J. Wind Eng. Ind. Aerodyn. 102 (2012) 22–35

Contents lists available at SciVerse ScienceDirect

Journal of Wind Engineeringand Industrial Aerodynamics

0167-61

doi:10.1

n Corr

E-m

wpang@

journal homepage: www.elsevier.com/locate/jweia

Three-dimensional probabilistic wind-borne debris trajectory model forbuilding envelope impact risk assessment

Michael Grayson a,n, WeiChiang Pang b, Scott Schiff b

a Civil Engineering Department, Clemson University, 123 Lowry Hall, Clemson SC 29634-0911, USAb Civil Engineering Department, Clemson University, 110 Lowry Hall, Clemson SC 29634-0911, USA

a r t i c l e i n f o

Article history:

Received 24 July 2011

Received in revised form

14 December 2011

Accepted 4 January 2012Available online 28 January 2012

Keywords:

Debris Trajectory

Debris impact

Hurricane

Building envelope

Risk assessment

05/$ - see front matter & 2012 Elsevier Ltd. A

016/j.jweia.2012.01.002

esponding author.

ail addresses: jmgrays@g.clemson.edu (M. Gra

clemson.edu (W. Pang), schiffs@clemson.edu

a b s t r a c t

This paper presents a probabilistic debris trajectory model adapted from current 6-degree-of-freedom

(6-DoF) deterministic models, in which the aleatoric (inherent) uncertainty is explicitly considered in

the proposed probabilistic model. While the inherent randomness in the debris flight trajectory is

irreducible due to the wind turbulence, variation in wind direction, gustiness of the wind event, and

so forth, the proposed probabilistic model seeks to address these uncertainties through Monte Carlo

simulations with the appropriate statistical distributions applied to the governing equations of motion

of the debris. Calibration of the probabilistic debris trajectory model is performed through an analytical

and visual comparison of the simulated data to wind tunnel test data. Reasonable agreement is

observed between the simulated and the wind tunnel test debris landing locations, thus confirming the

applicability of the probabilistic wind-borne debris model. The proposed probabilistic model provides

an effective method for predicting the variation of debris trajectories in a three-dimensional (3D) space,

which is imperative when performing regional building envelope impact risk assessments in which a

large amount of debris sources and targets must be considered in the simulation.

& 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Wind-borne debris created by strong wind events, particularlyhurricanes, is a major source of damage to the built environment.According to Minor (1994), the envelope of a building mustremain intact during strong wind events to prevent the internalpressurization of the building, which can lead to an increase infailure of buildings and a subsequent increase in the injection ofadditional debris into the wind stream. The cumulative damageimposed by debris may lead to breaching of the building envelopeand thus cause further damage to the interior contents ofbuildings (Wills et al., 2002). Extensive studies of buildingperformance by Minor (1994) concluded that wind-borne debrisis a principal cause of building envelope breaches during a strongwind event.

According to reports from the Federal Emergency ManagementAgency’s (FEMA) mitigation assessment team, the most commonwind-related damage observed in light-frame wood constructionwas roof sheathing panel failures, which was a significant con-tributor to the total damage to buildings during HurricanesAndrew, Charley, Ivan, and Katrina (FEMA, 1992, 2005a,b, 2006).

ll rights reserved.

yson),

(S. Schiff).

Current state-of-the-art wind-borne debris flight models thatseek to analyze the flight trajectory of roof sheathing panelstypically employ one of two methods in the assessment ofbuilding envelope impact risk. The first method, based on theinnovative research of Tachikawa (1983), utilizes simplifieddimensionless equations to de-couple the basic equations ofmotion (e.g., Tachikawa, 1988; Holmes, 2004; Lin et al., 2006,2007; Holmes et al., 2006; Baker, 2007); however, many of thesubsequent studies based on this method typically only investi-gate the two-dimensional (2D) motion of debris in a uniform,horizontal wind. While 2D debris flight models are easy toimplement, these 2D models cannot be used to realisticallyrepresent the motion of debris during actual wind storms.The second method involves the implementation of a 6-degree-of-freedom (6-DoF) model to describe the debris motion in athree-dimensional (3D) space (e.g., Twisdale et al., 1996; Richardset al., 2008; Noda and Nagao, 2010). However, due to a lack ofaerodynamic data for most debris shapes, Twisdale et al. (1996)employed a ‘random orientation 6-degree of freedom’ (RO 6-DoF)model to account for the orientation of the debris during flight,whereas the deterministic 6-DoF model presented by Richardset al. (2008) and Noda and Nagao (2010) employed experimen-tally determined force and moment coefficients to calculate theappropriate orientation of the debris during flight.

A current probabilistic debris trajectory model proposed by Linand Vanmarcke (2008) and Lin (2010) utilizes empirical data and

M. Grayson et al. / J. Wind Eng. Ind. Aerodyn. 102 (2012) 22–35 23

numerical simulations to obtain the distribution of the debrislanding positions and the conditional distribution of the horizon-tal momentum. Due to the lack of statistical data of debristrajectories in hurricanes currently available, the model reliesheavily on numerical simulations to provide the estimates for thedebris trajectory distributions; therefore, it is important to con-tinue to develop numerical models, such as the proposed 6-DoFprobabilistic debris trajectory model, that can be utilized toincrease understanding of the complex phenomenon that isdebris flight.

Wind-borne debris trajectories are further complicated by theflow patterns that develop as a result of the interaction of thewind field with structures. A lack of understanding of the non-linear relationship of this wind/structure interaction has resultedin an emphasis on physical modeling (Yu and Kareem, 1996).However, due to the limitations of current technology andcomputational capacities to adequately measure and model thewind speed along a complete debris trajectory, the actual windspeed and wind vector (direction) experienced by debris duringflight remains largely unknown.

While the numerical integration necessary in a 6-DoF debristrajectory model may be more computationally intensive to analyzewhen compared to the simplified dimensionless equation approach,the benefits, in terms of accuracy and the information available forbuilding envelope impact risk assessment, in the opinion of theauthors, justify the increased computational costs (Grayson, 2011).The objective of this study is to adapt current 6-DoF deterministicdebris trajectory models into a 6-DoF probabilistic debris trajectorymodel that can account for the aleatoric uncertainties that areassociated with physical wind-borne debris flight.

2. Deterministic trajectory model

There are several studies presented in current research literaturethat illustrate practical methods to implement a deterministic 6-DoFdebris trajectory model (e.g., Richards et al., 2005, 2008 and Nodaand Nagao, 2010). The following sections present a general determi-nistic debris trajectory model that utilizes the assumptions and datafrom current research as noted. This is done to adequately illustratethe process followed in the adaptation of a deterministic debristrajectory model into a probabilistic debris trajectory model.

2.1. Deterministic model coordinate system

In order to adequately develop a full 6-DoF debris trajectorymodel, it is necessary to define the appropriate coordinatesystems necessary to track all aspects of the debris trajectory(i.e., translational and rotational motion). The coordinate systems

Z

�Z

ZG

�Z

XG

XI

YG

Y�

I

ZG

�YZI

�YYI

Fig. 1. Transformation of the global translating, non-rotating axes to the debri

used to define this motion are the earth fixed axes (i.e., Xe, Ye, Ze),in which the center of gravity of the debris is defined by theposition vector X¼[x, y, z] and the velocity vector V¼[VX, VY, VZ]. Theglobal earth translating, non-rotating axes (i.e., XG, YG, and ZG), whichremain congruent to the earth’s fixed axes while moving with thedebris, and the debris principal axes (i.e., XP, YP, and ZP), whichcoincide with the dimensions of the debris (i.e., lX, lY, and lZ) suchthat lXrlYrlZ.

An understanding of the orientation of the debris principalaxes (i.e., XP, YP, and ZP) with respect to the global translating,non-rotating axes (i.e., XG, YG, and ZG) is necessary for thetransition of a deterministic trajectory model to a probabilistictrajectory model. As illustrated in Fig. 1, the general orientation ofthe debris principal axes in relation to the global translating, non-rotating axes is defined using the ‘‘pitch–yaw–roll’’ convention ofthe Tait–Bryan angles (yZP, yYP, yXP), which leads to the followingtransformation equation:

XP

YP

ZP

8><>:

9>=>;¼ TðyX ,yY ,yZÞXG ¼ ½RXðyXÞ�

T ½RY ðyY Þ�T ½RZðyZÞ�

T

XG

YG

ZG

8><>:

9>=>; ð1Þ

where T is the global transformation matrix which relates thetranslating non-rotating axes to the debris principal axes, XG isthe global translating, non-rotating axes vector, and RX, RY, and RZ

are the rotation matrices as follows:

RXðyXÞ ¼

1 0 0

0 cosyX �sinyX

0 sinyX cosyX

264

375 ð2Þ

RY ðyY Þ ¼

cosyY 0 sinyY

0 1 0

�sinyY 0 cosyY

264

375 ð3Þ

RZðyZÞ ¼

cosyZ �sinyZ 0

sinyZ cosyZ 0

0 0 1

264

375 ð4Þ

In order to define counterclockwise rotation positive withrespect to the debris principal axes, the transpose of the rotationsmatrices is taken since the rotation of the coordinate system isactually clockwise to a unit vector representing the globaltranslating, non-rotating axes. Therefore, the global transforma-tion matrix from Eq. (1) becomes:

T¼

cosyY cosyZ cosyY sinyZ �sinyY

sinyXsinyY cosyZ�cosyXsinyZ sinyXsinyY sinyZþcosyXcosyZ sinyXcosyY

sinyXsinyZþcosyXsinyY cosyZ cosyXsinyY sinyZ�sinyXcosyZ cosyXcosyY

264

375ð5Þ

XI

�Y

XP

ZI

�XZP

XPYI

YP�X

�X

s principal axes using the rotation order of (a) pitch, (b) yaw and (c) roll.

M. Grayson et al. / J. Wind Eng. Ind. Aerodyn. 102 (2012) 22–3524

2.2. Deterministic model methodology

The 6-DoF deterministic debris trajectory model utilized forthis study requires that the initial physical conditions of the plate(e.g., X0, V0, etc.) serve as the initial conditions of the beginningtime step. Once the initial conditions are set, the relative velocitycomponents of the debris with respect to the global translating,non-rotating axes are calculated:

UG ¼ V�W ð6Þ

where V is the debris velocity vector, [VX, VY, VZ], and W is theuniform, horizontal wind vector, [WX, 0, WZ]. In order to definethe flow angles of the debris (i.e., angle of attack and tilt angle)the relative debris velocity with respect to the global translating,non-rotating axes must be transformed into the relative velocitycomponents along the debris principal axes, (UP), by means of theglobal transformation matrix from Eq. (5):

UP ¼

UXP

UYP

UZP

8><>:

9>=>;¼ TðyX ,yY ,yZÞ

UXG

UYG

UZG

8><>:

9>=>; ð7Þ

The angle of attack (e) and the tilt angle (g) are the observedangles between the relative debris velocity vector, and the debrisprincipal YPZP-plane and the debris principal XPZP-plane, respec-tively, as illustrated in Fig. 2.

Results from wind tunnel tests performed by Richards et al.(2008) illustrated that the force coefficients (CF) and momentcoefficients (CEM) are a function of the angle of attack, the tiltangle, and the plate geometry (G), in this case the side lengthratio, which is defined as the ratio of the two debris dimensionsperpendicular to the principal debris axis of interest with thelarger dimension as the numerator. Several current studies(Baker, 2007; Holmes et al., 2006; Kordi et al., 2010; Lin, 2005;Lin et al., 2007; Martinez-Vazquez et al., 2009; Scarabino andGiacopinelli, 2010) agree with the classification and utilization ofthe force and moment coefficients as a function of the angle ofattack and the plate geometry; however, the effect of the tilt angleon the force and moment coefficients, which is presumablycaused by the attachment of vortex structures to the leadingedges of the plate at certain flow angle combinations, has onlyrecently been explored through wind tunnel testing (Richardset al., 2005, 2008; Richards, 2010). A recent study by Noda andNagao (2010) has provided experimental results to assert that theforce and moment coefficients are a function of more than oneflow angle based on the relationship between the relative debrisvelocity vector and the debris principal axes; however, theirdefinition of the flow angles is fundamentally different from

ZP

XP

YP

lYlX

lZ

V

|UP|

W

Ze

Xe

Ye

γ ε

Fig. 2. Definition of the flow angles with respect to the debris principal axes.

Richards et al. (2008) albeit appropriate within their respective6-DoF debris trajectory models.

Variations of the force and moment coefficients at a relativewind speed and direction are most likely second-order effects,and are assumed not to contribute significantly to the debristrajectory; however, due to the dependence of the force andmoment coefficients on the calculation of the angle of attack andthe tilt angle within the deterministic debris trajectory model, itwas assumed appropriate that the randomness required for aprobabilistic debris trajectory model would be incorporated intothe deterministic model at these points. Eqs. (8) and (9) providethe basis for the transition of the deterministic debris trajectorymodel into the probabilistic debris trajectory model. Since thedeterministic flight model algorithm is unable to account forthe random debris fields typically observed after wind storms, theflow angles are modeled as random variables to account forthe stochastic nature of the debris trajectory; therefore, it isassumed that the deterministic debris trajectory model willprovide the mean value of the flow angles (i.e., mean angle ofattack (e) and mean tilt angle (g)) in order to sample the flowangles from a continuous statistical distribution:

e¼ sin�1 UXP

9UP9

!ð8Þ

g¼ tan�1 UYP

UZP

� �ð9Þ

The sampled flow angles are used to select the appropriateforce and moment coefficients from the experimental databasecompiled by Richards et al. (2008). This provides the necessaryinformation to calculate the force applied to the plate along thedebris principal axes as follows:

FP ¼ CF ðe,g,GÞ1

2ra9UP9

2Ar )

FPX

FPY

FPZ

8><>:

9>=>;

¼1

2ra9UP9

2

CFXðe,g,GÞ

CFY ðe,g,GÞ

CFZðe,g,GÞ

lY lZ

lX lZ

lX lY

8><>:

9>=>; ð10Þ

where FP is the external force applied to the plate along the debrisprincipal axes, CF is the force coefficients as a function of the angleof attack (e), the tilt angle (g), and the plate geometry (G), ra is theair density, and Ar is the reference area of the plate, which isusually a projected frontal area, but since the dimensions of theplate always lie along the debris principal axes, it is the areaperpendicular to the principal debris axis of interest (Holmes,2007; Noda and Nagao, 2010; Richards et al., 2008). The momentsabout the debris principal axes are handled in a similar fashionwith the following equations:

MP ¼MEþMD ð11Þ

where MP is the applied moment vector about the debris principalaxes, ME is the external applied moment vector defined as:

ME ¼ CEMðe,g,GÞ1

2ra9UP9

2Vr )

MEX

MEY

MEZ

8><>:

9>=>;

¼1

2ra9UP9

2

CEMXðe,g,GÞ

CEMY ðe,g,GÞ

CEMZðe,g,GÞ

ðlZAYþ lY AZÞ

ðlZAXþ lXAZÞ

ðlXAYþ lY AXÞ

8><>:

9>=>; ð12Þ

where, CEM is the external applied moment coefficients as afunction of the angle of attack (e), the tilt angle (g), and thegeometry (G) of the plate, which is defined as the ratio of thetwo debris dimensions perpendicular to the principal debris axis

M. Grayson et al. / J. Wind Eng. Ind. Aerodyn. 102 (2012) 22–35 25

of interest with the larger dimension as the numerator, ra is theair density, 9UP9 is the magnitude of the relative velocity vector,and Vr is the debris reference volume (Noda and Nagao, 2010).In addition to the external moments applied to the plate, Richardset al. (2008) states that without some form of damping, the plateswould continue to rotate without bound; therefore, a dampingmoment vector, MD, has been included in the external appliedmoments:

MD ¼ CDM1

2ra 9UP9þ

1

29x9

ffiffiffiffiffi‘2

q� �Vrx

ffiffiffiffiffi‘2

q)

MDX

MDY

MDZ

8><>:

9>=>;

¼1

2ra

9UP9þ 12 9oX9

ffiffiffiffiffiffiffiffiffiffiffiffiffil2Yþ l2z

q� �� �ðCDMXoX

ffiffiffiffiffiffiffiffiffiffiffiffiffil2Yþ l2Z

qðlY AZþ lZAY ÞÞ

9UP9þ 12oY

ffiffiffiffiffiffiffiffiffiffiffiffiffil2Xþ l2Z

q� �� �ðCDMYoY

ffiffiffiffiffiffiffiffiffiffiffiffiffil2Xþ l2Z

qðlXAZþ lZAXÞÞ

9UP9þ 12 9oZ9

ffiffiffiffiffiffiffiffiffiffiffiffiffil2Xþ l2Y

q� �� �ðCDMZoZ

ffiffiffiffiffiffiffiffiffiffiffiffiffil2Xþ l2Y

qðlXAYþ lY AXÞÞ

8>>>>>>>><>>>>>>>>:

9>>>>>>>>=>>>>>>>>;ð13Þ

where CDM is the damping moment coefficient vector defined byRichards et al. (2008) as:

CDM ¼

CDMX

CDMY

CDMZ

8><>:

9>=>;¼

�0:05

�0:185

�0:185

8><>:

9>=>;, ð14Þ

where ‘ is the debris reference length, and x is the angularvelocity vector:

x¼oX

oY

oZ

8><>:

9>=>;¼

yX

dþ yY

dsinðyZÞ

yZ

dsinðyXÞþyY

dcosðyXÞcosðyZÞ

yZ

dcosðyXÞ�yY

dsinðyXÞcosðyZÞ

8>>>><>>>>:

9>>>>=>>>>;

ð15Þ

where oX, oY, and oZ are the angular velocity components aboutthe XP, YP, and ZP debris principal axes, respectively.

The database of force and moment coefficients utilized withinEqs. (10) and (12) are based on discrete values obtained fromwind tunnel tests performed at the University of Auckland(Richards et al., 2008), in which the angle of attack and the tiltangle were incremented in specific intervals between 01 and 901during testing to develop a force and moment coefficient data-base. Due to the discrete nature of the experimental database, alinear interpolation is performed between the discrete values inorder to determine the force and moment coefficients for valuesof the angle of attack and the tilt angle other than those collectedduring testing.

The forces along the debris principal axes must be transformedinto the forces along the global translating, non-rotatingaxes using the inverse of the global transformation matrix fromEq. (5):

FG ¼

FGX

FGY

FGZ

8><>:

9>=>;¼ TðyX ,yY ,yZÞ

�1

FPX

FPY

FPZ

8><>:

9>=>; ð16Þ

This is necessary to determine the acceleration, and by exten-sion the velocity, of the debris as follows:

€X ¼1

mFG�gj ð17Þ

where m is the mass of the debris, g is the acceleration due togravity, and j is the unit vector along the global translating, non-rotating YG-axis, [0, 1, 0].

It is not necessary to transform the moments along the debrisprincipal axes to the global translating, non-rotating axes sincethe angular accelerations and velocities, and the rate of change ofthe Tait–Bryan angles can be calculated along the debris principal

axes using Euler’s equation for rigid body dynamics. Not trans-forming the moments about the debris principal axes to theglobal translating, non-rotating axes essentially reduces thenumber of calculations required as the mass moment of inertiavector (I) is constant along the debris principal axes:

LP

d¼MP�x� LP ð18Þ

where LP is the angular momentum of the debris, which is definedas:

LP ¼ Ix)LPX

LPY

LPZ

8><>:

9>=>;¼

IXX

IYY

IZZ

oX

oY

oZ

8><>:

9>=>; ð19Þ

I¼

IXX

IYY

IZZ

8><>:

9>=>;¼

mðlY2þ lZ

2Þ=12

mðlX2þ lZ

2Þ=12

mðlX2þ lY

2Þ=12

8>><>>:

9>>=>>; ð20Þ

and MP is the applied moment vector about the debris principalaxes as defined in Eq. (11), and x is the angular velocity vector asdefined in Eq. (15). Due to the coupling that is present in the basicequations of motion based on Newton’s second law of motion,Eqs. (15), (17), and (18) are solved numerically using the ModifiedEuler’s Method. These solutions are then utilized to solve for thesolutions of the next time step, and the process is repeated untilthe debris impacts the ground or another object (e.g., structure,vehicle, tree, etc.).

2.3. Time step considerations

The use of a numerical method such as the Modified Euler’sMethod requires small time step increments to ensure that theapproximation provided by the method is a reasonable assess-ment of the actual solution to the problem. However, a probabil-istic debris trajectory model that is utilized within a largersimulation framework to assess building envelope failures mustbe simulated thousands of times to ensure that the simulation is areasonable assessment of the physical situation. There must be abalance between computational intensity and mathematical pre-cision; therefore, a time step sensitivity study was performed onthe deterministic debris trajectory model to determine an appro-priate time step that would provide a reasonable amount ofprecision, since at this point the mathematical accuracy of thesolution is limited by the deterministic debris trajectory model,yet be efficient in a large scale debris simulation. Fig. 3 illustratesthe results of the time step sensitivity study, in which sixparameters (i.e., debris flight time, longitudinal position, long-itudinal velocity, lateral position, lateral velocity, and verticalvelocity at impact) were tested to determine the relative errorinduced into the system with an increase in time step above theinitial value of 0.002 s:

eRE ¼x�x0

x0¼

x

x0�1 ð21Þ

where eRE is the relative error for a specific parameter representedby the variable x.

Fig. 4 illustrates the square root of the sum of the squares(SRSS) error (Eq. (22)) obtained from the individual parametererrors in Fig. 3, and establishes the decision to choose 0.03 s as thetime step length for the probabilistic debris trajectory modelbased on the total error at 0.03 s being relatively close to10 percent for the deterministic model.

etotal ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffie2

t þe2xþe2

yþe2z þe2

uþe2vþe2

w

qð22Þ

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050

0.2

0.4

0.6

0.8

1

1.2

1.4

Tot

al E

rror

Time Step Duration (s)

0° 15° 30° 45° 60° Average

Fig. 4. Total error and the average total error for all wind directions based on an

increasing time step duration within the deterministic debris trajectory model.

The initial time step duration was 0.002 s. All values taken at ground impact.

0 0.01 0.02 0.03 0.04 0.050

0.025

0.05

0.075

0.1

Deb

ris

Flig

ht T

ime

Err

or

Time Step Duration (s)

0 0.01 0.02 0.03 0.04 0.050

0.025

0.05

0.075

0.1

Lon

gitu

dina

l Pos

ition

Err

or

Time Step Duration (s)

0 0.01 0.02 0.03 0.04 0.050

0.025

0.05

0.075

0.1

Lat

eral

Pos

ition

Err

or

Time Step Duration (s)

0 0.01 0.02 0.03 0.04 0.050

0.025

0.05

0.075

0.1

Ver

tical

Vel

ocity

Err

or

Time Step Duration (s)

0 0.01 0.02 0.03 0.04 0.050

0.025

0.05

0.075

0.1

Lon

gitu

dina

l Vel

ocity

Err

or

Time Step Duration (s)

0 0.01 0.02 0.03 0.04 0.050

0.025

0.05

0.075

0.1

Lat

eral

Vel

ocity

Err

or

Time Step Duration (s)

0° 15° 30° 45° 60° 3% Error 5% Error

Fig. 3. Sensitivity of the deterministic debris trajectory model to the duration of the time step. The initial time step duration was 0.002 s. All values taken at ground impact.

M. Grayson et al. / J. Wind Eng. Ind. Aerodyn. 102 (2012) 22–3526

where et is the relative error for the debris flight time,ex,ey, and ez are the relative error for the debris landing position,eu,ev, and ew are the relative error for the debris landing velocity,and etotal is the SRSS error.

3. Probabilistic trajectory model

3.1. Transition from deterministic to probabilistic

Limited experimental data on the statistical distribution of thedebris flow angles during flight required that certain assumptionsbe made at the outset of the transition to a probabilistic debristrajectory model. It was assumed that the flow angles hadan equal chance of falling on either side of the mean as calculated

by the deterministic debris trajectory model; therefore thiswarranted the assumption that the appropriate statisticaldistribution was a normal distribution for both flow angles;therefore:

e¼Nðe,seÞ ¼Nðe,COVeUeÞ ð23Þ

g¼Nðg,sgÞ ¼Nðg,COVgUgÞ ð24Þ

where N(.) represents the normal distribution, e and g are themean angle of attack and tilt angle, respectively (Eqs. (8) and (9)).s is the standard deviation, and COV is the coefficient of variationof the flow angles. The COVs are introduced to characterize theinherent randomness of debris flight due to wind turbulence,gustiness, roof local velocities, the building wake, etc.

3.2. Coarse parametric study

In order to determine appropriate coefficient of variationvalues to be used to calculate the standard deviation that wouldbe used in conjunction with the mean values of the flow angles tosample from a normal distribution, a coarse parametric study wasperformed by varying the values of the COV for the angle of attackand the tilt angle in increasing increments of 0.1 from a minimumCOV equal to 0.1 up to a maximum COV equal to 2 for five winddirections: 01, 151, 301, 451, and 601. The values for the winddirections and the debris initial conditions input into the debristrajectory model (Table 1) were deliberately chosen to permitcalibration of the probabilistic model through a direct comparisonto test data from research on the trajectories of sheathing panelsunder high winds (Visscher and Kopp, 2007; Kordi et al., 2010).The change in the wind direction as described by Kordi et al.(2010) was equivalent to changing the initial yaw (yY0) of the roofsheathing panel (Fig. 1) within the probabilistic debris trajectorymodel. The initial position values provided in Table 1 are inreference to the geometric centroid of the roof sheathing panel.Since the scaled house model in Kordi et al. (2010) was rotated torepresent a change in wind direction within the wind tunnel, theassumption was made in the probabilistic debris trajectory model

M. Grayson et al. / J. Wind Eng. Ind. Aerodyn. 102 (2012) 22–35 27

that the geometric centroid of the sheathing panel remained inthe same location with each subsequent change in the winddirection.

The scaled house model roof pitch was 4:12, which providedan angle from the horizontal of 18.41, but based on the definitionof the pitch angle (yZ) in Fig. 1, the initial pitch of the roofsheathing plate implemented in the probabilistic debris trajectorymodel was actually 901–18.41¼71.61. The uniform horizontalwind velocities (WX) reported in Table 1 represent the equivalentfull-scale 3-sec gust failure wind speed (UH) at the mean roofheight reported by Kordi et al. (2010), and were reported to fit aGumbel distribution (assumed to be a Type I Smallest extremevalue distribution), which was taken into account within theprobabilistic debris trajectory model. Debris flight times aregenerally on the order of a few seconds once flight has beeninitiated (Holmes, 2004; Lin et al., 2007); therefore, it is reason-able to assume that the wind speed at the initiation of flight, inthis case the release of the roof sheathing panel into the windfield after failure of the fasteners, remained constant for theduration of the debris flight (Lin, 2005).

A sensitivity study of the random flow angles sampling ratewas carried out to assess how often it would be required to

-20 0 200

20

40

60

80

100

120

Lon

gitu

dina

l Pos

ition

(m

)

-20 0 20

Lateral

Wind Tunnel Data Deterministic Initia

Fig. 5. Results from the sensitivity study of flow angle sampling for a wind direction o

debris flight, (b) the flow angles sampled every 0.3 s, (c) the flow angles sampled once ev

and end of every time step (time step¼0.03 s).

Table 1Initial conditions of a 1.2 m�2.4 m�12.7 mm (side length ratio¼2) roof sheath-

ing panel in the probabilistic debris trajectory model.

Wind speed (WX) Initial debris

location

Initial debris

orientation

Mean

(m/s)

Std. dev.

(m/s)

Gumbel

distribution

X0

(m)

Y0

(m)

Z0

(m)yZ0

(deg.)

yY0

(deg.)

yX0

(deg.)

mn bn

56 2.2 55.0 1.7 0.6 7.3 3.7 71.6 0 0

45 1.4 44.4 1.1 15

41 1.6 40.3 1.3 30

40 1.5 39.3 1.2 45

38 1.4a 37.3 1.2 60

a Designates assumed value not obtained from Kordi et al. (2010).

sample the flow angles (i.e., the angle of attack and the tilt angle)to adequately match the physical data. Four sampling rates wereinvestigated within the sensitivity study: (1) sampling the flowangles once at the initiation of debris flight, (2) sampling the flowangles once every 0.3 s, (3) sampling the flow angles at thebeginning of every time step, and (4) sampling the flow anglesat the beginning and end of every time step. Fig. 5 shows resultsof the sensitivity study for a wind direction of 451. It is evident inFig. 5 that lower sampling rates of the flow angles were unable tocapture much of the data that was less than the value of thedeterministic model results for the longitudinal position, or thevariability of the physical data. Therefore, it was decided tosample the random flow angle values used to select the experimentalforce and moment coefficients at the beginning and end of each timestep (0.03 s) throughout the flight of the debris. Choosing thissampling rate for the study ultimately means that the flow anglesare not indicative of the actual physical situation encountered duringthe debris flight. However, the intent of the probabilistic debristrajectory model is not to model the exact physical situation of debrisflight, but rather to represent the entire physical system (i.e., thedebris trajectory, and the influence of the building and variations ofthe wind field on the debris trajectory) to provide a model thatadequately represents the physical results that occur when debris isreleased from a structure.

Utilizing the flow angle sampling rate identified in the sensi-tivity study, the ultimate goal of the coarse parametric study wasto identify the combinations of the flow angle COVs that mini-mized the SRSS error between the simulated and full-scaleexperimental debris landing location statistics:

etotal ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffie2mþe2

sþe2l

qð25Þ

where etotal is the SRSS error between the mean (m), standarddeviation (s), and the coefficient of skewness (l) of the simulateddata and the test data, respectively. The mean, standard deviationand the coefficient of skewness were calculated as follows:

m¼ 1

n

Xn

i ¼ 1

xi ð26Þ

-20 0 20

Position (m)

-20 0 20

l 0.3 s 0.03s Begin and End: 0.03 s

f 451 to the source building. (a) The flow angles sampled once at the initiation of

ery time step (time step¼0.03 s) and (d) the flow angles sampled at the beginning

M. Grayson et al. / J. Wind Eng. Ind. Aerodyn. 102 (2012) 22–3528

s¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

n�1

Xn

i ¼ 1

ðxi�mÞ2vuut ð27Þ

l¼ð1=nÞ

Pni ¼ 1 ðxi�mÞ3

ð

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1=nÞ

Pni ¼ 1 ðxi�mÞ2

qÞ3

ð28Þ

where n is the number of data points, i is an index, m is the mean,s is the standard deviation, and l is the coefficient of skewness ofthe data. It should be noted that for the coarse parametric study,Eq. (25) is reduced to calculate the total error for only the firsttwo terms in the equation, namely, the mean and standarddeviation of the landing location in the longitudinal direction(along the Xe-axis), and in the lateral direction (along the Ze-axis).

The upper limit COV value was chosen as 2 due to thephenomenon witnessed from a preliminary parametric sweep todetermine the individual effect of the angle of attack and the tiltangle upon the total error (etotal) as seen in Fig. 6. It is evident,especially for the cases when the initial yaw (i.e., wind direction)is equal to 01 and 151, that as the COV utilized to sample the flowangles approaches 2 the total error increases significantly, andtherefore, within this study it was desirable to not only select COVvalues for the angle of attack and the tilt angle that minimized thetotal error, but that also minimized the COV values of the angle ofattack and the tilt angle themselves.

Fig. 7 illustrates several of the approximating surfaces fromthe coarse parametric study that were plotted to visually selectspecific ranges of the COV values of the angle of attack and the tiltangle for inclusion in the fine parametric study. The darkest areasof the surfaces provide the minimum values of the total error.Further evidence of the instability of the total error as the COV ofthe flow angles increases is apparent in Fig. 7a, in which the winddirection is 01 (i.e., perpendicular to the initial position of thesheathing). The decrease in the instability of the total error withan increase in the wind direction (i.e., initial yaw) could beattributed to the reduction in the normal force on the plate atthe higher wind directions.

3.3. Number of simulations required

To determine the number of simulations required to providesufficient data to determine the COV for the flow angles, the COVof the longitudinal landing impact position for a wind direction

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

1

2

3

4

5

6

7

8

9

10

11

COVTilt Angle

Tot

al E

rror

0° 15°

Fig. 6. Effect of (a) the COV of the tilt angle on the total error with the COV of the angle

at each tested wind direction.

equivalent to 01 was calculated and plotted against the number ofsimulations. Simulations were performed until the COV of thelongitudinal landing impact position for a 01 wind directionstabilized to the calculated mean of the COV to at least within a3% error of the mean. It can be seen from Fig. 8 that the COV of thelongitudinal landing impact location reaches equilibrium with themean value of the COV within 3% error at a minimum of 1500simulations. It was assumed that the number of simulationsrequired for the COV of the longitudinal landing impact locationto reach equilibrium extended to the remaining wind directionswithin the probabilistic debris trajectory model.

3.4. Fine parametric study

The fine parametric study consisted of establishing a range ofthe COV for the flow angles for each wind direction tested withinthe simulation (Table 2). These ranges of the COV were deter-mined from the surface plots that were created in the coarseparametric study; however, for the fine parametric study thenumber of simulations was increased from 100 to 1500 simula-tions as determined in the number of simulations study. The fineparametric study also included the skewness of the data (Eq. (28))in the calculation of the total error, in addition to the mean andstandard deviation from the coarse parametric study. It should benoted that it was not necessarily the combination of the COV ofthe flow angles that provided the overall lowest total error thatwere selected as the final values. A combination of the minimumCOV of the flow angles, a visual inspection of the landing locationplots, and a visual comparison of the debris trajectories weretaken into consideration to ensure that the debris trajectoriessimulated by the probabilistic debris trajectory model agree withthe trajectories reported in physical and numerical models(Richards et al., 2008; Kordi et al., 2010).

Fig. 9 displays the final results from the fine parametric study.To provide the full 1801 spectrum of COV values required withinthe probabilistic debris trajectory model, the assumption wasmade that the model would perform the same for the correspond-ing negative values of the wind direction (i.e. clockwise rotationabout the positive Y-axis from 01 to �901). Tests results fromKordi et al. (2010) at 751 and 901 determined that none of thepanels took flight after failure, but rather landed back on the roofof the house model; therefore, since ground impact locationswere unavailable for analysis, the COV for wind directions greater

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

COVAngle of Attack

30° 45° 60°

of attack¼0 and (b) the COV of the angle of attack with the COV of the tilt angle¼0

100 250 500 750 1000 1250 1500 1750 2000 2250 2500 2750 30000.46

0.48

0.50

0.52

0.54

0.56

0.58

Number of Simulations

Coe

ffic

ient

of

Var

iatio

n

Longitudinal Landing COV

Mean Longitudinal Landing COV

3% Mean COV Error

Fig. 8. Required number of simulations based on the COV of the longitudinal landing impact position.

Table 2Range of the COV identified by the surface plots in the coarse parametric study.

Wind direction (deg.) Range (dimensionless)

Angle of attack COV Tilt angle COV

0 0.3–0.8 0.1–0.3

15 0.3–0.5 0.4–0.6

30 0.2–0.4 1.0–1.5

45 0–0.6 0.7–2.0

60 0.7–0.9 1.0–1.2

Fig. 7. Surface plots of the total error used in the coarse parametric study for wind directions of (a) 01, (b) 151, (c) 301 and (d) 451.

M. Grayson et al. / J. Wind Eng. Ind. Aerodyn. 102 (2012) 22–35 29

than 601 and less than �601 were assumed to be constant up toand including 7901. For wind directions in between those shownin Fig. 9, linear interpolation was used within the probabilisticdebris trajectory model to calculate the values of the COV of theangle of attack and the tilt angle.

4. Probabilistic debris trajectory model comparison

Generally, most wind-borne debris research considers only sub-systems of a building individually (e.g., roof sheathing, shingles, etc.)

M. Grayson et al. / J. Wind Eng. Ind. Aerodyn. 102 (2012) 22–3530

rather than testing the entire complex system as a whole (Surry et al.,2005; Kordi et al., 2010). This basic approach is insufficient in that itdoes not address the effect of the building aerodynamics, and the

-60 -45 -30 -150.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

Wind D

Coe

ffic

ient

of

Var

iati

on

Angle of Attack

Fig. 9. Final representation of the COV of the angle of attack and the tilt angle utiliz

parametric study.

0 5 10 150

2

4

6

8

10

12

Ver

tical

Pos

ition

(m

)

Wind

Longitudinal Po

Vertical slic

Model initial conditions

Fig. 10. Probabilistic debris trajecto

-30 -15 0 15 30 -30 -15

Lateral P

-30 -15 0 15 30

0

10

20

30

40

50

60

70

80

90

100

110

120

Lon

gitu

dina

l Pos

ition

(m

)

Wind Tunnel Data Determinis

WindWind Wind

Fig. 11. Comparison of simulated ground impact locations of a 1.2 m�2.4 m�12.7 mm

(b) 151, (c) 301, (d) 451 and (e) 601.

local velocities on the roof and in the wake on the flight of wind-borne debris (Kordi et al., 2010). Therefore, a ‘failure model’ has beenemployed by several recent research studies to take into account the

0 15 30 45 60

irection (°)

Tilt Angle

ed within the probabilistic debris trajectory model as determined from the fine

20 25 30 35

sition (m)

e taken at 20 meters

Vertical slice taken at 30 meters

Impact

ry model comparison locations.

0 15 30

osition (m)

-30 -15 0 15 30 -30 -15 0 15 30

tic Model Probabilistic Model

Wind Wind

roof sheathing panel to test data (Kordi et al., 2010) for wind directions of (a) 01,

M. Grayson et al. / J. Wind Eng. Ind. Aerodyn. 102 (2012) 22–35 31

influence imposed on wind-borne debris flight by the aerodynamicsof the building and the local roof and building wake velocities (Surryet al., 2005; Visscher and Kopp, 2007; Kordi et al., 2010). Typically,numerical models that utilize a uniform, horizontal wind flow areunable to account for the variations in the velocity field due to theaerodynamics of the building and the local roof velocities, and acurrent study by Kordi and Kopp (2011) observed that debris flightbehavior is significantly influenced by these variations. The probabil-istic wind-borne debris model developed in this study accounts forthese variations by sampling from a normal distribution with anappropriate COV of the flow angles that produce results comparableto experimental test data produced by the scaled debris flight tests ofVisscher and Kopp (2007) and Kordi et al. (2010). Fig. 10 illustratesthe locations identified for comparison between the wind tunnel dataand the simulated debris trajectories from the probabilistic model.

4.1. Model debris position comparison

The values of the COV of the angle of attack and the tilt angle wereverified by plotting the simulated ground impact locations with thetest data provided by Kordi et al. (2010) as seen in Fig. 11. The modesof flight identified by Visscher and Kopp (2007) and Kordi et al.(2010) (i.e., ‘‘translational’’, ‘‘auto-rotational’’, ‘‘3D spinning’’, and‘‘falling down’’) were not tracked in this study; however, it can beseen in Fig. 11 that regardless of the mode of flight illustrated duringthe simulated flight of the roof sheathing panels, there is reasonableagreement between the simulated ground impact locations and thattested by Kordi et al. (2010).

Table 3 provides a comparison of the impact location datastatistics of the test data (Kordi et al., 2010) and the probabilisticdebris trajectory model. As seen in Table 3, the simulated data isslightly overestimated or underestimated for each of the four data

Table 3Comparison of simulated impact location data statistics to wind tunnel

debris data.

Wind direction

(deg.)

Impact location statistics (m)

Test data (Kordi et al., 2010) Simulated data

Xavg Zavg Xs Zs Xavg Zavg Xs Zs

0 33 5 19 5 36 3 17 7

15 36 2 12 5 34 1 11 5

30 43 9 9 3 39 10 12 6

45 31 11 15 4 27 13 12 8

60 13 10 6 4 12 9 5 4

30 40 50 60 700

20

40

60

80

100

120

Lon

gitu

dina

l Pos

ition

(m

)

30 40 50 60 70 30 40

Wind Failur

Probabilistic Model

Fig. 12. Longitudinal debris impact location as a function of wind failure ve

statistics (i.e., the mean and standard deviation of the longitudinaland lateral data, respectively). As previously stated, the COV of theflow angles were selected to minimize this over/underestimation ofthe debris landing locations, and generally agree well with the windtunnel data provided by Kordi et al. (2010).

Fig. 12 shows the longitudinal debris impact locations as afunction of the wind failure velocity (flight initiation wind speed).This plot reiterates the findings in Kordi et al. (2010) that the variationin the mean wind velocity are not a dominant factor affecting thedistribution of the flight distances and the debris scatter.

Fig. 13 depicts a vertical slice of the sheathing panel flights atlongitudinal locations of 20 m and 30 m for wind directions of 01,151, 301, and 451. The plot of 601 is not shown since none of theroof sheathing panels attained a longitudinal position greaterthan or equal to 30 m. Generally, these plots agree well for thelateral positioning of the data; however, the probabilistic debristrajectory model exhibits an approximately 2 m increase in themaximum vertical range over the physical test data plotted byKordi et al. (2010). It should be noted that the debris trajectorydetermined by the deterministic model did not attain a long-itudinal position of 20 m in Fig. 13c, or a longitudinal position of30 m in Fig. 13b, c, or d.

It was imperative that the debris flight trajectories wereplotted and analyzed visually to ensure that the probabilisticmodel was providing reasonable results. Unfortunately, there arefew photographic examples of actual debris trajectories with thecorresponding flight data available; as a result, comparisons weremade to scale model photographs of roof sheathing panel failuresfrom Visscher and Kopp (2007), and the calculated deterministicdebris trajectories from Richards et al. (2008). Fig. 14 illustrates asimulated flight trajectory of a 1.2 m�2.4 m�12.7 mm thick roofsheathing panel. The setup within the probabilistic debris trajec-tory model is the same as presented in Table 1 for a winddirection of 01. The result of Fig. 14 compares well with theresults of Fig. 11a, as the majority of the ground impact locationsare contained within an approximately 15 m to 35 m range in thelongitudinal direction.

4.2. Model debris velocity comparison

Debris velocity at impact is important in determining eitherthe momentum or available energy of the debris at the time ofimpact. Typically, in many previous 2D debris flight trajectorymodels, the horizontal velocity is the parameter of interest in thedetermination of the impact speeds; however, as epistemicuncertainty has decreased with the advent of actual 6-DoFtrajectory models, debris impact research is capable of lookingat the magnitude of the resultant debris translational velocity

50 60 70

e Velocity (m/s)

30 40 50 60 70 30 40 50 60 70

Deterministic Model

locity for wind directions of (a) 01, (b) 151, (c) 301, (d) 451, and (e) 601.

-30 -20 -10 0 10 20 300

2

4

6

8

10

12

14

16V

ertic

al P

ositi

on (

m)

-30 -20 -10 0 10 20 30 -30 -20 -10 0 10 20 30 -30 -20 -10 0 10 20 30

-30 -20 -10 0 10 20 300

2

4

6

8

10

12

14

16

Lon

gitu

dina

l Pos

ition

= 3

0 m

-30 -20 -10 0 10 20 30

Lateral Position (m)

-30 -20 -10 0 10 20 30 -30 -20 -10 0 10 20 30

Probabilistic Model Deterministic Model

Lon

gitu

dina

l Pos

ition

= 2

0 m

Fig. 13. Vertical slice of sheathing panel flights for wind directions of (a) 01, (b) 151, (c) 301, and (d) 45 at (top) longitudinal position¼20 m and (bottom) longitudinal

position¼30 m.

0

5

10

15

20

25

30

35

-5

0

5

02468

1012

Longitudinal Position (m)

Ver

tical

Pos

ition

(m

)

Wind

Fig. 14. Simulated flight of a 1.2 m�2.4 m�12.7 mm (side length ratio¼2) roof sheathing panel in a 56 m/s uniform horizontal wind, and wind direction¼01.

M. Grayson et al. / J. Wind Eng. Ind. Aerodyn. 102 (2012) 22–3532

(umag) as defined by Kordi et al. (2010):

umag ¼ ðu2þv2þw2Þ

0:5ð29Þ

where u is the longitudinal (horizontal) component, v the verticalcomponent, and w the lateral component of the debris velocity.Kordi et al. (2010) then non-dimensionalized the magnitude

M. Grayson et al. / J. Wind Eng. Ind. Aerodyn. 102 (2012) 22–35 33

of the resultant debris velocity using the equivalent full-scale3-sec gust failure wind speed at the mean roof height:

umag ¼umag

UH

ð30Þ

Table 4 illustrates a comparison of the experimental andsimulated dimensionless magnitudes of the resultant debrisvelocities that occurred at ground impact. Generally, there isreasonable agreement between the experimental and the simu-lated data; however, there is slightly less variability in thedimensionless resultant velocities. This is most likely due to thefact that the wind speeds actually experienced by the plate inthe physical testing performed by Kordi et al. (2010) may notnecessarily be the same as the equivalent full-scale 3-sec failuregust speeds at mean roof height reported. Since the probabilisticmodel does not account for the boundary conditions of thebuilding, the assumption was made that the equivalent full-scale3-sec failure gust speed measured at mean roof height reported

Table 4Comparison of the experimental and simulated dimensionless magnitude of the

resultant debris velocities.

Wind direction (deg.) Kordi et al., (2010) Simulated data

ðumag Þavg ðumag Þs ðumag Þavg ðumag Þs

0 a a 0.59 0.13

15 0.73 0.21 0.67 0.15

30 0.58 0.24 0.61 0.10

45 0.43 0.20 0.65 0.13

60 a a 0.49 0.08

a Denotes values not reported by Kordi et al., (2010).

0 0.25 0.5 0.75 10

2

4

6

8

10

12

14

16

Ver

tical

Pos

ition

(m

)

0 0.25 0.5 0.75 1

0 0.25 0.5 0.75 10

2

4

6

8

10

12

14

16

Lon

gitu

dina

l Pos

ition

= 3

0 m

L

ongi

tudi

nal P

ositi

on =

20

m

0 0.25 0.5 0.75 1

Dimensionless Horizonta

Probabilistic: Horiz Vel Probabilistic: Res Vel

Fig. 15. Dimensionless panel velocities as a function of vertical position for wind direct

(bottom) longitudinal position¼30 m.

by Kordi et al. (2010) is the wind speed experienced by thesimulated plate. Therefore, there may be more variability in theactual wind field experienced by the plate in the physical teststhat the probabilistic model is unable to account for, which couldexplain the discrepancy between the variability of the physicaland the simulated data.

Fig. 15 illustrates the dimensionless velocity quantities as seenat a vertical slice of the sheathing panel flights at longitudinallocations of 20 m and 30 m for wind directions of 01, 151, 301, and 451.As with Fig. 13, the plot of 601 is not shown since none of the roofsheathing panels attained a longitudinal position greater than orequal to 30 m. The range of the simulated roof sheathing panel speedstypically agrees well with the physical scale model data reported byKordi et al. (2010) for the wind directions of 151 and 301; however,there appears to be less variability in the simulated panel speeds, anda slightly higher (�2 m) vertical position at both 20 m and 30 mlongitudinal positions similar to the results in Fig. 13. The simulateddata does exhibit the same upward trend for roof sheathing panelspeeds and increase in roof sheathing panel speeds with an increasein longitudinal position from 20 m to 30 m for wind directions of 151and 301 as reported by Kordi et al. (2010). These variations in panelspeeds and vertical location could be attributed to the vortices andbuilding wake effects experienced during the physical test as the roofsheathing panel left the house model. The probabilistic debristrajectory model only minimized the error associated with the debrislanding locations and neglected any influence that the debris velocitycould have imposed in selecting the COV of the flow angles; however,after comparing the simulated data to test data, the slight differencesin speed between the simulated and the physical data may not besignificant if enough simulations are performed in a typical impactstudy. It should be noted that the debris trajectory determined by the

0 0.25 0.5 0.75 1 0 0.25 0.5 0.75 1

l and Resultant Debris Velocity

0 0.25 0.5 0.75 1 0 0.25 0.5 0.75 1

Deterministic: Horiz Vel Deterministic: Res Vel

ions of (a) 01, (b) 151, (c) 301, and (d) 451 at (top) longitudinal position¼20 m and

0 0.25 0.5 0.75 10

0.1

0.2

0.3

0.4

0.5

0.6Pr

obab

ility

0 0.2 0.4 0.6 0.80

0.2

0.4

0.6

0.8

1

Cum

ulat

ive

Prob

abili

ty

0 0.25 0.5 0.75 1

0 0.25 0.5 0.75 1

0 0.25 0.5 0.75 1

0 0.25 0.5 0.75 1

Magnitude of the Dimensionless Resultant Translational Velocity

0 0.25 0.5 0.75 1

0 0.25 0.5 0.75 1

0 0.25 0.5 0.75 1

0 0.25 0.5 0.75 1

Lognormal PDF Lognormal CDF

Fig. 16. Probability distribution function (PDF) and cumulative distribution function (CDF) of the magnitude of the dimensionless resultant translational velocity of a

1.2 m�2.4 m�12.7 mm roof sheathing panel at ground impact for wind directions of (a) 01, (b) 151, (c) 301, (d) 451 and (e) 601 simulated within the probabilistic debris

trajectory model.

M. Grayson et al. / J. Wind Eng. Ind. Aerodyn. 102 (2012) 22–3534

deterministic model did not attain a longitudinal position of 20 m inFig. 15c, or a longitudinal position of 30 m in Fig. 15b, c, or d.

Kordi et al. (2010) observed that the dimensionless magni-tudes of the resultant debris velocities fit a lognormal distributionwhen measured at the scale model house eave height (debrisheaded down from point of maximum trajectory towards theground), and at ground impact. For comparison, Fig. 16 depictsthe dimensionless magnitude of the resultant debris translationalvelocity at ground impact for wind directions of 01, 151, 301, 451,and 601 fitted with both a lognormal probability density function(PDF) and the corresponding cumulative distribution function(CDF). The dimensionless magnitude of the velocities at groundimpact were fit with a lognormal distribution in most cases;however, a level of significance or goodness of fit was notreported by Kordi et al. (2010); therefore, assuming a level ofsignificance of 0.05, a Kolmogorov–Smirnov one-sample test (KS)confirmed that all cases, except for a wind direction of 151(Fig. 16b), were likely to come from a lognormal distribution.Upon closer inspection of Fig. 11b, there appears to be an area atapproximately 19 m that has a cluster of debris impacts thatcould explain the higher number of low dimensionless velocitiesat impact in the PDF of Fig. 16b. This could be attributed to someinstability in the force coefficients obtained at earlier intervals ofthe COV for a wind direction of 151.

5. Limitations of the probabilistic model

The complexity of debris trajectory, and the variable, relativelyunknown wind field experienced by debris during flight, whetherfrom building influences, wake effects or turbulence, warrants

using a stochastic approach to model wind-borne debris trajec-tories. The proposed probabilistic debris flight model is intendedto increase the existing body of knowledge in the area of wind-borne debris trajectory modeling. Limited data sets for modelcalibration require that assumptions be made as to the generalityof the proposed probabilistic model. Certainly the proposedmodel may not be applicable to every scenario encountered inpractice due to endless variations in building orientations, rooflayouts, and sheathing panel locations, but the proposed flightmodel is intended to improve upon current probabilistic model-ing. Until current technological limitations in wind speed mea-surement and data collection are addressed to allow for furtherrefinement of the current probabilistic wind-borne debris trajec-tory model, it is assumed that this model is applicable to otherscenarios encountered in which debris flight may occur.

6. Conclusions

A deterministic 6-DoF debris trajectory model has been mod-ified to perform as a probabilistic model in this paper. By adaptingthe model to provide results that reasonably match physical testdata, the aleatoric uncertainty has been addressed by incorporat-ing error usually associated with uniform, horizontal flow debristrajectory models into the coefficient of variation values used tosample the angle of attack (e) and the tilt angle (g) of the roofsheathing panel from a normal distribution.

Overall comparisons to current ‘failure model’ research arereasonable in terms of debris flight trajectories and velocities.The proposed probabilistic model thus provides an effectivemethod for predicting debris trajectories in a 3D space, which

M. Grayson et al. / J. Wind Eng. Ind. Aerodyn. 102 (2012) 22–35 35

is imperative when performing regional building envelope impactrisk assessment in which a large amount of debris sources andtargets must be considered in the simulation; therefore, thisprobabilistic debris trajectory model can be utilized to investigatebuilding envelope impact failures of residential structures due tohurricane wind hazards and to provide guidance on debris impactprotection. Though a true 6-DoF can be slightly more computa-tionally intensive to implement, the proposed probabilistic modelpresents the opportunity to obtain a greater knowledge of aninherently complex system.

Acknowledgments

This paper was prepared in part as a result of work sponsoredby the S.C. Sea Grant Consortium with NOAA financial assistanceaward NA10OAR4170073. The statements, findings, conclus-ions, and recommendations are those of the authors and donot necessarily reflect the views of the South Carolina Sea GrantConsortium or NOAA.

In addition, the authors would like to acknowledge Dr. PeterJ. Richards from the University of Auckland, New Zealand, andDr. Gregory A. Kopp and Dr. Bahareh Kordi from the University ofWestern Ontario for their invaluable assistance during the courseof this research.

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