Structure Damage Earthquke

8/3/2019 Structure Damage Earthquke

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NONLINEAR ANALYSIS TO EVALUATE DAMAGE OF

RC STRUCTURES DUE TO EARTHQUAKE

Hikaru NAKAMURA1

SUMMARY

A destructive damage which leads to the loss of human life shall be prevented

against a strong earthquake ground motions. In addition, from social and

economic points of view, livelihood and productive activities of inhabitants after

the earthquake should be restored smoothly with rapid restoration of the

structures. This paper presents the damage of concrete structures due to recent

earthquakes and the brief outline of the JSCE Standard Specifications for Seismic

Performance Verification published in 2002. Then two topics related with the

damage were discussed. First, the strain localization problems of nonlinear finite

element analysis were described and an advanced method based on non-local

continuum theory to obtain the reliable local strain was proposed. Second, theinfluence of the buckling on seismic performance was discussed.

Keywords: seismic performance verification; restoration; strain localization;

non-local theory; buckling; dynamic response.

INTRODUCTION

It was observed the destructive damages of many concrete structures in the TheHanshin-Awaji (Kobe) Earthquake in January, 1995. As the result, the JSCE Standard

Specification for 'Seismic Design' was established in 1996[JSCE 1996], in which the methods

for seismic performance verification of concrete structures were described. The specification

was revised in 2002 based on the concept of the performance based design and was published

as renamed 'Seismic Performance Verification'[JSCE 2002]. In the specification, the nonlinear

analysis, in principle, is performed to verify the seismic performance with modeling of the

structures and the ground.

In past two decades, nonlinear finite element analysis for concrete structures has advanced

remarkably with the developments of sophisticated constitutive models, robust numerical

1 Professor, Department of Civil Engineering, Nagoya University, JAPAN, e-mail: [email protected]


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algorisms and analytical theories. At present, the nonlinear analysis are used to simulate most

behaviors of concrete structures, such as the cracking, the stress flow, the load carrying

capacity, the deformation capacity, the failure behavior and so on. Moreover, it became a most

powerful tool to verify the required performance of concrete structures in performance based

design. Especially, seismic performance permit yielding of member and some damages

considering energy absorption for strong earthquakes. Then, the nonlinear analysis is useful tosimulate post yielding and further post peak behavior. Advanced nonlinear analysis is possible

to simulate dynamic behavior such as load displacement relationships, time response

displacement and so on even in post peak region. The restoration ability as well as dynamic

behavior is important in seismic performances. Then, it is desirable to simulate the local

damage occurred in the concrete structures. The damage of concrete structures is classified

with the spalling of cover concrete, the buckling of longitudinal reinforcement, the

compression failure of concrete and so on. The simulation of the behaviors is, however,

insufficient, especially relating with the local damage.

This paper presents the damage of concrete structures due to recent earthquakes and the brief

outline of the JSCE Standard Specifications for Seismic Performance Verification publishedin 2002. Then two topics related with the damage are discussed. First, the strain localization

problems[Bazant et al. 1998, Jirasek et al. 2001] of nonlinear finite element analysis are

described. In the problems, the influence on the reliability of numerical results related to the

mesh sensitivity in local continuum theory are discussed. Then, an advanced method based on

non-local continuum theory[Bazant et al. 1994, Borst et al. 1992] to obtain the reliable local

strain is proposed and the effectiveness is discussed. Second, the influence of the buckling on

seismic performance is simulated and is compared with test results. The necessity to consider

the buckling of re-bars are shown.

DAMAGE OF CONCRETE STRUCTURES DUE TO RECENT EARTHQUAKES

After the Hanshin-Awaji Earthquake on January 17, 1995, which was magnitude (Mj) of 7.2,

the concrete structures have been damaged several earthquakes in Japan as listed in Table 1.

In this chapter, the damage of concrete structures, especially the damage of railway and

highway bridges, due to recent earthquakes in Japan are presented[JSCE 2004].

Table.1 Information of recent earthquakesDate Magnitude(Mj)

Type Recorded max.

acceleration

Hanshin-Awaji Earthquake Jan. 17, 1995 7.2 inland 818 cm/s2

Geiyo Earthquake Mar. 24, 2001 6.7 intraslab 830 cm/s2

South of Sanriku-Oki

Earthquake

May 26, 2003 7.1 intraslab 1106 cm/s2

Tokachi-Oki Earthquake Sept. 26, 2003 8.0 inter-plate 972 cm/s2

Niigata-ken Chuetsu

Earthquake

Oct. 23, 2004 6.8 inland 1722 cm/s2

Noto-Hanto Earthquake May 25, 2007 6.9 inland 945 cm/s2

Niigata-ken Chuetsu-Oki

Earthquake July 16, 2007 6.6

inland

813 cm/s

2


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South of Sanriku-Oki Earthquake on May 26, 2003

The South of Sanriku-Oki Earthquake on May 26, 2003, occurred at Off Miyagi Prefecture

and the magnitude was 7.1. It was intraslab type earthquake in The Pacific Ocean plate. The

epicenter was deep and it was about 71km. The recorded maximum acceleration was more

than 1100cm/s2

. The feature of the earthquake ground motion is that short-period wave isdominant. The severe damages in 5 one story RC elevated bridges of Tohoku Shinkansen

were observed. The feature of these damages was that the end columns which is shorter than

intermediate columns were mainly damaged.

Photo.1 shows damaged Dai-san Otagi Bridge R2 of Tohoku Shinkansen which was

constructed in 1977 to 1978. This is four bay one story RC frame elevated bridge. The end

columns have severe condition for shear failure, since they are shorter than intermediate

columns. Two of the end columns failed in shear with the spalling of cover concrete, while

others were observed diagonal cracks. The damage due to flexure, however, were hardly

observed.

Tokachi-Oki Earthquake on September 26, 2003

The Tokachi-Oki Earthquake on September 26, 2003, occurred at Kushiro-Oki (southeast

offshore of Hokkaido) and the magnitude was 8.0. Tsunami was also observed. It was typical

inter-plate type earthquake. The recorded maximum acceleration was about 1000cm/s2.

Around the focal area, an earthquake of magnitude 8.2 occurred on March 4, 1952, and an

earthquake of magnitude 7.9 occurred on May 16, 1968 at south of this area. The feature of

the earthquake ground motion was that long-period wave is dominant and the duration time is

long. A fire of the oil storage tank caused by the sloshing occurred and the effect of the

long-period wave was paid to attention. The concrete structures were mainly damaged in

superstructure and substructure of railway and highway bridges.

Toshibetsu-gawa railway bridge of Nemuro Line was constructed in 1966 to 1969. This is 13

spans girder PC bridge of 416m bridge length with RC piers of circular section. This bridge

was damaged around the supports at The Kushiro-Oki Earthquake in 1993. Photo.2(a) shows

damaged pier in which the spalling of concrete cover and the buckling of longitudinal re-bars

were observed. Photo.2(b) shows the damage of the floor slab at the end of girder which

occurred due to the collision between girders. Similar damage of bridge piers were observed

in Uroho-gawa railway bridge. This is 4 spans PC bridge of 128m bridge length with RC piers

of circular section. Photo.3 shows a damaged pier in which the spalling of concrete cover and

the buckling of longitudinal re-bars were observed at cut-off section of the longitudinalre-bars.

Photo.4 shows damaged Chiyoda highway bridge which consists of middle bridge of 5 spans

warren truss constructed in 1954 and side bridge of 5 spans PC girder constructed in 1966.

The flexural failure in the piers of PC girder bridge as shown in Photo.4(a) and the punching

shear failure at a support of warren truss bridge due to horizontal force from anchor as shown

in Photo.4(b) occurred. The feature of the damage of the piers was the flexural cracks, the

spalling of concrete cover and the buckling of longitudinal re-bars. Similar failure at support

was already observed in The Hanshin-Awaji Earthquake.

Niigata-ken Chuetsu Earthquake on October 23, 2004


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The Niigata-ken Chuetsu Earthquake on October 23, 2004, occurred at Mid Niigata Prefecture

and the magnitude was 6.8. It was caused by inland active fault. The depth of the epicenter

was about 13km. The recorded maximum acceleration was more than 1500cm/s2. The

recorded maximum acceleration for this earthquake is much greater than that for The

Hanshin-Awaji Earthquake. The feature of the earthquake was that many aftershocks

including some magnitude 6 class events have been following. Aftershocks were distributedalong the northeast and southwest direction with a length of about 30km. Several concrete

structures of railway and highway bridges were damaged.

Photo.5 shows the damage of Uono-gawa bridge of Joetsu Shinkansen, which is three spans

box girder PC bridge. In 2P and 3P with circular RC section of 6.5m diameter, the spalling of

concrete cover and buckling of longitudinal re-bars were observed at the mid height. Lateral

ties at the location were found to detach. The failure occurred at the location of cut-off section

of the longitudinal re-bars. Three steel arcs with central angle little more than 120 degree

were used to make circular ties. No hooks were provided in the lateral ties.

Photo.6 shows the damage of Dai-san Wanazu bridge R1 of Joetsu Shinkansen, which is threebay one story RC frame elevated bridge. One of the end columns failed in shear as shown in

Photo.6, while another one showed diagonal cracks. The end columns have severe condition

for shear failure, since they are shorter than intermediate columns. This failure is the same as

the one observed in The South of Sanriku-Oki Earthquake. These were designed by the same

specifications for Shinkansen published in 1970 and 1972. It is noted that some columns of

elevated bridge of Joetsu Shinkansen were strengthened by the steel jacketing before the

earthquake as shown in Photo.7. As for them, no damage was observed and the effect of

strengthening was confirmed.

JSCE STANDARD SPECIFICATION FOR CONCRETE STRUCTURES-2002

SEISMIC PERFORMANCE VERIFICATION

Outline of Seismic Performance Verification

The provisions concerning the seismic performance of concrete structures had been specified

as one chapter of the JSCE Standard Specification for Design until 1991 version. A similar

form was planned to be adopted at the revision in 1996. However, it was necessary to presentthe specification based on a new design concept, since concrete structures were damaged

destructively in The Hanshin-Awaji Earthquake 1995. Therefore, the specification for Seismic

Design was established in 1996, in which the methods for seismic performance verification of

concrete structures was described basically.

In Seismic Design(1996), the modeling and the analytical method of the structures were not

described enough. In Seismic Performance Verification(2002), these items were enhanced

based on the knowledge of seismic performance and the advancement of the analytical

technique afterwards. Especially, the items of (1)Loads (earthquake ground motion in

verification), (2) Evaluation for the effect of ground, (3) Verification technique(analytical

method) and (4) Structural details were enhanced. Moreover, it was systematized that themore reasonable seismic performance verification becomes possible. In order to indicate their


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contents clearly, the specification 'Seismic Design' was renamed as 'Seismic Performance

Verification'. Fig.1 shows the general procedure to verify the seismic performance based on

Seismic Performance Verification(2002).

Seismic Performance

The required seismic performance for concrete structures was defined as described in Table 2.

When setting seismic performance of a structure, it should be better to take its response

characteristics into account according to the magnitude of an expected earthquake, as well as

its behaviors during the earthquake and restoration ability after the earthquake. Needless to

say that a destructive damage which leads to the loss of human life shall be prevented against

a strong earthquake ground motion that has a rare probability of occurrence within the

lifetime of a structure. In addition, from social and economic points of view, deterioration in

functions of the structure should be avoided as much as possible, and livelihood and

productive activities of inhabitants after the earthquake should be restored smoothly.

Therefore, the seismic performances are related to the necessity of repair and/or strengthening

and to the serviceability after an earthquake. The seismic performances are decided by thecombination with serviceability, restoration ability and safety.

Table 2 Definition of seismic performance

Seismic

Performance 1

Function of the structure during an earthquake is maintained, and the

structure is functional and usable without any repair after the earthquake.

Seismic

Performance 2

Function of the structure can be restored within a short period after an

earthquake and no strengthening is required.

Seismic

Performance 3

There is no overall collapse of the structural system due to an earthquake

even though the structure does not remain functional at the end of the

earthquake.

Seismic Performance 1 means that the residual deformation of a structure after an earthquake

remains sufficiently small. The performance is related to serviceability.

Seismic Performance 2 is that load carrying capacity does not deteriorate after an earthquake.

In general, this performance may be thought to be satisfied in case that each constitutive

member of a structure does not fail in shear during an earthquake and response deformation

does not reach its ultimate one. In case of structures such as wall type ones in which

earthquake force acts mainly as in-plane force, large deformation capacity cannot be expected.

In the design of such structures, it is necessary to provide sufficient shear capacity so that

brittle failure may not occur. The performance is strongly related to serviceability andrestoration ability.

Seismic Performance 3 requires that whole structural systems do not collapse due to self and

imposed masses, earth pressure, hydraulic (liquid) pressure, and so on, even if the structure

becomes un-restorable after an earthquake. In case of concrete structures, generally Seismic

Performance 3 can be satisfied if each constitutive member has enough safety against shear

failure. In some structural systems, however, displacement of the whole structure becomes

excessive, and the additional bending moment as well as longitudinal displacement increase

due to the self weight. These may lead to self-overturning or collapse mechanism. The

performance is related to safety.


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Verification technique(analytical method)

The performance verification techniques of concrete structures that used a nonlinear analysis

have improved with the advancement of analytical environment after The Hanshin-Awaji

Earthquake. It became possible to analyze the structure and the ground together. Analytical

method is applied by finite element method based on the background of advancement of suchtechnology.

Structures should be modeled as an assembly of member models in which columns and beams

are modeled as linear members and members with planar spread such as wall or floor are

modeled as planer members. Basically, the linear member is modeled using beam element and

planar member is modeled using plate or layered shell element. Strictly speaking, structures

should be modeled three dimensionally, because it is composed of members connected in

three dimensions. Two dimensional modeling of the structure may, however, be applied if

only the response behavior in a two dimensional is considered according to the direction of

input earthquake ground motion and characteristics of structural response. It is possible to

model a linear member using plate or layered shell element. In general, however, modeling byusing beam element is effective, since the computation time becomes shorter and accuracy

does not decrease much. The beam element with fiber technique can consider the material

nonlinearities easily in which the uni-axial stress-strain relationships of materials in each fiber

cell are applied. Shell structures such as tank structures should be modeled as an assembly of

shell elements for the whole structure, since the structure consists of one member. The

pull-out of re-bar at joint of members is usually negligible. It is, however, desirable to use a

joint element between members, when it is necessary to consider the effect of the pull-out of

re-bar if the diameter of the re-bar is relatively large compared with the member size.

In finite element method, mechanical model of materials are applied. Therefore, the

constitutive model of concrete, reinforcing bar, and soil are described with those hysteresis

and with notes to use. The hysteresis of concrete in compression and reinforcing bar are

shown in Fig.2 and Fig.3, respectively.

REQUIREMENT AGAINST NONLINEAR ANALYSIS TO VERIFY SEISMIC

PERFORMANCE

The nonlinear finite element analysis for concrete structures has advanced remarkably with

the developments of sophisticated constitutive models, robust numerical algorisms and

analytical theories. The importance of the nonlinear analysis has been increased in the

performance based design and it became a most powerful tool to verify the required

performance of concrete structures. Especially, seismic performance permit yielding of

member and some damage considering energy absorption for strong earthquakes. It was

described in previous chapter that the use of nonlinear analysis was already specified in JSCE

Standard for the seismic performance verification. At present, the seismic performance of

many structures are verified by the dynamic nonlinear analysis. Then, the safety during an

earthquake are mainly checked by the response displacement and the shear resistance capacity.

Seismic Performance 2 is generally satisfied, unless members fail in shear and unless thedisplacement reaches the ultimate displacement during an earthquake. These factors are


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defined from mechanical view points. On the other hand, the important view points come

from social and economic. The seismic performances are related to the restoration ability with

the necessity of repair and/or strengthening and to the serviceability after an earthquake.

Therefore, the restoration ability within a short period after an earthquake should be evaluated,

because the damage of the structures can not avoid from a strong earthquake. The examples of

repair after the earthquakes are shown in Photo.8 and Photo.9. Photo.8 shows repair processof a column of Tohoku Shinkansen damaged by The South of Sanriku-Oki Earthquake. The

damaged column was repaired by injection of epoxy resin, cross sectional restoration and

steel jacketing. Photo.9 shows a damaged and a repaired pier of Uroho-gawa railway bridge

after The Tokachi-Oki Earthquake. The damaged column was repaired by injection of epoxy

resin, cross sectional restoration and RC jacketing.

In order to evaluate the restoration ability, the damage of concrete structures as well as the

plastic deformation should be evaluated accurately. Although the plastic deformation which is

a index corresponding to structure behavior, is often identified with damage index indirectly,

the damage which is local behavior in the structures should be evaluated directly. The damage

of concrete structures is classified with spalling of cover concrete, buckling of longitudinalreinforcement, compression failure of concrete and so on. They will occur in a localized area

such as plastic hinge. Therefore, a requirement against nonlinear analysis for the restoration is

to make clear the localized area and the local strain as a damage index for the damage

evaluation.

Other requirements are listed such as a) the behavior of the damaged structures during an

earthquake and aftershocks, b) the seismic performance of the repaired structures, and so on.

The behavior of the structures depends on the type of earthquake. The earthquake ground

motion waveforms for inland type usually show large maximum acceleration and short

dominant period with short duration time. On the other hand, the waveforms for inter-plate

type usually show long dominant period with long duration time. We have to consider the

relationship between the dominant periods of the structure and ground motion wave. It is

noted that the dominant period of structure will change due to the damage during an

earthquake. The influence on the dominant period of the damage usually becomes remarkably

after the buckling of the longitudinal re-bars occur in plastic hinge. The change of the

dominant period is also important to verify the performance during aftershocks. The bucking

of re-bars have been often observed in the damaged structure. The buckling behavior,

however, have not been considered in nonlinear analysis.

From the discussion of above mentioned, the local strain evaluation based on non-local

continuum theory and the influence on the seismic performance of buckling will present innext chapters.

STRAIN LOCALIZATION PROBLEM AND ITS SOLUTION

Strain Localization Problem for Strain Softening Material

The localized area and the local strain as a damage index are required in damage evaluation.Therefore, the strain localization problem should be solve in the nonlinear analysis of


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concrete structures. This section discuss the strain localization problem for strain softening

material such as concrete, when the local continuum theory using unique stress-strain

relationship or mesh dependent stress strain relationship based on the fracture energy is

applied to finite element method.

a) Analytical ModelIn order to discuss the localization problem using the analytical results, the cantilever type RC

beam is analyzed by the beam element based on uni-axial stress state using the fiber technique

in which the member cross section is divided into many cells[Nakamura et al. 1991]. The

dimensions of RC beam are shown in Fig.4, which was designed with over reinforcements to

investigate compression failure behavior of concrete in compressive flexure mode. The

feature of the test is that the longitudinal local strain was measured by the deformed acrylic

resin bar embedded in specimen[Tatematsu et al. 1997].

The analysis is performed by changing the element size of 50mm, 200mm and 300mm. Then,

two types of stress-strain relationship with strain softening are applied. One is unique

stress-strain relationship independent on mesh size in local continuum theory(Local Analysis).This type has usually been applied to the analysis of concrete structures, since the stress-strain

relationship is regarded as material property of local point. It is assumed that the relation is

expressed by a second degree parabola before peak and a linear descending branch after peak

as written in Eq.(1). In this paper, u is assumed -0.012 as the constant value.

where, c'f is the compressive strength, co is the strain corresponding to c'f .

Another is stress-strain relationship based on the facture energy in the local continuum theory

(Fracture Energy Analysis)[9]. The compressive fracture energy is considered in the hatched

area based on its definition as shown in Fig.5. Then, u is defined as

022 ./)'f/(G coelmcfcu = l and elml is element size. The compressive fracture energy is

defined by Eq.(2)[Nakamura et al. 2001].

The slope of the strain softening curve vary with the element size by getting the balance of

compressive fracture energy in strain localized element. Therefore, it represents mesh

dependent stress-strain relationship.

b) Problem of Strain Localization

Fig.6 shows the load-displacement curves obtained from the analysis and test. Fig.6(a) shows

the results of Local Analysis and Fig.6(b) shows the results of Fracture Energy Analysis. Fig.7

and Fig.8 show the longitudinal compressive strain distribution of concrete at the location of

25mm from the compression edge of cross section at the 90% of maximum load in pre-peak

region, 90% and 80% of maximum load in post-peak region.

The load-displacement relationships obtained from Local Analysis show the strong meshdependent behavior. That is, for smaller mesh size, the maximum loads and the displacements

( )

( )( ) ( )ucocou

u

cococo

c'f

=

=

022

'cfc f.G 88=

(1)

(2)


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corresponding to the maximum load become smaller and the brittle post-peak behaviors

appear. Moreover, the longitudinal compressive strain distribution also depends on the mesh

size after the peak point in which strain localize in a mesh and the localized strain show quite

larger values for smaller element size. It has been well known that the localization problem

occurs for post-peak behavior of strain softening material in local continuum theory. The

absorbed energy in the localized element become smaller for the smaller element size, whenunique stress strain relationship independent on mesh size is applied. A method to solve the

problem is to use stress strain relationship based on fracture energy.

The load-displacement relationships obtained from Fracture Energy Analysis show the similar

behavior independent on mesh size. This is reason that the absorbed energy in localized

element become constant due to adjustment of the softening behavior of stress strain

relationship based on fracture energy. That is, the softening curve become milder in

proportion with the mesh size for smaller mesh size. Therefore, reliable load-displacement

relationships are obtained independent on mesh size. The longitudinal compressive strain

distributions, however, depend on the mesh size similar with the results of Local Analysis as

shown in Fig.8. It is difficult to solve the mesh sensitivity problem of the local strain as far asthe local continuum theory is applied in finite element method for strain softening material.

The local strain values have a close relationship with the local damage of structures and

provide important information for the restoration ability of RC structures. Therefore, more

advanced method based on nonlocal type constitutive law is presented in next section to

obtain reliable numerical solution decreasing the mesh sensitivity of both displacement and

strain.

Integral Type Nonlocal Constitutive Law

a) Basic Concept

In local continuum theory as discussed in previous section, stress at a local point is given by

only the mechanical information at the point. On the other hand, nonlocal constitutive laws

assume that the local state of material at a given point may not be sufficient to evaluate the

stress at the point. This assumption allows including characteristic length scale in the nonlocal

formulation. In case of strain field, nonlocal formulation consists in replacing local strain )(

by its nonlocal strain )(x characterizing the strain softening of material[Bazant et al. 1988].

The nonlocal response of strain is defined as

where, ),( xW is the normalized nonlocal operator, defined as

),(0 x is weight function which has symmetric function around the point x , )(xVr is the

representative volume for the nonlocalization space, defined as Eq.(5).

where, x is the arbitrary coordinate of received point in the analytical field. , are the

coordinate of a points in the representative volume. The normalized nonlocal operator

),( xW is determined by satisfying the following normalized condition.

= V )(dV)(),x(W)x(

)x(V

),x(),x(W

r

0=

= Vr )(dV)x()x(V 0

1=V )(dV),x(W

(3)

(4)

(5)

(6)


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Fig.9 shows the outline of nonlocal procedure in one dimensional problem. The local strain

distribution is assumed as shown in Fig.9(a) in which strain localize in a discrete area and has

constant value in other area. The nonlocal strain of Fig.9(b) is obtained by considering the

local strain distribution, the weight function and the characteristics length*

l . At point A, the

nonlocal strain is same as the local strain, since the strain value is constant in the

characteristics length. At point B and C, the localized strain area is included in thecharacteristics length and the nonlocal strain show the different values from the local strain.

That is, the nonlocal strain become smaller than the local strain at point B and become larger

than the local strain at point C. The nonlocal strain distribute continuously based on the

procedure and the localization problem is avoided without the strain jump in continuous

mechanics.

b) Damage Based Material Model For Concrete

A scalar damage evaluation law is applied to compressive and tensile stress field of concrete,

respectively. The stress-strain relationship of concrete is controlled by the damage function,

)( , which is function of the nonlocal strain and has range 0 to 1 as shown in Eq.(7). The

feature of the nonlocal constitutive model is to have the nonlocal parameter besides the

local parameter . For damage evaluation in the compression, the damage function isdefined based on the Saenz model[Saenz 1964], as written by Eq.(8). In tension, the damage

function is defined by Eq.(9), which is derived from the tension stiffening model[Nakamura et

al. 1993].

where, a ,b , c are material parameters defined by the initial elastic modulus 0E , the secant

elastic modulus for peak point and a point in post peak curve (f

,f

), c , t is nonlocal strain

in the compressive and tensile field, respectively. 0c is strain corresponding to compressive

strength. tf is tensile strength. 0t is strain corresponding to tf .

c) Nonlocal Parameters of Concrete in CompressionAs described above, the characteristics length, the stress-nonlocal strain relationship and the

weight function should be defined in the integral type nonlocal constitutive law. Then, the

characteristics length and the stress-nonlocal strain relationship may be regard as inherent

material properties. Although several models have proposed for concrete in tension such that

the characteristics length is three times the maximum aggregate size[Bazant et al. 1988], the

nonlocal parameters in compression have not been defined clearly. The compressive behavior

is, however, more important than tensile behavior for concrete structures.

Author studied about strain localization behavior in compression in detail by uni-axial loading

test for the parameters of the size, the shape and the compressive strength of the concrete

cylinders and the aggregate grading[Nakamura et al. 2001]. Fig.10 shows an example of the

failure behavior of the test specimens with different length. For shorter specimen, the failure

001 E))(.( =

3

000

1

101

+

+

=

c

c

c

c

c

c

c

cba

.)(

))((E

f.)(

ttt

tt

00 20020101

+=

(7)

(8)

(9)


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behavior is observed to extend into the entire specimen. On the other hand, the failure

behavior is localized in a particular length for longer specimens. Using these test results, the

nonlocal parameters in compression are identified by the inverse analysis[Kwon 2006].

The stress-nonlocal strain relationship using Saenz model is defined by comparing with the

test result of the cylinder specimen having the diameter of 150mm and the height of 150mm,since the strain distribution was uniform over the height even in post peak behavior and the

stress-average strain relationship was identified with the local relationship as the material

property. Then, parameter of Saenz model to define the softening behavior is identified with

(f

,f

)= ( c. 40 , c. 40 ). Fig.11 shows the comparison between the test and the model of the

stress-strain relationship and a good agreement can be seen from the figure. The

characteristics length and the weight function are identified by the comparing with the test

results for longer specimens which show the localized behavior in a particular zone of the

specimens. Based on the inverse analysis, it is defined that the characteristics length in

compression is 250mm and the weight function is rectangular shape. Fig.12 show the load-

displacement relationship and local strain distribution along the specimen height for the

specimen having the diameter of 150mm and the height of 600mm. The result obtained from

the nonlocal constitutive law can evaluate accurately the localized strain distribution as well

as the stress-strain relationship.

d) Applicability to RC Beam

The Integral type nonlocal constitutive law using the nonlocal parameters in compression is

applied to the RC beam of Fig.4. Fig.13 shows the load-displacement relationship obtained

from the analysis with different mesh size and test. Fig.14 shows the longitudinal compressive

strain distribution of concrete at the location of 25mm from the compression edge of cross

section at the 90% of maximum load in pre-peak region, 90% and 80% of maximum load in

post-peak region. The load-displacement relationships are independent on mesh size and showthe similar behavior with the results of Fracture Energy Analysis as shown in Fig.6(b), though

a unique stress strain relationship is used. Moreover, the longitudinal compressive strain

distributions is also independent on mesh size. The results show the similar strain distribution

and progress of strain with test results. It is understood that the model can evaluate a realistic

and an unique displacement and strain distributions. This indicate that the local damage of

concrete structures become possible to evaluate analytically and it provides us useful

information for restoration.

INFLUENCE ON SEISMIC PERFORMANCE OF BUCKLING OF RE-BARS

Buckling Model

Several stress strain relationships of reinforcing bar considering the buckling have been

proposed[Monti et al. 1992, Gones et al. 1997, Nakamura eta al. 2001]. In this paper, the

model proposed by Tanoue et al. is used in which the hysteresis behavior after the buckling

was modeled accurately[Tanoue et al. 2002]. The model before buckling is assumed Tri-linear

model considering isotropic and kinematic hardening as shown in Fig.15 . The buckling stress

is calculated theoretically by Euler's theory for the elastic buckling and Engesser-Karman'stheory for the plastic buckling. The stress in compression after buckling gradually decrease


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depending on the slenderness ratio. It is noted that the model shows strain softening behavior

after buckling and stain localization problem also occur. Therefore, the softening branch is

adjusted considering the element size as same as fracture energy model of concrete. The

curves to tension stress after the buckling change from the convex shape to the reversed S

shape curve according with increase of the buckling displacement. Fig.16 shows the outline of

the model in increasing and decreasing history. The buckling length is defined by the Eq.(10)proposed by Asazu et al.[Asazu et al. 2000]. In the equation, Lp is buckling length, E0 is

elastic stiffness,I0 is geometrical moment of inertia and C2 is constant value(=2.7).

Influence of Buckling in Shaking Table Test of RC Bridge Pier

Nonlinear analysis used beam element with fiber technique is applied to simulate the Shaking

Table Test of RC bridge pier carried at PWRI[Kawashima et al. 1993]. The outline of thespecimen is shown in Fig.17. Input acceleration wave and time history response displacement

at the top of the pier are shown in Fig.18.

Fig.19 shows the time history response displacement at the top of the pier obtained from

analysis for the case that the buckling is considered and is not considered. Fig.20 shows the

load displacement relationships. In the case that the buckling is not considered, tri-linear

model as shown in Fig.15 is used for re-bars. Fig.21 shows Fourier spectrum of time history

response displacement. In the test, the spalling of concrete and the buckling of re-bars at the

bottom part of the pier were observed. The feature is that the time history response

displacement had several large amplitude even after the maximum input acceleration. The

response displacement was excited for the small acceleration of less than 100 gal. The load

displacement relationship showed the change of the hysteresis loop to the inversed S shape

curve after the maximum response displacement. Moreover, the response period was

dominant in the range of 1.3-1.7(sec), though the eigen period of the pier in elastic was

0.52(sec). It is understood that the buckling strongly influenced to the change of the hysteresis

loop and the dominant response period.

In the analytical results in which the buckling is not considered, the response displacement is

small and the hysteresis loop shows large energy absorption behavior. These results are quite

different with test results. The difference can also be understood by the Fourier spectrum. The

dominant response period and the amplitude are obviously smaller than the test one. On theother hand, in the analytical results in which the buckling is considered, the buckling occur at

10.82(sec) in a side and 11.48(sec) in another side. The response displacement increase after

the buckling and the large displacement response continue with comparatively long period

even in small input acceleration. After the buckling of the re-bars, the hysteresis behavior

change to the inversed S shape in which the stiffness, restoration force and energy absorption

ability are deteriorated. The difference with the case that the buckling is not considered,

appear in the Fourier spectrum clearly. The dominant response period and the amplitude

become large and they are identical with the test results. The buckling behavior influence

remarkably to dynamic response and it should be considered in nonlinear analysis for the

evaluation of seismic performance and the damage accumulation for the ground motion wave

after maximum acceleration and aftershocks.

np IECL 004

2=

6440 =I(10)


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CONCLUDING REMARKS

The seismic design in Japan changed greatly after The Hanshin-Awaji Earthquake. The

damages of most concrete structures in the Earthquake were caused by the shear failure. As

the result, the structural details were greatly revised and large plastic deformation has been

required in flexural failure mode. Moreover, the performance based design has been adopted,and the nonlinear analysis, in principle, is performed to verify the seismic performance.

Fig.22 shows the typical damage events on skeleton curve of a concrete member. Although

these are the limit values for the required performance and some events can be simulated

reasonably by the advancement of nonlinear analysis, several events remain to be evaluated

yet.

The damage can not be prevented for a strong earthquake. Then, functions of the structure

should be restored as soon as possible smoothly from social and economic points of view. To

do so, restoration ability after an earthquake is important. That is, it is important to make clear

the relationships among damage, restoration ability and seismic performance after an

earthquake as well as the performance during an earthquake. The damage is closely related tolocal behavior in members and structures. Typical observed damage of concrete structures

after recent strong earthquakes were diagonal shear cracks, spalling of concrete cover,

buckling of longitudinal re-bars as shown in this paper. The damage, spalling of concrete

cover, buckling of longitudinal re-bars are events in localized area such as plastic hinge.

Therefore, the localized behavior in concrete structures should be simulated by nonlinear

analysis to evaluate damage.

In this paper, the problem to simulate the localized behavior were discussed. The method

based on local continuum theory which is generally used can not avoid mesh sensitivity for

post peak behavior. When unique stress strain relationship independent on mesh size is

applied, the load displacement and strain distribution depend on the mesh size. When mesh

dependent stress-strain relationship based on fracture energy is applied, reliable

load-displacement relationships can be obtained independent of mesh size. Strain distribution,

however, depend on the mesh size. On the other hand, the integral type nonlocal constitutive

law can avoid the strain localization problem and evaluate the local damage such as localized

area and strain as well as displacement of concrete structures accurately.

The influence on seismic performance of buckling of longitudinal re-bars were discussed.

When the stress strain relationship considering the buckling is applied, the behavior after the

buckling can simulate reasonably. The buckling greatly influence the seismic performance

and the hysteresis behavior change to the inversed S shape in which the stiffness, restorationforce and energy absorption ability are deteriorated. As the results, the dominant response

period and the amplitude become large. Therefore, the buckling behavior should be

considered in nonlinear analysis. It will provide the knowledge about damage accumulation

for the ground motion wave after maximum acceleration and aftershocks.

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SB spalling of cover

SC crack width >

SD crack lwidth

Documents

Structure Damage Earthquke