**Transactions of the 17****th ****International Conference on ****Structural Mechanics in Reactor Technology (****SMiRT 17) **

**Prague, Czech Republic, August 17 –22, 2003**

** Paper # K04-5 **

**Study on the Soil-Structure Interaction Model under Vertical Earthquake **

**Motions**

Noriyoshi Ogawa1), Atsushi Tachibana1), Yasuaki Fukushima2), Yoshinori Mihara2)

1) Chubu Electric Power Co., Inc., Nagoya, Japan 2) Kajima Corporation, Tokyo, Japan

**A**

**BSTRACT**

Multi-degree-of-freedom (MDOF) parallel model has already been proposed and introduced to aseismic design for some nuclear power plant (NPP) buildings in Japan as one of the various soil-structure interaction models under horizontal earthquake motions. However, there are no previous references or research works on MDOF parallel model under vertical earthquake motions. Therefore, to establish the analytical method using proposed MDOF parallel vertical model, the following topics are theoretically and practically studied in this paper.

1) Theoretical application of MDOF parallel vertical model

2) Seismic simulation analysis with actual observation record under vertical motions 3) Proposal of MDOF parallel vertical model for aseismic design under vertical ones

**KEY WORDS: nuclear power plant building, reactor building, seismic response analysis, vertical earthquake motion, **
soil-structure interaction, evaluation of foundation input motion, dynamic soil stiffness, horizontally layered soil,
MDOF parallel model, observation record, simulation analysis, simple model for aseismic design

**1. INTRODUCTION **

MDOF parallel model illustrated in Fig.1 has already been proposed and introduced to aseismic design for some nuclear power plant buildings in Japan as one of the various soil-structure interaction models under horizontal earthquake motions [1]. It will also be necessary to consider dynamic vertical earthquake motions for near future aseismic design in Japan, although the current one is based on static vertical seismic force. However, there are no previous references or research works on MDOF parallel model under vertical earthquake motions. Considering such a

background, MDOF parallel vertical model shown in Fig.2 is theoretically and practically studied in this paper.

First, to evaluate MDOF parallel vertical model theoretically, it is

compared to one- and three-dimensional wave propagation theory from the viewpoint of the evaluation on foundation input motion through soils, dynamic soil stiffness due to building vibration and application to horizontally layered soils. Next, using actual vertical earthquake observation record in a nuclear power plant site, seismic simulation analysis is carried out with proposed MDOF parallel vertical model for the BWR reactor building and surrounding soils. As the result of that, it is shown that MDOF parallel vertical model simulates the dynamic response characteristics of observation records better than the conventional model.

Finally, based on the seismic simulation technique of MDOF parallel vertical model, more design-oriented model is proposed with emphasis on the maximum dynamic response of buildings.

**2. THEORETICAL EVALUATION ON MDOF PARALLEL VERTICAL MODEL **

The evaluation on foundation input motion and dynamic soil stiffness is very important to construct a soil-structure interaction model under vertical earthquake motions. From this viewpoint, proposed MDOF parallel vertical model is compared to one-and three-dimensional wave propagation theory for an assumed homogeneous soil. Through the

Fig.1 MDOF parallel model (Horizontal)

Fig.2 MDOF parallel model (Vertical)

comparison, influence over distance between vertical soil nodes, area of the model, boundary conditions and also application to horizontally layered soils are studied to evaluate the proper model.

**2.1 Evaluation on Foundation Input Motion **

Adequate distance h between each vertical node of soil columns is evaluated to obtain the response similar to one-dimensional wave propagation theory. Properties of an assumed homogeneous soil are S-wave velocity Vs=700m/s,

P-wave velocity Vp=2000m/s and soil density 20.6kN/m3. Also, amplitudes of transfer function from 500m-depth level to

ground surface are compared between MDOF parallel vertical model and one-dimensional wave propagation theory.

Here, vertical distance h of each node is calculated by Eq. (1) and the values of parameter α are 2, 5, 10 and 20 for

parametric study.

h = Vp / (α ×fmax) (1)

α : Constant regarding the wavelength of P wave

fmax : Maximum frequency to be considered (=20Hz)

The soil column model for α=5 is shown in Fig.3 and comparison of the results are shown in Fig.4. Therefore, for

500m-depth model, the result by α=5 agrees well with the result by one-dimensional wave propagation theory up to

10Hz which is almost upper limit of dominant frequency of vertical response of NPP buildings. Also, the result by

α=20 agree better with it up to 20Hz.

**2.2 Evaluation on Dynamic Soil Stiffness **

Besides foundation input motion, the evaluation on dynamic soil stiffness, i.e. impedance is also important to construct a proper soil-structure interaction model. Thus, impedances by Tajimi vibration admittance theory [2]

based on three-dimensional wave propagation theory and by MDOF parallel vertical model are compared, exciting the corresponding building foundation of the model. Homogeneous soil properties same as before are assumed. MDOF parallel vertical model shown in Fig.5 is used in this comparison, which is the detailed

axisymmetric model for α=20 with

four soil columns in the area of 4 times equivalent foundation radius. The shear stiffness between soil columns is evaluated from the surface area at midpoint and S-wave velocity Vs. Impedances by the theory and MDOF parallel vertical model are compared in Fig.6. Both real and imaginary parts of

impedance by MDOF parallel vertical model mostly match those by Tajimi vibration admittance theory.

Based on the detailed MDOF parallel vertical model well simulating the theoretical result, parametric studies in the following are carried out regarding model depth and boundary condition for side and bottom of the model.

For the influence on model depth, impedances by 100m-depth, 300m-depth and 500m-depth MDOF parallel vertical

model are computed, which is based on the simple model for α=5 with two soil columns in the area of 1.5 times

equivalent foundation radius. Comparison between these and also the 500m-depth detailed model results is shown in Fig.7. All the results by simple models above generally show enough agreement with the result by detailed model. But, going into detail, the result by 100m-depth simple model is different from that by detailed model. Therefore, at least 300m depths should be needed in MDOF parallel vertical model.

For the influence on side boundary, impedances by three MDOF parallel vertical models with and without viscous

side boundary are computed. Basic one has a viscous side boundary for α=5 with two 300m-depth soil columns in the

area of 1.5 times equivalent foundation radius. Second one has a free side boundary for α=5 with two 300m-depth soil

# of Nodes

26

500 m

### ～

### ～

### ～

### ～

h=20m

0 5 10 15

0 2 4 6 8 10

1D Wave Propagation Theory

Proposed Method (α=2)

Proposed Method (α=5)

Proposed Method (α=10)

Proposed Method (α=20)

Amp

. of

A

cc.

T

rans

.

F

u

nc

.

(G

L0

.0

M(

2E

)/

GL

-5

0

0

M(

EF

))

Frequency (Hz)

0 0.5 1 1.5

10 12 14 16 18 20

1D Wave Propagation Theory

Proposed Method (α=2)

Proposed Method (α=5)

Proposed Method (α=10)

Proposed Method (α=20)

A

m

p

. of

A

cc.

T

ran

s.

F

u

nc

.

(G

L0

.0

M(

2

E

)/

G

L

-5

0

0

M

(E

F

))

Frequency (Hz) Fig.3 Example of soil column model

(a) 0 to 10Hz

(b) 10 to 20Hz

columns in the area of 1.5 times equivalent foundation radius. Third one

has a free side boundary for α=5 with four 300m-depth soil columns in

the area of 4.0 times equivalent foundation radius. Comparison between these and also the detailed model results is shown in Fig.8. Therefore, when free side boundary is applied, MDOF parallel vertical model had better consider the larger area such as 4.0 times equivalent foundation radius.

For the influence on bottom boundary, impedances by three MDOF parallel vertical models with and without viscous side/bottom boundary are computed. Basic one has a free side and viscous bottom boundary for

α=5 with four 300m-depth soil columns in the area of 4.0 times

equivalent foundation radius. Second one has a free side and fixed

bottom boundary for α=5 with four 300m-depth soil columns in the area

of 4.0 times equivalent foundation radius. Third one has a free side and

fixed bottom boundary for α=5 with four 500m-depth soil columns in

the area of 4.0 times equivalent foundation radius. Comparison between these and also the detailed model results is shown in Fig.9. When fixed bottom boundary is applied, apparent differences between them are observed due to the reflect wave at bottom. Therefore, although slightly improved by deeper model, viscous bottom boundary is recommended for MDOF parallel vertical model.

Finally, from the viewpoint of evaluation on dynamic soil stiffness, it is recommended that simple MDOF parallel vertical model has a free

side and viscous bottom boundary for α=5 with four 300m-depth soil

columns in the area of 4.0 times equivalent foundation radius.

-1 1012 0 1 1012 2 1012 3 1012 4 1012 5 1012

0 5 10 15 20

Admittance Theory(Real) Admittance Theory(Imag.) Proposed Method(detailed[α=20],Real) Proposed Method(detailed[α=20],Imag.)

Dy

nam

ic

C

o

m

p

le

x

S

o

il

S

p

rin

g

(N/m)

Frequency (Hz)

-1 1012 0 1 1012

2 1012 3 1012 4 1012 5 1012

0 5 10 15 20

Simple[α=5],Depth=100m (Real) Simple[α=5],Depth=100m (Imag.) Simple[α=5],Depth=300m (Real) Simple[α=5],Depth=300m (Imag.) Simple[α=5],Depth=500m (Real) Simple[α=5],Depth=500m (Imag.) Detailed[α=20],Depth=500m (Real) Detailed[α=20],Depth=500m (Imag.)

D

ynam

ic

C

o

m

p

le

x

S

o

il Sp

ri

n

g

(

N

/m)

Frequency (Hz)

-2 1012 0 2 1012 4 1012 6 1012

0 5 10 15 20

Simple[α=5], Bottom:depth 300m. w/ viscous BC(Real) Simple[α=5], Bottom:depth 300m. w/ viscous BC(Imag.) Simple[α=5], Bottom:depth 300m. w/ fix BC(Real) Simple[α=5], Bottom:depth 300m. w/ fix BC(Imag.) Simple[α=5], Bottom:depth 500m. w/ fix BC(Real) Simple[α=5], Bottom:depth 500m. w/ fix BC(Imag.) Detailed[α=20], Bottom:depth 500m. w/ viscous BC(Real) Detailed[α=20], Bottom:depth 500m. w/ viscous BC(Imag.)

D

y

n

am

ic

C

o

m

p

le

x

So

il

S

p

ri

n

g

(N

/m

)

Frequency (Hz)

Fig.5 MDOF parallel vertical model

500

（Side:Viscous Boundary） （Bottom:Viscous Boundary）

48.5 24.25 48.5 72.25

194

h=5m

（α=20）

60.625 36.375 60.625

（1.0ｄ）

（0.5ｄ）

（1.0ｄ）

（1.5ｄ）

（4.0ｄ）

～ ～ ～

～ ～～ ～～

～ ～ ～

～ ～～ ～～

Unit: m

Fig.6 Comparison of impedance Fig.7 Influence on model depth

Fig.8 Influence on side B.C. Fig.9 Influence on bottom B.C.

-1 1012 0 1 1012 2 1012 3 1012 4 1012 5 1012

0 5 10 15 20

Simple[α=5], Side:1.5d w/ viscous BC(Real) Simple[α=5], Side:1.5d w/ viscous BC(Imag.) Simple[α=5], Side:1.5d w/ free BC(Real) Simple[α=5], Side:1.5d w/ free BC(Imag.) Simple[α=5], Side:4.0d w/ free BC(Real) Simple[α=5], Side:4.0d w/ free BC(Imag.) Detailed[α=20], Side:4.0d w/ viscous BC(Real) Detailed[α=20], Side:4.0d w/ viscous BC(Imag.)

Dyn

am

ic

C

o

m

p

lex So

il S

p

ri

n

g

(

N

/m)

Frequency (Hz)

**2.3 Application to Horizontally Layered Soil **
For the purpose of application to actual
soil conditions, foundation input motion and
dynamic soil stiffness are evaluated using not
assumed homogeneous soil but actual
horizontally layered soil properties. Table 1
shows the horizontally layered soil properties.
Detailed MDOF parallel vertical model above
is compared to one-dimensional wave
propagation theory for evaluation of
foundation input motion and thin layered
element method [3] for impedance evaluation.
Comparisons for evaluation of foundation
input motion and impedance are shown in

Fig.10 and Fig.11. Similar to the results in homogeneous soil, results by MDOF parallel vertical model in horizontally layered soil agree with the result by one-dimensional wave propagation theory and also thin layered element method.

**2.4 General Concept of MDOF Parallel Vertical Model **

It is clear from the results of section 2.1 to 2.3 that proposed MDOF parallel vertical model could be used as one of the soil-structure interaction models under vertical earthquake motions. Also, Fig.12 shows the general concept and evaluation procedures of proposed MDOF parallel vertical model: evaluation on distance between vertical soil nodes, mass of soil nodes, axial and shear stiffness of soils and boundary conditions etc.

**3. SEISMIC SIMULATION ANALYSIS UNDER VERTICAL MOTIONS **

To investigate practical validity of MDOF parallel vertical model, seismic simulation analysis is carried out with the model for the BWR reactor building and surrounding soils, using an actual vertical earthquake observation record in a nuclear power plant site. Vertical observation record of Enshunada (Omaezaki) earthquake, which occurred in October 11th, 1997, is used for this simulation analysis. The earthquake is reported as JMA magnitude Mj4.9, focal distance from the observation site 43km and epicentral distance from the observation site 23km.

**3.1 Analytical Conditions **

Soil-structure interaction model consists of soil part and BWR reactor building part is used in this simulation analysis. The soil part is detailed MDOF parallel vertical model described in Chapter 2 shown in Fig.13 and the building part is multi-degree-of-freedom model shown in Fig.14. Vertical seismic response model of the building properly consider the axial stiffness of RC structural members and bending/shear stiffness of steel roof trusses. Elastic modulus of RC is derived from the test results of concrete pieces. Damping constants of RC and steel are assumed to be 5% and 2% respectively. The detailed evaluation of the building refers to [4]. Soil properties in Table 1 are used. For reference purpose, conventional vertical soil spring model with embedment [4] is also evaluated. The model is explained such that soil under foundation is evaluated as a vertical spring based on three-dimensional wave propagation theory and embedded soil surrounding building is also evaluated as vertical side spring based on Novak’s method [5]. Then, the stiffness of dynamic soil spring is assumed to be constant independent of frequency and the damping of it is assumed to be proportional to frequency.

Level S-wave

Velocity P-wave

Velocity Density

(m) (m/s) (m/s) (kN/m3)

0.0 - - -

--4.0 180 400 0.3730 17.7 0.05 1m×4

▽ -23.0 670 1500 0.3754 20.6 0.05 3m×1,4m×4

-25.0 670 1500 0.3754 20.6 0.05 2m×1

-35.0 750 1780 0.3921 20.6 0.05 5m×2

-46.0 850 1780 0.3523 20.6 0.05 3m×2,5m×1

-92.0 850 1920 0.3781 20.6 0.05 3m×2,5m×8

-500.0 850 2100 0.4020 20.6 0.05 4m×2,5m×80

Lower Level of Base Mat

Dividing vertical soil columns Damping

Constant Poisson's

Ratio

0 5 10 15

0 5 10 15 20

1D Wave Propagation Theory [-23m(2E)/-500m(E+F)] Proposed Method (Detailed) [-23m(2E)/-500m(E+F)]

A

m

p

. of

A

cc.

T

rans

. F

u

nc

.

(G

L

-2

3

M/

G

L

-5

0

0

M)

Frequency (Hz) -1 1012 0

1 1012

2 1012

3 1012

4 1012

0 5 10 15 20

Thin Layered Element Method(Real) Thin Layered Element Method(Imag.) Proposed Method(detailed,Real) Proposed Method(detailed,Imag.)

Dy

n

amic

C

o

m

p

le

x

S

o

il

S

p

rin

g

(N/m

)

Frequency (Hz)

Table 1 Properties of horizontally layered soil

Fig.10 Evaluation of foundation input motion

Input motion to MDOF parallel vertical model is estimated from 2E wave (E: incident wave) at GL-500m obtained by one-dimensional wave propagation theory with observation record at GL-25m in free field. Fig.15 illustrates the evaluation flow for input motion to MDOF parallel vertical model.

477m

48.5 24.25 48.5 72.75

194

60.625

157.625

36.375 60.625

（1.0ｄ） （0.5ｄ） （1.0ｄ） （1.5ｄ）

97.0

（4.0ｄ）

23m Side : Viscous

Boundary

Bottom : Viscous Boundary Free filed

（Soil Columns）

～

～ ～～ ～～ ～～

～ ～

Fig.12 General concept and evaluation procedures of proposed MDOF parallel vertical model

Fig.13 Soil part of MDOF parallel vertical model

Fig.14 Building part of MDOF parallel vertical model

Fig.15 Evaluation flow for input motion to the model
Shear stiffness of soil （K_{s}）

Vertical distance of soil nodes（h）

max

×

*p*
*V*
*h*

*f*

α =

l

mid s s

A G

K =

Notes) This definition is derived from a 1st order axisymmetric isoparametric element.

mid

A

Weight of soil （w) Axial stiffness of soil Axial stiffness of soil （（KKNN))

h A E

K s

N=

: Maximum freq. (=20Hz) : Constant regarding wave length : P-wave velocity

max

*p*
*V*

*f*

α

( = 5 to 20)

hA

w=γ

Base mat

Near area

r

h About 3 times r

～
～
～
～
～_{～}

～ ～ ～ ～ ～ ～ ～ ～

Bottom : Viscous boundary

～ ～ ～ ～ ～

S

id

e bo

u

n

dar

y

:

F

ree

～ ～ ～ ～

About 6 to 10 times r Intermediate

area Far area

h W

d 0

l

h W

d 0

l

h

d 0

l

h

d 0

l

d 0

h

2

l

2

l

d 0

h

2

l

2

l

: Area

: Distance of soil nodes : Density of soil

A h

γ

(=πl l( +2 ))*d*
: Area

: Distance of soil nodes : Density of soil

A h

γ

(=πl l( +2 ))*d*

: Young’s modulus of soil

A h Es

: Distance of soil nodes

: Area(=πl l( +2 ))*d*
: Young’s modulus of soil

A h Es

: Distance of soil nodes

: Area(=πl l( +2 ))*d*

: Distance between soil columns : Shear modulus of soil

A G

mid s

: Area(=π*h*(l+2 ))*d*

l : Distance between soil columns : Shear modulus of soil

A G

mid s

: Area(=π*h*(l+2 ))*d*

l

F E

2E Proposed model (MDOF parallel)

-500m

Bottom : Viscous bounadry Side : Viscous

boundary

Observation

1D wave propagation

theory

Bld. model

-25.0m

Comparison Simulation Observation

**3.2 Analytical Results **

**(1) Acceleration Time History **

Fig.16 shows the acceleration time history of observation record at GL-25m in free field and also comparison between acceleration time histories of observation records and analytical results by MDOF parallel vertical model at 3rd floor inner box wall, end and center of roof truss. Results by MDOF parallel vertical model show mostly good agreements with the observation records.

**(2) Acceleration Response Spectra **

Fig.17 shows the comparison between acceleration response spectra (damping constant h=5%) of observation records and analytical results by MDOF parallel vertical model and conventional vertical soil spring model at 3rd floor inner box wall, end and center of roof truss.

For the 1st mode appearing around 0.2 second, results by not only MDOF parallel vertical model but also conventional vertical soil spring model show mostly good agreements with the observation records. But response values by conventional vertical soil spring model are greater than those by MDOF parallel vertical model.

For the 2nd mode appearing around 0.06 second, results by MDOF parallel vertical model show good agreements with the observation records. On the other hand, results of conventional vertical soil spring model can’t show even the peak configuration of 2nd mode itself. Since the mode excited by the surrounding soils upper foundation is almost corresponding to such a 2nd mode, it is concluded that MDOF parallel vertical model can properly evaluate the influence on the surrounding soils upper foundation. -10 -5 0 5 10

0 2 4 6 8 10 12

Free field : GL-25m Observation A cc eler ati o n ( 1 0 -2m/ s 2) Time(s) -10 -5 0 5 10

0 2 4 6 8 10 12

3rd Fl. Inner Box Wall Observation A cc eler ati o n ( 1 0 -2m/ s 2) Time(s) -10 -5 0 5 10

0 2 4 6 8 10 12

3rd Fl. Inner Box Wall Calculation(Proposed) A cc eler ati o n ( 1 0 -2m/ s 2) Time(s) -15 -10 -5 0 5 10 15

0 2 4 6 8 10 12

End of Roof Fl. Observation A cc eler ati o n ( 1 0 -2m/ s 2) Time(s) -15 -10 -5 0 5 10 15

0 2 4 6 8 10 12

End of Roof Fl. Calculation(Proposed) A cc eler ati o n ( 1 0 -2m/ s 2) Time(s) -100 -50 0 50 100

0 2 4 6 8 10 12

Center of Roof Fl. Observation A cc eler ati o n ( 1 0 -2m/ s 2) Time(s) -100 -50 0 50 100

0 2 4 6 8 10 12

Center of Roof Fl. Calculation(Proposed) A cc eler ati o n ( 1 0 -2m/ s 2) Time(s) 0 10 20 30 40 50 0.1 1 Observation Proposed Method

Conventional Method (w/ embedment)

Acc eler at io n ( 1 0 -2m/s 2) Period (s)

3rd Fl. Inner Box Wall Damping constant h=0.05

0.05 2 0 10 20 30 40 50 0.1 1 Observation Proposed Method

Conventional Method (w/ embedment)

Ac ce le ra ti o n ( 1 0 -2 m/ s 2 ) Period (s) 0.05 2

End of Roof Fl. Damping constant h=0.05

Fig.16 Comparison of acceleration (a) Input motion (GL-25m)

(b) Observation (3F I.W.)

(c) Analysis (3F I.W.)

(d) Observation (End of roof)

(e) Analysis (End of roof)

(f) Observation (Center of roof)

(g) Analysis (Center of roof)

Fig.17 Comparison of response spectra (a) 3F Inner Box Wall

**4. PROPOSAL OF MDOF PARALLEL VERTICAL MODEL FOR ASEISMIC DESIGN **

Based on the detailed MDOF parallel vertical model verified by seismic simulation analysis, using basic design earthquake motion and response analysis model for design, more design-oriented model is proposed with emphasis on the dynamic vertical responses of buildings up to 10Hz which is almost upper limit of dominant frequency of them.

**4.1 Evaluation Conditions **

Fig.18 shows a vertical response analysis model of a BWR reactor building for design and Fig.19 shows basic design vertical earthquake motion. Using the design model and earthquake motion, the vertical responses by more design-oriented model which is suggested in Chapter 2 are compared to those by the detailed MDOF parallel vertical model verified by seismic simulation analysis. All analytical cases are shown in Table 2. For reference purpose, conventional vertical soil spring model without embedment is also evaluated.

**4.2 Evaluation Results **

Generally speaking, response evaluation of roof truss is most important for reactor buildings under vertical earthquake motions. Thus, to propose a simple MDOF parallel vertical model for design, response values are evaluated of interest in maximum response shear force and bending moment of roof truss. Comparisons of maximum response shear force and bending moment are shown in Fig.20 and Fig.21. First, response values by conventional vertical soil spring model are apparently greater than those by all MDOF parallel vertical models. Second, for the comparison

0 100 200 300 400 500

0.1 1

Observation Proposed Method

Conventional Method (w/ embedment)

Ac

ce

le

ra

ti

o

n

(

1

0

-2m/

s

2)

Period (s)

0.05 2

Center of Roof Fl. Damping constant h=0.05

-600 -400 -200 0 200 400 600

0 20 40 60 80

Ac

ce

le

ra

ti

on

(

1

0

-2 m/

s

2 )

Time(s)

MAX:300cm/s2

Fig.17 Comparison of response spectra (c) Center of roof truss

Fig.18 Building model for design

Fig.19 Basic design vertical earthquake motion

Table 2 Analytical cases for design model

Case # # of soil

columns Depth α

*) _{Embment} _{Side BC} _{Bottom BC}

#1 4 500m 20 W/ Viscous Viscous

#2 4 300m 5 W/ Viscous Viscous

#3 4 300m 5 W/ Free Viscous

#4 4 300m 5 W/ Free Fix

Conventional _{－} _{－} _{－} W/O _{－} _{－}

between MDOF parallel vertical models, listed in order of increasing response values are the detailed one (#1), the

simple one which has a free side and viscous bottom boundary for α=5 with four 300m-depth soil columns in the area

of 4.0 times equivalent foundation radius (#3) and the simple one which has a free side and fixed bottom boundary for

α=5 with four 300m-depth soil columns in the area of 4.0 times equivalent foundation radius (#4). The difference

between viscous condition with free field (#2) and free (#3) on side boundary doesn’t apparently shown.

Following the results above and also in section 2.2, more design-oriented model such as #3 is adequately proposed with emphasis on the dynamic vertical responses of buildings up to 10Hz which is almost upper limit of dominant frequency of them.

**5. CONCLUSION **

In this study, MDOF parallel vertical model illustrated in Fig. 2 is proposed similar to horizontal one. As the results of various evaluations, MDOF parallel vertical model can simulate more accurate vertical responses than conventional one from the viewpoints of not only wave propagation theory but also actual seismic simulation analysis.

Based on the detailed MDOF parallel vertical model verified by seismic simulation analysis, more design-oriented model shown in Table 3 is adequately proposed with emphasis on the dynamic vertical responses of buildings up to 10Hz which is almost upper limit of dominant frequency of them.

**REFERENCES **

[1] NUREG/CR-6241: Technical Guidelines for Aseismic Design of Nuclear Power Plants, June 1994

[2] Tajimi, H. et al.: Research on Dynamic Spring of Foundation Bottom-ground According to Elastic Theory (Part1, Part2), Summaries of Technical Papers of Annual Meeting, Architectural Institute of Japan, October 1975, pp.395-398

[3] Tajimi, H. and Shimomura, Y.: Dynamic Analysis of Soil-structure Interaction by the Thin Layered Element Method, J. Struct. Constr. Eng., Architectural Institute of Japan, No.243, May 1976, pp.41-51

[4] Kuno, M., Motohashi, S., Mizuno, J. et al.: Simulation Analyses of Measured Seismic Vertical Response of a Nuclear Reactor Building (Part1, Part2 and Part3), Summaries of Technical Papers of Annual Meeting, Architectural Institute of Japan, September 1994, pp.1691-1696

[5] Novak, M. et al.: Dynamic Soil Reactions for Plane Strain Case, Proc. of ASCE, EM4, 1978, pp.953-959 # of soil columns

4

(with the area of about 4 times of base mat width)

Depth

300m

(about 3 to 5 times of base mat width)

Distance between vertical soil nodes

Vp/100 (Vp : P-wave velocity)

Side B.C. Free

Bottom B.C. Viscous boundary Recommendable features of proposed MDOF parallel

vertical models for design

Fig.20 Comparison of maximum shear force Fig.21 Comparison of maximum bending moment

Table 3 Features of proposed model for design 0

3000 6000 9000 12000

0 5 10 15 20

Proposed(#1,500m,α=20, Side: viscous, Bottom:viscous)

Proposed(#2,300m,α=5, Side: viscous, Bottom:viscous)

Proposed(#3,300m,α=5,Side: free, Bottom:viscous)

Proposed(#4,300m,α=5,Side: free, Bottom:fix)

Conventinal(w/o embedment)

M

ax

im

u

m s

h

ea

r

fo

rc

e(

k

N

)

Distance from end of roof truss(m)

0 20000 40000 60000 80000

0 5 10 15 20

Proposed(#1,500m,α=20, Side: viscous, Bottom:viscous)

Proposed(#2,300m,α=5, Side: viscous, Bottom:viscous)

Proposed(#3,300m,α=5, Side: free, Bottom:viscous)

Proposed(#4,300m,α=5, Side: free, Bottom:fix)

Conventinal(w/o embedment)

A

b

s.

of m

ax

imu

m

b

e

nd

in

g

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