Indian Journal of Engineering & Materi als Sciences Vol. 7, February 2000, pp. 29-34

Long-term deflections of reinforced concrete beams

Terezia Niirnbergerova, Martin Krizrna & Jan Hajek

Institute of Construction and Architecture, Slovak Academy of Science, Dubravski:i cesta 9, 84220 Bratislava, Slovak Republic

Received 14 April 1999; accepted 30 December 1999

The effects of different long-term loading levels beyond cracking limit on the defl ections of the reinforced concrete beams are investigated. Six identical beams were subjected to a sustained load at three levels over a period of about 200 days. The arrangement of measurements enabled the separation of the effect of bending moments from that of shear forces. The results show that the defl ection curves at the two arbitrary time intervals are similar in spite of intensive crack development. The linear relationship between the initial and time­dependent deflections can be assumed . This linear relationship also applies for the shear-induced deflections.

The serviceability of concrete structures has been one of a major concern of the des igners for a long time. Its importance is increas ing with the avail ability of hi gh quality materials (concrete and reinforcement). The higher quali ty and the more precise calculations as well , result in the des ign of more slender and therefore more economical structures. However, the reduction of the sti ffness of a structure directly influences its deformat ions. From thi s point of view, the practical importance of the correct prediction of the deflections is indi sputable . As the long term contribution to the total deflecti ons exceeds the short­term contributi on, the accuracy of the pred iction of the defl ection is primarily influenced by the accuracy of the determination of the long-term and/or cycle dependent deflection s 1,2 .

Verificati on of the methods of calculati on requires a suffic ient amount of experimental data . Therefore, attention has been paid to the investigations on time­dependent de formations under long-term sustained load and the deformations under repeated load as well. Investigations on these deformations are reported by Corley and Sozen" Dilger and Abe le4

,

Jaccoud and Favre5, Pitonak6

, Ding Daj un et af. An increase in the deflecti on of an e lement

subj ected to the sustained load is influenced mainly by: the creep of concrete in compress ion ; time­dependent changes in tension stiffening; deterioration of bond between the concrete and reinforcement ; the shrinkage of concrete; increas ing crack widths and formation of new primary cracks. The analytica l mode ls tak ing into account all of these processes are, as a rule , very complicated and inconvenient for

practical calculations. These mode ls often require detailed information as input data which might not be known at the time of the design of a structure, and give results which sometimes differ from those obtained in tests.

According to Corley and Sozen3, the total

deflection of a reinforced concrete beam may be divided into three components, viz.: defl ection resulting from instantaneous strains, deflection resulting from creep strains, and defl ection resulting from shrinkage strains. Thi s can be expressed by Eq . ( 1 ):

.. . ( I )

where a' tll is the total defl ection in a di stance x from the left support of the beam, aill is the initi al deflection in the same point at the time of load application , a sh is the defl ection due to shrinkage, f31 is the multiplier of the initial defl ection giving the deflection increment

due to creep. If coefficient f31 does not depend on the load level at a given instant, then Eq . ( I ) represents a straight line. In this case, the hypothesis of linearity of deflections can be assumed. To verify this hypothesis, inves tigations8 on the long-term loaded s labs are reported from the Institute of Construction and Architecture in Brati s lava, confirming the assu med linearity of deflections of s labs. But the question about the long-term behaviour of the beams wi th flanged cross sec tions has remained unresolved.

Therefore, the tests on beams subjected to susta ined load with different load levels have been carried out. The dimensions of the beams are devised in such a way that the shear-induced deflect ions can

30 INDI AN 1. ENG. MATER. SCI., FEBRUARY 2000

not be neglected. A method of measuring the deformations enabling to separate the bending- and shear-induced deformation s from the total deformations has been used.

Experimental A series of six identical reinforced concrete [

beams was subjected to the long-term load . The only vari able parameter was the intensi ty of the sustained load. The cross-section of the beams is shown in Fig. I and the elevation in Fig. 2. The overall length of the beams was 4.15 m and their span was 3.6 m. The aggregate of the river alluvia was used for the concrete. The particle size distribution for 0-4 mm, 4-8 mm, and 8- 16 mm was 720, 300, and 725 kg, respectively, that is 41, 17, and 42% by weight. Portland cement PC I 42.5 in the amount of 470 kg.m-3

was applied and the water to cement rati o was 0.4. The average va lues of concrete properties at the time of testing are given in Table I . The beams were

2.010 /I;. easured points (1 0425) / _t ... i~

11016

I. 240 I, ..f "

o co ...:r

Fig. I-Cross-seet ion of the beams

reinforced in the tension zone with 11 hi gh-bond deformed bars (0 = 16 mm). The stirrups were made of high-bond wire (0 = 8 mm, spacing 180 mm). The average properties of both reinforcements were: yield stressj,y = 491 .7 MPa, tensile strength Ir = 582 MPa, modulus of elasticity E, = 199 GPa.

The beams were placed in pairs during the long­term tests. The concentrated force in the middle of the span of a beam was produced by the hydraulic jack with (he incessant check on the value of the loading force. Three levels of sustained load were chosen on the bas is of the preliminary short-term test, namely 0.35 (beams A), 0.50 (beams B), and 0.65 (beams C) of the short-term failure limit which was 500 kN. There were two beams loaded on each level. All the beams were loaded gradually using deflection rate control until the required loading leve l was reached except the beam B I where the simulated moving load9

was applied. The defl ections in five poi nts over (he span and

strains of the continuously linked-up measurement bases at the compressed and tensioned edges, as well as at the crossing diagonals were registered and cracking was observed durin g long-term loading at regular time intervals. The bases shown in Fig. 2 form a "truss" consisting of "struts" and "ties" enabling the calculation of the deflections by using a method based on Williot-Mohr translocati on polygons. This method offered the possibility of separati ng the deflecti on due to shear from those due to bending.

Results and Discussion The relationship between the short-time deflections

and the loading force obtained during the gradual loading to the required load-term loading level is shown in Fig. 3. In thi s figure the resu lts of al l tested beams except the beam B I (a mov ing load has been

F

>/ I Y /' I / ,

',/ I ,/ 1)/ '- ./ ;\ T, I" Y

/ '. I /, /" / "

Jt.. _ ___ ~---~ ____ ~--- ----~-----~---~- --

~ X 36p

3600

4150

Fig . 2- Side- view of the beams; a-network of blses for mechanica l measurement o f deformation s

NURNBERGEROV A el al.: LONG-TERM DEFLECTIONS OF REINFORCED CONCRETE BEAMS 31

350

300

, .& I "

; • . / *

250

Z 200

~

... 150

100

50

0

x ... x · .. x- .--x

- ~~)(.xX . • j«

/ • . - ,'_. :){)t . .... j .. •...

X a,tot,exp . a,n,exp . 8,sh,exp -;- ...... ...... a,l o1,STN

_ .. .. - ll ,fI ,STN

--"'T -o r ..... - -a.n.Ee2 - - - a,sh,STN

0 4 6 8 10 12 14

E S U> <!

a" (mm)

Fi g. 3-Shorl-tim~ defl ection vs loading force rel ationship

I

-12 82

-O.B

-0.4

0.0

0,4

. ', :: . ~ ::.., .. ~

: :: a :: : >:~ : :-:-~-·. ~

10

-0- 1= 0

.- . -. -.- t= za

.... 0.... -0--1=207 --0 - 1:::498

O.B -'---'-------~----------

ri g. -I- Deformati ons along the comp ressive (das hed lines) and tensioned (full lincs) zones o f the heam 82

1.6

82 1.2

08 •. &: E

..... /:: .. o ' s 0.< .. /. -<!

00

-0.4

-08

::~ .' "' .....

.-' _0 - :

.. 0 -·1 .... . · 3

: ~:> . .. : :: ~ .: ~ • 0 .• ····.:: ..•

····. · :&::1 ' Q .~.

"\ '.

\~:'~ I

10 t1

Fig. 5- Deformations in the crossing d iagonals of the hea m 82: descending diagonals fro III the left end o f the heam (dashed lines); ascending diago nals ( ful l li nes)

E .S-

15 ;; ..

18

16

14

12

10

8

6

2 o t= 0

.. t· 28

• t = 207

•

o+---~--~--~--~----~~

6 8 10 12

ain,lol (mm)

Fig. 6-Relati onship o f the total time-dep~ndent deflecti on G'um

vs total init ial deflect ion Gill.IO'

4

a in,sh (!TIm)

• ····· i

·4/ . / .ilil

o t = 0

.. t = 28

• t = 207

Fig. 7-·Relationship of the shear-i nduced time-dependent deflection a,,..,, vs shear-indu ced ini ti al deflecti on aill . .<"

Table I- A verage values o f mechanical properties

Beam

A I A2 B I 82 CI C2

Age

(days)

30 40 31 4i{ 42 53

Cube strength

;;'11/ (MPa)

55.94 58.04 54.34 6 1.89 6 1.84 60.06

Prism Modulu s Tens ile strength o f elasticity strength

fflll! £, 1;, (M Pa) (GPa) (MPa)

47 .56 44.92 2.43 39.70 39.31 2.48 47 .60 35.83 2.40 54.12 .HU6 2.56 49.67 38.7X 2.56 50.94 38.31) 2.52

32 INDIAN 1. ENG. MATER. SCI., FEBRUARY 2000

applied in this beam) are plotted . It can be seen that the shear-induced deflections (full triangles) by higher forces starting at the loading force F = 150 kN are approximately one third of the total deflections (crosses). The dashed lines are the deflections calculated according to the Slovak Standard lO

, the full line represents the bending-induced deflections calculated according to the ENV II

. The deflections have been calculated using the average values of materials properties.

The deformations at the compressive (dashed lines) and tensioned (solid lines) zones of the beam B2 are plotted in Fig. 4. The figure shows similarity (proportionality) between the deformations measured at the time of the sustained load application and those after some period of the duration of the load (28, 207, and 498 days in the figure) . Fig. 5 depicts the deformations obtained by measuring the crossing diagonals where the dashed lines represent the descending diagonals from the left end of the beam B2, and the solid lines are for the ascending diagonals . Also, the shear-induced deformations show the similarity between their initial and time-dependent va lues.

Fig. 6 shows the relationship between the time­dependent total deflection aluOI (after 28 and 207 days of loading) and the initial total deflection ail/,/III at the application of sustained load. Every point on the graph represents a pair of values (ail/.to" aIUIII) of all the deflections (i n tenths of span) resulting from the calculation usi ng Williot-Mohr translocation polygons for all the beams. It can be seen that all the points lie practically on a straight line no matter to what spot and loading level the points belong. Similar

Table 2-Paramcter hi of straight line regression of the time­dependent deflections vs initi al dellection

Deflections

Total deflection

Bending- induced detl ection

Shear-induced detlection

Time of loading

(days)

28 207 28

207 28

207

1.2504 1.4634 1.2415 1.4432 1.2682 1.5082

Table 3-Parameter c of the time fUllction of the coefficient /3 Detl ecti on Parameter

c

{/IO/ 0.746

a j7 0.7 12

a xh 0.82 1

relationships for the shear-induced deflections are plotted in Fig. 7 and could be drawn also for bending­induced deflections .

It follows from the above mentioned evaluations that the arbitrary values of the initial deflection a in

depend linearly on the values of air. i.e. the long-term deflection belonging to them. This means that the initial deflection curve and the deflection curve after time t of sustained loading are affine-similar. Since this affinity exists between the initial deflection curve and the deflection curve at the arbitrary time interval , it holds between the two arbitrary time intervals, too . It should be emphasised that this affinity exists despite of the time-dependent development of cracks. The variations of coefficient hi of the straight line regression is given in Table 2. It is evident that the coefficients of the straight line regression of the total (bending- and shear-induced) deflect ions do not differ substantiall y.

Fig. 8 shows the variation over time of the increment of the coefficient hi of the straight line regression (the creep factor). The symbols represent the experimental value of total (crosses), bending­(full circles), and shear-i nduced (triangles) deflections, and lines express the regression functions given by the Eq. (2):

f31=C{I-ex p[-((t2-tl)/(tr)J2J} ... (2)

where tl is the age of the concrete at the sustained load application, t2 the age of the concrete in the time considered, and tr is the retardation time l 2

. The Eq . (2), with the retardation time of Ir = 200 days is the basic formula for calculating the strain increase due to creep according to the Slovak Standard 10. The same retardation time has been appli ed for the regression curves and seems to approximate the test results quite satisfactorily . The parameter c for all the deflections is given in Table 3. These results confirm the hypothesi s that the deflections at the arbitrary time

. 0.6 ..---~-----------;------,

0.,4

0. 0.3

0.2 -. .. -r-- . --- ---: x 5a.tot

0,1 .----.--- --- ... -- - .- , . ... -. .... ,.. i'·· . BR,n - --

Q. o i-----+---~--~----T-----j 50 100 150 200 250

Time (days)

Fig. 8- Time development of the increment of the creep coefficient /3,

NURNBERGEROV A el al.: LONG-TERM DEFLECTIONS OF REINFORCED CONCRETE BEAMS 33

1.0 ,---------,-----; .-----,------,

i , Q.B --~~.--. ~ .. -.j. - ---- - ... ~-~. _HH ___ • _ _ ".HH. _ "i-·---------t-

.:..~ 6 ... :\~&: .. ~_=~.: .. .:.c.~~\.·.A!t.'_~ • .1' . . .... _ ... -... , .... , ; f

li 0 6 . ,,­

~.

_ • L .. ~ __ _

,. 0.4 . - .. ~ _ _ __ ... J _~_ . ,

r-b_M-A~ .• 6~~1~r-~~"-~~,3.: 0, 2 · .. - - - • _ .. Affll1'l.ot -6- A fsh/ftot

, • B2 tfIIfIot -0- 8 2 fsMloI J _ . _ emm.et -0-- C fsMto1

0.0 +----..----...-----r--~.c:.;....,.__--'-="__l

o 50 100 150 200 250

Time (days)

Fig. 9-Time development of the quoti ents a,(a,o, and a,/a,o'

interval of the sustained load depend linearly on the initial deflecti ons.

As the parameter c of the Eq. (2) in Table 3 does not differ significantl y for the indi vidual defl ecti ons, it implies that the ratio of the shear-induced deflec ti on to the total deflecti on does not depend on the time of durati on of the sustained load. This can be seen in Fig. 9, where the time development of the quotients apia/fl/ and a ,Ja'fI' is shown. In thi s figure, the average va lues of the quotients of the bending- and shear­induced defl ect ions to the total defl ecti ons are plotted, respecti vely. It is evident, that these quotients do not change with time and are not influenced by the load level (all the load levels, viz. 175 kN, 250 kN and 325 kN were above cracki ng limit).

Another questi on connected with the long-term behav iour of rein forced concrete elements is the position of the neutral ax is. It is assumed that the neutral ax is lowers with time 13

·' 4

. Our observat ions during the long-term tests show that cracks are not closed with lime; on the contrary their widths grow and new cracks occur, imp lying thereby that the neutral ax is does not drop. Thi s phenomenon can be explained by the assumpt ion that creep takes place not onl y in the compressed but in tens ioned fibres , as well as in bond between the steel bars and concrete. The measurement of the deformations of the continuously lin ked-up measuremen t bases at the compressed and tensioned edges enables the calcul ati on of the depth to the neutra l ax is. The ti me deve lopment of the relat ive dep th to the neut ra l ax is is shown in Fi g. 10. The average va lue of the re lative depth to the neutral ax is of the two beams with the same leve l of sustained load at the mi dd le of the span is plotted in the Fi g. 10. It should be noted that the measured deformati ons contain a lso deformations due to shrinkage. The shrinkage induced deformat ions increase the deformati ons on the compressive edge, but decrease those on the tensioned edge. To thi s phenomenon, the slight drop

0.0 ,---,----,~~-~--...--~-_,_-,- -~--,

0.2 -...--- - '-f--··~-- · - "--

O.B -. - • • A

" 0 ·· B

-+-C 1 .0.L--'---------'------~----'

o 50 100 150 200 250 300 350 400 450 soc Time (days)

Fig. la-Time development or the relati ve depth to the neutral axis

of the relati ve depth to the neutral axi s un til approximately 200 days of loading observed in Fig. 10 could be prescribed.

Conclusions The reported in vesti gati ons show that the init ial

and long-term defl ec tions of rein fo rced concrete beams can be assumed to be dependent. The deflecti on curves of beams at the arbitrary time interval are affine similar. The rati o of the shear­induced deflections to the total defl ections does not depend on time. This leads to the conclusion that a simplified method for calculation of defl ecti on applying the assumption of the linearity of deflecti ons is poss ible.

Acknowledgement

The authors are gratefu l to the Slovak Grant Agency for Sc ience VEGA (Grant No 2/4086/99) fo r parti al Sltpp0l1 of th is work.

Nomenclature

am =initial deflection due to sustained load a" =Iong-term defl ecti on due to sustained load ajl =bending-induced deflccti C' n a,." =shear-i nduced de fl ecti on a:o, =total defl ecti on h, =coeffi cient or the straight line regression (creep ractor [3) c =parameter or the creep rJctor [3 l:', =Illodulus or elas ticity or concrete E" =Illod ulus or elas tici ty or steel F =Ioading force ];.", =cube strength ur conc rete ];., =tensil e strength or concrete 1"", =pri sm strength or concrete !.,. =yield stress or steel !.~ =tensi le strength or steel 1 =time I , =age of concrele at the appli cation of the ustained load I ] =age of concrete in the time considered I ,. =assumed retardation time (200 days) xIII =relative depl h to the neutral axis [3, =Illu ltipl ier of the in it ial defl ection ; creep factor

34 INDIAN J. ENG. MATER . SCI. , FEBRUARY 2000

References I Beeby A W, Mag Concr Res , 26 (1974) 16 1. 2 CEB , Bull " 1{onn , No. 235 ( 1997) I. 3 Corley W G & Sozen M A, ACI J, 63 ( 1966) 127. 4 Dilger W H & Abele G, in Deflection of concrete structllres

(AC I Publi cati ons) , SP-43 ( 1974) 487. 5 Jaecoud J P & Favre R, AnI/. ITBTP, No. 406 ( 1982) 66. 6 Pitonak A, Staveb. Cas, 40 ( 1992) 16 1. 7 Ding Daju n et a l. Staveb. Cas . 33 ( 1985) 489. 8 NOrnbergerov,l T & Hajek J, Constr Bllild Mater, 8 ( 1994)

147. 9 NOrnbergerova T , Krizma M , Bolha L & Shawkal S,

In dian J Eng Mater Sci, 5 ( 1998) I I I .

10 STN 73 1201. Design of concrete stmctllres, Praha, UNM 1987 .

II ENV 1992-1- 1: I : European prestandard: Design of structures--Partl: General rules and rules for bllildings , European Committee for Standardization , Bru ssels, 1991.

12 Neville A M, Dilger W H, & Brooks J J, Creep of plain and strLlc/ural conere/e, (Constru cti on Press , London ;md New York), 1983 .

13 Gilbert R I, Tilli e effecl.~ in conerete st rUC/ llres, (Elsevi er, Amsterdam-Oxfod-New York-Tokyo), 1988.

14 DebernardiPG , JStruct Eng, 115 ( 1989) 32.