京 都 大 学 防 災 研 究 所 年 報 第48号 C 平成17年4月
Annuals of Disas. Prev. Res. Inst., Kyoto Univ., No. 48 C, 2005

Advanced Techniques for Evaluating Vulnerability of Urban Infrastructure by Integrating
Multiple Evaluation Indexes

Tadanobu SATO, Hitoshi TANAKA, Masayoshi NAKASHIMA, Tomotaka IWATA,
Keiichiro SUITA, Sumio SAWADA, Shuji TAMURA,
Riki HONDA, Nobuyuki YAMADA** Disaster prevention Research Institute, Kyoto University
Synopsis
It is essential for disaster prevention to quantitatively evaluate the earthquake vulnerability of urban
infrastructures considering the duration of infrastructures in service, external loads to act on structureduring earthquakes, cost of damage and restoration, and the cost of maintenance and retrofitting. For thispurpose, we propose an integrated methodology that covers the prediction of strong ground motions for
scenario earthquakes, a full-scale test of a frame sub-assemblage including RC floor slabs, concrete
members reinforced by newly developed polyacetal-fiber (PAF) filaments, the horizontal subgrade
reaction of foundation piles during soil liquefaction and an efficient method to estimate a fragility curve
based on Subset Simulation method and MCMC (Markov Chain Monte Carlo).
Keywords: strong ground motion prediction, long period ground motions, seismic rehabilitation,
full-scale experiment, life cycle cost, fragility curve
1.

Introduction

To evaluate earthquake vulnerability of urban
infrastructures we have to develop techniques forevaluating (1) life time of infrastructures, (2) external input
to structure during event occurrences, (3) cost of damage
to infrastructures and restoration cost, (4) maintenance andretrofit cost. For this purpose we are conducting researches
on prediction of strong ground motions for scenario
earthquakes, a full-scale test of a frame sub-assemblage
including RC floor slabs, concrete members reinforced by
newly developed polyacetal-fiber (PAF) filaments, thehorizontal subgrade reaction of foundation piles during soilliquefaction and an efficient method to estimate a fragility
curve based on Subset Simulation method and MCMC(Markov Chain Monte Carlo).
In the next chapter, we are constructing a framework ofstrong ground motion prediction using a characterized

source model and underground structure model for
scenario earthquakes. The observed ground motion during
2004 off the Kii-Peninsula earthquake is used forvalidating the underground structure model Simulated
long-period ground motions fairly well reproduced theobservation and that model is applicable for long-period
ground motion simulation of coming Tonankai and Nankai
earthquakes.
In the third chapter, a full-scale test of a frame
sub-assemblage including RC floor slabs was conducted toobtain realistic data about the performance of composite
beams in deformation ranges much larger than thoseconsidered in seismic design. Finite element analysis was
also conducted to simulate progressive collapse ofcomposite systems. We propose a model that is capable ofcapturing the behavior of test structure before and after theconnection fracture with acceptable accuracy.
In the forth chapter, the strengthening effects ofreinforced concrete members by applying newly
developed polyacetal-fiber (PAF) filaments wasinvestigated. In addition, the horizontal subgrade reactionof foundation piles during soil liquefaction has also beenstudied.
The necessity of effective method to estimate low
failure probability is increased from the standpoint ofevaluation of life cycle cost, risk and probabilistic safetyassessment (PSA). In the last chapter, We propose an
efficient method to estimate a fragility curve by using
Monte Carlo simulation and limit seismic intensity, which
is defined as the minimum amplitude of input motion
causing damage to a structure. When the uncertainty ofseismic load is not so large, the tail part of the fragility
curve is important to estimate failure probability accurately.
For the effective estimation of the tail part, namely low
conditional failure probability, we use Subset Simulation
method. Samples are generated to search the failure region
in each subspace by using MCMC (Markov Chain Monte
Carlo). The proposed method is verified through a
numerical example with simple limit state functions, andthen a fragility curve of a steel pier of bridge is calculatedby the proposed method.
2.

Sufficient-quality ground motion data in the wide period
range from these events enable us to confirm the validity of
the underground structure model for long-period groundmotion simulation. Here, we show the characteristics ofobserved long-period ground motions and try to simulate
long-period ground motions in Kinki area to validate theunderground structure model.
2.1 Long-period ground motion characteristics of the2004 off the Kii-Peninsula earthquake

Strong Ground Motion Prediction for Design
Ground Motion

The M8-class subduction earthquakes have occurred
repeatedly in Nankai trough. The headquarters forearthquake research promotion reported that long-term
evaluations of occurrence potentials of the next Nankai and
Tonankai earthquakes at the trough are from 50% to 60%
within 30 years from 2005. During the 2003 Tokachi-Oki
earthquake (MJ8.0), many oil tanks in Tomakomai city,
about 250km far from the source region, were damaged bylong-period ground motions. Long-period ground motion is
generated by large earthquake and large sedimentary basin
effects. This shows that long-period ground motions attack
Osaka, Nobi, and Kanto sedimentary basins during the
Nankai and Tonankai earthquakes.
Two large earthquakes with MJ 7.1 and 7.4 rocked
Honshu from Kanto to Kyushu on September 5, 2004.
These events occurred off Kii peninsula in the shallower
part of the subducting Philippine Sea plate. Although these
events did not occur source regions of the hypothetical
Tonankai earthquake, strong motion records including
long-period ground motions were observed.

Fig.1 Distribution of pseudo velocity
response spectra during the mainshock.
(5%, 1s and 3s)

of long period ground motions. To explain thecharacteristics of long period ground motions, we try tosimulate long period groundmotions by the source and underground structure model.
2.2

Simulation of long-period ground motions
Fig. 2 shows example of observed ground motion at therock sites in Kinki area for the pre-event (MJ7.1). Even
nearest site to the epicenter, SNM, long duration groundmotion is recorded, even the source time duration is about
15s by waveform inversion result (Yagi, 2004). This meansthat underground structure between source region to the
SNM station is complex and modeling of undergroundstructure of this portion is very important for successful
simulation of this event records. Fig.3 shows our model area.
Osaka, Kyoto, and Nara basins and some portion of Nobibasin are included. Schematic cross section of the model isshown in Fig. 4. The cross section line AA’, that includes thesource and the Osaka basin, is marked in Fig. 3. In Fig. 4,
the source region of the Tonankai earthquake, that is theboundary between the subducting plate and the continental

Fig.1(continued) Distribution of pseudo
velocity response spectra during themainshock. (5%, 6s and 10s)
Fig.1 shows distribution of pseudo velocity response spectra(larger horizontal component, damping=5%) during themainshock. We show those of period of 1, 3, 6, and 10s. At
the period of 3s, large responses are observed in Osaka andNobi basins. At the 6s, coastal area in Osaka basin has
maximum response. On the contrary, at the 10s, maximum
responses are observed at Kanto coastal area. These
characteristics are thought to be occurred by effects ofpropagation-path and underground structure of the basins.
For example, roughly speaking, basin depth and shape ofbasins would control the predominant period and duration

Fig.2
Examples of observed ground
velocities(EW
component,
4-20s
band-pass-filtered).

Fig.3

Model areaplate, is also shown. There are sedimentary wedge at
subsurface portion between the Kii peninsula and the
hypocenter. This sedimentary underground structure would
affect to elongate ground motions at the rock sites in the Kii
peninsula.
Following to the studies (e.g. Kagawa et al., 2002;
Nakamura et al, 1997, Zhao et al., 1994, Furumura et al,
2003, Yamada and Iwata, 2004) three dimensional velocity
structure is constructed. Several interface depth contour
maps are shown in Fig. 5. Layer number can be referred toFig. 4. Source time function consists two continuous
triangles whose length is totally 15s, referring to Yagi (2004).
In Fig. 6, comparisons between observed and simulated
velocity ground motions are shown. All traces are
band-pass-filtered to 4-20s. Duration and apparent

Fig.7 Basin depth in Kansai area.

Fig.4 Schematic cross section of theunderground structure.

Fig.5 Depth contour map of interfaces of
our underground structure model.

Fig.6 Comparisons of observed andsimulated ground velocities.

Fig.8 Comparisons between observed andsimulated ground velocities at the basin sites.

predominant periods of ground motions are reproduced
well.
In Fig. 7, basin depth map in the Osaka, Kyoto, andNara region is shown. Comparisons between observed andsimulated velocity ground motions are shown in Fig. 8.
Stations are shown in Fig. 7. Grossly speaking, simulated
ground motions fairly well reproduced ground motion
duration and amplifications at the sediment stations.
Ground motions for the pre-event of the 2004 Off Kii
peninsula earthquake are fairly simulated both at the
propagation path and basin sites. Grossly speaking, theconstructed underground structure model can be used forlong period ground motion simulation in this area. However,insufficient estimations are occurred at several basin sites,
and to tune up the model is still needed.

3.2 Test program
The test structure was a two-story, two-bay by one-bay
steel moment frame. To ensure that plastic hinges form at
beam-ends, relatively strong cross sections were used forcolumns. Plans and elevations are shown in Fig. 9. Twoparallel planar frames placed in the longitudinal direction(Fig. 9) are nearly identical, but one frame (referred to as thecomposite frame) has a floor slab extending by 1.75 m in
the transverse direction on both sides of the beam, while theother frame (referred to as the bare steel frame) has no floorslab. This arrangement was deliberately adopted to directly
evaluate the effects of RC floor slab from the difference inresistance between the two frames. The structure was a
typical low-rise building designed following the post-Kobe

3.1 Introduction
In common construction of steel buildings, RC floorslab is always present. According to current design
specifications on composite structures, stiffness as well as
strength of steel beam increase due to the effects of RC floorslab (composite action).
To ensure life safety by the avoidance of structural

Without slab

With slab
Oil jack

Link
RC slab

2C 1

G1

1C1

Reaction wall

2C 2

G1

1C2

Pin

2C1

2C 1

1C1

CB

(c)

RC slab

B1

With slab

G1

Column 4
CB
B2

B4

G2
B3

B3

600
Strain gauge
300300(B56) G1400
(B54)Column 5
CB
B2

5,000

B1
(B65)
Column 6
B4
CB

Stud:
Diameter=16 or 19
Height=110

Mesh:
Diameter=6
150 150
75 90
165

B3

Reaction wall
(B45)

(B32)

G2
B3

Oil jack

Column 3

G1

1,750

G2

(B23)

1,750
5,250

Oil jack

Column 2
(B21)

G1

1C 1

Pin

Pin

Column 1
(B12)

G2

1C1

(a)
Without slab

2C 1

Deck PL:
Height=75
Sickness=6

5,000

4,500

10,000(b)

Fig.9

450 1,525 1,525
3,500

Effect of composite action on reserved capacity ofsteel building structures

1,750

3.

collapse under extremely large earthquakes, it is imperative
to characterize complete failure of structural members andsystems. This involves the investigation into the effects ofcomposite action on complete failure of steel beams. To thisend, a full-scale test of a frame sub-assemblage including
RC floor slabs was conducted to obtain realistic data about
the performance of composite beams in deformation ranges
that are far beyond those considered in contemporary
seismic design.
Finite element analysis was also
conducted to simulate progressive collapse of composite
systems.

(d)

Test specimen: (a) South Elevation; (b) Plan (numbering of beam-ends shown in parentheses);(c) East elevation; (d) Profile of RC floor slab; (unit: mm)
Comparison for local behavior

Japanese seismic design method for steel moment frames.
As shown in Fig. 9, braces were joined horizontally tothe column tops by high strength bolts through gusset plates.
Two quasi-static jacks were arranged for horizontal loading.
Each jack was placed at one end of each frame. An identicaldisplacement was applied by both jacks. Quasi-static cyclicloading with increasing displacement amplitudes was
adopted. The displacement was expressed in terms of theoverall drift angle, defined as the horizontal displacement at
the loading point relative to the loading height (i.e. 3.5 m).
Overall drift angles of 0.005, 0.01, 0.015, 0.02, 0.04, 0.06
and 0.1 rad were adopted. Three cycles were repeated foreach amplitude except for amplitude of 0.1 rad, in which
only one cycle was loaded. The structure was finally pushed
up to an amplitude of 0.13 rad, which corresponded the
maximum stroke of the jacks.
3.3 Test results
Fig. 10 shows the base shear versus total drift angle
relationships for the bare steel frame and composite frame.
The base shear was obtained from the load applied by thecorresponding jack. Fig. 10(a)-(b) are the plots of loading to
the amplitudes of 0.06 rad, whereas Fig. 13(c)-(d) are thoseplotting the data to 0.13 rad. Although somewhat

subjective, 0.06 rad was chosen as the maximum rotationthat steel moment frames may experience under large
earthquake loads considered in contemporary seismic
design. Drift angles greater than 0.06 rad were used for theevaluation of reserved capacity to collapse.
As shown in Fig. 10, the bare steel frame exhibits very
stable behavior until the negative loading of the first cycle at
0.06 rad drift angle when the first fracture [refer to Ÿ in Fig.
13 (a)] occurred on the bottom flange of beam-end B32 [seeFig.9(b)]. The composite frame also shows stable behavior
till the third cycle of positive loading at 0.04 rad drift angle
when the first fracture [refer to ź in Fig. 13 (b)] developedon the bottom flange of beam-end B45 [see Fig. 9(b)]. All
fractures exhibited a brittle mode. The first fracture
occurred earlier in the composite beam relative to the baresteel beam. Obviously, the presence of RC floor slab
increased the strain demanding of bottom flange, leading to
the advance of fracture. As evidenced from Fig. 13(a)-(b),the maximal strength of composite frame is 38% (positive)
and 29% (negative) greater than that of the bare steel frame.
The effect of RC slab is significant. After the loading of
0.06 rad, all four connections of composite frame failed by
the fracture of bottom flanges. All four connections of thebare steel frame failed after the deformation range of 0.1 rad.
1166

Base shear (kN)845

Base shear (kN)

800

800

-0.06

0.06

-0.06

0.06

Drift angle(rad)Drift angle
(rad)

-800

-800
Without RC slab

-911With RC slab

(a)

-1174

(b)

Base shear (kN)
1000

Base shear (kN)
1000

-0.12

0.12

-0.12

0.12
Drift angle(rad)Drift angle
(rad)

-1000
(c)

Fig.10-1000

Without RC slab

With RC slab(d)Global performance of each frame: (a) bare steel frame (to 0.06 rad); (b) composite frame (to 0.06 rad);(c) bare steel frame (to 0.13 rad); (d) composite frame (to 0.13 rad)

Ratio
2.0

1.0

a

0.0
0.005

0.01

Fig. 110.015 0.02
0.04
Amplitude (rad)

c
d

f

h

g
b e
0.06 0.1 to 0.13

Comparison of maximal strength

All fractures occurred in the bottom flanges also in the baresteel frame.
The ultimate strength of the composite and bare steel
frames for both positive and negative loading are comparedin Fig. 11 with respect to the loading amplitude. Symbols, Ɣand ż, indicate the points of beam bottom flange fracture
for the composite or bare steel frames, respectively. As
evidenced from Fig. 11, the strength of the composite frame
is significantly greater than those of the bare steel frame
until the second cycle in the 0.04 rad amplitude at which thefirst beam fracture occurred in the composite frame. In the
0.005 rad amplitude, which is considered as thedeformation level in small to moderate earthquakes, thestrength ratio is about 1.5. For the amplitudes exceeding
0.005, the structure was no longer in the elastic range andcracks developed on the RC floor slab. The strength ratio
was reduced to about 1.25. After the first fracture in thecomposite beam (during the positive loading of the third
cycle in the 0.04 rad amplitude), the two frames had nodifference in the property. After the third fracture in thecomposite beam (during the positive loading of the firstcycle in the 0.06 rad amplitude), the composite frame wassmaller in the properties than the bare steel frame, which
had not experienced any fracture. The strength of thecomposite frame is 80% of those of the bare steel frame. In
the 0.06 rad amplitude, the ratio remains lower than unity.
The ratio of strength drastically increased in amplitudes
beyond 0.1 rad when all connections fractured by thefractures of the last two connections that had survived in thebare steel frame. The residual strength of bare steel frame
was achieved through the negative bending connections
where the separated bottom flange contacted the column
faces. While the residual strength of composite frame wasachieved through both negative bending connections andpositive bending connections in which compressive andtensile forces were sustained by concrete slab and beam top

Fig. 12

FE model of test structure

flange. Composite floor slab beneficially maintained thestrength of composite frame.
3.4 Modeling of test structure
A general-purpose commercial FE package DIANA
8.1 was adopted to develop the numerical models. To
faithfully reproduce all the complicated boundary
conditions, the entire test structure including two frames
was considered. The FE models are shown in Fig. 12.
The steel beams and columns shown in Fig. 12 aremodeled using two-node straight beam elements based on
the Bernoulli theory. The RC floor slab with profiled sheet
is modeled by four-node shell elements having anequivalent thickness. Concrete is idealized using the totalstrain formulation following the coaxial stress-strain
concept (known as the rotating crack model) in which thestress-strain relationship are evaluated in the principal
directions of the strain vector. The steel reinforcement meshis simulated by an embedded grid with equivalent thickness.
The stud bolts are taken into account as the horizontal shear
transfer mechanism in composite action. In this study, a line
interface element with uncoupled stiffness in both thehorizontal and vertical directions is employed to represent
the shear studs.
3.5 Numerical results
Fig. 13 shows the hysteresis curves for the cycles of
0.005 to 0.04 rad. The dashed and solid lines are theanalytical and experimental curves. Correlation between theexperimental curves and analytical curves for the composite
frame is excellent, with the difference in the stiffness not
greater than 10%, and the difference in the maximum
strength less than 1.08 % (positive) and 0.52% (negative). In
reference to Fig. 13(a), correlation between the test andanalysis for the bare steel frame is also good.

3.6 Simulation of fracture
Since the proposed composite structure model is proved
to be able to capture the test behavior before fracture occurs,
the model is extended to mimic the fracture behavior. Basedon the observations from the test and available function in
DIANA, a joint model for capturing the effects of weld
fracture and subsequent nonlinear response of theconnection is proposed.
The joint model consists of 8 rigid links modeled asbeam elements and 4 line interface elements. Each pair ofrigid links is connected by a 4-node line interface element.
The four line interface elements represent top flange, upper
part of the web, lower part of the web and bottom flange
having corresponding height and out-of-plane thickness. A
discrete brittle cracking constitutive relation is adopted forthis line interface element. The whole joint model is like a
zero-length beam having same dimensions of the adjacentbeam.
Prior to fracture, the combination of joint model andadjacent beam elements works like a normal ductile
connection. After a fracture occurs, the position where thejoint model locates becomes a real hinge.
Each joint model is adopted for eight connections. The
Base shear (kN)connections are separated into two groups, the interior andexterior connections to assign different fracture stresses. The
fracture stresses are determined so that the first fracture to
the exterior connections and the first fracture to the interior
connections would be the same in the test and analysis. The
fracture stresses are selected as 325N/mm2 and 360 N/mm2
for the exterior and interior connections respectively.
Considering the yielding stress of 312N/mm2 obtained fromthe associated coupon test for beams, the adopted fracture
stresses are within a reasonable range.
Fig. 14 shows the relationship between the shear forces
versus the overall drift angle for the composite frame. The
analysis results show good correlation between the test andanalysis. Note that before the fracture occurred, the structure
experienced degradation of stiffness and strength due tocumulative damage at the beam ends. This effect was notable to be simulated by the present model.
3.7 Conclusions
The presence of RC floor slab beneficially increases thestiffness and strength of composite beam while imposes
higher strains demanding on the bottom flange of composite
beam. That would result in premature failure of composite
Base shear (kN)

800

800

0.04

-0.04

-0.04

0.04

Drift angle
(rad)
-800

Drift angle
(rad)

DIANA
Test

-800

(a)

Fig.13

DIANA
Test

(b)

Comparison for global performance of each frame: (a) bare steel frame (to 0.04 rad); (b) composite frame (to 0.04 rad)Base shear (kN)

Base shear (kN)

800

800

-0.04

0.04

-0.06

0.06
Drift angle(rad)Drift angle
(rad)
-800
(a)

DIANA
Test

-800

DIANA
Test

(b)

Fig. 14 Comparison for composite frame behavior including fracture (a) to 0.04 rad; (b) to 0.06 radbeam connections. Even after all connections failed byrupture of bottom flanges and webs, the structure still havecertain lateral resistance. The proposed model is capable ofcapturing the behavior of test structure before and after theconnection fracture with acceptable accuracy.
4.

Strengthening effects of reinforced concrete
columns with polyacetal fiber filaments

In this study, the strengthening effects of reinforced
concrete members by applying newly developedpolyacetal-fiber (PAF) filaments was investigated.
Profitable mechanical properties of the PAF are hightensile strain capacity with adequate strength and stiffness,high shear resistance and light weight. The PAF filaments
as a reinforcing material have significant advantages notavailable from other high strength materials, such as steel
and carbon fiber which are currently used for reinforcing
of concrete members. These advantages include ability to
deform in an acute angle and with a very small radius atroom temperature without implement. Moreover The PAF
filaments do not need to be impregnated unlike other fibers
because of its high tensile strain capacity of 8%. The
strengthening effects of concrete members obtained byapplying PAF filaments were investigated by conducting
lateral loading tests on 5 reinforced concrete columns. The
main objectives of the tests were to evaluate the shear
capacity determined by bond failure and the ductility of
PAF applied columns and to investigate the prediction forboth capacities. The test results showed high validity of the
PAF filament as a transverse reinforcement, especially forenhancement of the ductility. Meanwhile the shear
behavior due to bond failure for columns was modeled by
the truss and arch analogy based on the theory of plasticity.
The model was verified against the experimental results. It
was shown that this model can predict satisfactorily theshear capacity by bond of columns reinforced by not only
the PAF filaments, but also steel hoops. A prediction
formula for the ductility of columns was proposed by amodification of previous formula by other researchers. The
experimental results were found to be in good agreementwith the predicted values. This study has been conducted
mainly by Cikara IIBOSHI.
The mechanical properties of materials are given in
Tables 1 and 2. Each column specimen has a square crosssection of 300 mm and a effective height of 900 or 1,200mm as shown in Figure 15. Unit weight of PAF sheet used2

in the test is 528 g/m . Specimens No. 1 ~ 5 and No. 6 ~ 16
have shear span ratio of 1.5 and 2.0, respectively. The
major objective of these tests is to investigate the influenceof transverse reinforcement with PAF, on shear andductility for each shear span ratio. The loading ofspecimens was carried out by means of applying
horizontal and axial forces simultaneously. Specimens
were loaded statically in the horizontal direction. The axial
load was applied constantly throughout the experiment.
During the tests, displacements of the loading point andstrains in the reinforcing bars and PAF sheet were
measured.

Fig. 1Fig.
Test
15 Specimen
Test specimen

From the tests, the following conclusions were obtained:
1) Shear capacity and ductility of existing RC columns
can be significantly enhanced by strengthening with PAF
sheet.
2）The sheet strengthened column shows the same behavior
in both maximum loading capacity and hysteresis loop, as a
RC column with equivalent steel reinforcement. Shearstrength and ductility of columns confined by PAF can bepredicted by proposed equations.

Table1 Mechanical
Properties
ofof
Reinforcement
Table
1 Mechanicalproperties
reinforcement

Fig.16 An example of the test results
Fig.2
F
ig.2 An Example of the Test Results

TableTable
2 Details
2 Detailsofof Specime
specimens ns3）For strengthened columns with PAF without removal offinished mortar, shear strength and deformability are notaffected.
4）Using slits with 90 mm length and 10 mm width,columns with wing walls can be effectively confined by
PAF sheets.
5.

Contribution of Viscous Force to Horizontal
Subgrade Reaction of Piles during Soil
Liquefaction

5.1 Objectives
To design pile foundations, the horizontal subgrade
reaction of a pile is important factor. Therefore, thehorizontal subgrade reaction of piles during soil liquefaction
has been studied. Dynamic horizontal subgrade reaction

consists of an elastic force component depending on
displacement and a viscous force component depending on
velocity. The elastic force is generally dominant; therefore,
many researchers concentrate on studying p-y curves, where
p is soil resistance and y is pile deflection. On the otherhands, basic knowledge of the viscous force remains limited.
This study investigates the viscous force during soilliquefaction based on pile top vibration tests using alarge-scale shear box.
5.2 Pile top vibration tests
The pile top vibration tests were performed at NIED
(National Research Institute for Earth Science and Disaster
Prevention) in Tsukuba, Japan. The test was jointly
conducted by NIED, Tokyo Institute of Technology,
Kajima Corp., Taisei Corp., Takenaka Corp., Nippon SteelExciter

Pile

Sand

0.5m

Table
3 1.
Test
cases
Table
Test
cases

Strain gage
Pore Pressure
Accelerometer

Pile top vibration test

4.0m

PWPB
(Pile)

Boiling

1.5m

6.0m

PWL1
(Ground)Pile cap

GravelEx0

Before boiling

Ex1

1 minute after boilingEx2

2.5 minutes after boilingEx3

4 minutes after boiling

12m

Fig.1: Model
Fig.17
Modelsetup
setup

Corp. and Tokyo Soil Research Corp. The large-scalelaminar shear box (6m high, 12m long, and 3.5m wide)
including a soil-pile-structure system is diagrammed in Fig.
17. An exciter was mounted on the footing which was
supported by 2x2 steel piles. A boiling was performed at
GL-4m, the top of the gravel layer. Four cases of pile top
vibration tests were carried out before and after the boiling(Table 3). The eccentric moment of the exciter was
98kNm at frequency in the 4- to 10-Hz range.

where D is the pile displacement, P is the modulus ofelasticity and P' is the coefficient of viscosity. The firstterm in the right-hand side corresponds to the elastic force
and the second term corresponds to the viscous force. P
and P’ can be estimated from the following formula.

P

Pc
5.3 Test Results
The relation between the subgrade reaction and the pile
displacement, and that between the subgrade reaction andthe pile velocity at a height of 4m are shown in Fig. 18.
The relation between the pile displacement and thesubgrade reaction is an oblate ellipse, while the relation
between the pile velocity and the subgrade reaction is anellipse that approaches a circle in Ex0. This indicates the
subgrade reaction depends on the pile displacement in
non-liquefied soil. The subgrade reaction is, in contrast,
proportional to the pile velocity, indicating that the subgrade
reaction depends strongly on the pile velocity in Ex1.
5.4 Elastic force and viscous force
The relation between the pile displacement and thesubgrade reaction can be drawn as an inclined elliptical orbit.
The inclined ellipse consists of an inclined line component
and a non-inclined ellipse component as shown in Fig. 19.
The inclined line corresponds to the elastic force (Pd) andthe non-inclined ellipse corresponds to the viscous force (Pv).
The horizontal subgrade reaction can be expressed by thefollowing equation, according to the Voigt model ,P

P  D  Pc 

dD
dt

(1)

P1
Da

P2

Z  Da(2)(3)

where Z is circular frequency, Da is the amplitude of the pile
displacement, P1 is the subgrade reaction with D = Da, andP2 is the subgrade reaction with D = zero (Fig.19).
The estimated elastic force and the estimated viscous
force are shown in Fig. 20. In Test Ex0, the amplitude ofthe elastic force is larger than that of the viscous force. On
the other hand, the amplitude of the elastic force is far
smaller than that of viscous force in Test Ex1.
Figure 21 shows the relation between the excess porewater pressure ratio (u/Vv’) and Pv/P. Pv/P is between 0.1
and 0.2 when the excess pore water pressure ratio is lessthan 0.5. Pv/P increases with an increase in the excess porewater pressure. When the excess pore water pressure ratio
is greater than 0.9, the contribution of the viscous force isgreater than 50%.
5.5 Conclusion
The horizontal subgrade reaction of a pile during soilliquefaction was back-calculated based on pile top vibration
tests using a large-scale shear box. The subgrade reaction
was resolved into an elastic force component and a viscous
force component by back analysis using the Voigt model.

Subgrade Reaction(kN/m) Subgrade Reaction(kN/m)

4(b) Ex0 8-10s(a) Ex0 8-10s

Subgrade Reaction
0

P1
P2
-4
0.2

(c) Ex1 8-10s

Da(d) Ex1 8-10s

Disp.

Pv
0

P

Fig.3: Schematicof relation
relationbetween
betweenFig.19
Schematic ofsubgrade reaction and displacement
subgrade reaction and displacement

-0.2

-0.5

0

0.5-25

Pile Displacement(mm)025Pile Velocity(mm/s)

Pd,Pv(kN/m)

5

100
Ex0
Ex1
Ex2
Ex3

(a) Ex0

PV/P(%)

Pd,Pv(kN/m)

Fig.2: Subgrade reaction with pile displacement
Fig.18Subgrade reaction with pile displacement andand pile velocity at height of 4m
pile velocity at height of 4m

0
Pd

-5
0.3

Pv

50

(b) Ex1

0

0

-0.3
8

Pd

Pv

9
Time(s)10

Fig.20force
andandviscousforceforce
viscousFig.4:Elastic
Elasticforce

the contribution of the viscous force to the horizontal
subgrade reaction increases with an increase in the excess
pore water pressure. When the excess pore waterpressure ratio is greater than about 0.9, the contribution of
the viscous force is greater than 50%.
6.

Pd

Fragility Curves of Bridge Pier Using Subset
Method

6.1 Introduction
Necessity of probabilistic safety assessment (PSA) has
been recognized for important facilities such as nuclear
power plant in Japan. An effective reliability method is
also needed for life cycle cost estimation considering riskthat is the product of failure probability and impact of a
failure event. In seismic PSA, seismic load and strength ofa structure are often estimated separately. Seismic load ismodeled based on probabilistic concept, which usually
yields to a seismic hazard curve, whereas the structure
strength with uncertainties is modeled by a fragility curve.

0

0.2
0.4
0.6
0.8
1
Excess Pore Water Pressure Ratio (u/Vv')

Fig.21
Contributionof viscous
force
subgradereaction
reactionContributionof viscous
force
totosubgradeFig.5:

We propose an efficient method to estimate a fragility
curve by using Monte Carlo simulation and limit seismic
intensity, which is defined as the minimum amplitude of an
input motion causing damage to a structure, in other words,
input motion level at where safety factor is 1.0.
If the uncertainty of seismic load is not so large, thetail part of the fragility curve is important to estimate
failure probability accurately. For the effective estimationof the tail part, namely low conditional failure probability,
we use Subset Simulation proposed by Au and Beck
(2002). In this method, samples are generated to search the
failure region in each subspace by using Markov Chain
Monte Carlo (hereafter referred as MCMC) (Gilks et al.,1996).
The proposed method is verified through a numericalexample with simple limit state functions, and then afragility curve of a steel bridge pier is calculated by the
proposed method.
6.2 Estimation of Failure Probability with Limit
Seismic Intensity

In general a limit state function defines the failure
region as follows,g S , x R  z0

1 n
( j)
¦ 1  FS R( z 0 , x R )
n j1

(7)

FR (S)1 n( j)¦U S  R(z0 , xR )
nj1

(8)

(1)

where, S is a structural parameter with uncertainty, which
is sensitive to the limit state, xR and z0 are the otheruncertain parameters and threshold value to define thefailure. The inverse function R(z0, xR ) is defined to
calculate S which satisfy g(S, xR)=z0, provided xR and z0
being given deterministically. Assuming monotonic
increase or decrease of function R(z0, xR ), the failure
region can be expressed as follows,
R(z0, xR)<S, or R(z0, xR)>S

A fragility curve can be defined as conditional failure
probability, provided seismic intensity (PGA). An
approach to estimate a fragility curve is to perform MCS
based on Eq.(1). Figure 22 shows an illustration of MCS
for given PGA. This is a simple way to estimate a fragility
curve but we need to perform MCS many times. The
proposed method is illustrated in Figure 23. In this method,for each generated xR(j) by a single MCS the inverse
function R(z0, xR(j)) is calculated and then fragility curve isobtained by using Eq.(8). Although the estimation of R(z0,xR(j)) needs longer computation time than to evaluate theordinary limit state value g(S, xR), the proposed method is(2)

f S ( S ) f xR ( x R )Safety Factor FS

The R is defined as limit intensity. For simplicity, we
assume R<S for failure zone in following discussion.
Assuming independence between S and the otherparameters, the probability density function (hereafterreferred as f or pdf) of the parameters is expressed by,f (S , x R )

Pf

(3)

1.0

The failure probability can be obtained as follows,³g  z0f ( S , x R )dSdx R§ f³ ¨© ³R ( z0 , x R )

³

R S

f S ( S ) f x R ( x R ) dSdx R·
f S ( S ) dS ¸ f x R ( x R ) dx R¹§

³ ¨© ³RS·
f S ( S ) dS ¸ f x R ( x R ) dx R¹

³ ^1  F R z , x `f ( x )dx
S

0

R

xR

R

R

(4)
The failure probability for S conditioned by R<S can be
expressed.

³

R S

f (S, xR )dxR

³ U(S  R(z , x )) f (x )dx
0

R

xR

R

(5)
Here, U(x) is step function, =1 when x is non-negative, 0when x is negative. It corresponds to a so-called fragility
curve used in seismic PSA, when S is seismic intensity
such as SI, PGA, or PGV.
In this study, Monte Carlo simulation is used for the
integration of Eq.(4) or (5). The probability density
function of xR is approximated by using samples
generated according to the f(xR ) and Dirac’s delta
function.

f xR ( xR )

1.0

1 n
G ( xR  xR( j ) )
¦
nj1

0.0

R

PGA

Fig. 22 Illustration for Evaluating a Fragility Curvein General

Variation of Safety Factor
Safety Factor FS

FR (S)

PGA

Failure Probability

Pf

1.0

(6)

Substituting Eq.(6) into Eq.(4), we obtain Eq.(7), whichcalculates the failure probability. In same way, Eq.(8) is
derived to estimate fragility curve.

PGA

Design Earthquake

Variation of Limit Acc.

Fig. 23 Estimated fragility curve (cdf) and pdf of thegenerated samples

more efficient in most cases because a fragility curve canbe obtained by the single MCS.

)x

6.3 Outline of Subset Simulation

The proposal density function q(x'|xk) can be any arbitrary
function. In our study, it is assumed to be a uniform
distribution with mean xk+1 as defined byProcedure of Subset Simulation
The procedure of Subset Simulation (Au and Beck,
2002) is summarized as follows.
(a) Generate nt samples according to pdf(x) and calculate
limit state function value.
(b) Sort the samples according to the order of its value
such as z1, z2,..., zn in which z1 is the smallest value. The
subspace Fk+1 is defined with the following equation forgiven ns.

Fk 1

^x z ( x )  Ck 1`, Ck 1

znS  znS 12(9)(c)Generate nt samples in the subspace with MCMC
(Gilks et al, 1996).
(d)If the number of samples reaches failure zone becomeslarge enough, then stop the calculation, otherwise go backto the step 2.
The failure probability can be calculated with the numberof samples in failure zone nf in subset k,x) 1 ( u )q(u ' / u k )P ( z  0)(10)pdf ( x / Fi )otherwise

pdf ( x ) I Fi ( x )P ( Fi )(14)

in which IFi(x) is indicator function, =1 when x is insubset Fi, otherwise =0. Defining I(0,1)(u) as uniform
distribution in region (0, 1), Eq.(7) can be written,

pdf (u ) I Fi (u )P( Fi )

pdf (u / Fi )I ( 0,1) (u ) I Fi (u )P( Fi )

(15)
In this study, the uniform distribution in Eq.(13) is used as
the proposal distribution, therefore the acceptance ratio canbe simplified as follows.
D( x k , x' )­° S( x ' ) q ( x k x c) ½°
min ®1,¾°̄ S( x k ) q ( x c x k ) °¿^distribution S(x) can be generated by MCMC.
(a)Set an arbitrary initial value x0 and k=0 (k being acounter)(b)Generate x' according to the proposal distribution q(x'|xk)(c)Calculate the acceptance ratio­° S( xc)q( xk xc) ½°
min®1,¾°̄ S( xk )q( xc xk ) °¿u k  a  u '  u k  a (13)`­° P ( Fi ) I ( 0,1) (u ' ) I Fi (u ' ) 2 a ½°
min ®1,¾°̄ P ( Fi ) I ( 0 ,1) (u k ) I Fi (u k ) 2 a °¿min 1, I ( 0 ,1) (u ' ) I Fi (u ' )MCMC with uniform distribution
The outline of Markov Chain Monte Carlo is describedbriefly. Samples according to arbitrary target

D( xk , xc)

­1
° ;
® 2a
°̄ 0;

The pdf of subset Fi is given as follows,k§ ns · n f¨¨ ¸¸© nt ¹ nt

(12)

(16)
In the case of high dimensional problem, it is not efficient
to update all variables at the same time because theacceptance ratio becomes very low. In the algorithm called
single-component Metropolis-Hastings (Gilks et al, 1996),
each variable is updated separately. Eq.(16) contains two
kinds of criteria I(0,1)(u’) and IFi(u’). For I(0,1)(u’), theprocedure of single-component Metropolis-Hastings isapplied. The parameter “a“ has a strong influence on theefficiency of the simulation, several values of “a“ are usedsequentially in the simulation.

(11)
6.4 Numerical Example of Push OverAnalyses(d)x' is accepted as xk with probability D , otherwise xk
becomes xk+1(e)k= k +1, go to step 2
In our method, the every variable is firstly generatedusing uniform distribution which is denoted as u. The u isthen transformed into any type of distribution withcumulative distribution function (hereafter referred as cdf)

(1)Estimation of Fragility Curve of an Existing Steel Pierof Bridge
We apply the proposed methodology to an actual steel
frame bridge shown in Figure 24 and calculate a fragility
curve of this structure in which we consider several
uncertainties of the structural parameters including themass of upper structure, yield stress, thickness of plate andso on. We divide the bridge into 6 parts and consider ayield stress for each part. Limit acceleration (PGA, peakground acceleration) is used as limit seismic intensity.
Though three limit states, bending, shear and deformation,
are considered in an actual design, we consider only
bending mode for simplicity. We followed the ordinary
design procedure of a bridge pier in Japan (Design ofHighway Bridge Foundations in Japan, 2003) which isconsisted of the following three steps: 1) Perform
two-dimensional push over analysis and obtain relation
between applied load and bending deformation. 2)
Estimate applied load to cause failure based on energy
conservation principle and limit stage design point. 3)
Limit acceleration is calculated based on the applied load,
the predominant period and weight of the bridge.
Based on the energy conservation principle wecalculate seismic load to cause the structure failure,
however, it has been shown that this principle tends tooverrate the maximum response displacement for steel
structures in comparison with the result of nonlinear
dynamic analysis. Some researchers study this reason andshow that it is based on the difficulty to define yield pointand damping characteristic of steel structures. Because thisoverrating brings designed structures in the safer region,
we use this principle even though there are still
controversial opinions. We, however, take this uncertainty
into account to calculate the fragility curve.
Basic parameters of the Pier Model
Young’s Modulus of steel and concrete are 2.0×
8
10 (kN/m2)and 2.0× 107(kN/m2), respectively. Damping
constant of them are 0.03 and 0.02. The caisson foundation
is modeled by a mass with rotation and horizontal springs.
Mass of upper structure is loaded at node 2, 6, 10, 15(Figure 24), and they are 21551.18(kN), 915.32(kN),

Fig.24 The model of steel bridge pier985.88(kN), 19763.66(kN), respectively. Stress strain
relationship of steel is assumed to be expressed by theBi-linear model and we calculate M-φ relation of the box
type steel pier element to obtain equivalent bending
rigidity of beam element based on the Guide line ofanti-seismic reinforcement for existing steel bridges(Hanshin Expressway Public Corporation, 1997).
Furthermore we take into corrosion effect account on the
reduction of thickness of steel plate used for bridge pier.
Uncertainties to be considered(a)Mass of upper structures
From observation data for several bridges, we assume
the ratio between observed value and design value isTable 4 Uncertainties Considered in the Example
Mean(µ)Standard
Deviation(ı)

C.O.V

Distribution

1.03

0.02

0.019

Normal1.15

0.1265

0.11

Lognormal

1.00

0.05

0.05

Normal

1.0×Calc.

0.1µ

0.1

NormalUncertainties
Mass of upper structure
observed value / design value
Strength
Yield stress Mm
observed value / design valueofThickness of plate Fmcomponent
observed value / design value
Ground factor

Soil Springdistributed as normal distribution with the mean 1.03 andstandard deviation 0.02.
(b)Strength of component･Yield stress: We define Mm=ıy/Fy. ıy is observed
yield stress of actual structures and Fy is designed value.
From 1122 data, we assume Mm is distributed as
lognormal distribution with the mean 1.15 and c.o.v. 0.11.
･Thickness of steel plate: We define Fm=Sm/Sn. Sm is
observed cross-section factor of actual structures and Sn isdesigned value. From data, we assume Fm is distributed as
normal distribution whit the mean 1.00 and c.o.v. 0.05.
(c)Foundation part
Because variation of soil parameters is not easy toevaluate for defining equivalent spring constants offoundation model we assume soil spring factor is
distributed as normal distribution with c.o.v. 0.1.
(d)Energy conservation principle
The energy conservation principle tends to overrate themaximum response displacement for steel structures.
From investigation (by Kitahara, 2001), we assume thatthe ratio between maximum response displacement
derived by this principle and that obtained by nonlinear
10.1

1.0E-2 - 1.0E-0
1.0E-3 - 1.0E-1
1.0E-4 - 1.0E-2
1.0E-5 - 1.0E-3
30000sample

C.D.F

0.01
0.001
0.00010.00001

0.000001
200 250 300 350 400 450 500 550 600

acc(gal)Fig.25 Estimated fragility curve (cdf)(9 uncertainties)
1

P.D.F

0.8

0.6
0.4

dynamic analysis is 1.42 (mean) and its c.o.v. is 0.329.
(e)Corrosion
10 years have passed since the target bridge wasconstructed. From observation data, we assume the mean
decrement of the thickness of plate caused by corrosion is
0.255mm and its standard deviation is 0.4.
(f)Ultimate strength
Ultimate strength is assumed to be distributed. From
results of buckling experiment, we assume ultimate
strength is distributed as normal distribution, c.o.v. 0.05.
Criterions
Because there are many elements in the bridge piermodel we define the critical condition (failure) of the pieras one of element’s curvatures exceeds the threshold value.
At this time, the displacement of top of the pier (node 8 in
Figure 24) is defined as the final horizontal displacement
and the horizontal load as final horizontal load to cause
failure. Using them, we estimate the applied load to cause
failure based on energy conservation principle and we
obtain limit acceleration dividing this lord by the mass ofsuperstructure.
Numerical Example
In this case, we consider 9 uncertainties as summarizedin Table 4 and assume ultimate strain of 10İy (İy is yieldstrain) as critical condition of pier (Guide line ofanti-seismic reinforcement for existing steel bridges, 1997).
So the curvature defined when strain exceeds 10İy isthreshold value. First, we perform the ordinary MCS toobtain the fragility curve with 30,000 samples. Then we
estimate the fragility curve by Subset Simulation with 100
samples in each subspace. The computation time is lessthan 1/70 of ordinary MCS. We adopt the parameter
nt=100, ns=10, and the ten types of proposal density
function using different values of parameter “a”. The
obtained fragility curves are shown in Figure 25. An
accuracy of MCS is represented by c.o.v. į.When the
failure probability PF is small, the number of samples Nįrequired to achieve given į is expressed by,

NG

0.2
0
200 250300 350 400 450

500 550 600

acc(gal)Fig.26 Distribution of samples (pdf)

1  PF
1|
2
PF G
PF G 2(18)

As we use 30,000 samples for ordinary MCS, we canestimate PF =10-3 with an accuracy of į=0.2. In Figure 25,
the fragility curve obtained by using Subset Simulationagrees well with the fragility curve obtained by the

800

HKNS

400

400

acc(gal)200

acc(gal)

JMANS

600

0
-200

200
0

-200
-400
-600

-400

-800

0

5

10

15

20

25

30

0

5

10

15

20

25

30

time(sec)time(sec)a) Time history of acceleration observed at Higashi Kobeb) Time history of acceleration observed at JMA Kobe

(NS)

(NS)

Fig.27

Input motions
1

ordinary MCS up to the level of 10-3. Additionally, thedistribution of the generated samples (pdf) is shown in
Figure 26. This result shows that the samples are wellgenerated from the tail part of the former subset.

Higashi Kobe

0.01

C.D.F

6.5 Comparison of Fragility Curves Obtained by
Pushover and Dynamic AnalysesConsidering an actual steel frame bridge shown in
Figure 24 and uncertainties in Table 4, we estimate fragility
curves by dynamic analysis using two different input
motions which are shown in Figure 27, a)Higashi Kobe(NS), b)JMA Kobe (NS), which are observed during the
1995 Hyogoken Nambu earthquake in Japan. We use
Rayleigh damping for nonlinear structural analysis
assuming 0.5% damping constant for first and second
mode. We assume bi-linear hysteric characteristics for eachbeam element to express nonlinear M-φ relationship. We
conduct dynamic analysis for each sample bridge pier andcalculate the limit acceleration by changing peak
amplitude of acceleration time history. Estimated fragility
curves are shown in Figure 28. It should be noticed thattwo fragility curves obtained by dynamic analysis aredifferent each other although the process to obtain the limit
acceleration are same for both cases. This causes by thedifference of time history shape which is strongly
controlled by the phase spectrum. In this figure, a fragility
curve estimated by pushover analysis based on energy
conservation principle is also shown. To obtain this curve
we consider the uncertainties not only given in Table 4 but
uncertainty of the energy conservation principle. We cansee fragility curves obtained by dynamic analysis are on
the right side of that obtained by push over analysis. Thistendency is consistent with the Kitahara’s finding that theenergy conservation principle tends to overestimate anultimate displacement of steel bridge piers about 1.42

JMA Kobe

0.1

0.001

1.0E-2 - 1.0E-0
1.0E-3 - 1.0E-1
1.0E-4 - 1.0E-2
1.0E-5 - 1.0E-3

0.00010.00001
0.000001
200

400

600

800

1000 1200 1400

acc(gal)Fig.28 Estimated fragility curves (cdf)times with standard deviation of 0.467 of which data
distribution width is from 0.7 to 2.8 times. (Kitahara 2001).

6.6 Conclusion
An efficient method based on MCS was proposed toestimate a fragility curve. The concept of Subset Simulation
was adopted when the tail part of the fragility curve
becomes important. Since efficiency of MCMC plays animportant role in the Subset Simulation several method
were proposed to improve the efficiency by using such as
hit and run algorithm (Gilks et al, 1996). We applied
proposed method to actual bridge piers considering several
uncertainties of the structure. The efficiency of this method
was demonstrated by numerical examples.
Acknowledgements

We use data of K-NET, KiK-net, PARI strong motion
network, and CEORKA.
References

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多次元評価指標の統合化による都市施設地震脆弱性診手法の高度化

佐藤忠信・田中仁史・中島正愛・岩田知孝・
吹田啓一郎・澤田純男・田村修次・
本田利器・山田伸之**

京都大学防災研究所
要旨

都市域に存在する多くの構造物の地震時における脆弱性を合理的に評価するためには，構造物の共用期間，
地震時に作用する外力，それに起因する被害コストや復旧にかかるコスト，さらには補強や維持管理にかかるコストも考慮に入れる必要があると考えられる．本研究では，このような点に鑑み，シナリオ地震による強震動予測や，RC床板を含む鉄骨フレームの実物大実験，新たに開発されたアセタール樹脂繊維 (PAF)によって補強されたコンクリート部材の変形特性，液状化地盤内の基礎杭にはたらく水平地盤反力の特性，さらに，サブセット法とマルコフ・モンテカルロ法に基づいてフラジリティ曲線を求めるための効率的な方法に関する統合的な研究を実施した．
キーワード：強震動予測，長周期地震動，耐震補強，実大実験，ライフサイクルコスト，フラジリティ曲線