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JRDN 2011 Quantitative Risk Assessment for Rockfalls for buildings - 1

Quantitative risk assessmenent for buildings due to rock-

falls : some achievements and challenges

Jordi Corominas & Olga-Christina Mavrouli1

1 Department of Geotechnical Engineering and Geosciences, Technical University of Catalonia (UPC),

Barcelona, Spain; e-mail: jordi.corominas@upc.edu, ph. +34 93 4016861

Résumé

An application example of Quantitative Risk Assessment due to rockfalls for a developed area is presen-

ted in this paper. The methodology aims at the calculation of the risk for buildings which are situated at

the bottom of a rockfall prone slope and may be impacted by rock blocks and of the global risk by aggre-

gation of the results. It can be applied at either site-specific or local scales and it is analytical. Frequency

of the rockfall events has been obtained from historical records and dendrochronology. The probability of

a rockfall reaching the developed area is obtained by trajectographic modelling. A key issue is the consi-

deration of fragmental rockfalls. Falling rock masses are expected to break apart after first impacts on

the ground, leading to individual blocks that will follow independent paths. Different impact energy le-

vels may lead to four potential damage states: 1. non-structural damage, 2. local structural damage, 3.

partial collapse, and 4. extensive to total collapse. For every building, the risk is expressed in terms of the

annual probability of loss and it is the sum, for all rockfall magnitudes, of the products of the rockfall

frequency with the conditional probability of a rock block reaching the building with a certain kinetic

energy sufficient to cause a specific state of damage and its associated vulnerability. The details of the

proposed methodology are presented here through an application example in the Principality of Andorra.

Mots clefs: Rock falls, Quantitative Risk Assessment, Pyrenees.

1. Introduction

Quantitative Risk Assessment (QRA) is progres-

sively becoming a requirement for the administra-

tions in charge of landslide risk management.

QRA aims to provide objective evaluation of risk

in a reproducible and consistent way, avoiding the

use of ambiguous terms, and thus favouring the

comparison of risk level between distant loca-

tions. The QRA may provide information on the

potential loss (i.e. in €/year) due to a potential

hazardous event thus allowing the interpretation

based on risk acceptability criteria. The QRA

results can be used by administrative authorities

for urban planning and/or mitigation measure

purposes, as well as by insurance companies for

the application of their policies.

The methodologies for the QRA due to landslides

which are used globally vary according to the

type of mechanism, the applied scale, and the

available input data. In what concerns the rockfall

risk, several important contributions to the field of

the QRA, have been made by Hungr et al. 1999;

Bell and Glade, 2004; Roberds, 2005; Corominas

et al. 2005; Agliardi et al. 2009; Li et al. 2009.

The objective of this communication is to present

a methodology for the quantification of the risk

for buildings which are located at the bottom of a

slope and are exposed to rockfalls (Corominas

and Mavrouli, 2010). The proposed methodology

takes into account the fragmental nature of the

rockfalls and the structural characteristics of the

impacted buildings. It is analytical and it includes

individual sub-procedures allowing their refine-

ment according to information available, the scale

of work, and the desired degree of the detail. The

local conditions are taken into account including

the topographical relief and the limits of the built

area.

First of all, the methodology with its sub-

procedures is presented through an application

example of the Andorra Principality. At the end, a

discussion is made on its possibilities and limita-

tions.

The study area is a slope situated next to the ur-

ban area of Santa Coloma, in the Principality of

Andorra, located in the east-central Pyrenees

2 – COROMINAS & MAVROULI

(Figure 1). It experiences a relatively high rate of

rockfall activity and has been the object of several

studies on rockfall hazard during the last years

(i.e. Copons, 2004; Copons et al. 2005; Coromi-

nas et al. 2005). The outcropping rock consists of

densely fractured granodiorite and was shaped by

Pleistocene glaciers that after their retreat gener-

ated the steep slopes of the valley. The intense

rockfall activity has produced thick talus deposits

which have been partly developed mainly during

the 70s and 80s.

Fig. 1 Partial view of the study area of Santa Co-

loma, Principality of Andorra. Potential sources,

trajectories, stop points and volumes of some re-

cent rockfall events are shown.

2. Methodology

2.1 General procedure

For the QRA of rockfall threatened developed

areas, an integrated analytical methodology is

proposed here, for application at site-specific sca-

le. The general equation of the rockfall risk is

given as follows :

R= λ(Ri)xP(D:Ri)xP(S:T)xV(E:S)xC (1)

Where

R: expected loss due to rockfall

λ(Ri): frequency of a rockfall of magnitude i

P(D:Ri): probability of a rockfall reaching the

element at risk

P(S:T): the temporal spatial probability of the

element at risk

V(E:S): vulnerability of the exposed element at

risk to impact by a rock fall of magnitude i

C: value of the element at risk

The terms λ (Ri) and P(D:Ri) represent the hazard,

P(S:T) the exposure and V(E:S) the vulnerability.

Equation 1 allows the calculation of the risk due

to the occurrence of a single rockfall size only. To

obtain total risk, all potential rockfall sizes must

be considered. On the other hand, in the case that

elements at risk consist of buildings, the damage

capability of the rockfall is given by its velocity

or kinetic energy rather than by its size. Conse-

quently equation 1 must be substituted by the

following equation to obtain the expected annual

risk:

[ ]

xC))xV(RR:)xP(E(R=(P)R

ijiji

j

1=j

i

1=i

ΣΣ

λ

r

(2)

Rf(P): expected annual loss to the property due to

rockfall, relative to the value of the building;

λ

(Ri): annual frequency of a rockfall with a mag-

nitude “i”;

P(Ej:Ri): probability of a rockfall reaching the

building with a kinetic energy. The latter is calcu-

lated as a function of the magnitude (volume) “i”

and the velocity “j”. The kinetic energy levels are

those leading to the respective damage states (de-

fined in table 6).

V(Rij): vulnerability of the building for a rockfall

of magnitude “i” and velocity “j”;

C: value of the building.

The temporal spatial probability of the element at

risk P(S:T) for static elements such as buildings

is 1. Consequently, it is not considered in equa-

tion 2.

2.2 Frequency of rockfalls

λ

(Ri)

Frequency of rockfalls can be calculated by

means of statistical analyses of rockfall records.

(i.e. Hungr et al., 1999; Dussauge-Peisser et al.,

2002; Guzzetti et al., 2003).

Unfortunately, the availability of such records is

restricted to a few road and railway maintenance

offices and national park services. Historical

records are often too short in comparison with the

time scale of large rockfall events. In the case of

Santa Coloma, the available rockfall record cov-

ers a span of time of about 50 years but it is com-

plete only for the last 15 years when the Andorran

administration established a systematic inventory

of all the rockfall events occurring in the area.

This inventory covers exclusively rockfall events

larger that 1m3 that were noticed by the inhabi-

tants of the area and by annual surveys with heli-

copter flights. The rockfall series has been com-

pleted by intensive dendrogeomorphological ana-

lyses of damaged trees (Moya et al. 2010) and has

JRDN-11 Quantitative Risk Assessment for Rockfalls for buildings - 3

allowed extending the rockfall record to the last

40 years. The average annual frequency λT for all

rockfall sizes is 0.5 events per year. This figure

must be considered a minimum value because the

occurrence of small-size rockfall events without

producing impacts on trees cannot be absolutely

disregarded.

The magnitude (volume) - frequency relation of

the inventoried rockfalls in the entire Santa Colo-

na area is shown in table 1 and it is assumed to be

the same for the study site.

λ

(Ri) is the product of

λT with the relative frequency of each volume

class. Volume corresponds to that measured at the

source.

Source

volume

(m3)

Number

of events λ(Vi) λ(Ri)

≤ 5 14 0.667 0.333

10 4 0.19 0.095

25 2 0.095 0.047

150 1 0.048 0.024

Table 1. Rockfall events observed in Santa Colo-

ma area. λ(Vi) is the relative frequency of each

volume class and λ(Ri) its annual frequency

However, the volume to consider in the trajecto-

graphic analysis is not an evident issue. A rockfall

may involve the displacement of a single or sev-

eral blocks. It may also begin by the detachment

of a more or less coherent rock mass that after the

first impact with the slope face splits into several

pieces. The latter is the case of a fragmental rock-

fall which is characterized by the independent

movement of individual rock fragments after de-

tachment from a rock face (Evans and Hungr,

1993). The fragmentation mechanism is not cur-

rently included in trajectographic models and may

strongly affect the reliability and validity of the

results. The detachment of large rock masses

without considering their fragmentation after the

first impacts on the ground will give unrealistic

travel distances in excess of what should be ex-

pected. The important effect of the number and

mean size of fragmented rocks on the hazard due

to a single event has been discussed by

Jaboyedoff et al. (2005), who proposed the em-

pirical evaluation of the latter.

Table 2 shows the average distribution of block

sizes from several rockfall events of the Santa

Coloma area inventoried during the last decade.

Both the number and size of the blocks increase

with the volume of the detached rock mass. The

assumption made in the methodology presented

here is that the frequency of falling blocks of a

given size has to be increased by adding the fre-

quency of the blocks of the same size produced

by fragmentation of larger rockfall events.

Block

size

(m3)

Volume of the rock mass detached at

the source area (m

3

)

5

10

25

150

1

1

2

4

12

2.5

1

1

1

8

10

0

0

1

2

30

0

0

0

1

Table 2. Number of fragmented blocks of each

size class for different volumes of the rock mass

detached at the source area.

Thus the frequency of rockfalls of a defined block

size “s” is given by the following expression:

[ ]

)(R)xN(R=(Rs)

isi

i

1=i

Σ

λλ

(3)

where

λ (Rs) is the frequency of blocks of “s” size

λ(Ri) is the frequency of rockfalls of “i” volume

Ns(Ri): the number of blocks of size “s” per every

rockfall of volume “i”

Table 3 shows the annual frequency of each block

class in Santa Coloma area calculated using equa-

tion 3.

Block size (m3) λ

(Rs)

1

1.000

2,5

0.667

10

0.095

30

0.024

Table 3. Frequency of different block sizes in the

area of Santa Coloma area calculated from Eq.3

2.3 Trajectographic analysis P(Ej:Rs)

For each range of block volume, a three-

dimensional probabilistic trajectory analysis has

been performed with ROTOMAP32 to define the

percentage of possible rockfall paths reaching

each exposed building with a given level of ki-

netic energy. This level is defined by the potential

damage states caused to the buildings (Mavrouli

and Corominas, 2010a) which are: non-structural

damage, local damage, partial collapse or exten-

sive to total collapse. The thresholds of the E that

distinguish between the damage states are calcu-

lated by the analysis of the response of the ex-

posed structure to the block impact. More infor-

mation on this is given at section 2.4.

4 – COROMINAS & MAVROULI

ROTOMAPS32 code provides different rockfall

paths from different initial velocities, respective

directions and exact locations of the rockfall

sources. Some rockfall sources produce paths

that have higher probability of affecting some

buildings than others and this is taken into ac-

count in the analysis. The potential range of ki-

netic energy E of the rock blocks reaching the

buildings is also calculated. The blocks that reach

a particular building are classified into groups

with respect to their E, and the probability of each

group is evaluated.

The model was calibrated to comply with histori-

cal rockfall events data (Corominas and Mavrouli,

2010). The results were considered acceptable

when the stop points from the simulation ap-

proximated those of the real events. In the case of

the blocks of 30 m3 size, the velocity of impact

onto the buildings is not known and the restitution

coefficients and limit angles were calibrated

through successive trials to reproduce the path of

the block. However, the obtained velocities from

the calibration indicate extremely high levels of

E, the reliability of which should be validated

with the back-analysis of future rockfall events.

After the calibration, the trajectory analysis was

performed. Given the detailed work scale, the risk

in this example is evaluated using Equation 4 and

the probability of reaching directly a target build-

ing with a certain E was obtained by:

T

nΕ

sj n

)R:P(E =

(4)

where:

nE: number of block paths reaching any particular

building with a certain E;

nT: total number of block paths

Fig. 2 Rockfall paths for blocks of 2.5 m3 size.

The obtained values for the P(Ej:Rs) are shown in

Table 4. Additionally, Figure 2 presents an exam-

ple of all the potential paths produced by a block

of 2.5 m3, their associated rockfall sources and the

potentially affected buildings. The total number

of simulations for every magnitude class was

1500.

Build-

ing Block

volume

(m3)

Kinetic Energy (KJ)

<14 14-28 >28

A 1 0.003 0.008 0.008

2.5 0 0 0.013

10 0 0 0.020

30 0 0 0.104

B 1 0.001 0.003 0.002

2.5 0 0 0.004

10 0 0 0.001

30 0 0 0.037

C 1 0.003 0 0.004

2.5 0 0 0.011

10 0 0 0.029

30 0 0 0.060

D 1 0 0 0

2.5 0 0 0

10 0 0 0

30 0 0 0

E 1 0 0 0

2.5 0 0 0

10 0 0 0

30 0 0 0.015

F 1 0 0 0

2.5 0 0 0

10 0 0 0

30 0 0 0.033

G 1 0 0 0

2.5 0 0 0

10 0 0 0

30 0 0 0.017

Table 4. Probability of reaching the building with

an energy P(Ej:Rs) as calculated in the trajecto-

graphic analysis

JRDN-11 Quantitative Risk Assessment for Rockfalls for buildings - 5

The used thresholds that may lead to different

damage states that, for this example, were ob-

tained during the sub-procedure that is described

in section 2.4 (Table 6). They are: < 14 kJ for

non-structural damage, 14 – 28 kJ for local or

partial structural damage and > 28 kJ for exten-

sive to total collapse.

2.4 Vulnerability of the exposed building

The exposed elements here are considered to be

the buildings which are situated at the bottom of

the slope. A single structural typology is consid-

ered: a 2-storey frame reinforced-concrete frame

(RC) structure.

The vulnerability is quantified considering the

potential repair cost of the building, with respect

to its reconstruction value. To evaluate it, the

step-by-step procedure for the response of RC

buildings to single rock impacts on their basement

column(s), proposed by Mavrouli and Corominas

(2010a), is used. It is an analytical methodology

that can be adapted to various structural typolo-

gies, for the assessment of the physical damage in

case of loss of structural and non-structural ele-

ments of a building, taking into consideration the

potential of a cascade of failures (progressive

collapse) which extends to a part or to the entire

building, due to the initial loss of the element.

For simple regular frame RC frame buildings the

damage extent and the potential of progressive

collapse depends on the number and damage of

the struck element(s) for a rockfall impact of a

given E, and their importance for the overall sta-

bility of the building. Potential damage can be

classified into four damage states: a. non-

structural damage of the infill walls, b. local

structural damage, c. partial collapse and d. exten-

sive to total collapse.

In the worked example, it is considered the same

building typology as used in Corominas and Mav-

rouli (2010b) and is shown in Fig.3. The results of

the analysis for the considered building, which is

composed by two frames (three columns) at its

exposed façade, 3 frames perpendicularly to it,

along its length, and 5 m infill walls in-between

the columns, indicate the conditions that lead to

the proposed damage states. The four following

initial damage scenarios are investigated: loss of a

central column, of a corner column, an infill wall

and two or more central or corner columns per-

pendicularly to the exposed façade, depending on

the impact location and the kinetic energy that

determines the capacity of a block to destroy one

or more columns. The considered scenarios are

unfavourable regarding the direction of the rock

blocks perpendicularly to the exposed façade and

are considered here from the safety side.

The proposed vulnerability is calculated as the

sum of the products of the probability of encoun-

ter of the rock block with a structural or non-

structural element and the associated RRC:

)RRC x(P=)V(R kke,

k

1=k

ij Σ

(5)

where,

V(R ij): vulnerability for a rock bloc with a magni-

tude i” and velocity “j”;

Pe,k: encounter probability of a rock with a possi-

ble structural and non-structural element of the

building “ k” that may be struck by a rock block

of magnitude “i”;

RRCk: relative recovery cost that corresponds to

the damage of one or more structural and/or non-

structural element(s) of the building “k” by a rock

block of magnitude “i” and velocity “j”.

To calculate the probability of each impact loca-

tion, the following Equations are used:

sinψ a dl

c

+

= P

ec

(6)

sinψ a dl

1

c

+

=n

P

s

(7)

sinψ a dl

1

w

w

+

=n

P

(8)

where:

Pec: the probability of encounter with any exposed

column;

Ps: the probability of encounter with a specific

column;

Pw: the probability of encounter with an infill

wall;

n: the number of projected columns on a line ver-

tical to the rock path;

lc: column width;

lw: infill-wall width;

a: distance between centers of columns;

d: rock block diameter;

ψ: angle between the rock path and the façade

plane.

Using Equations (6) to (8) for the given building,

the probability of encounter with a non-structural

or a structural member is given by Table 5. The

impact location is expected to occur exclusively

in the structural elements present at the first level

of the building.

6 – COROMINAS & MAVROULI

Fig. 3 Typical structural typology of the area

Building

m3 Central

column

Corner

column

Any

column

A, B, C, D,

E, F, G

1

9.91E-02

1.98E-01

2.97E-01

2.5

1.27E-01

2.53E-01

3.80E-01

10

1.88E-01

3.77E-01

5.65E-01

30

2.62E-01

5.24E-01

7.86E-01

Table 5. Probability of encounter with a non-

structural or a structural member

The RRC expresses the cost of the repair in rela-

tion to the value of the building. It is calculated in

function of the physical structural and non-

structural damage, translated into economical

cost, for every potential location of the impact

(Mavrouli and Corominas 2010b).

For the considered building, the RRC is provided

by Table 6, for every scenario (impact location

and kinetic energy sufficient to cause the loss of

one or more elements)

Damage state Damaged

element E

(kJ)

RRC

No damage Any col-

umn

< 14 0

Non-structural

damage External

infill wall 0.01

Local struc-

tural damage

Central

column

14 - 28 0.2

Partial struc-

tural collapse

Corner

column

14 - 28 0.4

Generalised

damage Two or

more col-

umns > 28 1

Table 6. Conditions leading to every damage state

and associated RRC

Considering these, the vulnerability is calculated

in function of the block diameter and kinetic en-

ergy as shown in Table 7.

Building m E (kJ)

< 14 14 - 28 > 28

A, B, C, D,

E, F, G

1 1.00E-02 1.09E-01 3.07E-01

2.5 1.00E-02 1.37E-01 3.90E-01

10 1.00E-02 1.98E-01 5.75E-01

30 1.00E-02 2.72E-01 7.96E-01

Table 7 Vulnerability V(Rij) for every possible

impact energy

2.5 Calculation of the relative risk

For every building, the relative risk to its value is

calculated here, using Eq. (2) just substituting the

rockfall frequency λ(Ri) by the block size fre-

quency λ (Rs) as discussed in section 2.2. The

results are presented in Table 8.

The global risk for an area is then evaluated by

summing up the products of the relative risk for

all the exposed buildings with their values:

C)*(P)R( R(P) r

Σ

=

where

R(P): global risk for an area;

Rr(P): relative risk for a building;

C: value of the building

The total relative risk for the entire area is the

sum of the relative risk for all buildings and equal

to 2.16E-02.

In buildings A, B, and C, the risk is higher for

small rock sizes (1 and 2.5 m3). This is due to the

higher frequency associated to them. Instead,

buildings E, F, G are affected by the largest

blocks only. This is because, for these particular

cases, only blocks of 30m3 are able to reach the

building locations with the required energy level

to produce damage (see table 4). It has to be taken

into account that these are results of a simulation

and they should be validated with real cases.

Building D has no risk because none of the mod-

elled trajectories passes through the building loca-

tion.

JRDN-11 Quantitative Risk Assessment for Rockfalls for buildings - 7

Buildi

ng m3 Rr(P) for every

magnitude i

Total Rr(P) for

the building

A

1

1,32E-03

7,86E-03

2.5

3,46E-03

10

1,09E-03

30

1,99E-03

B

1

9,12E-04

2,69E-03

2.5

1,04E-03

10

3,64E-05

30

7,01E-04

C

1

1,26E-03

6,78E-03

2.5

2,78E-03

10

1,60E-03

30

1,15E-03

D

1

0,00E+00

0,00E+00

2.5

0,00E+00

10

0,00E+00

30

0,00E+00

E

1

0,00E+00

2,92E-04

2.5

0,00E+00

10

0,00E+00

30

2,92E-04

F

1

0,00E+00

6,36E-04

2.5

0,00E+00

10

0,00E+00

30

6,36E-04

G

1

0,00E+00

3,31E-04

2.5

0,00E+00

10

0,00E+00

30

3,31E-04

Table 8 Relative risk for every building Rr(P)

3. Conclusions

The calculation of risk using the proposed meth-

odology is quantitative and may be expressed in

terms of annual loss

The proposed procedure takes into account the

fragmentation on the detached rock masses which

otherwise would have produced longer runout

distances and higher impact energies in the trajec-

tographic analyses. The increase of the number of

blocks of small size caused by the fragmentation

of the detached rock mass has been included in

the assessment of the frequency of the different

block sizes. However, this may not prevent an

underestimation of their kinetic energy and runout

distance. It is thus necessary a validation with

further rockfall events.

In what concerns the vulnerability the methodolo-

gy includes a weighted vulnerability that takes

into account the encounter probability of the

block with key structural and non-structural ele-

ments and the subsequent damage. Thus, the vul-

nerability can be integrated into the risk equation.

The worked example includes only a particular

structural building typology and it is necessary to

add other typologies before generalizing the pro-

cedure.

The methodology that has been presented here

may be used for the calculation of the risk for a

building that is impacted at its basement by a sin-

gle block fragmented rockfall, as well as for the

calculation of the global risk for a built area, by

aggregation.

The application example was carried out at site-

specific scale. This analysis indicated that not all

the exposed buildings have the same impact pro-

bability; instead rockfalls follow preferential

paths towards some of them. As a result the risk

for each building is different even though the vul-

nerability and their location with reference to the

topographic elevation under the rockfall source

are the same. This can be useful for the optimiza-

tion of the cost/benefit relationship of protection

measures.

Acknowledgements : This work has been per-

formed within the projects Safeland, funded by

the European Union (7th Framework Program)

grant agreement 226479 and Big Risk, funded by

the Spanish Ministry of Science and Innovation,

contract number BIA2008-06614. Partial support

was given to the second author by the European

Reintegration Grant for the project RISK-LESS,

grant agreement 268180

The authors appreciate Julien Godefroy’s assis-

tance with the calibration of the trajectory model.

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