クリープによる鉄筋コンクリート柱の安定性解析

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Creep on S

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Reinforced Concrete Column

Zhang HENGPING

，

T

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AOKI and S

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INOMA

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クリープによる鉄筋コンクリート柱の安定性解析

張

恒 平 @ 青 木 徹 彦 @ 猪 股 俊 司

In this paper the second-order巴ffectis evaluated by taking into account th influence of creep

correnponding to th巴workingloads. The moment-curvature r巴lationsat time t are calculated

from the equ日ibriumequations for axial load and moment， and the assumption of linear strain distribution (plane-section assumption). In this case， the stress-strain diagram for concret巴IS

modi自巴dby multiplying the short-term strains by a factor(1

+

co)， in order to take account the effect of creep. A program for determining N -M -l/R relationships is developed， and a rational approach is proposed to investigate the creep stability failur巴ofreinforced concrete columns

1. INTRODUCTION A column subjected to eccentrically applied axial load will deflect laterally. This deflection may signifi cantly affect the distribution and magnitude of inter -nal forces and consequently also the load bearing capacity of the column. If the load is sustained， the deflection will increase further due to creep of com cret巴 Under sustained external forces， th巴initialdef!ec

tion， and consequently the moment， is magnifi巴ddue

to cr巴ep.If the external forces are relatively small in

comparison with the bearing capacity of the column，

the creep deflection will terminate at some finite value and an equilibrium stat巴isr巴achedbetween external and internal forces at all sections constitut ing the column. Furthermore， at each section， th巴 curvature due to internal stress is then equal to the curvatur of the de日巴ctionline. At high external forces， the creep def!ection may increase continuously until the column fails. Depend ing upon the column geometry (mainly slendern巴ss

and eccentricity)， th巴 failu巴 load may be reached either at material failure (typical for shorter columns) or at stability fa日間 (typicalfor more slender col -umns). Material failue is associated with crucking of the concret巴orexcessive yielding of the reinforec 巴-ment. When failure is due to instability， the material is not loaded to the limit of its strength This paper deals with the e任ectof creep on the behaviour of column subjected to sustained loading and proposes a rational approach to investigat巴the cr巴epstability failure of the column (unit rnrn)

2

5

2

5

25 25 (a) beam (b) co1umn E工E・1 Cross Sectional Dimensions of RC Mernbers ロ 一一一一一一一一一一一一一一一一」ー-o 0.1 0.2 0.3 円.4 0.5 0.6 0.7 E Strain ( % ) Fig.2 Stress-StraエnDiagram for Concrete Subュectedto Axial Cornpression

2. N司M-l/RRELATIONS WITHOUT EFFECT OF

CREEP

The numerical program for calculating N -M-l/R relations is almost the sam巴asthat for the inelastic

bending analysis of RC beams. The program devel -op巴din this study was asc巴rtainedby comparing the

results of RC beam bending tests. The cross sectional dimensions of specimens in the test ar巴shownin Fig.

1 (a) and the sam巴dimensions were used for the

numerical analysis. Fourte巴n cylindrical specimens

(D=100mm， h=200mm) of the same concrete mate. rial as used for beams were tested， yielding the stress.

strain diagrams shown in Fig. 2. The deformed steel bars used in the beams were all of the same steel grade with SD 35 (nominal yield strength 350Mpa) and size D 13 (nominal cross sectional area 1.267cm2)

Twelve specimens were used for the tensil巴test.The

idealised str巴ss.straincurves indicated in Fig. 3 were

applied to each test result in order to determine the material constants in the figures. These data were summarized in Table 1 and 2， in which M is th巴mean

average and S the standard deviation

In the calculation of N時M.1/Rr巴lations，the t巴n守

sil巴stressin the concrete members was neglected and

the law of reservation of plane was assumed. By first assummg curv且turel/R and the assigning a certain

value for one extrem巴日berstrainε1， the opposi te extreme fiber strainε2 can be found by trial， given that the axial stress resultant N has to be zero. The flow chart for th巴calculationsof N.M.1/R is shown in

Fig.4.

Numerical calculations for beams w巴reperform

ed using material data M and M士2Sfrom Tables and compared with the results of the beam bending t巴st.

The solid curves in Fig. 5 indicate numerical results and the round marks indicates th巴testresults， which

coincided with the M十2Scurve， being 6% above the M curve.Itmay be said that a close agreement was observed for the whole M.1/R curve b巴tweenthe test

results and those obtaied from calculation. One con ceivable reason for the slightly greater moment found in the test result might be the increasing strength of the steel bars embedded in the concr巴teof the beam

Zhang HENGPING， Tetsuhiko AOKI and Shunji INOMATA 144 σ 印 2UHUω σ Table 1 Results of Concrete Compression Test(百=14) M E ( X10-4 ) l/R M -l/R Curve by Test and Calculatユon Table 2 Resu!ts of Steel Bar Tension Test(日=12) εsu 66.3 0.0 i1+2S 日 Test

。

(x 工0 εcu Ec

。

3.06 13.8 O.口8 0.4 30 (x 10") εst 18.5 0.0 20 Curvature ~ 2.32 0.15 σsu l笠呈) 524 9.7 10 F工g.5 σsy i笠心 388 9.6 M(tf'm) M s 山 口 ω E 0 2 国 口 ﹂ 門 司 ロ ω 同 M 山 ハ U T よ 中 ム 官 μ p u p μ F U T M PL-nlupApu s R O R o F D S F E y n b y L S O D A L ロ PAr-MM 川 A M U -F U T上 D A 了 ム S γ し u p u v A A n F P T A + 0 P N V y J A 工 H C T U σ H L L H . G A A T_工 ム C 工 I I X A E X X A = 口 い い い A A A A M E D A ， •.. E ・ 1 u 悶 H N N N Ac， As CROSS SECTIONAL AREA CONCRETE AND STEEL BAR， RESPECTIVELY Calculatlon Flow Chart of N-M司l/RRelat工on Fig園 4

Nu=Ac.O:七y十As.σ' y [lJ

where the notation Ac and As ar巴us巴das indicated in

Fig. 4. N on-dimensionalized external axial force N / N u changes at step 0.1 from 0.1 to 0.8 in this example Although the maximum moment incr巴aseswith N /N u

up to 0.3， rotational capacity decreas巴smarkedly in

spite of the small addition ofaxial force. For axial force N/Nu beyond 0.3， both maximum moment and rotational capacity decrease In order to ta ke account the e百ectof creep on N-M-1/R relations， the stress-strain diagram for con crete is modi五edby multiplying the short-term strains by a factor (1 + q¥)， as shown in Fig. 7， where q¥ is creep coe日cient

The creep coefficientゆ (t，to) can be determined from

q¥(t， to)=βa(tol+q¥ d.βd(t-to) +φf[βf(t)βf(to)J 3. EFFECT OF CREEP ON N-1¥ιl/R RELATIONS

where: βa(tO)/Ec28=rapid initial d巴formationwhich is develop巴dduring the first day aft巴rthe load has been imposed， ゆd.βd(t-to)/Ec28=recoverable part of the delayed deformation (delayed elasticity) assumed to be independent of aging in its development and characterized by a constant value of the coe伍ci巴ntq¥d， φf[βf(t)βf(to) J /EC28 = irreversable delayed deformation 白(ow)which is very much affected by the age at which loading commences

tす to二 age of concrete at the moment under consideration and of loading， respective -ly，

βd，βf = function corresponding to the develop -ment with time of the delayed elastic strain and the ftow strain， respectively The creep coef五cientdepends upon the ambient humidity， the dimensions of the elem巴nt，and the time

t and to. In order to allow the recording of the model in the computer calculation， analytical expressions shown in the “CEB Manual on Structural Eff巴ctof

Time -dependent Behaviour of Concrete (1984)川 )ar巴

adopted in this paper

A computed M-1/R diagram for a given normal force is shown in Fig. 8.

The column is divided in a finite number of elements， and the deftection of巴ach巴lementis approx

-imated by a circle. The curvature of segment is 4. ITERATION PROCEDURE FOR DETERMINA-TION OF STABILITY Example of numerical calculation of N-M-1/R relations for column sections (Fig. 1 (b) Ac二 400cm'， As二 1Z.16cm2)are presented in Fig. 6. The material constants used in calculation are the same as those used for the b巴am.The basic axial force N u in Fig. 6 is so chosen that the uniform strain CCy occurring in the member exactly corresponds to maximum con -crete str巴ngthσCy.At this moment the stress in the steel bars becomes the yield strength σSy. N u can then be expressed as in Eq.(1) ) D ι e 1 ノ e E r j 円 C 工 、 r r

、

t b 一 ¥ t 、 ヤ ム 唱 q 、 ¥ 河 川 、 ¥ ¥ A L 口 ¥ L F ¥ 、 / V M ¥ E

o

10 2.0 30 l/R Fig. 8 Example of N-M-l/R

Curves Tak工ng into

account of Creep(t=∞)

St士ess - Strain Diagram

of Concre七eAffected by Creep Curvature Fi呂.6 N -M -l/R Cu工vesfor Various Axial Force N/Nu (N~N~

。

:

口

2

、

e = 5 cm .

「ア二

St士a工n 工nstantenuous / / / 町 六 / ノ

Eσr-

i¥'ー/一 円 i

ム

/ '

の 1 / ~ I

1

1

ι一ームー」 O

J

E大(1+中) 7 日(七f'm) 3 2 F工E・ い ] 口 ω E O 呂 田 口 ﹂ 門 司 口 ω 同 H 3 σ _cy 以 ロ ω 目 。 Z 回口同甘口 ω 同

146 Zhang HENGPING， Tetsuhiko AOKI and Shunji INOMATA determined by interpolation， finding the point on the previously calculated N -M-1/R curve that corre -sponds to th巴externalmoment at the center of the segment. Fig. 9 shows one part of the column deformed to an approximate arc with local coordi nate x-y. The global coordinates for the column are 巴xpressedas X and Y. Using coordinates (Xi， Yi) to de白nepoint A on the one end of the arc， and coordinate (Xi十ムXi，Yi十

ムYi)to d巴自nepoint B sep旦ratedfrom A by the arc lengthム1，we find th巴followingrelation ; ] ' 斗 [ 、_{. 1} 1 } l i_， r i f 1 1 1 1 1 ' b o ん 八 ω r h l t 喝 、 I l l_・ -1 1 1 1ノ αl l α r 1 1 1 1 L 一 、₁ t 1 7 1 E J

。

b A r ' t ' J 4 、 l ' l l、 、 l I B -' S / α α n s a 工 O S C α αE S 工 O B c -f 1 l i l l -、

一

-j

xy

A U 八 ハ ] r i l l J J 、_{1 1} 1 1 1_、 where b is the x-coordinate of point B on the x y axis AssigningL ABCニ

e

and th巴radiusof curvature R，。 and b can be expressed as follows :

。

二

6.1，(l/R)， b

二

6.1・e/2=(ム1)'(l/2R) [5J Therefore， if the curvature l/R of this arc is obtainable from the N -M-1/R r巴lation，b may be determined from Eq. (5) andムX，ムY from Eq.(4)

Assembling the segment into the column again in such a way that the inclination on the tangents of adjacent segments becomes巴qual，the shape of the column can be determined. The external moment at each segment may be calculated from the column's deft巴ctionand degree of eccentricity， and the corre

sponding curvature l/R may be found from the N-M-l/R curves. Since each segment has a slightly di妊巴

rent curvature from the form巴rstage， the column's shape after reassembling will also become slightly di任erent.Repeating this procedure， the column's shape will tend to an converge to an equilibrium position 占(cm) F 斗 1 J 口 O J 明 以 U ω J [ 刷 ω 口 Y y (Xi+ムXi，Yi+ムY工) ィP A

，

I (Xj_

、

:

;

-

;

I ノ / l l t .¥"

。

_X F工E・ 9 Calc:ulat工onof Column Deflec工on 5. NUMERICAL EXAMPLES

Using above mensioned procedur巴， the column

deftectiono versus time t relation is calculated. Fig 10 shows one example of numerical analysis for a RC column with the cross section illustrated in Fig. 1 (b)，

lenght 3 m and eccentricity e=5cm. Wh巴naxial force

N /N u was changed from 0.2 to 0.3 by step 0.02， it was found that at N/Nu二 0.26 the column remained stable， but that at N /Nu二 0.28it b巴cameunstable Accordingly， N/Nu was divid巴dinto four values in

this region and theo - t relation was onc巴agam

investigat巴d，giving the broken lines in Fig. 10

From these results， the critical axial force N/Nu of this column may be assumed to lie betw巴en0.270

and 0.275. 1n this way， if the relation between time and column de司ectiondependant cre巴pis calculated

and the maximum sustainable axial load of a column with eccentricity can be ascertained STABLE 巴 =5cm N/Nu=0.2 500 1000 Time(days) 1500 2000 t J u u n a n o p ・ 1 n h o s e -n 十 ﹂ 円 U c ・ l e t

- a

5 L 1よ e E D R n ↑ ι m u E l m

o

-戸し巾ム η u ー ム σ 白 ' l 門 E

6. CONCLUDING REMARI五S

The e佐ectof creep on the stability of reinforced concrete columns has been analyzed. by a rational approach. The method assumes that concrete creep strain is proportional to stress under warking load conditions and the stress-strain diagram at time t can be obtained by modifying the short-term diagram which strain values are multiplied by (1

+

<t).It can be used with any chosen functions expressing the time variation of concrete creep.

The column examples analyz巴dshow that ac

curate evaluation of stability of reinforced concrete columns cannot be done without accounting for th巴

巴任ectof concrete creep

References

(1) Kato. M， Aoki. T， Fuwa. A，"Experimental study of ultimate bending bearing capacity of RC continu -ous beams"， Annual Sympo. of ]SCE Chubu Branch，

March. 1985

(2) CEB Manual on Structural Effect of Time dependent B巴haviourof Concret巴，1984