Engineering Structures 31 (2009) 999–1009

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Engineering Structures 31 (2009) 9991009

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Engineering Structures

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Analytical study on energy consumption and damage to cylindrical and I-shapedreinforced concrete shear walls subjected to cyclic loading

Naohiro Nakamura a,, Naohiko Tsunashima b, Tomio Nakano c, Eizaburo Tachibana d

a Research & Development Institute, Takenaka Corporation, 1-5-1, Ohtsuka, Inzai, Chiba 270-1395, Japanb Nuclear Power Division, The Kansai Electric Power Co., Inc., Osaka, Japanc Newjec Inc., Osaka, Japand Osaka University, Osaka, Japan

a r t i c l e i n f o

Article history:

Received 6 March 2008

Received in revised form

17 December 2008

Accepted 17 December 2008

Available online 16 January 2009

Keywords:

Reinforced concrete

Shear wallEnergy consumption

3D-FEM

Nonlinear analysis

a b s t r a c t

An analytical study using the nonlinear finite-element method was conducted for reinforced concreteshearwalltestsundercyclic loading to estimate thedamage from theaspect of energy consumption.First,the validity of the analytical method was confirmed by studying the loaddisplacement relationship, theconditionof thecracking, and a comparison between thetotal strainenergy of theanalysisand theloadedenergy of the test. Next, the distribution of the energy consumption was investigated, and the divisionbetween rebars and concrete was studied. It was determined that energy consumption is an effectivemethod for estimating damage of shear walls.

2008 Elsevier Ltd. All rights reserved.

1. Introduction

The finite element method (FEM) is useful for structuralanalyses of reinforced concrete (RC) structures. Many studies,following the first conducted by Scordelis et al. in 1967 [1],have resulted in improvements in the constitutive models anddevelopment of solutions to static cyclic loading and dynamicproblems [e.g., [2,3]]. Furthermore, to validate these methods,numerous experiments were performed [e.g., [4,5]], and manyresearchers continue to improve the accuracy and practicality ofthe methods [e.g., [69]].

The authors investigated the ultimate behavior of nuclearpower plant buildings at their failure stage, using five times

the design-level ground motion [10], and the effects of thesoilstructure interaction [11] by conducting seismic responseanalyses using detailed nonlinear FEM models of the buildings.

In the seismic design of nuclear power facilities in Japan,nonlinear seismic-response analyses of buildings have beenperformed mainly using lumped-mass system models and theshearstrain of each storey has been used as a damage assessmentindex. When FEM is used, local damage to each part of the buildingcan be understood in detail. However, estimating damage to theentire building is necessary for seismic design.

Corresponding author. Tel.: +81 476 47 1700; fax: +81 476 47 7744.

E-mail address: [email protected] (N. Nakamura).

Damage estimation using the amount of energy consumed canbe considered for this purpose. In order to estimate the damage

to an RC structure, a number of investigations based on energyabsorption ability have been carried out recently [e.g., [12,13]]. Afragility assessment method in which consumed energy is used asan index has been proposed as well [e.g., [14,15]].

However, the results of these studies were obtained mainly

from beams and columns that rupture because of bending. Themechanisms of energy consumption and the energy absorptionability in the case of shear failure are not well understood yet.There are few examples of energy estimation with regard to shear

walls that are greatly affected by shear deformation. This energy

estimation is important, especially for nuclear power facilities. Inparticular, the distribution and allotment of energy consumptionin shear walls have not always been investigated satisfactorily

because they are difficult to measure experimentally.

In RC nonlinear analyses, the damage conditions are estimatedgenerally using figures of concrete cracks, rebar yielding condi-tions, etc. They provide important information about the damagecondition. However, the information is completely separated be-

tween concrete andrebarat each position.On theotherhand, sinceenergy consumption is a common and unified index, it is an effec-tive estimation tool for surveying the total damage conditions. Itcan elucidate how the energy inputted to the specimen by the im-

posed loads is shared between the concrete and rebar, and clarify

0141-0296/$ see front matter 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.engstruct.2008.12.013
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1000 N. Nakamura et al. / Engineering Structures 31 (2009) 9991009

Fig. 1. Outline of layered shell element for reinforced concrete.

Fig. 2. Uni-axial stressstrain before cracking.

how the shared energies are distributed, absorbed, and connectedto different failure modes.

As a basic study on the energy assessment of cylindrical andI-shaped shear walls (hereafter referred as RC shear walls),the energy consumption and damage to the RC shear wallssubjected to cyclic loading performed in simulation analyses inpast experiments are investigated, in this study.

First, the results obtained from the experiments of theloaddisplacement relationship and cracking are compared withthe analytical results to confirm the accuracy of the analysis. Next,the relationship between rotational anglesand hystereticabsorbedenergy (hereafter referred to as consumed energy) is investigatedto confirm thevalidityof thehysteresis modelsused in theanalysis.

Furthermore, the distribution of the energy consumed in the testspecimen is investigated. In particular, by dividing the burden ofrebars and concrete, the characteristics of the energy consumedfor each are investigated and compared with the damageconditions.

2. Analysis method

In this study, simulations of tests are performed using the3D nonlinear analysis method. The validity and efficiency of theanalysis program used in this study were confirmed by simulationanalyses of many shear wall specimens, i.e., cyclicloading tests [16]and dynamic shaking table tests [17,18]. An outline of the analysismethod is shown below.

The RC wall is modeled using layered shell elements fabricatedby substituting layers for rebars and concrete under considerationof the antiplane bending (see Fig. 1). The nonlinearity of materialsis considered for the in-plane stressstrain components, and theout-of-plane shear component is dealt with as linearity.

2.1. Modeling of concrete

The uniaxial stressstrain relationship of concrete beforecracking was approximated from a tri-linear curve, as shown inFig. 2. As shown in Fig. 3, before cracking, the concrete is anelasto-plastic body, which conforms to the DruckerPrager yieldcondition.

The cracked concrete is expressed using the smeared crack

model. The first crack occurs when the main tensile stress reachescr at each Gaussian point in each concrete layer. Once a crack

Fig. 3. Crack and yield surface.

Fig. 4. Envelope curve of stressstrain of concrete.

occurs, the stressstrain relationship of the point can be expressedas follows:

1212

=

E1(1) 0 0

0 E2(2) 00 0 G12(12)

1212

(1)

where 1 and 1 are the stress and strain, respectively, in thedirection perpendicular to the crack surface; 2 and 2 are the

stress and strain, respectively, in the direction parallel with thecrack surface; and 12 and 12 are the shear stress and strain,respectively, along the crack surface.

Note that in this equation,the directionof thesecond crack is setat right angles to the direction of the first crack. Fig. 4 indicates theenvelope curve of the stressstrain relationship for both directions(11 and 22) usedin Eq. (1). The curve was determined basedon Fafitis and Shar [19] and Stevens et al. [20].

Moreover, the compressive stiffness and strength in thedirection parallel with the crack surface is reduced by the constantcoefficient , based on the following equation proposed byNaganuma [21]:

= 0.74 c2/255. (2)

In this study, the value of was set as 0.63 for all specimens.This value corresponds to c2 = 29 N/mm2. The hysteresis


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Table 1

Re-bar arrangement for single surface of shear wall for test specimens. (All re-bars

are arranged for both surface of the wall.)

Specimen Vertical direction Transversal direction

C-1 D6a @ 9 (degree)b D6 @178 (mm)

C-2 D6 @ 4.5 (degree)b D6 @89 (mm)

I-1 D10 @82 (mm) D10 @82 (mm)I-2 D10 @61 (mm) D10 @61 (mm)

a D6 means the deformed bar whose equivalent diameter is 6 mm.b The angle at the circumference.

(Lc and Ls) of the layered shell element, where U = Uc + Us.Furthermore, in the case of the increment value ofi being i , theincrement value of the concrete and that of the rebar in the bearingload (Pci and Psi) at each loading stage can be obtained byEq. (10). As a result, the loaddisplacement relationship, in whichthe composite shell element is divided into the concrete and therebar, can be illustrated.

Uc =

Nk=1

Lc

j=1

tj

djT

j

Ak

Us =

Nk=1

Lsj=1

tj

djT

j

Ak

(9)

Pi =U

i, PC i =

UC

i, PS i =

US

i. (10)

3. Outline of tests subjected to studies

Investigations related to the static cyclic loading tests forcylindrical shear walls and I-shaped shear walls [25] are carriedout.

With regard to the former, two specimens C25-A2-12 andC25-A2-24 (hereafter, C-1 and C-2), which were obtained fromthe horizontal loading test for the cylindrical shear walls shown

in reference [26], are subjected to the investigation. Theirreinforcement ratios are different from each other.

As for the latter, two specimens 36-M8-30 and 48-H8-30(hereafter, I-1and I-2) are investigated. They were obtained from aseries of tests shown in reference [27]. Their concrete compressivestrengths are similar, but the reinforcement ratios are different.

Figs. 8 and 9 illustrate the shape of each specimen. In C-1and C-2, the displacement for calculating the rotational angle wasmeasured at the position on the 90 direction web surface and50mm below the lower end ofthe loading slab. InI-1 and I-2,it wasmeasured on the center of the loading direction and at the lowerend of the loading slab. These positions are shown in Figs. 8 and 9.Table 1 shows the reinforcement pattern of each specimen. Table 2lists material characteristics, the scale of the tests, and an outline

of the test results. The loading patterns of the test are shown inFig. 10(a) and (d). The reinforcement ratios of these specimens arehigh because the specimens are modeled as parts of nuclear powerfacilities.

3.1. Failure behavior of cylindrical specimen

This test was conducted to estimate the effects of the concretestrength and the reinforcement ratio upon its loaddisplacementrelationshipof thecylindricalRC shear wall. Thescale of specimenswas 1/38 for shear walls of a real nuclear power building. Thereinforcement ratio was determined so that the specimen canshow shear failure.

Regarding specimen C-1, bending cracks occurred on the flange

surface (the surface where the center directions are 0

or 180

inFig. 8) of the cylindrical wall near R = 0.1 103 for the rotational

Fig. 8. Test specimens shape for cylindrical shear walls (C-1, C-2).

angle. In the vicinity of R = 0.3 103, shear cracks occurredon the web surface (the surface where the center directions are90 in Fig. 8) of the cylindrical wall. In the case of R = 2 103,the transverse rebars at the center of the web and the verticalrebars at the bottom of the flange on the tensile side yieldedalmost simultaneously. Then, with an increase in the rotationalangle, the number of yielding points increased. After that, thetest specimen showed high deformability. The compression failureof concrete at the bottom on the compressive side of the flangesurface was not discernible. In the vicinity of R = 40 103,the central part of the web surface of the cylindrical wall swelledconspicuously toward the outside and the specimen reached shear

failure.Test specimen C-2 behaved in almost the same manner as

C-1. However, at theend of thetest,compression failure of concretebegan to occur in the area at almost 60 from the compressiveedge center in the direction of 180, as shown in Fig. 8. Thecompression failure spread horizontally toward the compressiveedge, and the specimen completely collapsed at R = 17 103.

3.2. Failure behavior of I-shaped specimen

As for test specimen I-1, diagonal cracks at the area peripheralto the web wall as well as bending cracks at the bottom of theflange wall occurred in the proximity of R = 0.3 103. NearR = 0.7 103, shear cracks occurred at the lower part of the web

wall, and the stiffness of the specimen decreased slightly. Then,as the displacement amplitude increased, the number of cracks


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N. Nakamura et al. / Engineering Structures 31 (2009) 9991009 1003

(a) Elevation. (b) Plan.

Fig. 9. Test specimens shape for I-shaped shear walls (I-1, I-2).

(a) Loading pattern for specimens C -1, C-2. (b) Speci men C-1 .

(c) Specimen C-2. (d) I Loading pattern for specimens I-1, I-2.

(e) Specimen I-1. (f) Specimen I-2.

Fig. 10. Time history of strain energy for entire body and re-bar (Internal strain energies are used for all lines).

Table 2

Material properties and outline of results of test specimens.

Specimen name C-1 C-2 I-1 I-2

Specimen no. C25-A2-12 C25-A2-24 36-M8-30 48-H8-30

Model scale 1/38 1/38 Not defined Not defined

Shear span ratio 1.0 1.0 0.8 0.8Elastic modulus of concrete (N/mm2) 2.26 104 2.06 104 2.36 104 3.49 104

Compressive strength of concrete (N/mm2) 23.0 25.2 39.3 41.8

Reinforcement ratio 1.2%, 0.6% 2.4%, 1.2% 1.16% 1.56%

Elastic modulus of re-bar (N/mm2) 2.07 105 2.07 105 1.93 105 1.93 105

Yield stress of re-bar (N/mm2) 324 324 296 296

Average axial stress (N/mm2) 0 0 1.96 1.96Maximum strength (kN) 414 639 1901 2264

Rotational angle for maximum load (rad) 41 103 17 103 10.0 103 6.0 103

Failure mode Shear bond Shear compression Shear slip Shear slip

Load for failure (kN) 414 639 1754 2156

Rotational angle for failure (rad) 41 103 17 103 12.2 103 10.0 103

increased. In the case ofR = 2.6 103, both the vertical rebars

at the web part and the transverse cracks at the bottom of the

flange on the tensile side yielded. Compression failure began tooccur at the bottom on the compressive side of the web wall at

R = 8 103. The specimen reached the maximum load near

R = 10 103 and ruptured in the shear sliding failure.

Test specimen I-2 behaved in almost the same manner as I-1.The compression failure began to occur at R = 6 103 and the


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(a) Cylindrical shear wall. (b) I-shaped shear wall.

Fig. 11. Analysis model.

Fig. 12. Cracking conditions of specimen C-1 at failure stage. (a) Test web part (on the side of 90 ), (b) Analysis web part (on the side of 90 ), (c) Test flange web part (on

the side of 0), (d) Analysis flange web part (on the side of 0 ).

specimen reached the maximum load at almost the same time. Itruptured in the shear sliding failure at R = 8 103.

4. Outline of simulation analyses

Simulation analyses are performed for each test and theaccuracy and validity of the analyses are investigated.

Fig. 11 illustrates the analysis model for each specimen. Theloading slab of the cylindrical specimen was made using linear

elements and those of the I-shaped model were made with a rigidbody. In the analyses, the enforced displacement was given to theloading points at the ends of both specimens.

It was confirmed that each specimen can be analyzed in a rangeup to the vicinity of the failure rotational angle. In this paper, inorder that the consumed energy at the failure stage may easily becompared after loading up to R = 15 103 for the cylindricalspecimen or up to R = 10 103 for the I-shaped specimen, theelastic strain was released by decreasing the loading value to 0 asthe final step for the analyses.

Figs. 12 and 13 compare the cracking conditions of C-1 andI-1 between the tests and the analyses. Table 3 explains themarks used in the figures. At both the web and flange parts, thecracking conditions obtained from the analyses correspond quite

well to the results of the tests. The colored parts indicate the partswhere the strain in a direction perpendicular to the cracks exceeds

Table 3

Marks in figures for cracking condition (Figs. 12 and 13).

Cracking (Closed at the final step)

Cracking (Opened at the final step)

Strain is greater than the steel yielding strain after cracking.

Axial crash after cracking.

Strain perpendicular to the crack is greater than the steel yielding strain.

the yield strain of the rebars. The analysis results of the yieldconditions of the rebars correspond quite well to the test results.

Furthermore, the analysis results of C-2 and I-2, which are nearly

equivalent to those of C-1 and I-1, correspond quite well to the test

results.

Fig. 14 shows the comparison in the loaddisplacement

relationship of each specimen between the tests and the analyses.

Although slight differences are discernible in the loop of theI-shaped specimen, the analysis results correspond quite well to

the test results as a whole.

Thus, it can be thought that these analyses could simulate thetests well. With regard to the relationship between the rotational

angle R and the horizontal displacement (mm), R was set as/1150 for the cylindrical specimens and /1400 for the I-shaped

specimens.


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(a) Test. (b) Analysis.

Fig. 13. Cracking conditions of specimen I-1 at failure stage.

(a) Specimen C-1. (b) Specimen C-2. (c) Specimen I-1. (d) Specimen I-2.

Fig. 14. Comparison in loaddisplacement relationship.

5. Study on energy consumption in analyses

5.1. Comparison of strain energy

Fig. 15 shows the comparison in the relationship betweenthe rotational angle and the strain energy. The external energyobtained from Eq. (7) is used for the tests and the internal strainenergy computed using Eq. (8) is used for analyses. Fig. 15(a)and (b) illustrates the analysis results up to the final stage, whichdirectly correspond to the test results, from R = 10 103

for the rotational angle through the point where the loading valuedecreases to 0.

Overall, the analysis values are almost equivalent to the testresults. However, the former is slightly smaller than the latter.In particular, some differences for the cylindrical walls (C-1 andC-2) are seen in Fig. 15(a) and (b), while the loaddisplacementanalysis results seems to correspond well to the test results shownin Fig. 14(a) and (b). From these tendencies, it is considered thatthe area of the analysis hysteresis loop, which corresponds to theconsumed energy, is almostaccuratein thelarge strainregions,butit is smaller than that of the test in the small strain regions.

5.2. Consumed energy of rebars and concrete

The time history of strain energy for each specimen wasillustrated in Fig. 10 with the loading pattern. The broken linesindicate the strain energy absorbed by the rebars, and thedifference between the solid and broken lines implies the strainenergy absorbed by the concrete. These values were calculatedusing Eq. (9). The lateral axis in each figure indicates the analysisstep.

As for the cylindrical specimens, most of the input energy is

absorbed by the rebars for both C-1 and C-2. In particular, even inthe stages after the two hundredth analysis step where the energy

consumption amount increases, the increment in the energy isabsorbed mainly by the rebars, and the consumption amount ofenergy absorbed by concrete does not increase to a high degree. Itis considered that the aforementioned characteristics are relatedto the high ductility and the failure mode led by the yielding of the

rebars of the specimens.On the other hand, almost all the energy input to the I-shaped

specimen is absorbed by the concrete from the earlier stage, andthe quantity absorbed by the rebars decreases. It is considered thatthis phenomenon corresponds to the low ductility and the failuremode led by the crushing of the concrete of the specimens.

Moreover, the reinforcement ratio of specimen C-2 is two timesthat of specimen C-1. The reinforcement ratio of specimen I-2 is34% greater than that of specimen I-1, andthe elasticity modulus ofthe concrete is large. In the analyses, both the maximum strengthand total consumed energy for C-2 and I-2 increase more thanthose for C-1 and I-1. The rates of increase in the strength andenergy are almost similar. Furthermore, the distribution rates ofthe consumed energy for the rebars and concrete are therefore the

same.As can be considered from the characteristics shown in Fig. 5,

the amount of energy absorbed by the tensile side of the concreteis small for all the specimens. In these analyses, the amount ofenergy absorbed by the shear stiffness shown in Fig. 6 on thecracking surface was also small. As a result, most of the energy wasconsumed due to the compression side on the cracking surface.

Fig. 16 shows the loaddisplacement relationship obtainedfrom the allotted loads of the rebars and the concrete, whichhave been computed using Eq. (10). In particular, the hysteresischaracteristics of the rebars and the concrete for test specimenC-1 are clearly presented. Moreover, in the proximity of the areawhere the concrete displacement reaches the peak, a part in whichthe loaddisplacement relationship reverses can be seen. It is

considered that this is due to the effects of a rapid variation in theload allotment of the rebars and the concrete.


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Fig. 15. Comparison in rotational angle of memberenergy relationship (External energy is used for Test, internal strain energy is used for Analysis).


(e) Specimen C-1. (f) Specimen C-2. (g) Specimen I-1. (h) Specimen I-2.

Fig. 16. Separation of loaddisplacement relationship.


Fig. 17. Vertical distribution of final consumed energy.

5.3. Distribution of consumed energy

Fig. 17 shows the vertical distribution of the consumed energy

in the final step. It is observed that the amount of consumedenergy in the rebars for all of the specimens is small at their upper

parts and is large near their lower parts. It is considered that this

is caused mainly by the effects of the bending moment. On the

contrary, the consumed energy in the concrete shows a relatively

uniform distribution in a vertical direction. This distribution isconsidered to correspond to the shear distribution.


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( a) Consume d e ner gy dis tr ibutio n of e ntir e body. (b) Con sumed en er gy distribution o f r e- bar.

(c) Consumed energy distribution of concrete.

Fig. 18. Consumed energy distribution of cylindrical test specimen (web surface).

The amount of consumed energy at the lowest parts of thecylindrical specimen is the maximum, but the amount in the

parts second from the lowest of the I-shaped test specimen is themaximum. It is considered that this is because the flange wallrestrains the side-bottom of the web wall, and the failure occurredat the slightly higher position for the I-shaped specimen.

Figs. 18 and 19 show the contour with regard to the consumedenergy of the cylindrical specimens and the I-shaped specimens,respectively. The consumed energy is shown as the amount perunit volume of the RC, and the total consumed energy, i.e., theconsumed energy of the rebars and the consumed energy of theconcrete, are compared to each other. The consumed energy of therebars concentrating to the flange part corresponds to the bendingfailure mode. A tendency, in which the consumed energy of theconcrete that is generatedat the web part corresponds to the shearfailure mode, can be seen.

For the cylindrical specimens, the energy of the concreteconcentrates in the lower part of the web center. However, thedegree of energy concentration is fairly small and the energyexpands widely over the entire web part. This corresponds to thedamage conditions illustrated in Fig. 12. It can be considered thatthis relatively uniform energy distribution is related to the highductility of the cylindrical test specimen.

On the contrary, the energy in the concrete for the I-shapedspecimen concentrates in the small area at the side-bottom of theweb where the compression failure occurs. This corresponds to thedamage conditions shown in Fig. 13.

5.4. Relationship between consumed energy and damage condition

From above results, Table 4 summarized the relationshipbetween the consumed energy and the damage condition.

Interesting correspondences can be seen between them while thestudied cases were limited.

In RC nonlinear analyses, the damage conditions are estimatedgenerally using figures of concrete cracks, rebar yielding condi-tions, etc. The condition of the consumed energy is thought to beeffective as a means to make a survey of the total damage condi-tions. Moreover, more appropriate design of shear walls might bepossible by studying and controlling the consumed energy condi-tion.

6. Conclusions

In this paper, in order to estimate the damage to RC shearwalls using the consumed energy simulation analyses of the cyclicloading tests were performed. In the simulation analyses, thedistribution of consumed energy at each part of the specimen as

well as the allotment of rebars and concrete was investigated.Furthermore, the progress of the energy consumption in thespecimen was analyzed. As a result, the following points werededuced:

(1) The analyses satisfactorily simulate the tests and the validityof the analysis method was confirmed.

(2) The allotment of the consumed energy to rebars and concretewas calculated. The allotment of the loaddisplacementrelationship was also calculated. Concerning the studiedspecimens, it was shown that the effects of the rebars weregreater for the cylindrical walls (C-1 and C-2) and the effects ofthe concrete were greater for the I-shaped walls (I-1 and I-2)from these calculations. However, these tendencies were notconfirmed because the studied cases were limited.

(3) A tendency in which the consumed energy distribution ofrebarsis connectedwith thebending failure andthe consumed


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( a) Cons ume d e ne rgy distribution of en tire body. ( b) Cons ume d e ner gy dis tribution of re -bar .

(c) Consumed energy distribution of concrete.

Fig. 19. Consumed energy distribution of I-shaped test specimen.

Table 4

Summaries of results.

Shape of shear w all Energ y consumed mainly in: Failure mode Concentration of consumed energy in concrete Ductility

Cylindrical Rebars Bending Small High

I-shaped Concrete Shear Large Low

energy distribution of concrete is connected with the shearfailure was discernible.(4) The concentration degree of energy of concrete for the

cylindrical specimen is smaller than that for the I-shaped testspecimen. The energy of concrete for the cylindrical specimentends to distribute over the entire surface of the web. It isthought that this is related to high ductility.

(5) It was confirmed from the above that the consumed energy iseffective as an index for estimating the damage to RC shearwalls. Moreover, it can be said that contour figures are usefulas a means to easily understand the analysis results.

Acknowledgments

In this study, some parts of the Research on Rationalization ofNuclear Power Plant Facilities trough Adoption of High-StrengthReinforcing Bars, which was carried out under the sponsorship oftenJapaneseelectric companies, were used. Theauthors would liketo express their thanks to all the persons involved.

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