further validation and likely
modification, based on new data about debris trajectories. It is
notable, however, that the Poisson-model-based approach developed in Sections 2 and 3 would apply regardless of the types
of probability distributions adopted for the trajectory parameters.
5. Application to hurricane damage risk analysis
The proposed debris model may be incorporated into hurricane risk analyses. The wind condition can be characterized in
terms of a mean (or gust) wind velocity around each source
house, but it is generally thought sufficient, and common in
practice, to assume a uniform wind velocity over the whole
study region; the local turbulence effects may, after all, already
have been partially incorporated into the probability distributions. The model parameters needed, as input, for each house
are house longitude x and latitude z, house area A, the impact
resistance threshold ζ , the vulnerable fraction q, and the mean
number of each type of generated debris λ, for the given wind
condition. Locations and areas of residential houses are usually
available as deterministic data. The values of ζ and q may perhaps be treated as random variables, based on information about
the building stock in the study area, if site-specific data are not
available. The types and characteristics of house-source debris
may likewise be assigned at random, based on information on
local building material supplies. The damage to house components that could become debris may be estimated, according to
the structural vulnerability studies in the literature. The mean
percentage of damage for each component, and thus the value
λ for each type of debris from each house, can then be derived.
Monte Carlo simulation then may be carried out to obtain estimates of the debris-damage risk corresponding to a given wind
velocity.
We demonstrate this approach by an example. The
site chosen is in Brevard County, Florida, with 2200
houses clustered in a relatively separated and independent
development. We studied the probability of damage on house
windows by roof covers and sheathings; S = Si = 2 in
this case. The mean wind speed used was 49 m/s (U.S. 1min average wind strength of a Saffir–Simpson Category 2
Hurricane), blowing from North and constant over the study
region. Data on house locations and areas were obtained from
Brevard County’s Property Appraiser’s Office. The type of
cover materials (e.g., shingles or tiles) was randomly assigned
to each house, based on the information about the local building
stock. The characteristic data of 45 types of shingles, 31 types
of tiles, and 11 types of sheathings were collected from local
527
construction material manufacturers. The types of cover and
sheathing debris were then uniformly assigned to each house.
The number of covers and sheathings on each house were
estimated, based on the structural area and roof cover density.
The mean percentage of damage to roof covers was estimated
to be 5% and to roof sheathings 1%, for the wind speed in this
example, based on the simulation results from a componentbased pressure-induced damage model (IHRC and FIU [9]).
Noting that some damaged materials may remain attached to
the roofs and not fly, we assumed that half of the damaged
roof covers and sheathings became windborne debris. (Debris
generation will be the subject of future study). The impact
resistance capacity of a typical glass window for each house
was randomly assigned, assuming a Gaussian distribution with
mean 0.025 kg m/s and standard deviation 0.0025 kg m/s.
The window fraction for each house was uniformly assigned a
number between 0.1 and 0.2. The most likely flight distance and
impact speed were estimated using the empirical expressions in
Lin et al. [4] (included in the Appendix) for plate-type debris.
Monte Carlo simulation was carried out to estimate the debris
risk for this region. On the 3D map (of a part of the study
region) shown in Fig. 1, the height of the bar at each house
location represents the estimated number of debris impacts
(1a) and the probability of window damage (1b) of the house,
with the values varying from house to house. Although most
houses are expected to be hit by less than 10 debris items, some
houses that are located in the middle of a relatively dense area
may suffer many more impacts than others. This can also be
seen from the histogram, for all houses in the study region, of
the mean number of impacts in Fig. 2(a). The histogram of
the probability of damage in Fig. 2(b) is much more evenly
distributed among the houses, since it also involves the effect
of the impact momentum.
In strong and long-lasting hurricane winds, debris penetration not only damages building envelopes, but also induces internal pressurization, approximately doubling the net loading
on roofs. Consequently, failed roofing structures become new
debris sources, starting a ‘chain reaction’ of failures. The proposed debris-damage risk model can be used in conjunction
with a pressure-induced damage model (see, e.g., IHRC and
FIU [9] and Vickery et al. [10]) to estimate the cumulative structural damage in such situations. The model of pressure-induced
damage, which generates debris, and the model of debris damage, which affects internal pressure (causing more damage and
debris), may be interactively applied to a given residential area.
Given a storm time-history (expressed in terms of evolving
temporal-mean wind speeds, e.g., 3 s gust, and directions), the
probability of debris-induced damage can be computed for each
house at every time step (e.g., at 15 min intervals), preceded by
calculations of pressure damage, and the effect of the debris
damage can be simulated by updating the building characteristics (as well as the internal pressure) at the next time step.
The debris model presented herein lends itself to fast and efficient computation, since it involves only explicit calculations
and matrix operations, and the house parameters (e.g., impact
resistant capacity, vulnerable fraction, and roof types) need to
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N. Lin, E. Vanmarcke / Probabilistic Engineering Mechanics 23 (2008) 523–530
(a) Mean number of debris hits.
(b) Probability of window damage.
Fig. 1. Analysis of debris-damage risk for a residential area in Brevard County, Florida.
Fig. 2. Histograms of debris-damage risk for a residential area in Brevard County, Florida.
be assigned only once (by means of Monte Carlo simulation)
for a storm time-history.
6. Conclusions and comments on further work
A debris-damage model, intended for incorporation into
more comprehensive hurricane risk analysis, is proposed, based
on Poisson random measure theory. Combined with newtechnology tools such as GIS and Google Earth, this model
can be applied to the study of debris-damage risk (under strong
wind conditions typifying a hurricane) to a particular house,
a residential development, a postal-code region or some other
jurisdictional area.
Further research is needed to validate and further improve
the stochastic debris trajectory model. Whereas mutual
statistical independence of the random variables involved has
been assumed, alternate treatments of statistical dependence are
under investigation. Also, a stochastic model of clustered debris
generation is expected to be developed as an extension of this
study.
Acknowledgments
The research reported herein is supported by a grant
from Baseline Management Corporation (BMC) to Princeton
University. The first author gratefully acknowledges the
instruction and advice on Poisson modeling from Professor
N. Lin, E. Vanmarcke / Probabilistic Engineering Mechanics 23 (2008) 523–530
Erhan Cinlar of the Department of Operations Research and
Financial Engineering at Princeton University.
for spheres,
K x ∗ ≈ 0.248(K t ∗ )2 + 0.084(K t ∗ )3
− 0.1(K t ∗ )4 + 0.006(K t ∗ )5 ;
Appendix
529
(A.5)
and for rods,
This appendix concerns the empirical model of horizontal
debris trajectory developed by Lin et al. [4,5], and its use in
stochastic debris trajectory modeling. Extensive simulations of
trajectories of three typical types of debris were carried out in
the Texas Tech University wind tunnel, under a wide range of
wind speeds and experimental settings. Debris type is mainly
defined by the debris shape and characteristic dimension, as
these govern the debris aerodynamics. Although the trajectories
of a certain type of debris showed great variation in the vertical
direction, their horizontal components showed definite patterns,
dependent mainly on a non-dimensional parameter, later called
the Tachikawa Number by Holmes et al. [11]. Based on this
observation, Lin et al. [4] established empirical expressions,
in dimensionless form, to describe the horizontal trajectory
of plate-type debris, and Lin et al. [5] developed similar
expressions for compact-type (i.e., cubes and spheres) and rodtype debris. Numerical simulations by Holmes et al. [6] and
Lin et al. [5] showed, in general, satisfactory agreement with
the experimental data and this explicit model.
The basic dimensionless variables used in the expressions
given below are:
gx
(dimensionless horizontal displacement),
U2
gt
(dimensionless time),
t∗ =
U
u
(dimensionless horizontal debris velocity),
u∗ =
U
ρa U 2
K =
(Tachikawa Number),
2gh m ρm
x∗ =
(A.1)
where U is the mean wind speed, u is the debris horizontal
flight speed, x is the flight distance, t is time, h m is the debris
thickness, ρm is the debris density, ρa is the air density, and g
is the acceleration of gravity.
The horizontal flight distance was expressed as a function of
flight time as follows:
K x∗ =
1
C(K t ∗ )2 + a(K t ∗ )3 + b(K t ∗ )4 + c(K t ∗ )5 + · ·(A.2)
·,
2
where the values of the coefficients (C, a, b, c, . . .) depend
on the shape of the debris. Least-square fitting of large
experimental data sets yields these empirical equations, for
plates,
K x ≈ 0.456(K t ) − 0.148(K t )
∗ 2
∗
∗ 3
+ 0.024(K t ∗ )4 − 0.0014(K t ∗ )5 ;
(A.3)
for cubes,
K x ∗ ≈ 0.405(K t ∗ )2 − 0.036(K t ∗ )3
− 0.052(K t ∗ )4 + 0.008(K t ∗ )5 ;
(A.4)
K x ∗ ≈ 0.4005(K t ∗ )2 − 0.16(K t ∗ )3
+ 0.036(K t ∗ )4 − 0.0032(K t ∗ )5 .
(A.6)
The horizontal velocity is expressed as follows as a function
of flight distance:
u ∗ = 1 − e−
√
2C K x ∗
,
(A.7)
with the value of coefficient C the same as in Eq. (A.2), namely,
C = 0.911 for plates, 0.809 for cubes, 0.496 for spheres, and
0.801 for rods.
In modeling stochastic debris trajectories (Section 4) in
residential areas, the most likely flight distance (ds,i ) may be
estimated from Eq. (A.3) for roof tiles, shingles and sheathings;
from Eq. (A.4) or (A.5) for roof gravel; and from Eq. (A.6) for
2 × 4 timbers. The flight time of an item of debris (until it hits
the ground) is observed in the experiments to be a function of
the debris vertical trajectory, and it depends on the height of
the source building and the debris initial condition on the roof
(e.g., initial support configuration, initial angle of attack, etc.).
Thus, logically, the probability distribution of ds,i varies with
both debris type and the characteristics of the source building,
as is also suggested by the subscripts. However, establishing
the specific distribution of ds,i for each building may be too
expensive in practice, and currently unsupported by available
data. In studies of residential areas containing mostly similar
houses, we assume that the most likely flight time is 2 s, with a
standard deviation of 0.4 s, for every building and debris type.
This assumption is based on observations from damage surveys
and full-scale experiments (see Lin et al. [4]) that debris items
generally fly in the air for 1–3 s. The flight time distribution may
be assumed to be truncated Gaussian or lognormal. The flight
time statistics could also be made to depend on debris types
and house types (e.g., characterized by roof shapes, number of
stories) in future studies of windborne debris risk.
The most likely horizontal impact speed ū s,i| j can be
estimated for each type of debris by Eq. (A.7), for each
pair of source (i) and target ( j) houses in the study region,
by substituting the flight distance variable with the physical
distance between the two houses.
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