A damage model for the assessment of storm damage to buildingsPatrick Heneka a,b,∗, Bodo Ruck aa Institute for Hydromechanics, University of Karlsruhe, Kaiserstrasse 12, 76128 Karlsruhe, Germanyb Center for Disaster Management and Risk Reduction Technology, University of Karlsruhe, Hertzstrasse 16, 76187 Karlsruhe, Germany
a r t i c l e i n f o
Article history:Received 28 February 2008Received in revised form26 May 2008Accepted 4 June 2008Available online 17 July 2008
This paper presents a new model for the calculation of number and monetary amount of damage toresidential buildings exposed to winter storm winds. The damage model is derived based on physicalevidence and logical assumptions and is embedded in amathematical framework. Themodel is calibratedwith damage data from German winter storm events of the last decades. It is shown that the model iscapable of reproducing the total damage as well as the spatial distribution of damages.
Each year, extreme storm events such as winter storms,hurricanes and tornadoes cause widespread damage to persons,buildings, nature and provoke intensive indirect damages dueto infrastructure and network outages. Recent events as wellas statistics of the reinsurance industry ([24,19]) demonstratethe large damage potential and show the importance of areliable assessment in order to quantify the risks. Therefore,besides reasonable storm hazard calculation, the knowledge of thevulnerability of the affected structures to storm damage is of greatimportance. With this contribution, we want to propose a newvulnerability (damage) model for damage assessment of buildingstructures. Building damage is not the only part of the overall stormdamage, but represents a large proportion and is usually insuredand therefore well recorded. For these reasons, a storm damagemodel for residential buildings is developed.Damage surveys of past storm events (e.g. [3,27,23,1]) give an
overview of the typical damage patterns to building structures.Direct building damage occurs most frequently to roofs, walls,claddings and openings. Indirect building damage due to wind-borne debris is dominated by broken trees.In order to quantify building damage, a non-dimensional
damage ratio DR (also denoted as damage index or loss ratio) was
∗ Corresponding author at: Institute forHydromechanics, University of Karlsruhe,Kaiserstrasse 12, 76128 Karlsruhe, Germany. Tel.: +49 721 6082203.E-mail address: [email protected] (P. Heneka).
introduced in the 1970s and first used by Hart [8] and Leicesterand Reardon [16]. The DR is defined as the ratio betweenmonetaryrepair costs to the total value of the building. The latter is oftendefined as reconstruction costs. Between the lower border of 0 –no damage – and the upper border 1 – total building damage – thedamage ratio is verbally related to the physical damage patterns[8,16,5,2]. In Table 1, a comparison of the assignment of damagepatterns is given which also points out the regional differences inthe estimation of repair costs. For example, the half loss of roofsheeting is equivalent to 0.25%–0.8% damage ratio in Europe [5]and to 5%–20% in Australia [16,2].The aim is to find out the equivalentwind speedswhich result in
these damage patterns. It is obvious that due to the inhomogeneityof the existing building stock the statements are valid in an averagesense and not for individual buildings. A commonway is to developa function for a certain amount of buildings in a spatially definedunit.Mesoscale assessments of storm damage aim at loss determina-
tion on the spatial unit of postal-code zones or municipalities witha country-wide extent. Damage assessment on the scale of singlebuildings requires a huge amount of specific data and is generallynot feasible for a large spatial extent. For mesoscale damage as-sessment, a variety of models have been developed which can besummarized in qualitative and quantitative models. The latter aresubdivided into empirical, theoretical and stochastic models.Qualitative models describe the consequences of extreme wind
speeds by means of their visual effects of natural phenomena onstructures. Examples are the Beaufort scale, the Torro scale [17],the Fujita scale (e.g. [7]) and Saffir–Simpson scales. These models
3604 P. Heneka, B. Ruck / Engineering Structures 30 (2008) 3603–3609
Table 1Assignment of verbal damage descriptions and damage ratios
Damage description Damage ratio (%) as reported by[5] [8] [16] [2]Lowerbound
Upperbound
Lowerbound
Upperbound
Lowerbound
Upperbound
Lowerbound
Upper bound
No damage 0.01 0.05 0 0.5 0 0Light damage to roof tiles 0.05 0.1 1 5Roofs partly uncovered, light damage to structure 0.1 0.25 0.5 1.25 5 5Half loss of roof sheeting, some structural damage 0.25 0.8 5 20Severe damage to roofs, loss of roof sheeting 0.8 3 1.25 7.5 10 10Loss of roof structure, some damage to walls 3 10 7.5 65 15 20 20 60Severe damage to structure, some collapses 10 30 20 25Loss of all walls, 30 90 65 100 50 65 60 90Collapse of some buildings 60 100 75 90Total collapse of all buildings 80 100 100 100 100 100 90 100
are often applied inversely, which means that wind speeds areestimated based on the observed damage effects.Quantitative models calculate building damage in relation to
available meteorological and structural information such as windspeeds, storm duration, and building type.Some functions were acquired empirically by fitting simple
functions to damage data [26,4,19]. The functions have thedisadvantage that extrapolation to higher wind speeds thanalready observed is not satisfactory, as it is not based on physicalprocesses. As an example, Munich Re proposed a power of 3 for theincrease of damagewithwind speed based on data of the European1990 storm series and corrected it to the power of 4–5 for the 1999storm series where higher wind speeds occurred.In contrast to empirical approaches, some authors chose to
construct deterministic models [28,21]. However, these projectsonly refer to the US and need large amounts of specific buildinginformation that is not available on a large scale. Sill andKozlowski [25] propose a model to assess hurricane damage tobuildings. It is based on logical assumptions and provides someinteresting approaches which are used for the construction of ourmodel. Pure stochastic models have been developed by Rootźenand Tajvidi [22] and Katz [12] under the assumption that theoccurrence of storm events can be modelled by extreme valuefunctions.More details of different damage functions are given by Heneka
and Ruck [9] and Watson and Johnson [29].To summarize the differences of the models, two most crucial
factors can be pointed out: the trend of damage increase withwind speed and the implemented additional parameters. Therelationship between wind speed and amount of damage ismodelled either by potential, exponential, or composed functionsand no consensus about the ‘‘right’’ trend has been reached.Certainly, regional differences in the building stockmay also resultin different trend curves.Secondly, the models differ in the parameters which are
included in addition to wind speed which is obviously the mostimportant. Some models also consider storm duration, buildingtype, and surrounding surface roughness as important factors.However, due to the limited data available, most damage modelspublished do not have additional parameters.In this paper, we propose a new way to model storm damage
to buildings which at least is an alternative to the first unresolvedproblem of damage trends.
2. Storm damage model
2.1. Exact formulation of storm damage
In the following, v denotes the maximum wind speed during astormevent in the surroundings of a building and refers to a 3 s gust
Fig. 1. Mean damage function for buildings.
in 10 m above ground level. This building suffers from damage, ifv is higher than a wind speed vcrit. The latter is a wind speed, atwhich damage to a building occurs for the first time and is namedas critical wind speed. Atwind speeds v higher than vcrit, damage isexpressed by a damage increase function g(v). Maximum possiblebuilding damage is reached at wind speeds higher than the totalwind speed vtot. For every single building, the damage ratio G istherefore written in sections as
G(v) =
{0, v < vcritg(v), vcrit ≤ v < vtot1, vtot ≤ v.
(1)
The qualitative trend of G for a single building is plotted in Fig. 1.The total monetary loss is obtained by multiplying Gwith the totalvalue or the reconstruction costsW of the building.Summation of all single monetary losses results in the total loss
For an exact solution of this equation one would have to knowthe function G with its variables vcrit, vtot and g(v), the maximumwind speed v as well as the total value W for every building(!).It is theoretically possible – with a certain effort – to determineW and v for every building. The damage functions G wouldhave to be determined for every single building by deterministicapproaches [28,20,21,6] which is practically impossible due tothe large amount of buildings and the great diversity of buildingstructures.However, known values are the number of buildings N within a
postal-code zone or municipality, the total value ΣW within thisarea and wind speeds v during storm events on a 1 km × 1 kmraster by numerical wind field simulation.
P. Heneka, B. Ruck / Engineering Structures 30 (2008) 3603–3609 3605
Fig. 2. Built-up areas with the borders of a zip code zone or a municipality.
2.2. Model derivation
Due to the lack of data, assumptions have to be made toconstruct a fully applicable damage model. As the distribution ofbuildings and damageswithin a spatial unit is unknown, themodelis used to calculate the total number and amount of damage in aspatial unit. Themain assumptions for the derivation of the damagemodel are listed below:
1. All buildings have an equal valueW = ΣW/N . Consequently,the weight of the damage ratios for the summation is equal.
2. The calculatedwind speeds are averaged over the built-up areasof the spatial unit and are therefore applied for all buildings(Fig. 2).
3. Critical and total wind speeds of all buildings within a stock(postal-code zone, municipality) can be described with suitableprobability distribution functions f (vcrit) and f (vtot). Generally,every function can be used.
4. The function for damage increase g(v) is valid for all buildingswithin an area and represents therefore an averaged damagepropagation.
The distribution function for critical wind speeds f (vcrit)simulates the inhomogeneities of the building’s resistance towind loads. Damage will generally not start at a common windspeed. Moreover, some buildings will suffer at lower wind speedsthan others and some will resist even very high wind speeds.For the same purpose, Sill and Kozlowski [25] are using atriangular function over the square of the wind speeds. In Fig. 3,a distribution function for the critical wind speeds is plotted forexample. Integration of the function f (vcrit) with the upper limitv (maximum wind speed) results directly in the ratio of affectedbuildings CR (claim ratio) for which the critical wind speed wasexceeded and damage occurred.
CR(v) =∫ v
0f (vcrit) dvcrit. (3)
The integral is equal to the cumulative density function of f (vcrit)and plotted as the grey area in Fig. 3.The damage ratio of buildings in an area is calculated as follows
and illustrated in a graphical way in Fig. 4. A small proportionof buildings f1 with high vulnerability suffers high damage ratiosG1 up to the wind speed v. A larger proportion of buildings fnis damaged near the maximum wind speed v and each of thosetherefore suffers a smaller damage ratio Gn. Summation of theseproportions yields
Fig. 3. Probability density function of critical wind speeds.
Fig. 4. Graphical presentation of damage calculation.
For n → ∞, one receives the integral for the calculation of thedamage ratio DR for a number of buildings:
DR(v) =∫ v
0f (vcrit)G(v) dvcrit. (5)
Consequently, Eqs. (3) and (5) are used to calculate damage andclaim ratio of a building stock when hit by a wind speed v during astormevent. Up to this point, the derivation of the damage functionwas kept as general as possible to ensure a maximum of choicesfor the probability distribution functions f (vcrit) and f (vtot) as wellas for the damage increase functions g(v). Within the presentedmathematical framework, all types are possible and the followingconsiderations will help to choose suitable functions.Distribution function for critical wind speedsThe critical wind speed of a building is determined by
the weakest construction detail and, therefore, depends on theconstruction design of the structure [20]. Additional factors likeconstruction quality, age and maintenance also play an important– but unknown – role. We will have to face a large spreading ofcritical wind speeds in a building stock which shall be simulatedby means of a suitable distribution function.As there is no further evidence about the shape of the
distribution function, we use the Normal distribution functionwith the shape parametersµcrit and σcrit to describe this variation.Introduction of the Normal function in Eq. (3) and multiplicationwith the total number of buildings N results in the number ofdamaged buildings at a maximum wind speed v
Buildings(v) = N∫ v
−∞
1
σcrit√2πexp
(−(vcrit − µcrit)
2
2σ 2crit
)dvcrit.(6)
Hence, the number of damaged buildings is solely described by theshape parameters µcrit and σcrit.Damage increase function for buildingsIt is often discussed whether the increase of damage with
higher wind speeds can be based on physical assumptions insteadof pure empirical evidence [15,14]. From investigation of the
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European 1990 storm series [18] a power 3 relationship was found(damage ∼ v3) which was traced back to the fact that theavailable kinetic energy of a flow is a cubic function of windspeed. This may be true for observations of single buildings butnot for a building stock where the increased monetary damageis mainly a result of the increased amount of damaged buildingsand not of a significantly higher average damage. In the proposedmodel, these 2 mechanisms are treated separately; the damageincrease denotes this of a single average building while the abovementioned distribution of critical wind speeds is valid for anamount of buildings.The damage increase function is assumed to be proportional
to the power α of a wind speed and writes with the boundaryconditions g(vcrit) = 0 and g(vtot) = 1 as:
g(v) =(v − vcrit
vtot − vcrit
)α. (7)
Here, for α = 2 the average damage of a single building would beproportional to the wind force, for α = 3 to the kinetic energy ofthe wind flow.Distribution function for total wind speedsAnalogous to the critical wind speeds, the total wind speeds
are also described by a distribution function. Generally, these twofunctionswill not be independent fromeach other (correlatedwithcorrelation coefficient ρ) and can be written as a 2-dimensionalprobability distribution function f (vcrit, vtot, ρ). For reasons ofsimplicity and no further evidence, we assume that the total windspeeds also follow a Normal distribution and are fully correlated(ρ = 1)with the critical wind speeds. Consequently, vtot for everysingle building can be written as
vtot = vcrit +1v. (8)
As a result, the difference between total and critical wind speed isconstant for all buildings within an area. Substitution of Eqs. (7)and (8) in (5) and multiplication with the total building value Wresults in the damage at a given wind speed v
Damage(v)
= W∫ v
−∞
(v − vcrit
1v
)α 1
σcrit√2πexp
(−(vcrit − µcrit)
2
2σ 2crit
)dvcrit. (9)
Besides the shape parametersµcrit andσcrit, damage is additionallydependent on1v and α.Again, the damage increase function and distribution functions
chosen forvcrit and vtot are relatively simple approaches and, hence,subject to inaccuracies. However, within the proposed modelequations (3) and (5), any function can be used.
2.3. Implementation of further parameters
To describe damage in more detail, it would be desirable tohave different damage functions for different types of buildings ordifferent exposures. The influence of these additional parametershas to be quantified. For example, Sill and Kozlowski [25] andKhanduri andMorrow [13] proposed additional curves for differenttypes of buildings, Schraft et al. [24] for storm duration.In our model, µcrit, σcrit, α and 1v will have to be evaluated to
best fit the damage data available which is classified by additionalparameters.
2.4. Uncertainties of damage assessment
The analysis of past storm events show a large variation ofdamages when plotted against wind speed which is not entirely
Fig. 5. Illustration of uncertainty of damage functions.
explainable with the available information [4,11]. These variationshave to be treated as random variation. It is therefore necessaryto at least quantify these variations in order to obtain an estimateof the uncertainty of the damage assessment. The introduction ofartificial uncertainty is a way of dealing with this problem in apractical sense. This is realised by a simple add-on to the modelwhich consists of a random variation of the shape parameter µcritfollowing a distribution f (µcrit). Here, also a Normal distributionis used with the mean µµcrit = µcrit and standard deviation σµcrit.With Monte Carlo simulations for each wind speed a variation forthe damage is obtained (Fig. 5). The determination of σµcrit wasperformed bymeeting the percentiles of themodelled damage andthe observed damage, e.g. the 16th and 84th percentile.
2.5. Remarks to the model parameters
Theparametersµcrit,σcrit,α and1v are needed in order to applythemodel. Generally, there are twoways of obtaining these values:(1) determination of the parameters in an analytical way and (2)calibration to available wind speed and storm damage data of paststorm events. In our case damage and wind speed data of 4 stormevents are available. We therefore use both approaches: After thedetermination of reasonable starting values for the 4 parameters,the calibration procedure was performed as follows in order tominimize the difference between calculated and observed totalbuilding damage:
• Calculation of number of buildings anddamage for every postal-code zone and storm.• Summation of results for all postal-code zones for every storm.• Calculation of difference to observed overall damage.
The mean of the normal distribution µcrit is equal to thewind speed where exactly 50% of the buildings in an area aredamaged. Hence, given a suitable observation database of pastbuilding damages, this value can be directly determined. Silland Kozlowski [25] propose 48 m/s for this wind speed; frompublications of the Munich Re [19] a value of 45–50 m/s is read.The standard deviation σcrit can roughly be estimated based on thefollowing thoughts: First significant damage to buildings occurs atwind speeds approximately 20–25 m/s. Taking into account thatfor a normal distribution of critical wind speeds, approx. 0.1% ofthe values lay belowµcrit−3σcrit, we deduce values of 7–10m/s asa starting estimate for the standard deviation.The mean of the total wind speeds µtot represents the wind
speed where 50% of the buildings suffer total damage (DR = 1).Wind speeds of this magnitude are not expected for winter stormsin Germany and are even rarely observed in tornadoes. However,an approach to obtaining this wind speed is a comparison withF5 tornado damage where values of 120–130 m/s are observed
P. Heneka, B. Ruck / Engineering Structures 30 (2008) 3603–3609 3607
Fig. 6. Damage model 1 (left) and 2 (right) and observed storm damage.
Table 2Suggested parameters for residential buildings in Germany
Parameter Description Model 1 Model 2Absolute values Relative values
µcrit Wind speed where 50% of the buildings are damaged 50.5 m/s (σ = 2.5) 1.31 (σ = 0.04)σcrit Standard deviation of critical wind speed distribution 7.8 m/s 0.20α Increase of damage with wind speed 2 21v Difference in wind speed between start of damage and total damage 70 m/s 1.85
for these damage states [5]. Taking these values a first roughestimate forµtot, the difference1v between vtot and vcrit calculates70–90 m/s. There is also no evidence about the shape parameterα for the damage increase function but, for reasons of modelconsistency, the above mentioned values indicate that α = 2. Ahigher α would have the consequence that the mean value for thetotal damage distribution µtot is expected already at wind speedsof 70 m/s which is not a realistic value for the German buildingstock.Summarizing, with knowledge of building damage and corre-
spondingwind speeds, starting values of themodel parameters canbe determined in a rational way.
3. Application to winter storms in Germany
The model is used to calculate the number and monetaryamount of damages caused by winter storms in Germany. Damagedata on a postal code base are available for 4 storm events withdifferent severities for the German state of Baden–Württembergsituated in the South-West of Germany. The maximum gust fieldsof these past events have been calculated in a raster with ahorizontal resolution of 1 km × 1 km by the IMK, University ofKarlsruhe [10].In a previous publication [11] it was found that the number
and monetary amount of damage is rather a function of a relativewind than of the absolute wind speeds which occur during stormevents. The relative wind speed is hereby defined as the ratio ofthe maximum gust during a storm event and the 50-year windgust. The 50-year wind speed is denoted as wind climate and isexceeded with a probability of 2% in a time period of one year.This consideration ofwind climate has the effect thatwind damage
occurs solely if the local wind climate is exceeded. In regionswith high wind climate the building structures are used to alsowithstand higher gusts, thus damage is not as high as in otherregions with lower wind climate.In order to show the consequences for the damage calculation,
we use both approaches to run the model. Model 1 is run withabsolute wind speeds, model 2 with relative wind speeds. For theavailable damage and wind data, the parameters suggested for thedamage model for residential buildings in Germany are listed inTable 2. The values fit very well to the theoretical considerations ofSection 2.5.Both the damage data as well as the damage model are plotted
in Fig. 6, for the two approaches, respectively. The plots show thedamage ratio (above) and the claim ratio (below) in respect tothe absolute gust speed (left) and relative gust speed (right). Thedots show wind speeds and damage for single postal-code zones.Bothmodels represent verywell themean increase of damagewithwind speed.As the data points scatter verymuch, an uncertainty assessment
is given. In each case, 16% of the data points lay below and abovethe slash-dotted curves and consequently more than 2/3 of thedata points lay between these borders. The size of the area istherefore a measure for the uncertainty of the model; the smallerthe better the model is capable of explaining the data. At this pointit becomes clear that the damage model is solely representing theaverage damage trend without taking into account further localdifferences or uncertainties.The overall damage is calculated by the summation of the
number and amount of damage for all postal-code zones in thestate and comparedwith the observed damage (Table 3). The errorsrange from 0% to 40% for the number of affected buildings and
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Table 3Observation and model calculations for four past winter storm events
from 10% to 70% for the monetary damage. Ignoring the weakestevent of 1986 which generally caused little damage, model 2is better capable to calculate storm damage of the 3 strongestevents. Especially the number of affected buildings is sufficientlyreproduced with a maximum error of 7% for the 1999 event. Thisholds also for the monetary damage where the maximum error is15% compared to 40% for model 1.The reasons for the insufficient calculation of the 1986 event
is of a general nature: At lower wind speeds where only veryfew damage per spatial unit is observed, an inaccurate calculationresults easily in large errors.A comparison of the spatial distributions of storm damage for
winter storm Lothar is shown in Fig. 7. The amount of monetarydamage per postal-code zone is represented by black bars andthe average wind speeds are marked in colour. By eye, a bettercorrespondence of model 2 to the observed damage patterns isvisible, especially in the western, southern and central parts of thestate. This impression is proven by the calculation of the spatialcorrelation coefficients where we get 0.87 for model 2 and 0.61for model 1. For the number of affected buildings and all the otherwinter storms, the correlation coefficients are listed in Table 4. Itis obvious that model 2 reproduces the spatial distribution for allevents much better than model 1.
4. Conclusions
The most important conclusions of this work are listed below:
• A damage model was developed which is embedded in amathematical framework and which offers space for arbitraryimprovements. It is now possible to run this model with amini-mumof available data. The advantage of themodel compared toempirical damage functions is that the extrapolation for higherwind speeds is based on a set of logical assumptions instead ofbest-fit functions.• The model was calibrated with damages of past storm eventsin Germany and was able to reproduce the damage numbers. Itis shown that damage functions based on relative gust speeds
Fig. 7. Residential building damage per postal-code zone inMil. e for winter stormLothar in 1999.
(model 2) are better capable to calculate damage than thosebased on absolute gust speeds (model 1) as the precision
P. Heneka, B. Ruck / Engineering Structures 30 (2008) 3603–3609 3609
and especially the spatial correlation coefficients are higherthroughout.• The developed storm damage model represents the averagewind–damage relationship of residential buildings withinpostal-code zones or equivalent spatial areas. However, thediscrepancy betweenmodelled and observed damage for singlezones may be huge as at this stage no further influences likebuilding structure, age or storm duration are considered.• The derived set of model parameters is used to recalculatethe damage of all storm events that have been available.Changes of the model parameters result in different overalldamage predictions while the spatial correlation coefficientsare mainly insensitive to changes. However, compared to thegeneral uncertainty of the model, the sensitivity of the resultsto changes of the model parameters is low. This indicates thatthere should be no larger problems to determine the modelparameters to further building data.• Although calibrated to data of a specific region, model 2 withits model parameters is applicable also elsewhere as it uses thewind climate as a proxy for local vulnerability of buildings (seealso [11]). This holds only for regions where the constructionof buildings is similar to the calibration region which is at leasttrue for Germany andWestern Europe. However, due tomissingdamage data of other regions, this applicability could not yet befinally proven.
Acknowledgements
The authors would like to thank the SV GebäudeversicherungStuttgart, Germany, for the provision of stormdamage data and theInstitute forMeteorology and Climate Research of the University ofKarlsruhe for the provision ofmeteorological data. Thiswork is partof the project ‘‘Risk map Germany’’ of the Center for Disaster Man-agement and Risk Reduction Technology (http://www.cedim.de), ajoint venture of theGeoForschungsZentrumPotsdam (GFZ) and theTechnical University of Karlsruhe (TH). We thank the GFZ Potsdamand the University of Karlsruhe for financial support.
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