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Material:

Response Analysis of Civil Engineering StructuresSubjected to Earthquake Motions

Toshio Iwasaki

Ground Vibration Section, Public Works Research Institute, Ministry of Construction

2-308, Mitsuwadai Heights, 5-29 Mitsuwadai, Chiba City, Chiba Prefecture, Japan

[Published June, 1974]

1. Introduction

The Niigata Earthquake, measuring a magnitude of

7.5 on the Richter scale, hit the northwestern part of

Honshu, Japan, on June 16th, 1964. The epicenter was

under the sea about 55 km north from Niigata city, and

the hypocentral depth was 20 to 30 km. The earthquake

brought about severe damage to various engineering

structures in the alluvial plain near the mouth of theShinano River and the Agano River. Especially in the

vicinity of the mouth of the Shinano River where loose

sand layers plus a high water table exist, many modern

structures such as reinforced concrete buildings,

highway bridges (see Fig. 6, for example), harbor

structures, etc. sustained heavy damage due to

unexpectedly large deformations and settlements. This

particular earthquake emphasized the importance of the

dynamic effects of earthquake motions, as well as the

effects of liquefaction phenomena of the ground soils.

Two sets of strong motion accelerographs, installed

in the basement floor and the roof of a heavily inclined

4-story apartment building located along the Shinano

River, triggered the complete acceleration records of the

earthquake (see Fig. 1). The peak accelerations at the

basement are about 150 gals in the lateral direction, and

50 gals in the vertical direction. It was noted that these

accelerations were not so large, when considering the

severeness of the damage.

In order to clarify the causes of the damage to the

various structures, extensive investigations including

surveys on structural damage characteristics, soils and

dynamic response analyses, were carried out. The

acceleration records obtained from the apartment

building were utilized in the dynamic analyses of someof the other damaged structures. From these extensive

investigations many valuable lessons were learned.

Among them the following three seem most important.

1) To properly grasp the dynamic effects of earthquakes

on structures and to assess their earthquake-

resistance, dynamic response analyses are essential,

as well as conventional pseudostatic calculations.

2) Since liquefaction of saturated sandy soils affects

considerably the stability of structures, the

liquefaction effects are required to be taken into

account in the earthquake-resistant design for

structures constructed on soft saturated sandylayers.

3) Also particular attention should be given to the

design of structural such as procedures of member

connections, arrangements of details, reinforce-

ments, etc.

The present article was prepared to cover the first

item in the above mentioned, so this article is dealing

with the procedures of analyzing the dynamic behavior

of structures during earthquakes, and includes sometypical examples of dynamic response analyses for

various civil engineering structures.

Section 2 introduces the outline of procedures of

response analysis, and discusses seismic forces to be

considered as inputs in the analysis. Sections 3 through 5

describe several examples of response analyses on

highway bridges, earth structures, and submerged

tunnels.

As this article deals only with procedures and

examples of dynamic response analysis for civil

engineering structures, the reader is required to refer to

the section on pertinent materials for practical designprocedures.

Iwasaki, T.

274 Journal of Disaster Research Vol.1 No.2, 2006

Fig. 1. Strong-motion earthquake records during theNiigata Earthquake of June 16; (A) Roof, (B) Basement of

4-story apartment building.

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seismic behavior of long-period structures can be

reasonably evaluated using a response spectrum curve.

This method was applied to the aseismic design of the

San Francisco-Oakland Bay Bridge completed in 1936.

G. W. Howner, R. R. Martel and J. L. Alford [3]

analyzed several strong ground motions recorded in the

U.S.A., and obtained response spectra for a linear

singledegree-of-freedom system with viscous damping.

Furthermore, G. W. Housner [4] averaged several

response spectra of major ground motion records, and

obtained average spectrum curves. The average spectrum

curves are shown in Fig. 2-5 of [5]. Fig. 2 indicates

spectrum curves of magnification factors (ratios of the

response absolute accelerations to the maximum ground

acceleration), which are obtained from Housner’s

average spectra, by taking the maximum groundacceleration of 120 gals (or 4 ft / sec

2 ). Housner also

presented a procedure in which dynamic response

analyses of multi-degree-of-freedom systems can be

carried out employing the response spectra.

T. Takata, T. Okubo and E. Kuribayashi [6] alsoproposed average spectrum curves for the same linear

system by analyzing 20-component strong-motion

records obtained in Japan, including a record during the

Niigata Earthquake of June 16, 1964. Fig. 3 represents

the result of the average spectrum curves. It is noted that

the values of (magnification factor) in Fig. 3 are

greater than those in Fig. 2 for systems with non-zero

damping. The spectrum curves shown in Fig. 3 have

been applied to response analyses of several highway

bridges and other structures in Japan.

T. Katayama [7] computed 70-component Japanese

strong-motion records and clarified the effects of themagnitude of earthquakes and the magnitude of ground

Iwasaki, T.


Fig. 4. Amplification factor spectrum for four kinds of ground condition (after E. Kuribayashi, T. Iwasaki, K. Tuji).

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accelerations on the characteristics of linear spectrum

curves. S. Hayashi, H. Tsuchida and E. Kurata [8] also

computed 61-component strong-motion records obtained

at 21 stations during the Tokachi-oki Earthquake of May

16, 1968 and its aftershocks. They then discussed the

effects of ground conditions on the characteristics of

linear spectrum curves.

E. Kuribayashi, the author, Y. Iida and T. Tuji studiedseveral factors which are thought to affect the

characteristics of linear spectrum curves. These were the

effects of the magnitude of earthquakes, the magnitude

of ground accelerations, epicentral distances, and ground

conditions. As a result of the studies four different curves

shown in Fig. 4 are proposed. These curves are

corresponding to four different subsoil conditions from

rocky grounds to soft alluvial grounds.

In applying the response spectrum method to

dynamic response analysis, seismic conditions and

subsoil conditions at the site under consideration should

be taken into account carefully. It seems reasonable to

utilize average spectra rather than spectra from a certain

seismic record, because average spectra implies mean

properties of various seismic motions. For example,

average response spectrum curves shown in Fig. 2

through Fig. 4 can be employed. When applying Fig. 4,

the analyst may select one of the four spectrum curves

depending on the ground condition at the construction

site.

2.4. Idealization of Structures

In order to facilitate the response analysis of a

structure, it is inevitable to idealize the structure and tobuild up its analytical system suitable for the response

analysis. Two types of model systems are usually

employed for the dynamic response analysis of civil

engineering structures [10].

(a) Continuous System: Idealize a structure (see

Fig. 5(A), for example) as an assembly of

continuous members, as shown in Fig. 5(B).

(b) Lumped-Mass System: Idealize a structure as a

lumped-mass system as shown in Fig. 5(C). A finite

element system can be regarded as one of

lumped-mass systems.

In idealizing a structure for the response analysis, thefollowing may be indicated.

1) In view of force-deformation relationship of

structural members and surrounding soils, model

systems can be classified into two types: linear

systems and nonlinear systems. Although linear

systems are normally employed for simplicity,

nonlinear systems are sometimes formed for

structures subjected to considerably strong motions

where the response analysis beyond the elastic range

is required.

2) In forming analytical model systems it is advised to

take into account the effects of soil-structureinteractions for structures with foundations

embedded or placed on relatively soft soil layers.

3) As for damping capacities of structures, it is advised

to refer to measured damping factors for similar

existing structures, or to damping factors for similar

structures previously analyzed.

3. Response Analysis of Bridges

3.1. Outline

In reasonably assessing earthquake-resistance of

major bridges recently constructed in Japan, dynamic

response analysis is frequently carried out, in addition to

the pseudo-static design adopting the conventional

seismic coefficient method or the modified seismic

coefficient method in accordance with the structural

dynamic response. This trend was brought about by the

fact that newly constructed bridges sustained drastic

damage during the Niigata Earthquake of June 16, 1964.

This emphasized the importance of dynamic effects of

earthquake motions as described in section 1, and also

that several highrise bridges have been erected along the

expressway projects by the Japan Highway Public

Corporation since 1965. Moreover, the trend was

expedited after new specifications for earthquake-

resistant design of highway bridges stipulated in 1971. In

the specifications response analysis shall be adopted for

highway bridges such as tall ones which are required to

conduct detailed investigations for earthquake-

resistance.

Table 1 shows some of the highway bridges onwhich dynamic analyses have been carried out. In the

table, an outline of the analyses are summarized briefly.

A few of these analyses are mentioned in the following.

3.2. Showa Bridge

The Showa Bridge completed in May, 1964 just one

month prior to the Niigata Earthquake, is on the Niigata

prefectural road No.546 and across the Shinano River at

the point several kilometers above the river mouth. The

bridge is located about 55 kilometers south of the

epicenter. The ground conditions are of sandy soils,

comparatively loose near the left-bank andcomparatively dense near the right-bank.

Response Analysis of Civil Engineering Structures

Journal of Disaster Research Vol.1 No.2, 2006 277

Fig. 5. Idealization of bridge structure.

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The abutments are of pile bents (nine single-row piles

with a diameter of 609 mm and a length of 22 m), and the

piers are also of pile bents (nine single-row piles of the

same diameter and length of 25 m) with collar braces and

cap beams. The design seismic coefficient for the

substructures was 0.2 horizontally. The superstructures

are of 12-span steel composite girders with simple

supports. The total length is 303.9 m (= 13.75 +

10 @ 27.64 + 13.75), and the width is 24 m.

Due to the Niigata Earthquake the bridge sustained

very severe damage (see Fig. 6). The left-bank abutment

moved about 1 m toward the center of the river, and the

approach road subsided considerably. On the other hand,

the right-bank abutment and the approach road sustained

no significant damage. The first to fourth piers from the

left-bank tilted towards the right-bank. The permanent

deformations are 13 to 42 cm at the pier caps. The fifth

and sixth piers collapsed completely into the river bed.

The seventh to eleventh piers, however, suffered only

slight damage. Five girders, the third to the seventh from

the left-bank, out of a total of twelve girders, fell down

into the river bed (see Fig. 6). The sixth span fell down

on both its ends due to the failure of the fifth and sixth

piers which had supported the span.

To identify the causes of the damage, a dynamic

response analysis was conducted, in addition to otherextensive investigations including soil studies, the

measurement of dynamic properties of the remaining

spans, the survey on the permanent sets for the whole

structure and the deformed shapes of the embedded

piles.

In preparing a computer program for the response

analysis, a prototype of a bridge substructure was

considered, as one shown in Fig. 7, in order to

comprehensively utilize the program for general bridges

with substructures of pile foundations. Furthermore, the

prototype was idealized as an analytical system shown inFig. 8. In establishing the analytical system the

Iwasaki, T.


Table 1. Some examples of highway bridges on which dynamic analyses were conducted.

Fig. 6. General view of damage to the Showa bridge due to

the Niigata Earthquake of June 16, 1964.

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Several months after the occurrence of the earthquake

the bridge was reconstructed. The new bridge has the

substructures with double-row steel piles with wider pier

caps (see Fig. 9). For the new bridge two sets of

strong-motion accelerographs were installed on pier top

and ground to measure its dynamic response during

future earthquakes.

3.3. Yoneyama Bridge

The Yoneyama Bridge, completed in 1956 as a link inthe National Highway No.18, is situated across a deep

creek, in Yoneyama, Niigata Prefecture about 80 km

southwest of Niigata city. The bridge is a slightly curved

one with two high piers (the height is about 43 m), as

shown in Figs. 10 and 11. The substructures are steel

rigid frames with reinforced concrete footings on the

rocky ground. The superstructures consist of 3-span

continuous steel box girders (span length, 67 m, 93 m

and 67 m) with steel slabs and 2-span continuous steel

plate girders (span length, 2 25 m) with concrete slabs.Having highrise piers, the bridge was investigated for

stability against earthquakes by analyzing the dynamic

response, as well as the conventional earthquake-resistant design taking into account a horizontal seismic

coefficient of 0.2. In the dynamic analysis seismic

effects were evaluated by adopting the average response

spectra shown in Fig. 3. Seismic motions were applied

from two directions: longitudinal and transverse to the

bridge’s axis, and the maximum acceleration of theground motion was taken as 200 gals. The following

summarizes the assumptions and the results of the

analysis for the transverse direction that yielded the

critical state in the stability of the bridge.

The bridge was idealized as an analytical system

shown in Fig. 12. The system was built up on the basis of

the following assumptions:

1) Any girders and piers are substituted with uniform

members with the properties shown in Table 2.

2) Girders between nodal points 0 and 3, and between

3 and 5 are continuous. At nodal point 3, shearing

forces and bending moments can be transmittedthrough Pier 3. At nodal points 1, 2 and 4,

connections between girders and piers are made by

fixed shoes.

3) Points 0 and 5 through 9 move simultaneously in

phase during earthquakes. The bases of the four

piers are fixed perfectly on the footings at nodal

points 6 through 9.

At nodal points 0 and 6, three cases of end conditions

were considered: perfectly fixed, 90 percent fixed (or the

end rotations is restricted to 10 percent of perfectly free

end when subjected to bending moments), and 50

percent fixed (50 percent of perfectly free end).For the dynamic analysis, two cases of damping

Iwasaki, T.


Fig. 11. General view of the Yoneyama bridge.

Fig. 12. Analytical system for the Yoneyama bridge.

Table 2. Dimensions of girders and pier columns of the

Yoneyama bridge.

Fig. 10. General view of the Yoneyama bridge.

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ratios, 2 and 5 percent of the critical, were considered.

The mode-superposition method was employed to obtain

the maximum response. Two different ways were

utilized: one is the absolute sum of each nodal maximum

response (abbreviation in the figures is 11), the other is .

the square root of the square sum of each nodalmaximum response ( 11 2 ).

Figures 13 to 16 are the results of the analysis.

Fig. 13 shows the mode shapes of order from 1st to 5th.

Figs. 14 to 16 indicate the maximum displacements, the

bending moments, and the shearing forces, respectively.

Dotted lines in the figures denote the initial design

values obtained by adopting the conventional method

where a horizontal seismic coefficient of 0.2 was applied

to the weight of the superstructures. The design of the

Yoneyama Bridge was amended in the light of results

from the dynamic response analysis.In addition, three sets of strong-motion

accelerographs (see Fig. 11) were installed in 1966 after

the completion of the bridge, and its dynamic behavior is

being measured during actual strong earthquakes.

3.4. Sokozawa Bridge

The Sokozawa Bridge, completed in 1968 as a link in

the Chuo Expressway, and located on the Sokozawa

creek in Sagamiko town, Kanagawa Prefecture, about

50 km west of Tokyo. A general view of the bridge and

the dimensions of a typical pier (Pier 3) are shown in theleft of Fig. 17 and in Figs. 18 and 19. Both abutments are

of gravitytype reinforced concrete structures, and the

four piers are I-section steel framed reinforced concrete

structures with footings (cast-in-place concrete pile

foundations underneath the footings of Piers 1 and 4)

resting on the hard rock. Two piers (Piers 2 and 3) are

about 50-meter high, and the other two are about

30-meter high.

The superstructure consists of two continuous steel

truss girders: a two-span continuous girder and a

three-span continuous girder. Fig. 17 shows a stage in

construction of the truss girders. The superstructure is

hinged to the pier caps, and the longitudinal seismicforces exerted from the mass of the superstructure and



Fig. 15. Results of dynamic analysis – bending moment.

Fig. 16. Results of dynamic analysis – shearing force.

Fig. 17. The Sokozawa bridge under construction, the left

highrise Pier 3 (an excitor is seen atop) was testeddynamically.

Fig. 14. Results of dynamic analysis – displacement.

Fig. 13. Results of dynamic analysis – 1st to 4th mode

shapes of the Yoneyama bridge in the transverse direction.

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some parts of the piers are designed to be resisted by thetwo rigid abutments.

The bridge was designed in accordance with the

specifications proposed by the Committee on Highrise

Bridge Piers, Expressway Research Foundation. The

basic seismic coefficient was taken as 0.2 horizontal and

0.1 vertical at ground level. The design seismic

coefficient for the piers in the transverse direction were

increased by multiplying modification factors which

have values from 1.0 to 1.66 varying with the height of

the piers.

For the bridge two series of field dynamic

experiments were conducted. The first series were testsfor Pier 3 in 1967 before the erection of the

superstructure, and the second series were tests for the

overall structure in 1968 immediately after the

completion of the bridge.

The first series of the experiments for Pier 3 (isolated

pier seen in left of Fig. 17) consisted of steady state

forced vibration tests in the longitudinal direction by a

15-ton excitor and in the transverse direction by another

40-ton excitor, and step-function forced vibration tests

utilizing the propulsion of a rocket booster in the

longitudinal direction. Since the fundamental period of vibration of the pier was estimated comparatively long in

the longitudinal direction, a rocket engine which is

capable of generating a thrust of 2 t with a duration of

1 second was fixed on the pier cap for obtaining a free

damped vibration record of the pier after the release of

the thrust.

The results of the experiments for Pier 3 are tabulated

in Table 3, together with the theoretically calculated

ones. The resonant frequencies empirically obtained are

0.77 and 4.62 Hertz in the longitudinal direction, and

2.38 Hertz in the transverse direction. The damping

ratios are 0.6 to 1.0 percent of the critical in bothdirections.

Iwasaki, T.


Fig. 19. Pier 3 at the Sokozawa bridge.

Fig. 20. Locations of excitor and pick-ups.

Fig. 21. Test results for the whole bridge structure of the

Sokozawa bridge (transverse direction).

Fig. 18. General view of the Sokozawa bridge.

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The second series were steady state forced vibrationtests for the whole structure in the transverse direction

using an excitor set up on the slab of the midspan.

Fig. 20 indicates the locations of the excitor and twenty

transducers. Fig. 21(A) is an example of resonance

curves measured at the midpoint of the midspan. The

three lowest resonant frequencies were revealed to be

1.53, 2.38 and 2.63 Hertz, and the corresponding

damping ratios were 1.3, 1.7 and 1.7 percent of the

critical, respectively. Fig. 21(B) indicates the mode

shapes measured at the three resonant frequencies.

For the bridge, an extensive study of dynamic

response was also carried out for analyzing it’s dynamicproperties and seismic behavior. Fig. 22 illustrates the

analytical system for the response analysis in the

transverse direction. Six cases shown in Table 4 were

considered with varying beam-column connection

conditions and values of moments of inertia for

reinforced concrete sections. Natural frequencies andmode shapes obtained are illustrated in Table 5 and

Fig. 23 for 1st to 4th order. In these the results of the

field experiments are also indicated. Case 6 of the six

cases was found to be in comparative agreement with the

results of the field test.

For the case where ground motion with a maximum

acceleration of 200 gals is applied to a system having a

damping ratio of 2 percent of the critical, the maximum

bending moments were 22,500, 49,000, 57,000, and

28,000 t-m at the bases of the columns of piers P1

through P4. The maximum displacement was about

20 cm. These are the test results for Case 3 which isestimated to be the most reasonable in analyzing



Fig. 22. Analytical system for the Sokozawa bridge.

Table 4. Six cases considered in the analysis of the

Sokozawa bridge.

Table 5. Natural frequencies analyzed and resonantfrequencies from the field experiment.

Fig. 23. Comparison of mode shapes by analysis and by

experiment for the Sokozawa bridge.

Table 3. Test results for Pier 3 at the Sokozawa bridge.

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dynamic response during strong-motion earthquakes.

Since the quantities obtained were allowable, the

stability of the bridge against expected earthquakes was

assured.

3.5. Kanmon Bridge

The Kanmon Bridge, completed in November, 1973

as a link of the Kanmon Expressway, is a 6-lane highway

bridge crossing over the Kanmon Straits between the

Islands of Honshu and Kyushu in Japan. The bridge is a

3-span suspension bridge and is the longest in Japan at

present. Its total length is 1,068 m, having a center span

of 712 m and side spans of 2 178 m. Figs. 24 and 25 are

general views of the completed bridge.The two anchorages, 44 m wide, 55 m long and 40 m

high, are made up of reinforced concrete with large steel

frames to fix the cables. The tension on each cable is

12,500 t and the diameter is 667 mm. The base of each

anchorage, weighing about 140,000 t, is directly

supported by a rocky layer.

Each of the two piers supports the tower which exerts

a vertical force of 25,000 t. The Shimonoseki Pier, 40 m

wide, 20 m long, 14 m high and weighing 25,000 t, is a

huge footing made of reinforced concrete. the Moji Pier,

40 m wide, 20 m long, 30 m high, and weighing 50,000 t,

is a pneumatic caisson made of reinforced concrete. Thebases of the two piers reach to the bedrock.

Iwasaki, T.


Fig. 25. General view of the Kanmon bridge.

Fig. 26. Analytical model of the Kanmon bridge.

Fig. 24. The Kanmon bridge completed in November 1973,

the total length is 1,068 m, having a center span of 712 m.

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Each of the two towers, 134 m high and weighing

3,000 t, is made of steel frame with diagonal bracings.Before and after construction work began in 1968, the

following extensive investigations were carried out

regarding the stability against earthquake disturbances.

These were in addition to earthquake-resistant design

adopting the modified seismic coefficient method

considering structural response.

1) Observation of strong earthquake motions on both

sides of the Kanmon Straits starting in 1965.

2) Earthquake response analysis in 1969.

3) Dynamic field experiment of two piers in 1970.

4) Dynamic field experiment of one tower in 1971.

5) Static and dynamic field experiments on the wholestructure in 1973, and

6) Observation of ground motions and bridge responses

during strong earthquakes from 1973.

The outline of the dynamic analysis is described

below. Fig. 26 indicates that the analytical system is a

73-degree-of-freedom system. In the dynamic analysis

the characteristic value problem was first solved to get

natural frequencies and mode shapes, the modal response

for each mode was next obtained by adopting the seismic

record method and the response spectrum method, and

finally the resultant response was evaluated by the

mode-superposition method (or by superposing all themodal response participations). The following three

kinds of seismic inputs having a maximum acceleration

of 150 gals are considered.

1) Average spectra shown in Fig. 3,

2) Response spectra of the east-west component of

1962 Kushiro record shown in Fig. 27,

3) Time history of the east-west component of 1962

Kushiro record shown in Fig. 28.

Three damping ratios, 0, 2 and 5 percent of critical,

are taken into account in the analysis. Some of the results

obtained are indicated in Figs. 29 through 32. Fig. 29illustrates the 1st to 12th mode shapes, natural periods T

in seconds, and equivalent mass factor F in percent,

when subjected to transverse lateral excitation. In the

figure the upper mode shapes are for the cable, and the

lower mode shapes are for the girder, the towers, and the

piers.

Figures 30 through 32 show the maximum

displacements, bending moments, and shearing forces,

when the average response spectra mentioned above

were applied. In the figures h denotes the damping ratio

to the critical damping.

Figure 33 illustrates the time history of thetransverse response displacement of typical points of the



Fig. 27. Amplification factor spectrum of the east-west

component, 1962 Kushiro record.

Fig. 28. East-west component of the 1962 Kushiro record.

Fig. 29. Result of modal analysis for the Kanmon bridge

(1st to 12th mode shapes).

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bridge when subjected to the east-west component from

the 1962 Kushiro record. From the top to the bottom of

the figure are shown the followings:

1) Input acceleration record with maximum

acceleration of 150 gals,

2) Response displacement at the top of the Moji tower

in m,

3) Response displacement at the 2/8 point of the Moji

tower in m,


tower in m,


tower in m,

6) Response displacement at the base of the Moji tower

in m,7) Response displacement as the center of gravity of

the Moji Pier in m, and

8) Response rotation at the center of gravity of the

Moji Pier in radians.

It is found from the extensive analysis that the bridge

is sufficiently stable against earthquake disturbances

expected in the design.

After the completion of the bridge, more than twenty

pickups were installed to measure the motions of the

ground surface and underground, of the piers and the

abutments, and of the superstructure during strong

earthquakes. The location of the pickups is illustrated inFig. 25.

4. Response Analysis of Earth Structures

4.1. Outline

Earth structures have superior features in terms of

ease and cost of construction, therefore a large number

of earth structures have been constructed since ancient

times. Even at present numerous earth structures are in

existence and further construction of earth structures can

be expected for various important engineering works

such as highway banks, railway banks, dams, river

embankments, etc. Although a lot of earth structures

have been reported to have sustained seismic damage

during past earthquakes, design procedures and precise

analysis methods on earthquake-resistance of earthstructures are not sufficiently established. It seems that

special studies on dynamic effects of earthquakes on

those structures are required. It is supposed that one of

the reasons why the studies on earth structures are

behind those on other structures is that their detailed

analysis is very difficult because of the complex

properties of soil materials. After introduction of the

finite element method, however, for analysis of their

static and dynamic behavior, precise investigations could

be made considering the complicated properties of earth

structures (such as arbiter shape, non-linearity, and soil

property variation). In this section two typical examples

of dynamic analyses of earth structures idealized byfinite element systems are described briefly.

Iwasaki, T.


Fig. 31. Maximum response of bending moments ( 112 ).

Fig. 32. Maximum response of shearing forces.

Fig. 33. Time history of displacement of Moji tower and

Pier, subjected to E-W component of 1962 Kushiro record.

Fig. 30. Maximum response of displacement ( 112 ).

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4.2. Response Analysis of Earth Dams

R. W. Clough and A. K. Chopra [12] proposed a

powerful method of analyzing dynamic response of earth

structures utilizing the finite element procedure, and

presented an example of two-dimensional analysis for a

typical cross-section of an earth dam. The method

proposed, being very useful for the analysis of earth

structures, is outlined below.The equation of motion of the nodal points in the

finite element system subjected to seismic excitation, is

expressed in matrix form as

[ ] [ ] [ ] ( ) M r C r K r R t . . . . . . (1)

where [ ]K = the nodal stiffness matrix obtained by thefinite element procedure, [ ] M = the mass matrixassociated with the inertia forces in the system, and [ ]C =the viscous damping matrix.

The dots in Eq. (1) indicate differentiation with

respect to time. The load vector R t ( ) associated withthe seismic acceleration of the earth dam is given as

R t E V t E V t x g x y

g

y( ) ( ) ( ) . . . . . . (2)

in which

R t R t R t R t R t R t R t

E M

x y x y

n

x

n

y T

x

( ) ( ) ( ) ( ) ( ) ( ) ( )

1 1 2 2

1

0 0

0 0 0

2

1 2

M M

E M M M

n

T

y

n

T

(3)

M i indicates the mass lumped at the ith nodal point, and ( )V t g

x and ( )V t g

y represent the horizontal and vertical

components of the ground accelerations. In this analysisit is assumed that the entire base of the dam movessimultaneously as a rigid body. From Eqs. (1) and (2)

[ ] [ ] [ ] ( ) M r C r K r E V t E V x g x y g y

. . . . . . . . . . . . . . . . . . . . . (4)

The dynamic response of the structure was evaluated by

the modesuperposition method. To conduct the analysis,it was necessary to solve the characteristic value problem

[ ] [ ]K M n n n 2 . . . . . . . . . . (5)

for the undamped free vibration mode shapes, [ ] , andnatural frequencies, n . These mode shapes have thefollowing orthogonal properties

m

T

n

m

T

n

M

K m n

[ ]

[ ] ( )

0

0

. . . . . . . . (6)

and it is assumed that the damping matrix satisfies the

equivalent orthogonality condition

mT

nC m n[ ] , ( ) 0 . . . . . . . (7)

If the modal coordinates are transformed to the modeshape or normal coordinates as

r Y [ ] . . . . . . . . . . . . . . . (8)

in which Y = nodal amplitude vector, the coupledequations (Eq. (4)) can be reduced to a set of uncouplednormal equations by virtue of the orthogonality. Eachnormal response equation is expressed

( )*

*Y Y Y

P t

M n n n n n n

n

m

2 2 . . . . . . . (9)

using the notation

n

T

n n

n

T

n n n

n

T

n n n n

M M

K M

C M

[ ]

[ ]

[ ]

*

*

*

2

2

. . . . . . . . (10)

The generalized earthquake force in Eq. (9) is given by

P t E V t E V t n nT x

g

x

n

T y

g

y* ( ) ( ) ( )

. . . . . . . . . . . . . . . . . . . . (11)

As an example, the earthquake analysis of the

300-ft-high triangular dam section shown in Fig. 34 will

be described. It is assumed that this dam has side slopes

of 1.5 on 1; the material is homogeneous, isotropic, and

linearly elastic with a modulus of E = 5,700 kg cm2,

Poisson’s ratio = 0.45, and a unit weight = 2.08 t m 3 .

These properties are associated with a shear wave

propagation velocity of 300 m/sec. Damping was

assumed to be 20 percent of critical in each mode.

Although a system of simple geometry and homogeneity

was considered herein, arbiter geometry and material

property variations could have been treated with equalease.



Fig. 34. Finite element idealization of example earth dam

(after Clough and Chopra).

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The structural idealization consisted of 100 finite

elements with 66 nodal points, as shown in Fig. 34. Of

these nodal points, 11 were fixed to the base, thus the

remaining 55 provided the structure with 110 degrees of

freedom. The first 15 vibration mode shapes and natural

frequencies, computed by a standard eigenvalue

problem, are shown in Fig. 35, in which is given inradians per second.

This system was subjected simultaneously to twocomponents of the ground acceleration history recorded

at the El Centro Earthquake of May 18, 1940. The

north-south and vertical components are shown in

Fig. 36. The static stresses were also considered in the

analysis, because the static stress in an earth dam

represents a major part of the total stress state during an

earthquake. Thus, dynamic stresses are changes of stress

from the initial static condition.

The time history of stresses at four nodal points is

presented in Fig. 37. Each graph shows the variation at

the specific nodal points of both principal normal

stresses, the principal shear stress, and of the shear stresson a horizontal surface. The nodal point stresses were

obtained by averaging the stresses in the individual finite

elements associated with each nodal point. The relative

importance of the initial stress is clearly evident.

The distributions of stresses in the cross section at

various instants of time are illustrated by the stress

contours in Figs. 38, 39 and 40, showing the maximum

tensile (or least compressive), the maximum

compressive, and the horizontal shear stresses,

respectively. The top sketches in each figure show the

initial static state of stress. The middle and bottom are

the stress state at t = 2.0 seconds and t = 2.25 seconds.

These times are associated with a nearly maximumoscillation of stress conditions in the upper part of the

Iwasaki, T.


Fig. 36. Ground acceleration: EL centro earthquake, May

18, 1940 (after Clough and Chopra).

Fig. 37. Time history of stresses (after Clough and

Chopra).

Fig. 35. Free vibration mode shapes and frequencies (after

Clough and Chopra).

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cross section.

Concerning the finite element plane strain analysis

procedure mentioned above, it may be noted that (1)

compatibility is satisfied everywhere in the system, (2)

equilibrium is satisfied within each element, and (3)

equilibrium of stresses is not satisfied along the elementboundaries, in general, but the nodal force resultants are

in equilibrium.

From the research described above, it may be

concluded that the finite element procedure provides a

useful tool for dynamic response analysis of plane stress

or plane strain systems and the advantages of the

procedure with regard to the treatment of arbiter

geometry and material property variations aresignificant.

4.3. Response Analysis of Rock-Fill Dams

H. Watanabe [13, 14] proposed a procedure for

analyzing the dynamic response of finite element

systems subjected to strong earthquake motions in which

nonlinearity of materials can be taken into account. In

consideration of applying this procedure to the analysis

of rock-fill dams, cohesive soils and noncohesive soils

are idealized by the Maxwell-Kelvin model as shown in

Fig. 41(a), and by the Maxwell model as shown in

Fig. 41(b) [13].

The cross-section of a typical rock-fill dam is shown

in Fig. 42, where numerals 1 to 5 denote varieties of

materials. Fig. 41 illustrates the finite element system for

dynamic response analysis. The analysis was carried out

by applying the seismic motion of the north-south

component of the 1940 El Centro Earthquake with

reduced accelerations up to the maximum acceleration of

150 gals. Some of the results of the analysis are shown in

Figs. 44 to 47.

Figure 44 shows the time history of displacements at

four nodal points and the input ground displacement

which was obtained using the double integral of theoriginal acceleration record. Fig. 45 is the time history of



Fig. 38. Contours of major principal stress, 1 (scaling

factor 1) (after Clough and Chopra).

Fig. 39. Contours of minor principal stress, 2 (scaling

factor 1).

Fig. 40. Contours of shear stress on horizontal planes,

(scaling factor 10) (after Clough and Chopra).

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principal stresses at the center of four elements. In the

figure both the maximum and minimum principalstresses are illustrated. Fig. 46 indicates the

displacements of the entire cross-section at t = 3.40

seconds. Finally Fig. 47 shows the distribution of the

principal stress at that time. From the analysis the

following remarks have been derived:

1) Due to gravity, stresses in some portions of the

rock-fill dam are very large, however those in some

other portions are extremely small. The difference

seems to occur because of the nonlinearity of the

materials.

2) In cases where the dam is subjected to seismic

motions only in the lateral direction (or streamdirection), magnitudes of vertical deformation are as

much as those of lateral deformation.

3) The effects of the initial stress condition on the

earthquake resistance of the dam are significant, and

the dynamic response is largely controlled by the

soil properties.

4) The results of the analysis are in good agreement

with the results of model experiments conducted bythe same author using a large shaking table.

Iwasaki, T.


Fig. 42. Cross-section of rock-fill dam analyzed (afterWatanabe).

Fig. 43. Finite element system of the rock-fill dam (after

Watanabe).

Fig. 44. Time history of displacement at four nodal points

and the input ground displacement (after Watanabe).

Fig. 45. Time history of principal stresses at the centers of

four elements (after Watanabe).

Fig. 41. Visco-elastic models for rock-fill materials; (a)

Maxwell-Kelvin model, (b) Maxwell model (after Hatano

and Watanabe).

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5. Response Analysis of Submerged Tunnels

5.1. Outline

Construction of submerged tunnels is now popular in

Japan, and will become more frequent in the near future.

Since submerged tunnels are usually prefabricated on

land near the seashore and embedded in soft soil deposits

in bays or river mouths, their stability during earthquakes

is very important. In order to establish rational design

methodology in providing adequate resistance to seismic

disturbances, various investigations such as soils studies,

measurements and analyses of seismic behavior of soil

deposits, experiments and analyses on dynamic response

of these structures, etc. have been carried out in recent

years. This section will describe some typical examples

of dynamic response analyses on submerged tunnels

which are under construction or under consideration.

5.2. Dynamic Analysis for Cross-Section of

Submerged Tunnel

E. Kuribayashi and the author [15] conducted a

dynamic response analysis for a submerged tunnel

proposed across Yokohama Bay. A general side view

from the preliminary design is shown in Fig. 48. The

tunnel, a 6-lane highway tunnel, has a total length of

about 1,570 m and a cross-section of 8.5 37.4 m, and ismade of reinforced concrete with steel covering. The

dynamic behavior of three cross-sections was analyzed.

Fig. 49 shows the finite elements systems for sections A,

B, and C.

The following describes briefly the results for section

B. Assuming that the section is in plane strain and theshear wave velocity of the soil materials is 50 m/sec, the

fundamental period was found to be 2.6 seconds. Fig. 50

indicates the distribution of the maximum response

displacements and accelerations when the section was

subjected to the average response spectra shown in

Fig. 3. For the analysis a damping ratio of 10 percent of

critical, and the maximum input acceleration of 200 galswere considered.

In order to test the results of the analysis, dynamic

model experiments were also carried out by the authors,

employing a large shaking table.

5.3. Dynamic Analysis for a Longitudinal Section

of a Submerged Tunnel

S. Okamoto, C. Tamura, K. Kato and M. Hamada

[16] developed a procedure for analyzing dynamic

behavior of the longitudinal section of a submerged

tunnel. The tunnel analyzed consists of nine reinforced

concrete elements. Each element has a length of 110 mand cross-section of 8.95 37.4 m. The total length of



Fig. 48. Yokohama Bay undercrossing tunnel (proposed).

Fig. 49. FEM models for three sections.

Fig. 47. Distribution of principal stress, at t = 3.40 sec

(after Watanabe).

Fig. 46. Distribution of displacement of the entire

cross-section, at t = 3.40 sec (after Watanabe).

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the tunnel is about 1,035 m. The general views and soil

properties are shown in Fig. 51 [17]. As seen from the

figure the ground is made of soft silty alluvial deposits.

Fig. 52 shows the mathematical model for the dynamic

response analysis. In forming the model system, thefollowing assumptions were made:

1) Natural periods of the ground are not influenced by

the existence of the tunnel.

2) The tunnel can be treated as a beam resting on the

ground. The effects of the surrounding soils can be

represented by elastic or inelastic springs for the

motions in the transverse and the axial directions of

the tunnel. Damping effects of the soil-tunnel

system can be idealized as viscous damping with the

damping ratio of 10 percent of critical.

3) The shear deformation of the ground is considered

herein, and only the fundamental mode shape of thesurface soil layer is taken into account in evaluating

the displacements of the ground and the tunnel.

As for the inputs to the system, five different seismic

records, including the north-south component of the

1940 El Centro Record, were employed. Figs. 53 to 56

are the results of the comprehensive analysis. Fig. 53

shows the distributions of the maximum values of bending moments, shearing forces, axial forces, and

displacements developed in the tunnel when subjected to

the five seismic records which were adjusted in such a

way that the maximum value of each acceleration input

is equal to 100 gals at the bedrock. Fig. 54 indicates the

effects of hinges and flexible joints manufactured in the

tunnel on the dynamic response of the tunnel. In the

figure the following four cases are considered:

Case 1: No joints

Case 2: Hinge joints at points 32 and 13

Case 3: Hinge joints at points 28 and 13

Case 4: Flexible joints between eleven elementsFrom the study the effectiveness of hinge joints and

Iwasaki, T.


Fig. 52. Mathematical model of the tunnel (after S.

Okamoto, et al.).

Fig. 53. The response values of the tunnel to the

earthquakes (after S. Okamoto, et al.).Fig. 51. General view of the submerged tunnel analyzed.

Fig. 50. Results of a dynamic analysis for section B.

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flexible joints can be ascertained. Fig. 55 shows the

effects of the joints between the ventilation towers and

the tunnel ends. It is seen from the study that large

bending moments may be generated in the tunnel nearthe ventilation tower in the case where the tunnel ends

are fixed rigidly to the towers.

Figure 56 indicates the effects of inelastic properties

of the ground. Bi-linear characteristics of the soil

materials are assumed, and three values of yielding

displacements (1, 0.75 and 0.5 cm) are taken, as shown

in Fig. 55. It is found that stresses generated in the tunnel

will decrease considerably when inelastic properties of

soils are considered.

5.4. Dynamic Analysis of the Tokyo Bay

Submerged Tunnel

E. Kuribayashi, the author, and K. Kawashima [18]

also carried out a dynamic response analysis for a

submerged tunnel proposed as a 6-lane highway across

the central part of Tokyo Bay, together with its aseismic

design by means of a simplified procedure for

considering ground displacements. Fig. 57 illustrates the

general side view and the typical cross-section of one of

preliminary designs for the reinforced concrete tunnel. In

the design the total tunnel length is 3,340 m and the cross

section is 13 44.2 m. The ground at the constructionsite is soft silty soils. The average water depth is about

28 m. For dynamic analysis the following assumptionswere made:

1) The bedrock was taken at the depth of 65 m belowthe water surface (or 37 m below the sea bottom).

2) The rigidity of soils was determined by referring to

the results of the field seismic survey and by

considering the reduction of the rigidity with

respect to the magnitudes of strains expected during

strong earthquakes.

3) The damping ratio of the soil-tunnel system was

taken as 20 percent of critical.

4) As for the seismic inputs, an average response

spectrum shown in Fig. 4(A) and various seismic

records obtained underground were employed. The

maximum acceleration of the input was regarded as150 gals laterally and 75 gals vertically, at the level



Fig. 57. General view of the Tokyo-Bay-Crossing

submerged tunnel proposed.

Fig. 56. Effects of inelasticity of the subground on the

earthquake response of the tunnel (after S. Okamoto, et

al.).

Fig. 55. Effects of joints between the tunnel and the

ventilation tower on their earthquake response (after S.

Okamoto, et al.).

Fig. 54. Effects of hinges and flexible joints on the

earthquake response of the tunnel (after S. Okamoto, et

al.).

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of the bedrock.

Some of the results of the analysis are shown in

Figs. 58 and 59. Fig. 58 illustrates the distribution of the

maximum displacements relative to the bedrock, in three

directions: (a) longitudinal, (b) transverse, and (c)

vertical. The solid lines denote the results of the

simplified design procedure, and the dotted lines show

the results of the dynamic analysis taking the average

spectrum Fig. 4(A) as the input.Figure 59 illustrates the distribution of the maximum

values of various forces developed in the tunnel: (a) axial

forces, (b) bending moments in the lateral plane, (c)

shearing forces in the lateral plane, (d) bending moments

in the vertical plane, and (e) shearing forces in the

vertical plane. In the figure the solid lines denote design

procedure I where each wave length was determined so

as to produce the critical condition. The chain lines

indicate design procedure II where a wave length equal

to 4 times the depth of the surface soil layer was selected.

The dotted lines show the results of the dynamic analysis

considering the average spectrum Fig. 4(A) as the input.

From these figures it may be seen that the simplifiedprocedures, in which the ground displacements are

regarded as the input to the tunnel, give sufficiently

reasonable response values for both displacements and

forces generated in the tunnel during earthquakes

considered in the design.

References:[1] Strong-Motion Earthquake Observation Council, Tokyo, “The

Project for Observation of Strong-Motion Earthquakes and Its

Results in Japan,” Published by the National Research Center for

Disaster Prevention, Science and Technology Agency, Tokyo,

August, 1972.

[2] M. A. Biot, “Analytical and Experimental Methods in Engineering

Seismology,” Transactions of American Society for Civil Engineers,

1943, Paper No.2183.

[3] G. W. Housner, R. R. Martel, and J. L. Alford, “Spectrum Analysis

of Strong-Motion Earthquakes,” Bulletin of Seismological Society of

America, Vol.43, No.2, April, 1953.

[4] G. W. Housner, “Behavior of Structures during Earthquakes,”

Journal of Engineering Mechanics Division, Proceedings of

American Society for Civil Engineers, October, 1959.

[5] M. Watabe, “Aseismic Structural Systems for Buildings,” Part 2 of

Recent Progress of Earthquake Engineering, Technocrat, Vol.7,

No.1, January, 1974. (Republished in Journal of Disaster Research,

Vol.1, No.3, December, 2006.)

Iwasaki, T.


Fig. 58. Distribution of maximum displacement in threedirections.

Fig. 59. Distribution of maximum forces in the lateralplane and in the vertical plane.

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[6] T. Takata, T. Okubo, and E. Kuribayashi, “Earthquake Response

Spectra 1964 – Studies on Earthquake Resistant Design of Bridges,

Part I –,” Report of Public Works Research Institute, Ministry of

Construction, Japan, Vol.128, 1965.

[7] T. Katayama, “A Note on the Acceleration Ratio Spectrum of

Seventy Japanese Strong-Motion Earthquake Records,” Bulletin,

Faculty of Science and Engineering, Chuo University, Vol.12, 1969.

[8] S. Hayashi, H. Tsuchida, and E. Kurata, “Acceleration Response

Spectra on Various Site Conditions,” Proceedings, 3rd Japan

Earthquake Engineering Symposium, 1970 (in Japanese).

[9] E. Kuribayashi, T. Iwasaki, Y. Iida, and K. Tuji, “Effects of Seismic

and Subsoil Conditions on Earthquake Response Spectra,”

International Conference on Microzonation, Seattle, Washington,

1972.

[10] E. Kuribayashi, et al., “Earthquake Response Analysis and

Applications, Chapter 9 Bridges,” Edited by Japan Society for Civil

Engineers, 1973 (in Japanese).

[11] T. Iwasaki, J. Penzien and R. W. Clough, “Literature Survey –

Seismic Effects on Highway Bridges,” University of California,

Berkeley, Earthquake Engineering Research Report, No.72-9, 1972.

[12] R. W. Clough and A. K. Chopra, “Earthquake Stress Analysis in

Earth Dams,” Proceedings, American Society for Civil Engineers,

Engineering Mechanics, No.2, April, 1966.

[13] T. Hatano and H. Watanabe, “Dynamic and Static Coefficients of

Visco-elasticity and Poisson’s Ratios of Clays, Sands and Crushed

Gravels,” Transactions, Japan Society for Civil Engineers, No.164,

April, 1969, and also H. Watanabe, “Dynamic Analysis of

Visco-Elastic Systems by the Finite Element Method,” Transactions,

Japan Society for Civil Engineers, No.198, February, 1972 (in

Japanese).

[14] H. Watanabe, “Dynamic Analysis of Visco-Elastic Rock-Fill Dams

by the Finite Element Method,” Report, Second Technical Research

Institute, Central Research Institute for Electric Power Industry,

No.71009, November, 1971 (in Japanese).

[15] E. Kuribayashi and T. Iwasaki, “Effects of Soil Deposits on Seismic

Behavior of Prefabricated Highway Tunnels,” 5th World Conference

on Earthquake Engineering, Rome, Italy, June, 1973.

[16] S. Okamoto, C. Tamura, K. Kato, and M. Hamada, “Behaviors of

Submerged Tunnels during Earthquakes,” 5th World Conference on

Earthquake Engineering, Rome, Italy, June, 1973.

[17] Tokyo Harbor Undersea Tunnel Committee, Tokyo Expressway

Association, “Report of Earthquake-Resistance of Tokyo Harbor

Undersea Tunnel,” March, 1972 (in Japanese).

[18] E. Kuribayashi, T. Iwasaki, and K. Kawashima, “Dynamic Behavior

of a Subsurface Tubular Structure,” 5th Symposium on Earthquake

Engineering, Roorkee, India, November, 1974.


J l f Di t R h V l 1 N 2 2006 295

Name:Toshio Iwasaki

Affiliation:M.S. in Engineering, Chief, Ground Vibra-

tion Section, Public Works Research Institute,

Ministry of Construction

Address:2-308, Mitsuwadai Heights, 5-29 Mitsuwadai, Chiba City, Chiba

Prefecture, Japan

Brief Biographical History:1970- Senior Research Engineer, Public Works Research Institute,

Ministry of Construction

1973- Chief, Civil Engineering Section, IISEE, BRI, Ministry of

Construction

1975- Chief, Ground Vibration Section, PWRI, Ministry of Construction

Main Works:

“Literature Survey – Seismic Effects on High-way Bridges,” EERC,Report, No. EERC 72-1, University of California, Nov., 1972.

“Earthquake Resistant Design of Bridges in Japan,” Bulletin of PWRI,No.29, May, 1973.

“Effects of Soil Deposits on Seismic Behavior of Prefabricated HighwayTunnels,” Procedures of 5WCEE, Rome, Italy, June, 1973.

Membership in Learned Society

Japan Society of Civil Engineers

Japan Society for Soil Mechanics and Foundation Engineering

Seismological Society of Japan

Documents

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