JEE-2002-Vol-5-Issue-1

December 6, 2001 14:43 WSPC/124-JEE 00050

Journal of Earthquake Engineering, Vol. 6, No. 1 (2002) 1{15c Imperial College Press

DAMAGE DIRECTIVITY IN BURIED PIPELINES OFKOBE CITY DURING THE 1995 EARTHQUAKE

SHIRO TAKADA and NEMAT HASSANIy

Civil Engineering Department, Kobe University, Japan

KATSUMI FUKUDAz

Department of Science & Technology,Kobe University, Japan

Received 7 July 2000Revised 12 January 2001Accepted 27 March 2001

In this paper, we analysed the damage ratio in water distribution, gas and telecommu-nication pipelines buried normal and parallel to the fault direction using GIS database.We attempt to simulate strong ground motions in these two directions at near-eld con-sidering the S-wave radiation pattern. We have also analysed and compared the buriedpipes behaviour located normal or parallel to the fault direction in the ground withhorizontal or inclined bedrock, using a three-dimensional nite element approach. Bothstatistical and analytical results show that response and damage ratio of the pipes buriednormal to the fault direction are larger than those in parallel direction. It is also shownthat the eect of seismic directivity at near eld is large, while at far eld, the eectdue to inclination of the bedrock on the ratio of pipe damage in these two directions ispredominant.

Keywords: Damage directivity; pipeline damage; wave radiation pattern; seismicdirectivity; bedrock inclination.

1. Introduction

Lifelines suered extensive damage in the 1995 Hyogoken Nambu Earthquake and

the role of urban lifelines were much more closed up. Damage to the water and

gas distribution pipes in Kobe City, which is located near to the source fault,

happened in thousands of locations. The analyses of the recorded accelerations

show that the ground displacement and velocity were larger in normal direction

to the fault comparing to those in parallel [CIHAE, 1998; Mori, 1998]. There is

also a special characteristic in this city and that is the inclination of bedrock

from the mountainside to the seaside [CIHAE, 1998; Mori, 1998] (north to south).

ProfessoryAssistant ProfessorzGraduate Student

1


2 S. Takada, N. Hassani & K. Fukuda

Therefore, these two characteristics are supposed to have eective role on dam-

age directivity in buried pipelines during the earthquake in Kobe City. The GIS

based statistical analysis of data collected from the performance of buried pipes

in water, gas and telecommunication networks in this city showed that damage

in those pipes which were buried normal to the faults direction was conside-

rably more than those in the parallel direction. Also for studying the eect of

bedrock inclination on damage directivity, the generated earthquake waves, con-

sidering the radiation pattern of S-wave are used in the FEM analyses of pipe-

ground system with and without bedrock inclination using the ABAQUS computer

code.

2. Relation Between Damage and Burial Direction of Pipelines

The damage ratios of water, gas and telecommunication pipelines in two di-

rections of parallel and normal to the fault were statistically analyzed using

the GIS medium [Takada et al., 1999a; 1999b]. The four faults used for the

present analyses were not the main ones having a rapture origin. However surface

fault displacements along these faults were recognized by SAR analysis [Fujiwara

et al., 2000]. Here, the method that we have used for determining angles be-

tween pipes and fault is explained. Assuming an axis for the fault and using an

angle-step of 30, the pipe directions have been divided into six parts as shown inFig. 1.

Then considering the end points of each pipe in ve wards of the city, their

angles with the fault direction are calculated. In more general division the pipes in

directions 3 and 4 are assumed normal to the fault while the ones in directions

1 and 6 have been considered as parallel to the fault. Using such a classica-

tion, the pipe length, damage number and ratios have been determined for each

divided part, which we named them directions 1 to 6 to avoid complicated descrip-

tion in the paper. We have showed the directivity characteristics by concentrating

Fig. 1. Burial directions of the pipes.


Damage Directivity in Buried Pipelines of Kobe City 3

on directions 3 (as normal to the fault) and 6 (as parallel to the fault) [Fukuda,

2000].

2.1. Water distribution pipelines

The water distribution pipes in the ve wards of Nada, Higashinada, Chuo, Hyogo

and Suma which are located close to the four faults of Gosukebashi, Suwayama,

Egeyama and Suma in Kobe City are selected for the analysis. The materials of

pipelines are ductile iron with dierent joint type in more than 90% of total length in

Kobe City. Those pipes in the back ll lands which suered more from liquefaction

than being near to the fault are excluded in this analysis. Comparisons of directions

3 and 6 in dierent wards are given in Table 1 and Fig. 2.

The results show that the damage ratio in direction 3, which is normal to the

fault direction, is more than that in the area 6, which is parallel to the fault.

In those wards such as Nada, Higashinada and Chuo where the damage ratio is

large, the dierence between the ratios in two directions is also large, while in the

other areas, there is not so big dierences between the ratios in two directions.

Table 1. Comparison among damage condition of water distribution pipes in dierent wards ofKobe City.

The wards inDamage ratio

Kobe CityLength of buried pipes (km) Damage locations (location/km)

Direction 3 Direction 6 Direction 3 Direction 6 Direction 3 Direction 6

Nada and 152.05 228.83 150 166 0.99 0.73

Higashinada

Chuo 74.92 107.85 61 44 0.81 0.41

Hyogo 51.13 60.93 28 23 0.55 0.38

Suma 46.24 59.42 22 26 0.48 0.44

0.00

0.20

0.40

0.60

0.80

1.00

1.20

Dam

age

ratio

in w

ater

dist

ribut

ion

pipe

s(loc

ation

/km)

D i rec t ion 3Di rec t ion 6

N ada andHigash inada WardsGosukebashi Faul t

C h u o W a r dSuwayama Fau l t

H y o g o W a r dEgeyama Faul t

S u m a W a r dSuma Faul t

Fig. 2. Comparison of damage ratios in water distribution pipes for dierent burial direction.



Therefore, it is concluded that in those areas with big dierences in damage ratios

for two directions, the earthquake motion in the normal direction to the fault was

predominant.

2.2. Gas pipes

Considering the reliability degree of damage data of gas pipes in three wards of

Chuo, Nada and Higashinada the total damage location is decreased somewhat.

The same method of analysis in water distribution pipes, is used for nding the

relation between damage ratio and pipe directions against the Suwayama fault.

The length, damage location and damage ratio of gas pipes, are given for each

direction in Table 2. The materials of gas pipelines are mostly ductile iron and

steel pipe with screw joint.

The comparison between the damage ratios in directivity areas 3 and 6 are given

in Fig. 3. The damage directivity analysis was also done for the dierent types of

pipes. The results show that in the gas pipes damage ratio for the pipes in direction

Table 2. Comparison of damage condition of gas pipes indierent directions.

Direction Buried length Damage Damage ratio

(km) location (location/km)

1 201.89 33 0.16

2 228.67 55 0.24

3 813.05 292 0.36

4 143.82 34 0.24

5 231.12 35 0.15

6 1147.95 266 0.23

Fig. 3. Comparison of damage ratio for dierent directions.



3, is more than that in direction 6 (considering total whole lines). This dierence is

much greater (about 1.7 times) for the pipes with screw joints (considering damage

ratio in pipelines with screw joints).

2.3. Telecommunication pipelines

In a damage investigation, NTT performed the Street Mandrel Test for two vacant

conduits among several ones in each span between manholes. The judgement on

damage was done whether the Mandrel passes smoothly or not. The results showed

that the interruption rate has increased to 36% which is bigger than usual rate by

6% [CIHAE, 1998]. For telecommunication analysis, the damage data of pipelines

in Nada and Higashinada (the Nada, Mikage and Higashinada blocks) were used

against the Gosukebashi fault. Assuming 3 number interruption of Mandrel as the

damage to the pipeline, the interruption rate is calculated for whole length of the

pipeline. Figure 4 shows the comparison between the interruption rate in direction

3 and 6. It is shown that the damage ratio in normal direction to the fault is

somewhat greater than in the parallel direction without Mikage Block.

However, comparing to the water and gas pipes, there is dierent tendency for

damage dierence in block with less damage. Due to this reason, it is supposed that

there are some other causes for damage except earthquake that is necessary to be

included in the analysis of data. One of the main reason for interruption of Mandrel

is the influence of rust in the pipe accumulated during the past time.

One of the reasons for more damage in pipes buried in normal direction to the

fault is the eect of the predominant ground motion in this direction. Generally

speaking, the behavior of buried pipelines follows the ground deformation and the

longitudinal behavior is much predominant compared with perpendicular one due to

soil stiness in each direction. Since the motion in the normal direction to the fault

is predominant, the longitudinal strain in those pipes which are buried in this

direction is also predominant. Due to the pipe longitudinal severe compression and

tension, there is a lot of damage to the joints (such as pulling-out) and to the pipe-

body (such as buckling) both for water and gas buried pipes. However, since the

Fig. 4. Comparison of interruption percentage in dierent areas.



pipelines in Kobe City are buried also parallel to the bedrock inclination from the

mountains in the north to the sea in the south, there is possibility of topography

influence on the pipe damage in this direction. Therefore, the eect of bedrock

inclination is also investigated in this paper.

3. Introduction of Function of S-Wave Radiation Pattern

For analysis of damage directivity in pipelines, the earthquake waves are generated

through a statistical synthesis method. In order to make waves for small earth-

quakes (due to incremental fracture of the fault), the random phase is applied

to the spectrum model of the source fault using the Boor Statistical Simulation

Method [Mori, 1998; Boore, 1983]. The large earthquake wave is generated through

the summation of these small waves. It is not necessary to estimate the path of wave

propagation or ground characteristics in this method. Although the advantage of

empirical Green Function is not used very well, the best use of the propagation of

fault surface rupture and geometrical relation among fault and any arbitrary point

is the advantages of the method. By introducing the Radiation Pattern Function

in the relation between the location of epicenter and observation point, the same

earthquake is obtained in any direction at observation point. In this way, after

making ground motion for each short path in the fault, the radiation pattern of

displacement amplitude directivity is used and then the main earthquakes in nor-

mal and parallel direction to the fault are generated by summation of the small

quakes.

Hirasawa and Stauder [1965] have considered the characteristics of rupture pro-

pagation in a single direction of fault with vertical-horizontal displacements and

calculated the radiation pattern of the S-waves. Here, by assuming the unilateral

epicenter line of a vertical-horizontal fault as the fault rupture direction, the radia-

tion pattern of P - and S-waves are calculated in any arbitrary point. If the angle

between the rupture propagation direction and the line to the observation point is

Fig. 5. Radiation pattern of S-wave.



, the radiation pattern (Fig. 5) for S-wave is obtained through Eq. (1).

R =cos 2

R Vp=Vs

1 VrVs

cos : (1)

Here, R is the Radiation Pattern Function of the S-wave, Vr; rupture propagation

velocity, Vp and Vs; velocities of P - and S-waves, respectively and R is the distance

between the epicenter and observation point. As the analytical parameters it is

assumed that Vs = 3:0 km/s, Vr = 2:7 km/s, Vr=Vs = 0:9, Vp=Vs = 1:5 and

Vp = 4:5 km/s [Hirasawa and Stauder, 1965].

4. Simulated Earthquake Motions using Radiation Pattern

The resulted waves by the statistical synthesis method in an observation point

1.0 km far from the epicenter for any direction without considering the radiation

pattern is shown in Fig. 6.

-1000

-500

0

500

1000

1500

0 1 2 3 4 5 6time(sec)

The same wave in any direction

Acceleration(gal)

Fig. 6. Example of statically synthesised wave at ground surface (distance from the fault 1.0 km).

-1500

-1000

-500

0

500

1000

1500

0 1 2 3 4 5 6Time(sec)

Parallel to the faultNormal to the fault

Acceleration(gal)

Fig. 7. Wave at the ground surface using radiation pattern (distance from the fault is 1.0 km).



By introducing the Radiation Pattern Function, it became possible to estimate

the wave components in normal and parallel directions to the fault direction sepa-

rately as shown in Fig. 7. Here, the distance from the fault as shown in Fig. 8,

is measured from the end point of fault rupture to the observation point in the

normal direction to the fault. Figure 7 shows that for 1.0 km far from the fault,

the maximum acceleration in normal direction is 1.5 times larger than the one in

parallel direction to the fault. The relation between the maximum acceleration at

ground surface and the distance from the fault is depicted in Fig. 9 for ground

with horizontal bedrock. This gure clearly shows that the maximum acceleration

at ground surface in normal direction is much greater than the parallel one and the

former one is 1.5{5.2 times larger than the latter one.

Comparison among the results with those from the attenuation formula of

Fukushima and Tanaka [1990] shows that the maximum acceleration in normal

direction at a distance of 10.0 km to the fault is more than that obtained by the

formula. While in the parallel direction, by increasing the distance from the fault, it

decreases very fast, and for a distance of 2.0 km, it becomes less than that calculated

by the attenuation formula. The displacement hysteric paths are shown in Fig. 10

for dierent distances from the fault. The displacement amplitudes in the normal di-

rection are also larger than the ones in parallel direction to the fault. Figure 10 also

shows that the displacement amplitudes for normal to parallel direction increases

by increasing the distance from the fault. The next is determination of ground mo-

tion at underground bedrock. Figure 11 shows the velocity response spectra for the

Source fault

Distance from the fault

M=7.0L=40kmW=20km

Observation point

Fig. 8. Relation between presumption point and its distance from the fault.

dis tance f rom the faul t (km)1 0 0

1 0 0 0

2 0 0 0

1 1 0

Normal to the fau l tParal le l to the faul tFukush ima-Tanaka fo rmu la

Max

imum

acc

eler

atio

n at

gro

und

surfa

ce(ga

l)

2

5 0 0

5

Fig. 9. Relation between maximum acceleration at ground surface and distance from the fault.



(a) Distance from the fault: 1.0 km (b) Distance from the fault: 2.0 km

(c) Distance from the fault: 3.0 km (d) Distance from the fault: 4.0 km

Fig. 10. Path of a ground particle at dierent distances from the fault.

1

10

100

300

0.1 1 5

4km

10km

Period (sec)0.5 2

200

50


Waterguideline 90%Non-excessprobability

Waterguideline 70%Non-excessprobability

Sv (c

m/se

c) 8km6km

2km

(a) Normal to the fault

1

10

100

300

0.1 1 5

2km

4km

6km

8km

10km

Period (sec)0.5 2

200

50


Water guideline90%Non-excessprobability

Sv (c

m/se

c)

probability 70%Non-excess probability

(b) Parallel to the fault

Fig. 11. Velocity spectra obtained from numerically generated wave.



1

10

100

300

0.1 1 5

Higashi-Ohashi.y(GL-33m)Port Island.y(GL-83m)Kansai Electric Co.Research Center.y(GL-100m)

0.5 2

200

50

Period (sec)

Water guideline 70%Non-excess probability


Sv (

cm/s

ec)

(a) Normal to the fault

Period (sec)1

10

100

300

0.1 1 50.5 2

200

50

Sv (cm/sec)

Kansai Electric Co.Research Center.y(GL-100m)

Port Island.y(GL-83m)

Higashi-Ohashi.y(GL-33m)



(b) Parallel to the fault

Fig. 12. Velocity spectra from the waves recorded during Kobe Earthquake.

generated earthquakes in normal and parallel directions to the fault. The two bold

lines in the gure are the design velocity spectra, which are proposed in the Re-

commended Anti-Seismic Construction of Water Facilities (revised 1997) [JWWA,

1997] for design of underground structures against level 2 earthquakes in Japan.

These lines are corresponded to 90% and 70% non-excess probability. This gure

shows that the spectrum characteristics are clearly dierent for normal and parallel

directions to the fault and the one in the normal direction is much larger than that

in the parallel direction. At a distance of 6.0 km in the normal direction to the fault

and for periods more than 1.0 second, the velocity spectrum exceeds the 100 kine

(given for 90% non-excess probability), which shows the importance of considering

directivity in velocity spectra for seismic design of buried structures. This is also

clear from the Fig. 12 which depicts velocity spectra for two directions obtained

from recorded waves during Kobe Earthquake.

5. Performance Characteristics of Buried Pipes Network in

Dierent Distances from the Fault

5.1. Outlines and conditions of analyses

In this part, the generated waves by the Statistical Synthesis Method using Ra-

diation Pattern Function are used for increment distances from the fault as input

waves for response analysis of a 3D model of buried pipes in two directions using the

ABAQUS code. The generated waves in two directions are applied in two orthogo-

nal directions, normal and parallel to the faults, for comparison and investigation

of pipe seismic responses. The rupture length and width of the fault corresponded

to a magnitude 7.0 are obtained as 40.0 km and 20.0 km empirically. The waves at

the bedrock are generated for every 1.0 up to 10.0 km from the fault and applied

in direction 1 (parallel to the fault), direction 2 (normal to the fault) and both

directions simultaneously. The input earthquakes for distances 1.0 km and 2.0 km

are given in Fig. 13 as examples.



-600

-400

-200

0

200

400

600

0 1 2 3 4 5

Parallel to the fault

Maximum acceleration405.0gal

Maximum acceleration604.4gal

Accele

rati

on(g

al)

Normal to the fault

(a) Distance from the fault: 1.0 km

-600

-400

-200

0

200

400

600

0 1 2 3 4 5Time(sec)

Parallel to the faultNormal to the fault

Maxmum acceleration574gal

Maximum acceleration139gal

Accele

rati

on(g

al)

(b) Distance from the fault: 2.0 km

Fig. 13. Example of input ground motions.

Parallel to thefault buried pipe

100m100m

25m

1.5m 1.5mDirection 3

Direction 2iParallel to the fault j

Direction 1iNormal to the fault j

Burialdepth

Normal to thefault buried pipe

Burialdepth

(a) Case 1

Burialdepth

1.5m

Normal to thefault buried pipe

Direction 2Parallel to the fault j

Direction 1(Normal to the fault)

Direction 3

100m100m

20m

5m

20m

20m

20m

20m

20m

4m

Parallel to thefault buried pipe

1.5m

(b) Cases 2 and 3

Fig. 14. Model for analysis.

Table 3. Cases of analyses.

Case Ground Input ground motion

Case 1 Uniform Direction 1: Normal to the fault , Direction 2: Parallel to the fault

Case 2 Inclined bedrock Normal one was applied in directions 1 and 2

Case 3 Inclined bedrock Direction 1: Normal to the fault, Direction 2: Parallel to the fault

There are three cases of analyses, which are given in Table 3. The models of

the cases are also shown in Fig. 14. The length, width and depth of the models are

100.0, 100.0 and 25.0 m, respectively. The N -value is 10 and there is 45 bedrockinclination in the model for Cases 2 and 3. The Mohr{Coulomb criterion is used

for soil failure. The bottom of the models is xed while the other boundaries are

viscous. The pipe is a 150 mm welded steel with bilinear characteristics buried

in a depth of 1.5 m and simulated as a beam element in both directions.



Normal to the faul t

Paral le l to the faul t0.25

0.2

0.15

0.1

0.05

0

0.3

Max

imum

stra

in in

pip

e ax

ial d

irect

ion

0 1 2 3 4 5 6 7 8 9 10Dis tance f rom the fau l t (km)

Fig. 15. Relation between the maximum axial strain in the pipes and distance from the fault(Case 1).

1

1 .1

1 .2

1 .3

1 .4

1 .5

0 1 2 3 4 5 6 7 8 9 1 0Distance f rom the faul t (km)

Stra

in in

nor

mal

dire

ctio

n /

Stra

in in

par

alle

l dire

ctio

n

Fig. 16. Distance from the fault and ratio of axial strains of pipe (Case 1).

5.2. Results of the analyses and discussion

The relation between the pipes maximum strain and the distance from the fault

for the Case 1 is shown in Fig. 15. According to this gure, up to 7.0 km from

the fault, the strains in normal direction, are larger than those in parallel di-

rection. The relation between the maximum strain ratio of these two directions

and the distance from the fault is given in Fig 16. The ratio of strains in normal

to parallel direction in this gure for 1.0 to 7.0 km distances from the fault is

about 1.2 to 1.3 times, while for the distances more than 8.0 km, it decreases to

1.1 times.

The similar results are given in Fig. 17 for Cases 2 and 3. This gure also

shows that the maximum strain of pipe buried in normal direction are greater

than those buried in parallel. The ratio of maximum strains of normal to parallel

direction is shown in Fig. 18 for Cases 2 and 3. Figure 19 shows the influence ratio



fault

0

0 .05

0 .1

0 .15

0 .2

0 .25

0 .3

0 .35

0 .4

0 .45

0 .5

0 1 2 3 4 5 6 7 8 9 1 0

Case2 Para l le l to the faul t

Case3 Normal to the fau l t

Case3 Para l le l to the faul t

Dis tance f rom the fau l t (km)

Max

imum

axi

al st

rain

of p

ipes

(%)

Case2 Normal to the fau l t

Fig. 17. Relation between the maximum axial strains in the pipes and distance from the fault(Cases 2 and 3).

Dis tanse f rom the f au l t (km)

Stra

in in

nor

mal

to th

e fa

ult/

Stra

in p

aral

lel t

o th

e fa

ult

0 1 2 3 54 6 7 8 9 101 .8

2 .22 .42 .62 .8

33 .23 .43 .63 .8

1 .8

2 .22 .42 .62 .8

33 .23 .43 .63 .8 P [

X Q P [ X RC a s e 2C a s e 3

2

Fig. 18. Distance from the fault and the ratio of maximum axial strains in the pipes (Cases 2and 3).

of bedrock inclination (Case 2) in the total ratio of maximum strain of normal to

parallel direction, which includes the eects of both seismic directivity and bedrock

inclinations (Case 3 in Fig. 18. The strain ratio is dierent for any distance from

the fault and is larger in normal direction for any distance in Case 3, in which the

directivity is also included.

The results in Figs. 18 and 19 show that at near eld of the fault, the directivity

of ground motion (seismic directivity) is considerable. However for the distances



Dis tance f rom the fau l t (km)0.5

0 .6

0 .7

0 .8

0 .9

1

0 1 2 3 4 5 6 7 8 9 1 0

Effe

ct o

f bed

rock

incl

inat

ion

on p

ipe

strai

n ra

tioin

2 d

irect

ions

Fig. 19. Distance from the fault and the influence ratio of bedrock inclination on ratio of strainin pipe axial direction (Case 3).

more than 7.0 km from the fault (far eld), there is almost no eect of seismic

directivity and the eect of bedrock inclination becomes predominant.

6. Conclusions

Damage directivity in buried pipelines in Kobe City during the 1995 Hyogoken

Nambu Earthquake are investigated. For investigating the eect of seismic direc-

tivity and bedrock inclination eects at near and far elds, two orthogonal pipes

are analyzed in a 3D model of ground. The input waves for normal and parallel

direction are generated using a Statistical Synthesis Method considering Radiation

Pattern Function. The main ndings are as follows:

(1) The statistical analyses of the GIS based data in Kobe City show that the

water, gas and telecommunication pipelines, which were buried normal to the

faults suered much damage comparing to those in the parallel direction.

(2) Comparison among the results with those from the attenuation formula of

Fukushima{Tanaka shows that the maximum acceleration in normal direction

at a distance of 10.0 km to the fault is more than that obtained by the for-

mula. While in the parallel direction, by increasing the distance from the fault,

it decreases very fast, and for a distance of 2.0 km, it becomes less than that

obtained by the attenuation formula.

(3) The spectrum characteristics are clearly dierent for normal and parallel direc-

tions to the fault and the one in the normal direction is quite larger than that

in the parallel direction, which shows the importance of considering directivity

in velocity spectra for seismic design of buried structures.



(4) In the case of horizontal bedrock the ratio of pipe strain in normal to parallel

direction in the near eld is not increasing so severely (about 1.2 to 1.3 times)

and for the distances more than 8.0 km decrease to 1.1 times.

(5) At near eld of the fault, the directivity of ground motion is very eective.

However for the distances more than 7.0 km from the fault, there is almost no

eect of ground motion directivity and the eect of bedrock inclination is the

predominant one.

References

Boore, D. M. [1983] \Stochastic simulation of high-frequency ground motions basedon seismological models of the radiated spectra," Bull. Seism. Soc. Am. 73(6),1865{1894.

CIHAE [1998] Committee of Investigation on Hanshin-Awaji Earthquake, \Report onHanshin-Awaji disaster," JSCE, pp. 58{68 (in Japanese).

Fujiwara, S., Ogawa, S., Murakami, M. and Tobita, M. [2000] \Estimation of fault positionsof the 1995 Hyogo-ken Nanbu Earthquake using displacement gradient detected bySAR interferometry," JISIN 53(2), pp. 127{136.

Fukuda, K. [2000] \Study on dynamic behavior of buried pipelines and aseismic de-sign in the vicinity of active faults," Ph.D. dissertation, Kobe University, pp. 73{94(in Japanese).

Fukushima, Y. and Tanaka, T. [1990] \A new attenuation for peak horizontal accelerationof strong earthquake ground motion in Japan," Bull. Seism. Soc. Am. 80(4), 757{783.

Hirasawa, T. and Stauder, W. [1965] \On the seismic body waves from a nite movingsource," Bull. Seism. Soc. Am. 55(2), 237{262.

JWWA [1997] Japan Water Work Association, \Recommendation for anti-seismic con-struction of water facilities," JWWA, pp. 17{18 (in Japanese).

Mori, K. [1998] \A study about the strong motion estimation considering ground motionobservation near source fault and using asperity model," Master thesis, Civil Eng.Dept. Kobe University, pp. 32{56 (in Japanese).

Takada, S., Fukuda, K. and Kitada, T. [1999] \Damage directivity in buried pipelines,"The 4th Symposium on Urban Inland Earthquakes, pp. 511{514 (in Japanese).

Takada, S., Hassani, N., Fukuda, K., and Kitada, T. [1999a] \Damage directivity in buriedpipelines and ground motion characteristics at near eld," The 25th Conference onEarthquake Engineering, pp. 1053{1056 (in Japanese).



ON THE OPTIMAL LOCATION OF SENSORS INMULTI-STOREYED BUILDINGS

ALOKE K. DATTA, MANISH SHRIKHANDEy and DILIP K. PAULz

Department of Earthquake Engineering,University of Roorkee, Roorkee{247 667, India

[email protected]@rurkiu.ernet.in

Received 23 August 2000Revised 15 January 2001Accepted 30 May 2001

A new approach has been developed for nding optimal location of sensors in 3D multi-storeyed buildings. This approach is based on the compact probabilistic representationof acceleration response in terms of its covariance matrix. For a specied number ofsensors, the optimal location has been taken to be the one for which the computedcovariance matrix is closest to the exact covariance matrix of the random eld constitutedby the acceleration response process. It has been found that the determined sensorlocations match favourably with those predicted by earlier studies for the special case ofshear buildings. Further, the optimality of the determined sensor locations has also beenveried by identifying the system parameters from the time series data and comparingthem with those of the Finite Element Model.

Keywords: Optimal sensor location; system identication; structural health monitoring.

1. Introduction

The response of large, complex structural systems to external loads is usually

predicted by using a mathematical model developed with the help of a suit-

able discretization procedure, e.g. nite element method. The reliability of such

mathematical models can only be assessed by comparing the predicted response

with the response of the prototype structure recorded in eld conditions. The

mathematical model, if required, can be rened with the help of structural pa-

rameters estimated from eld measurements by using system identication tech-

niques [Friswell and Mottershead, 1995]. In addition to calibrating the analytical

model, the identied structural parameters are also very useful in damage assess-

ment or health monitoring, re-evaluating the original design, and for controlling the

vibrations. For this purpose, response has to be recorded at several locations in the

Research Scholar, on leave from Regional Engineering College, Durgapur, India.yAssistant Professor, Corresponding author.zProfessor.

17


18 A. K. Datta, M. Shrikhande & D. K. Paul

building/structure and it is desirable to install as many sensors as possible. The high

costs of data acquisition systems (accelerometers, recorders, etc.) and accessibility

limitation, however, often put a constraint on the number of sensors. Therefore, it is

important to determine the optimal location of sensor(s) in the building such that

selected locations maximize the information content with respect to the parameter

identication.

Shah and Udwadia rst formulated the problem of optimal location of sensors as

a statistical inference problem, wherein a norm of the covariance matrix of the esti-

mated parameters (from the time-domain records) is minimised [Shah and Udwadia,

1978]. The covariance matrix was computed by using a linear relationship between

small perturbations in a nite dimensional representation of system parameters and

nite sample of observations of the system time response. Another approach to solve

the optimal sensor location problem was presented by Udwadia [1994], in which the

optimal sensor locations were dened as those for which a norm of the Fishers

information matrix is maximised. A method for locating sensors in reinforced con-

crete structures to gain maximum information about the damage distribution has

also been presented on the basis of numerical simulation studies [Skjrbk et al.,

1996]. The method proposed by Udwadia was extended by Heredia-Zavoni and

Esteva [1998] to account for the uncertainty about the structural parameters and

the seismic ground motion excitation within the framework of Bayesian Decision

Theory [Goodwin and Payne, 1977]. The optimal locations were so selected that

the expected Bayesian loss function, expressed in terms of the Fishers information

matrix of the response records, is minimised. Recently, this approach has been fur-

ther extended to the case of optimal instrumentation of structures on flexible base

[Heredia-Zavoni et al., 1999].

All the above mentioned studies have focussed on the issue of locating sensors

in a structure so as to facilitate the best possible identication of a set of structural

parameters selected a priori. In spite of a rigorous formulation of the optimal sen-

sor location problems, the two approaches followed by Udwadia and Heredia-Zavoni

and his co-workers respectively, provide contradictory results in so far as the eect

of noise on optimal sensor locations is concerned. Moreover, these formulations may

not appeal to a majority of civil engineering community not trained in information

theory. On the other hand, the method proposed by Skjrbk et al. [1996], using

sub-structuring technique, does not guarantee convergence of the iterative proce-

dure. Therefore, we look for a simple and robust technique for solving the optimal

sensor location problem in this study. The excitation and response processes have

been modelled as stationary Gaussian with zero mean. The problem of optimal

sensor locations has been formulated in the form of compact probabilistic represen-

tation of random processes similar to the one considered by Masri and Miller [1982].

Two example cases (for dierent building congurations) have been considered for

illustration of the proposed approach.


On the Optimal Location of Sensors in Multi-Storeyed Buildings 19

2. Formulation

The equation of motion for a multi-degree-of-freedom (MDOF) system subjected

to earthquake excitation can be written as

Mx + C_x + Kx = Mrxg (1)where, M, C and K are the inertia, damping and stiness matrices of size N Nrepresenting the structural system, x is the N 1 vector of nodal displacements,r denotes the N 3 matrix of the rigid body influence coecients, xg is the 3 1vector of translational ground acceleration time histories, and a dot (_) over a

variable indicates the time derivative.

A 3-dimensional nite element model of multi-storeyed buildings is considered

in this study. It is assumed that each node of the the model represents a possi-

ble sensor location and the nodal accelerations (translational) correspond to the

sensor records. The ground acceleration process is assumed to be stationary Gaus-

sian with zero mean. Assuming that the structural system responds linearly, the

response process will also be stationary Gaussian with zero mean (neglecting the

eect of start-up transients). Since a Gaussian process is completely described by

the rst two spectral moments, it follows that a complete statistical representation

of the zero mean, Gaussian random process representing the response of a linear

system requires an accurate representation of the covariance kernel. Therefore, to

preserve the information content of the response process, its covariance needs to be

reproduced as faithfully as possible. The problem of determining optimal locations

of the sensors can be now recast in the form of determining the degrees of freedom

(DOFs) in the structural system for installation of a given number of sensors, say

m N , which provide the best estimate of the covariance kernel of the responseprocess.

As a rst approximation, let us assume the response process to be a zero mean,

ergodic Gaussian process. The sample realisation being characterised by concate-

nation of the response of all nodes for increasing the length of the time history

in order to get more stable estimates of the temporal statistics. For example, the

sample realization in the longitudinal direction is considered as the time history

obtained by appending the response in longitudinal direction of a node to the tail

of the response of previous node resulting into one long stretch of response time his-

tory in longitudinal direction. Similarly, realisations of the response time history in

other two orthogonal directions can be characterised. The 33 temporal covariancematrix of these three time histories represents the exact covariance matrix, C of theresponse process. Thus the ijth element of the temporal covariance matrix, Cij , isgiven by

Cij = 1N 1

NXk=1

(ai(k) i)(aj(k) j) ; i; j = 1; 2; 3 (2)

where, ai and aj denote the two orthogonal components of acceleration, N is the

total number of data points in the time history, and i(=1N

PNk=1 ai(k)) is the



temporal mean of the ith component of acceleration. Assuming that m number

of tri-axial sensors are available, these can be distributed among the N possible

locations in(Nm

dierent ways. For each such combination, sample realisations

can be characterised similarly by the concatenation of response of each of the m

selected nodes. The temporal covariance matrix of these realisations characterises

the approximate covariance matrix, C^ of the response process. The optimal locationsfor these m number of sensors will be those for which the dierence between some

suitable norm of C and C^ is minimised. In this study, the goodness of approximationof C by C^ is measured by using (i) the determinant norm, (ii) the trace norm, and(iii) the eigenvalues of these matrices.

The response records from the sensors thus located may be used for the system

identication studies. It will now be shown with the help of illustrative examples

that the response records corresponding to the derived optimal sensor locations

indeed lead to robust estimation of the system parameters.

3. Illustrative Examples

The optimal sensor locations have been derived for two multi-storeyed reinforced

concrete buildings shown in Figs. 1 and 2. One of these buildings is symmetric

in plan, while the other building, with asymmetry in both plan and elevation, is

typical of constructions on hill slopes. The column cross-section (300 mm 300 mm)

3.0m

3.0m

3.0m

3.0m

ELEVATION SIDE VIEW

3.0m

3.0m 2.0m

Fig. 1. Schematic diagram of 5-storeyed symmetric building.



ELEVATION SIDE VIEW3.

0m3.

0m3.

0m3.

0m3.

0m

5.0m 5.0m 5.0m 4.0m

1.5m

1.5m

1.5m

Fig. 2. Schematic diagram of 6-storeyed asymmetric building.

is assumed to be uniform throughout the height for both the buildings, whereas the

beam cross-sections are taken as 300 mm 300 mm and 300 mm 230 mm for thesymmetric and asymmetric buildings, respectively. The slab thickness is taken to

be 150 mm at all levels. The rst six modes were used in the mode superposition

for computation of the response of buildings to the earthquake excitations. The

damping has been assumed to be of Rayleigh type with the viscous damping ratio

of 5% in the rst and sixth mode. The resulting modal damping for intermediate

modes is approximately 5%, whereas the higher modes | with modal damping pro-

portional to the square of the natural frequency | are damped out of the response.

An in-house general purpose dynamic nite element analysis code [Kumar 1996]

has been used to compute the acceleration response at each node of the linear nite

element models of these buildings for two dierent types of earthquake motions,

viz., (i) Uttarkashi Earthquake of 21 Oct., 1991 in Western Himalayas recorded at

Bhatwari, 20 km from the epicentre, shown in Fig. 3, and (ii) Michoacan Earthquake

of 19 Sept., 1985 in Mexico recorded at CDAO station, 400 km from the epicentre,

shown in Fig. 4. While the Uttarkashi Earthquake records are representative of

near-eld motion due to a moderate size earthquake, the Michoacan Earthquake

time histories represent the far-eld motions due to a large (magnitude 8+) earth-

quake. No appreciable eect of the flexibility of the floor diaphragm was noticed in

the analysis. Hence the floor diaphragm is assumed to be rigid for all calculations

and the response of any one node on a floor is considered to be representative of

the floor response.



-0.30-0.150.000.150.30

0 5 10 15 20 25 30 35 40Acc

eler

atio

n (g)

Time (s)

N85E

-0.30-0.150.000.150.30

0 5 10 15 20 25 30 35 40Acc

eler

atio

n (g)

Time (s)

Vert

-0.30-0.150.000.150.30

0 5 10 15 20 25 30 35 40Acc

eler

atio

n (g)

Time (s)

N05W

Fig. 3. Time histories of Uttarkashi Earthquake.

-0.08-0.040.000.040.08

0 20 40 60 80 100 120 140 160 180Acc

eler

atio

n (g)

Time (s)

N00E

-0.08-0.040.000.040.08

0 20 40 60 80 100 120 140 160 180Acc

eler

atio

n (g)

Time (s)

Vert

-0.08-0.040.000.040.08

0 20 40 60 80 100 120 140 160 180Acc

eler

atio

n (g)

Time (s)

N90E

Fig. 4. Time histories of Michoacan Earthquake.



For each component of motion, the sample response time history is obtained by

concatenating the response of all the floors together. The covariance matrix com-

puted from these time histories are considered to be the exact representation of

the the zero mean, Gaussian random process characterising the building response.

Further, let us consider for the purpose of illustration, the optimal location of

two tri-axial sensors in these two buildings. The approximate temporal covariance

Table 1. Covariance matrix representation with two sensors (forUttarkashi Earthquake, Symmetric Building).

Combination j C^ j Tr[C^] Eigenvalues (101g2)

(104g6) (101g2) 1 2 3Exact 0.97 1.61 0.94 0.44 0.23

First + Roof 1.28 1.68 0.89 0.51 0.29

Second + Roof 1.99 1.97 1.07 0.57 0.33

Third + Roof 2.57 2.20 1.24 0.63 0.33

Fourth + Roof 3.52 2.48 1.41 0.72 0.35

First + Second 0.12 0.78 0.45 0.21 0.12

First + Third 0.23 1.02 0.62 0.26 0.14

First + Fourth 0.44 1.29 0.79 0.34 0.17

Second + Third 0.47 1.31 0.81 0.32 0.18

Second + Fourth 0.79 1.58 0.98 0.40 0.20

Third + Fourth 1.06 1.82 1.14 0.48 0.19

Table 2. Covariance matrix representation with two sensors (forMichoacan Earthquake, Symmetric Building).


(1017g6) (105g2) 1 2 3Exact 1.18 2.79 17.22 10.61 0.06

First + Roof 1.25 2.95 18.25 11.19 0.06

Second + Roof 2.01 3.42 21.06 13.04 0.07

Third + Roof 3.47 4.11 25.48 15.53 0.09

Fourth + Roof 5.56 4.89 30.47 18.34 0.10

First + Second 0.04 0.77 4.52 3.16 0.03

First + Third 0.21 1.46 8.93 5.66 0.04

First + Fourth 0.64 2.25 13.91 8.49 0.05

Second + Third 0.48 1.93 11.74 7.49 0.05

Second + Fourth 1.14 2.71 16.72 10.32 0.07

Third + Fourth 2.18 3.40 21.14 12.81 0.08



matrix, C^, computed from the concatenated records of these two sensors placed ondierent floors. The three norms, viz., determinant (jC^j), trace (Tr[C^]), and eigenval-ues (i), of C^ for dierent combinations of sensor locations are compared with thoseof C in Tables 1{4 corresponding to the four cases of two buildings excited by twodierent earthquake motions. It may be inferred from Tables 1 and 2 that for the

case of symmetric buildings, the records from two sensors located at either the rst

floor and roof or at the second floor and fourth floor provide the best approxima-

tion for the covariance matrix for both types of excitations. Further, the sensors

located either at the rst and second floor, or at the fourth floor and roof present

the two worst sensor placement strategies for this building. It may be noted that in

the case of symmetric building, response records from only two optimally located

sensors are sucient for a good estimation of the covariance matrix of the response

process. The maximum error in all the three norms of the covariance matrix is less

than 6% for the case of best location of sensors in the symmetric building excited

by Michoacan Earthquake motions. However, the maximum error is in the range of

5% to 20% in the case of Uttarkashi Earthquake. This wide variation in the errors of

approximation could be due to the short duration of Uttarkashi earthquake motion,

and presence of noise in Uttarkashi Earthquake records, which were derived from

analog traces [Chandrasekaran and Das, 1991]. These results for the identication

of optimal locations of sensors are in good agreement with the earlier published

Table 3. Covariance matrix representation with two sensors (forUttarkashi Earthquake, Asymmetric Building).


(106g6) (101g2) 1 2 3Exact 0.84 0.59 3.53 2.24 0.10

First + Roof 4.16 0.97 6.00 3.48 0.19

Second + Roof 2.12 0.99 6.45 3.42 0.09

Third + Roof 1.86 1.00 6.28 3.67 0.08

Fourth + Roof 3.59 1.21 7.24 4.77 0.10

Fifth + Roof 4.37 1.21 7.64 4.37 0.12

First + Second 0.06 0.15 0.96 0.39 0.15

First + Third 0.06 0.16 0.85 0.56 0.13

First + Fourth 0.48 0.36 1.77 1.72 0.16

First + Fifth 0.53 0.37 2.14 1.34 0.19

Second + Third 0.02 0.19 1.23 0.58 0.03

Second + Fourth 0.20 0.39 2.20 1.69 0.05

Second + Fifth 0.27 0.39 2.60 1.28 0.08

Third + Fourth 0.15 0.40 2.02 1.94 0.04

Third + Fifth 0.24 0.40 2.41 1.54 0.06



Table 4. Covariance matrix representation with two sensors (forMichoacan Earthquake, Asymmetric Building).


(1016g6) (104g2) 1 2 3Exact 12.52 2.68 225.20 42.79 0.13

First + Roof 60.81 4.24 357.84 65.98 0.26

Second + Roof 26.51 4.27 360.67 66.46 0.11

Third + Roof 22.97 4.38 370.75 66.69 0.09

Fourth + Roof 43.75 5.27 443.60 83.53 0.12

Fifth + Roof 91.15 6.78 571.86 106.25 0.15

First + Second 0.02 0.07 4.83 1.91 0.20

First + Third 0.07 0.17 14.52 2.50 0.18

First + Fourth 3.46 1.07 87.39 19.24 0.21

First + Fifth 21.56 2.58 215.67 42.03 0.24

Second + Third 0.02 0.20 17.47 2.84 0.03

Second + Fourth 1.05 1.10 90.31 19.64 0.06

Second + Fifth 8.45 2.61 218.53 42.47 0.09

Third + Fourth 0.83 1.20 100.31 19.92 0.04

Third + Fifth 7.15 2.71 228.57 42.72 0.07

results for shear buildings [Heredia-Zavoni and Esteva, 1998; Udwadia, 1994]. This

is not surprising since the optimality criterion used in earlier studies, e.g. maximi-

sation of a norm of Fishers information matrix, or minimisation of the Bayesian

loss function can be related to the accurate representation of the covariance matrix

of the Gaussian random process [Goodwin and Payne, 1977]. It may, however, be

noted that the derived optimal sensor locations for the symmetric building are not

applicable for asymmetric building. The rst and fth floors of the six storeyed

asymmetric building correspond to the best possible locations for two sensors as

can be inferred from Tables 3 and 4. Further, it should be noted that the response

records from only two sensors are not sucient for accurately representing the co-

variance matrix of the response process and more sensors should be used in the

case of asymmetric buildings.

The floor response data corresponding to the best location of sensors will now

be used for estimation of modal parameters of the building by using system identi-

cation techniques. It will be shown that data from the best sensor locations provide

a robust estimate of the model parameters vis-a-vis any arbitrarily selected pair of

response data.

3.1. Parameter estimation

The ground acceleration records and the building response computed at dierent

floors are assumed to be the input-output pairs for this identication study. In the



system identication terminology such systems are known as multi-input multi-

output (MIMO) systems and the system parameters may be identied by using

either parametric or non-parametric methods. However, parametric methods ex-

hibit better convergence properties in the case of aperiodic data sequences as com-

pared to the non-parametric methods [Ljung, 1987]. Therefore, the seismic response

of buildings as considered in the numerical examples is modelled as auto-regressive

sequence with external inputs (ARX model). The predictor of such a model is

characterised by a linear regression of model parameters on a known data set com-

prising of the past input and output sequences as:

y(t) + a1y(t 1) + + anay(t na)= b1u(t 1) + + bnbu(t nb) + e(t) : (3)

Equation (3) is a linear dierence equation relating the system response y(t) to

the past values of y(t), input u(t) and a random noise e(t). The model parameters

are then determined by minimizing the quadratic norm of the error between the

recorded and predicted response vectors [Ghanem and Shinozuka, 1995; Ljung,

1987; Shinozuka and Ghanem, 1995]. These parameters are then used to determine

the transfer function of the model, in z-transform, as

H(z) =

nbXk=1

bkzk

naXl=0

alzl

; a0 = 1 (4)

where, H(z) is the transfer function for the ARX model dened in Eq. (3). The con-

ventional transfer function in the frequency domain can be obtained by evaluating

H(z) on the unit circle. The poles, with positive imaginary part, of this transfer

function are used to determine the natural frequency and damping ratio [Ghanem

and Shinozuka, 1995]. In particular, the pole corresponding to the jth mode of

vibration is given as

zj = exphj!j + i!j

q1 2j

ti

(5)

where, j and !j respectively represent the damping ratio and the natural frequency

of vibration in jth mode, and t is the sampling interval of time histories. Alter-

natively, the natural frequencies can be identied from the resonant peaks in the

transfer function plots and the damping ratio can be estimated from the bandwidth

of transfer function in the neighbourhood of the natural frequency by the half-power

bandwidth method. Further, the mode shapes can be obtained by using quadrature

curve tting technique [Richardson, 1997]. In this technique, the peak values of the

imaginary part of the acceleration/force frequency response function, or transfer

function are taken as components of mode shapes at the sensor locations. In case

of velocity/force frequency response functions, the peak values of the real part are

taken as the mode shape components.



0 1 2 3 4 5 6 7 8 9 100

2

4

6

8

10Am

plitu

deFrequency response

0 1 2 3 4 5 6 7 8 9 101000

800

600

400

200

0

Frequency (Hz)

Phas

e (de

g)

Fig. 5. Sample transfer function of the ARX model for symmetric building. Solid curves: ARXmodel estimate, Dotted curves: Periodogram estimate.

The input-output pairs corresponding to the Uttarkashi Earthquake excitations

for both symmetric and asymmetric buildings are considered for the identica-

tion purpose. The identication of model parameters in this case is relatively more

dicult because of the short duration of records and presence of noise. Temporal

evolution of the model parameters have been monitored by using the data win-

dowing approach, wherein only a part of the data is used at a time. For each data

window of 5 s length the highest order stable ARX model is determined for the

input-output pairs by using MATLAB. The transfer functions were computed from

the identied parameters of the ARX model for the six data windows for dierent

combination of sensor locations. A sample of a transfer function for the case of

optimal location of sensors, i.e. rst floor and roof is shown in Fig. 5. The transfer

function estimate obtained from the non-parametric (periodogram) method is also

shown in the gure as dotted curve for comparison. It can be seen that the para-

metric modelling permits a ner resolution of resonant peaks which may be helpful

in estimation of damping by the half-power bandwidth method. However, the para-

metric transfer function tends to smoothen out some closely spaced peaks in the

transfer function as seen in the gure in the neighbourhood of 9.2 Hz. The phase

spectrum of the transfer function shown in the gure correspond to the unwrapped

phases [Childers et al., 1977; Tribolet, 1977] as computed by MATLAB routines. The

resonant frequencies of the system can be identied as 1.8, 5.3, and 9.2 Hz, which



compare favourably with the frequencies of the nite element model. The rst six

natural frequencies of the nite element model of the symmetric building were com-

puted as 1.92, 1.96, 2.80, 5.96, 6.20, and 8.46 Hz, respectively. It may be noted that

the translational modes of the symmetric building in longitudinal and transverse

directions are very closely spaced and it is dicult to distinguish such closely spaced

modes from the analysis of response data. Thus the rst identied frequency from

the model may be considered to be applicable for the rst and second mode of the -

nite element model, while the second identied frequency corresponds to the fourth,

and fth mode of vibration of the building. The third mode of vibration, a vertical

mode, could not be identied from the ARX model. Similar estimates of the natu-

ral frequencies were obtained from the model parameters identied from dierent

windows when the response records were those corresponding to the optimal sensor

locations i.e. either the rst floor and roof or the second floor and fourth floor.

However, the response records from other locations/floors did not yield consistent

frequency estimates for all windows, and in some cases the lowest mode could not

be identied at all. Thus the choice of the response records plays a very important

role in the identication process. The records from derived optimal sensor locations

are found to lead to robust estimates of the system parameters.

Similarly, the rst six natural frequencies of the nite element model of the

asymmetric building were determined as 1.97, 2.79, 2.84, 5.00, 5.37, and 6.39 Hz,

0 1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

Ampl

itude

Frequency response

0 1 2 3 4 5 6 7 8 9 101000

500

0

500

Frequency (Hz)

Phas

e (de

g)

Fig. 6. Sample transfer functions of the ARX model for asymmetric building. Solid curves: ARXmodel estimate, Dotted curves: Periodogram estimate.



respectively. The frequencies estimated from the identied ARX model of the system

using records from optimal sensor locations are 1.4, 3.4, 4.5, and 6.2 Hz. A sample

transfer function of the ARX model identied from the data is shown in Fig. 6 along

with the transfer function estimate by non-parametric method (shown as dotted

curve). These frequency estimates appear to be reasonable since it is not possible

to capture adequate information about the dynamic behaviour of the asymmetric

building by using only two sensors. It may be noted that though these frequencies

were identied from the models for most of the data windows, the peaks of the

transfer functions at lower modes were not very prominent. Further, for a viscous,

lightly damped system, the phase should change by approximately 180 in theneighbourhood of a natural model. However, the change in the phase spectrum at

4.5 Hz is much less than 180 indicating that this mode is very heavily damped inthe developed model. This is also apparent from the relatively flat nature of the

transfer function amplication at 4.5 Hz. This indicates that the data set used for

developing the model does not adequately capture the dynamic behaviour of the

system under consideration. Inclusion of response from one or two more locations

may improve the quality of the parameter estimates. Further, the non-parametric

transfer function contains several peaks which are not present in the smooth transfer

function of the parametric model. Authenticity of these additional peaks seen in the

non-parametric transfer function estimate can be ascertained only by considering

more response data from dierent locations for system identication, which will also

help in better modelling of phases. Moreover, as in the case of symmetric building,

the response records from other locations provide very inconsistent estimates of the

system frequencies identied from dierent data windows. Similar trends were also

observed in the case of parameter identication with response records for Michoacan

earthquake excitation.

4. Conclusions

A simple methodology for optimal installation of a specied number of sensors in

multi-storeyed buildings for system identication studies has been proposed. The

method is based on the concept of compact probabilistic representation of random

processes and is very easy to apply. The proposed methodology can also be used

to assess the adequacy of a specied number of sensors in capturing sucient in-

formation for system identication purposes. The methodology has been validated

by using it to derive the optimal locations of two sensors in two dierent types

of buildings. The derived optimal locations of two sensors in a ve-storeyed sym-

metric building compare favourably with the locations predicted by other sensor

placement methodologies. It has been found that the optimal sensor locations for the

symmetric building are dierent from those for the asymmetric building. Further,

the placement of sensors in the building are found to have an important bearing

on the reliability of the model parameters identied by using system identication

techniques.



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SIMULATION OF NONLINEAR SOIL-STRUCTUREINTERACTION ON A SHAKING TABLE

KAZUO KONAGAI and RAQUIB AHSAN

Institute of Industrial Science, University of Tokyo, 4-6-1 Komaba,Meguro-ku, Tokyo 153-8505, Japan

Received 23 May 2000Revised 12 January 2001

Accepted 28 February 2001

A new method to simulate soil-structure interaction eects in shaking table tests hasrecently been presented by the authors. In this method, analog circuits or digital signalprocessors are used to produce soil-foundation interaction motions in real time. Theirexpressions of soil-structure interaction motions are based on published rigorous formu-lations of impulse response functions or flexibility functions of foundations resting onor embedded in homogeneous or layered soils of semi-innite extent. In this paper themethod is further extended to take the \far eld" soil non-linearity into account. Anexample of non-linear soil-structure interaction simulation using the present method isalso described.

Keywords: Nonlinear soil-structure interaction; shaking table test; dynamic pile-group-head stiness; model experiment.

1. Introduction

When a structure is subjected to a ground excitation, it interacts with its sub-

structure, i.e. foundation and soil. In other words, the motion of the ground is

altered because of the vibration of the structure. This dynamic soil-structure in-

teraction is associated with the influx and eux of energy, which is generated by

the earthquake excitation and transmitted through the soil-structure interface. The

dierence between the influx and eux corresponds to the energy stored up within

the structure, and thus, is closely related to the extent of damage to it. If this inter-

action eect can be rationally simulated in shaking table tests, one may obtain the

information necessary to interpret the failure processes of prototype structures in

terms of energy. For shaking table tests conducted without taking the interaction

into account, the input energy is totally consumed by the structure, producing quite

conservative results. On the other hand, tests conducted using a physical ground

model to incorporate associated non-linearity have twofold disadvantages. Firstly,

the nite dimension of the ground model fails to provide radiation damping, and

thus, yields conservative results. Secondly the weight of the ground model causes

an extra burden on the performance of the shaking table, restricting superstructure

31


32 K. Konagai & R. Ahsan

models to a smaller size. Hence, if observation of the behavior of a superstructure

is the main concern, neither of the above methods seems to be satisfactory.

In order to incorporate the interaction between soil and structure without using

a physical ground model, Konagai and Nogami [1997, 1998] introduced a new

method for shaking table tests. In their method, appropriate soil-structure inter-

action motions are added to free-eld ground motions to simulate soil-structure

interaction eects. The expressions of soil-structure interaction motions are based

on published rigorous formulations of flexibility functions of foundations resting on

or embedded in homogeneous or layered soils of semi-innite extents. The method

considers radiation damping which, in general, causes the total damping of a soil-

structure system to be greater than that of the structure itself. Thus the incorpora-

tion of soil-structure interaction eects in a shaking table test leads to a reduction

in the demands on the capacity of the shaking table.

The method was initially developed with the assumption that soil behaves lin-

early. In the present paper, the method is extended to take \far eld" soil non-

linearity into account through an equivalent linear approach. The non-linearity

produced in the vicinity of foundations, which is usually associated with large strain

and separation between soil and foundation, is not considered in this study. Here,

the dynamics of the interaction between a superstructure and a pile-group, along

with the semi-innite soil, are modeled with a simple flexibility function. A com-

puter program, based on the Thin Layered Element Method [Tajimi et al., 1976], is

used to derive the flexibility function for the lateral sway of the pile-group. In this

method the dynamic soil parameters are varied in real time by means of a digital

signal processor. The method, on the one hand captures the non-linear soil behavior

of softening and rehardening during the course of an earthquake, and on the other

hand may allow testing of larger superstructure models by obviating the need for a

heavy physical ground model. A comparison of the results of a test incorporating

soil-structure interaction and those of a conventional test is also presented in this

paper.

2. Outline of the Present Method

In this study, a soil-structure system is divided into two substructures, the su-

perstructure and the unbounded soil extending to innity; the latter includes an

embedded foundation as illustrated in Fig. 1, and is represented by a shaking table.

The multi-step method is used to describe two primary causes of soil-structure in-

teraction: the inability of the foundation to match the free-eld deformation, and

the eect of the dynamic response of the superstructure on the movement of its

supporting soil-foundation system. In the lower substructure of soil, an earthquake

will cause soil displacements, fufg. The foundation embedded in this soil deposit,however, will not follow the free-eld deformation pattern. This deviation of the

displacements from the free-eld soil displacements, fufg, is denoted by fusg. Themass of the superstructure then causes it to respond dynamically, and the forces,


Simulation of Nonlinear Soil-Structure 33

Fig. 1. Two primary causes of soil-structure interaction.

fpg, transmitted to the lower substructure of soil and foundation will produce fur-ther deformation of soil, fuRg (inertia interaction), that would not occur in a xedbase structure. Thus, the displacements of soil, fug, are eventually expressed bythe following equation:

fug = fufg+ fusg+ fuRg : (1)Consider the case that a foundation has two degrees of freedom in sway and rocking

(x; ) at the base of its super-structure, as illustrated in Fig. 1. The interaction

forces, fpg(= fpx pgT ), from the superstructure cause the inertia interaction mo-tions, fuRg, in the frequency domain to be:(

uRx

uR

)=

"Hxx(s) Hx(s)

Hx(s) H(s)

#(px

p

)(2)

where "Hxx(s) Hx(s)

Hx(s) H(s)

#= [H] (3a)

is the flexibility (compliance) at the top of the foundation, and

s = i ! (3b)in which i =

p1 and ! is the excitation circular frequency.In the present method, the motion of a shaking table is controlled directly

following the actual process of soil-structure interaction. Figure 2 shows a schematic

view of the set-up of a shaking table test, in which a superstructure model is placed

directly on the table without a physical ground model. Soil-structure interaction

eects are simulated by adding appropriate soil-structure interaction motions to

free-eld ground motions on the shaking table. In the simulation, rst, the trans-

ducers at the base of the foundation pick up the signals of the base forces, px and

p in sway and rocking motions, respectively. These two amplied signals are then

applied to the circuits Hxx, Hx, Hx and H to produce outputs corresponding

to the soil-structure interaction motions, uRx and uR . The output signals are then



Fig. 2. Simulation of soil-structure interaction on a shaking table.

added to the signals of the base input motions, ufx + usx and u

f + u

s, to produce

signals of foundation motions, ufx + usx + u

Rx and u

f + u

s + u

R . The method is,

thus, based on the premise that ufx + usx and u

f + u

s are known beforehand as the

base input motions. Signals of the foundation motions are nally translated into

the shaking table motions by the shaking table controller.

The above method requires a device that generates signals corresponding to the

soil-structure interaction motions, and a digital signal processor (DSP) which is ca-

pable of producing a variety of transfer functions is used as this device. The transfer

functions to be realized on the DSP are designed from the analytical expressions for

stiness or flexibility functions of the foundations. Konagai et al. [2000] have shown

that np piles closely grouped together beneath a superstructure can be viewed as

a single equivalent upright beam whose stiness matrix is described with two sti-

ness parameters, EIp and EIG. The parameter, EIp is identical to npEpIp, whereEpIp is the bending stiness of an individual pile and np is the number of piles in

a pile-group. EIG is evaluated following the same procedure as that used for the

evaluation of the bending stiness of a reinforced concrete beam; in this analogy,

piles and the soil mass caught among them are respectively compared with rein-

forcing bars and concrete. In other words, EIG is assumed to be equal to the sum

of the Youngs-modulus-weighted products of all the elementary areas times their

distances squared from the centroid of the cross-section AG (Fig. 3). Careful exa-

mination of deflections of pile groups reveals that most piles are indeed flexible in

practice in the sense that they do not deform over their entire lengths. Instead, pile

deflections become negligible below their active lengths, La; the active pile length

is given as a function of EIp and the shear modulus of the surrounding soil, , as:

La = L0 =

sEIp

: (4)



remain on one plane

nLhj

wjj

R0 w1

np piles

Agcentroid

Fig. 3. Assumptions for evaluation of equivalent single beam.

The parameter in the above equation diers in dierent soil proles. Konagai,

Ahsan and Maruyama [2000] carefully examined the solutions of pile cap stiness,

Sxx, in sway motion by using the upright single beam analogy, and showed that

they are closely approximated by the following expression:

Sxx = ks + ia cs a2ms (5a)

where

a =! R0s

(5b)

with ! = circular frequency and s = shear wave velocity; R0 = the radius of

the equivalent upright beam, which is assumed to be identical to the radius of a

circle with the same area as the cross-sectionAG that includes all the grouped piles

enclosed by the broken line in Fig. 3. For a homogeneous soil,

ks =

2R0 +

2L0

; (5c)

cs = 2L0 (5d)

and

ms = L0

4: (5e)

Equation (5a) clearly shows that the flexibility function, which is the inverse of the

dynamic-stiness of a soil-pile group system is approximated by that of a simple-

damped oscillator with a spring K(= ks), a damper C(= R0cs=s) and a mass

M(= R20ms=2s ) (Fig. 4). For an inhomogeneous soil, soil stiness in Eqs. (5c){

(5e) will be a mode-weighted average of shear modulus over the active pile length,

and thus, will vary with time as the nonlinear feature of soil develops.



K

C

M

Fig. 4. Modeling of a pile-group as a simple-damped oscillator.

3. Control of Shaking-Table

The system illustrated in Fig. 2 is realised on condition that a shaking table loses no

time in faithfully producing its input motion. The motion produced by the shaking

table, however, is not exactly identical to the intended time history because the

ratio of the output to input amplitude of the shaking table system does not remain

the same over the desired frequency range. The performance of the systems transfer

function is also aected by the presence of models on the shaking table; this fact

may cause the motion of the table to further deviate from the intended time history.

A controller with the transfer function T normally performs like a low pass lter,

and experiments on the table are often conducted below its cut-o frequency. Below

this frequency there yet remains a time delay t between the produced motion and

the input signal. The eect of the time delay, described in the frequency domain

as T = ei!t, can be canceled by multiplying the flexibility function H by T1.Assuming that the performance of a soil-foundation system is approximated by that

of a simple-damped oscillator with a spring K, a dashpot C and a mass M (Fig. 4),

the flexibility function Hxx is expressed as:

Hxx =1

K !2M + i!C : (6)

Thus, the cancellation of the time-delay eects is achieved by

HxxT1 = e

i!t

K !2M + i!C : (7a)

For smaller values of !t, Eq. (7a) is rewritten as:

HxxT1 = 1

K !2(M M) + i!(C C) (7b)

where

M = C t (8a)C = K t : (8b)

Equation (7b) shows that the equivalent mass and the viscous damping coecient

are reduced by C t and K t, respectively. The reduced mass M M and



the damping coecient C C must be positive, calling for:M

M= 42

tct

t20< 1 (9a)

C

C=

t

tc< 1 (9b)

with

tc = C=K (T ime constant) (10a)

t0 = 2pM=K : (10b)

The above conditions [Eqs. (9a) and (9b)] are usually satised in reality for many

cases of soil-structure interaction, because radiation of waves from a foundation

leads the motion of the structure to be noticeably damped.

It is, however, necessary for the time delay to be minimised when Eqs. (9a) and

(9b) are not satised. One possible measure for reducing the time delay is to in-

crease the feedback gain of the servo-amplier of a shaking table. The increase of the

servo-amplier gain, however, leads to a decrease in the margin for unstable clat-

tering of the table that is caused by the noise echoing through the closed circuit of

the servo-amplier [Konagai et al., 1999a]. A form of adaptive control known as the

Minimal Control Synthesis (MCS) algorithm has recently been used on a shaking

table at the University of Bristol, UK [Gomez and Stoten, 2000]. The MCS algo-

rithm responds in real-time to dynamic changes experienced by a structure model

and tunes the control parameters to optimum values in an automated manner. The

main advantage of this method is that it does not require any prior information on

the plant dynamic parameters. Implementation of another new control method in

a shaking table system is currently being developed in the Institute of Industrial

Science, University of Tokyo [Yamauchi et al., 2000]. In this method, an H-innity

robust control scheme has been adopted in conjunction with an adaptive lter to

give the controller the ability to track changes in the controlled system characte-

ristics. Thus, one of the major impediments to implementing the present method,

namely that preventing reliable realisation of the input motion in shaking tables,

is being overcome.

4. Experiments

To provide a proper perspective on the usefulness of the present method, its

application to a particular prototype is described herein. The prototype is an ex-

pressway viaduct supported by pile foundations embedded in an alluvial soil de-

posit [Fig. 5(a)]. This soil-viaduct system is divided into two substructures: the

superstructure of the steel-pipe pier carrying the 666 tf weight of the viaduct

deck [Fig. 5(b)] and the lower substructure of soil, which includes the pile foun-

dation [Fig. 5(c) and Table 1]. A model of the upper substructure is mounted on a



shaking table whose motion expresses virtually the unbounded soil substructure.

Since the shaking table used in the experiment has only one sway degree of freedom,

discussions on rocking motions were excluded.

Table 1. Parameters for steel piles.

Ep (tf/m2) (t/m3) r0 (m) Thickness (m) Length (m)

2:1 107 7 0.406 0.012 50

Fig. 5. Prototype viaduct.



-0.4 -0.2 0.0 0.2 0.4-6

-4

-2

0

2

4

6

Late

ral f

orce

(M

N)

Relative displacement (m)Fig. 6. Force displacement relationship of pier (PC-15).

4.1. Superstructure model

Since the objective of the experiment was mainly focused on studying the change in

dynamic behavior of the structure due to the incorporation of soil-structure inter-

action eects, an exact physical model of the prototype structure was not necessary.

Therefore, instead of making such an exact model, an attempt was made to sim-

plify the dynamic features of the prototype. A typical pier of the viaduct sustaining

666 tf weight has a fundamental resonance frequency of about 1 Hz. The dynamic

force displacement relationship of the pier was calculated by using SFRAME [Li

and Goto, 1998] as shown in Fig. 6 [JSSC (Japan Society of Structure Construc-

tion) Report, 2000]. The force-displacement curve in Fig. 6 was approximated as a

bilinear one. One simple way to realise this bilinear force-displacement relationship

is to place a mass on top of a pedestal supported by a spring so that the mass slips

when the acceleration exceeds the level that the friction between the mass and the

pedestal can resist. Below this level the mass moves with the pedestal. If the spring

carrying the pedestal possesses a linear force-displacement relationship, stick and

slip motions of the mass yield a bilinear force-displacement curve.

In the present experiment, a mass of 6.75 kg was put on the crossbeam (pedestal)

of a steel frame. A teflon sheet was put on the crossbeam to realise a smooth

transition of the blocks motion from stick to slip and vice versa. A photograph of

the experimental set-up is shown in Fig. 7. The frequency of the steel frame with

the block mounted was 2 Hz. Since the frequency of the frame was two times that

of the prototype, the duration of the input motion as well as the time-constant

[Eq. (10a)] of the flexibility function was halved. From Fig. 6, it is found that the

pier carrying a mass of 666 tf shows plastic deformation beyond an acceleration

of about 600 cm/s2. But the average frictional coecient between the mass and



Fig. 7. Experimental setup.

the teflon sheet was found to be around 0.2. Therefore, the mass slipped when

acceleration exceeded about 200 cm/s2. Again the ratio of model displacement, umand prototype displacement, up can be expressed in terms of the ratio between

model frequency, !m and prototype frequency, !p and the ratio between model

acceleration, am and prototype acceleration, ap as,

um

up=!2pam

!2map: (11)

For the present case, therefore, the input motion was scaled down by a factor of

1/12.

4.2. Foundation input motion

The signal of the foundation input motion, ufx + usx, which is not aected by the

presence of the superstructure, must be obtained prior to the experiment, and for

this, the free-eld ground motion ufx was rst obtained.

The ground response analysis was carried out considering only vertically propa-

gating horizontal SH waves, and the presence of the flexible group-piles was ignored

in this analysis. Hence the soil prole (Table 2) was modeled as a one-dimensional

horizontally layered soil column having non-linear soil properties. For the analysis,

the Finite Element Method was adopted in the spatial domain assuming piecewise

linear variations of displacement for the soil slices and a Central Dierence Time

Marching Scheme in the time domain. A schematic illustration of the soil column

is shown in Fig. 8.

To express the non-linear stress-strain relationship of each layer of the soil co-

lumn, the Hardin-Drnevitch model was adopted in association with the Modied



Table 2. Ground prole at the site of prototype.

Shear Wave

Thickness Unit Weight Velocity Shear Modulus

h (m) (tf/m3) Vs (m/s) G0 (tf/m2)

3.8 1.9 144.0 4020

2.6 1.9 129.3 3241

12.2 1.4 159.8 3648

0.9 1.9 173.2 5816

11.5 1.4 159.5 3634

5.0 1.9 161.0 5026

6.0 1.9 188.2 6867

Fig. 8. One-dimensional soil column.

Masing rule. The backbone curve of the model is expressed in dierential form as,

d

d= G0

1

f

n(12)

where, and are respectively the shear stress and shear strain of soil; moreover

G0 is the initial tangent modulus and f is the shear strength of soil. The value of

the arbitrary constant n depends on the actual stress-strain relationship of the soil.

In this analysis the value of n was assumed to be 2. In order to take into account

the eect of the bedrock underlying the soil c

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