Reprinted from

Canadian I Journal of

Civil Engineering

I

Transverse reinforcement effect on cracking behaviour of R.C. members

S. H. RIZKALLA, L. S. HWANG, AND M.EL SHAHAWI

Volume 10 • Number 4 • 1983

Pages 566-581

/

1+ National Research Conseil national Council Canada de recherches Canada

Printed in Canada by K.O. Campbell Corporation

Reimpression du

Revue canadienne de genie civil

566

Transverse reinforcement effect on cracking behaviour of R.C. members

S. H. RrZKALLA, L. S. HWANG, AND M. EL SHAHAwrJ

Department of Civil Engineering, University of Manitoba, Winnipeg, Man., Canada R3T 2N2

Received December 3, 1982 Revised manuscript accepted May 30, 1983

Two extensive and independent experimental programs have been conducted to study the cracking behaviour of reinforced concrete members subjected to pure tension in the presence of transverse reinforcement. The first program involved the testing of eighteen reinforced concrete segments and was mainly designed to examine the applicability of the existing equations for predicting crack spacings and widths. The segments were reinforced in two directions and loaded in uniaxial tension beyond the yield stress of the steel. The measured average values of the final crack spacings were compared to the values presented by other researchers. Based on this comparison, a simplified and refined expression for prediction of crack spacing is proposed.

The second experimental program involved the testing of sixteen reinforced concrete segments, which were divided into two groups with different concrete covers. Within each group, all segments were identical in all parameters, except the spacing of transverse reinforcement. The program was designed to study the influence of transverse reinforcement spacing on crack behaviour. A methodology including proposed expression for predicting the crack spacing in reinforced concrete members subjected to axial tension with variable transverse reinforcement spacing is presented.

Keywords: cracking, crack spacing, crack width, membrane forces, reinforced concrete, tension, transverse reinforcement.

Deux programmes experimentaux independants importants ont ete poursuivis afin d'etudier la fissuration d'elements en beton arme soumis a une force axiale de traction et possedant un renforcement transversal. Le premier programme cciinprenait I'essai de dix-huit echantillons en beton arme, calcules de faISon a analyser I'applicabilite des relations existantes predisant l' espacement et la largeur des fissures. Les echantillons etaient renforces dans les deux directions et charges uniaxialement en traction en dessous de la limite d'elasticite de I'acier. Les valeurs moyennes des distances entre les differentes fissures furent comparees a celles obtenues par d'autres chercheurs. A partir de ces resultats, une expression simplifiee et rafinee est proposee pour la prediction de I'espacement des fissures.

Le second programme experimental comprenait I'essai de seize echantillons en beton arme separes en deux groupes comprenant differents recouvrements de beton. Dans chaque groupe, tous les echantillons etaient identiques 11 l'exception de I'espacement des armatures transversales. Le programme fut con,<u afin d'etudier I'influence de I'espacement des armatures transversales sur la fissuration. On presente une methode et une expression pour la prediction de l'espacement des fissures dans Ie cas d'elements en beton arme comprenant un espacement variable des armatures transversales et soumis 11 une force axiale de traction.

Mots-cles: fissuration, espacement des fissures, largeur des fissures, equilibre des membranes, beton arme, traction, renforcement transversal.

Can. J. Civ. Eng. 10, 566-581 (1983)

Introduction For many reinforced concrete structures, the problem

of tensile cracking is almost unavoidable. However, there are certain types of structures, such as tanks, silos, and nuclear reactor containment structures, where ten­sile cracks can pose serious problems. All such struc­tures are generally reinforced in two directions and mainly subjected to membrane tensile forces. There are many research reports on crack behaviour in reinforced concrete members. However, internationally, there is remarkably little agreement on theories or equations for predicting crack width and spacing. In addition, most of the experimental work to date has tended to concentrate on the behaviour of members reinforced in one direc­tion only. Very little work is available on crack behav-

'Present address: Civil Engineering Department, Queen's University, Kingston, Ontario.

[Traduit par la revue]

iour of members reinforced in two directions, which represent most' of the construction details of any structure.

To study the cracking behaviour of reinforced con­crete members subjected to pure tension in the presence of transverse reinforcement, two extensive and inde­pendent experimental programs, sponsored by the Natural Sciences and Engineering Research Council, have been conducted at the University of Manitoba. The first program involved the testing of eighteen rein­forced concrete segments and was mainly designed to examine the applicability of the existing equations for predicting crack spacings and widths in the presence of transverse reinforcement. The segments were rein­forced in two directions and loaded in uniaxial tension beyond the yield stress of the steel. The main parame­ters varied were the amount of concrete cover (c), con­crete thickness (t), and percentage of steel (p), as given in Table 1. For each specimen, strains, crack width, and

RIZKALLA ET AL. 567

TABLE 1. Variables considered in the experimental program

Concrete Concrete X-section Reinf. Transv. cover Steel thickness Specimen area bar bar spacing (in.) ratio (in.) number (sq. in.) size (in.)

First experimental program

5.0 TlA 60.0 #3 3.0 0.50 0.0147 7.0 T4A 84.0 10M 3.0

10.0 TIA 110.0 #4 3.0

5.0 T2A 60.0 10M 3.0 0.0207 7.0 T5A 77.0 #4 3.0

10.0 T8A 120.0 15M 3.0

5.0 T3A 55.0 #4 3.0 0.0294 7.0 T6A 84.0 15M 3.0

10.0 T9A 120.0 #6 3.0

5.0 TlB 60.0 #3 3.0 0.75 0.0147 7.0 T4B 84.0 10M 3.0

10.0 TIB 110.0 #4 3.0

5.0 T2B 60.0 10M 3.0 0.0207 7.0 T5B 77.0 #4 3.0

10.0 T8B 120.0 15M 3.0

5.0 T3B 55.0 #4 3.0 0.0294 7.0 T6B 84.0 15M 3.0

10.0 T9B 120.0 #6 3.0

Second experimental program

2A, 2B, 2C 2.0 4A, 4B, 4C 4.0

0.75 0.0147 7.0 6A, 6B, 6C 84.0 10M 6.0 #1 8.5 #2

#3 2.0 #4 4.0

1.50 0.0147 7.0 #5 84.0 10M 6.0 #6 10.5 #7

NOTES: 1 in. = 25.4 mm; 1 sq. in. = 645.2 mm2•

crack spacing were measured during testing. The mea­sured values of the final crack spacing were compared to the values based on equations presented by others (Beeby 1972, 1979; Leonhardt 1977).

The second experimental program involved the test­ing of sixteen reinforced concrete segments which were divided into two groups with different concrete covers. Within each group, all segments were identical in all parameters, except the spacing of transverse reinforce­ment, as given in Table 1. The program was designed to study the influence of transverse reinforcement spacings on crack behaviour. The proposed expression based on the first part of the study was applied and compared with results obtained from both experimental programs.

Test specimens

All segments in both programs were reinforced in two directions with deformed bars. Longitudinal rein­forcement was spaced at 3 in. (76 mm) centre to centre and extended 11 in. (280 mm) beyond each end of the specimen, as shown in Fig. 1. For the first program transverse reinforcement was provided by 10M bars spaced at 3 in. (76 mm) centre to centre on both faces for all the specimens.

In the second program, the sixteen segments were divided into two groups, with different concrete covers of 0.75 in. (19 mm) and 1.5 in. (38 mm) respectively. Within each group the selected spacings of 10M transverse bars were 2 in. (51 mm), 4 in. (102 mm), 6 in. (152 mm), 8.5 in. (216 mm), or 10.5 in. (267

(Jff_ ·c

,~; ,

RIZKALLA ET AL. 569

LOAD CELL BRACKET WASHER / SPACER

STRAIN GAUGES

LOAD CELL SOCKET

THREADED END

1----- STEEL REBAR

- •• - - --- DEMIC POINT

(203 mm) 8"

(203mm) 8" - - --- REINFORCED CONCRETE SPECIMEN

(203mm) 8" LVDT

FIG. 2. Test set up and instrumentation.

in the "air-dry" stage for at least an additional 21 days before testing.

Testing apparatus and procedure The loads were applied by using a 600 000 lb

(2670 kN) capacity universal testing machine. Load was transmitted from the loading machine to the speci­men by using specially designed end fittings, as shown in Fig. 2. The end fittings were very rigid in order to achieve uniformity of loads from the machine to each load cell. Each reinforcing bar was connected to a sep­arate specially designed load cell, which was used to adjust the load transferred to each reinforcing bar to provide uniformity of forces applied to the segment. The uniformity of loads in each reinforcing bar was achieved by tightening and loosening the load cells. After balancing the loads, testing of the specimen com­menced by applying an initial load of 3000 lb (13.34 kN) to ensure proper seating of the segment. This load was considered the "Zero" load condition for the test. The load was then applied in increments of 5000 lb (22.24 kN).

During testing, a continuous record of deformation over a 30 in. (762 mm) gauge length was made using a linear variable differential transducer (L VDT) (Fig. 2). Readings for mechanical strain gauges were also recorded. In addition, the number of cracks and their

widths were measured at regular intervals with a travel­ling micrometer-microscope, with an accuracy of 0.02 mm (Fig. 3). Two lines were marked on the seg­ment surface to measure the crack widths. One line was located directly over a reinforcing bar and the other midway between two adjacent bars. The widths of all cracks crossing these two lines were measured. The crack patterns were marked and numbered at the end of each increment. Testing was terminated when the load approached the yield point. The specimen was then unloaded and the final crack pattern was noted. The crack spacings were measured along the central line of the segment surface.

Governing equations The mechanism of cracking of axially loaded rein­

forced concrete members is suggested in many studies to depend on the bond failure between the concrete and steel (Beeby 1979; Goto 1971; Leonhardt 1977; Saligar 1936). As a different approach to the problem, Broms (1965) proposed a cracking mechanism based on an elasticity analysis of concrete stresses. He concluded that the crack spacing is directly related to the concrete cover (c).

Goto (1971) observed from his test results that when the member is subjected to tensile forces, internal cracks will occur behind the ribs of the deformed steel

568 CAN. 1. CIV. ENG. VOL. 10. 1983

12. 0" ---"~I (305mm)

t ...... • •. ~ ,·-···:·,······"'T

7..I~.I·::;·4·:i·: •• ::- ,' •.• 1.' ... ·.r.·

~!"; ." T J' ~.. • A: #,~ ,"

" I/. ..... . (76 mm) 3 : ... ;'.: ... ~

.1 .'.:1, ....... ',' J..-:!;:.' •...

::t;:":'. ''',' •..•

?~ :~~.:~;~~ ~.' ~ •• ,.\ ~l""

~ 3" ~ (76mm)

, A

,

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

(

30" (762 mm)

(

,_ ....

-..Jj..- d #3, IOM,#4, 15M or #6

FiG. 1. Reinforcement details of a typical specimen.

nun) and additional specimens without transverse reinforcement.

All concrete used in fabrication of the segments was designed for a nominal ultimate strength of 5000 psi (34.5 MPa) and was mixed in the laboratory. For each segment, six standard 6 in. x 12 in. (152 x 305 mm) cylinders were cast and tested to determine the com­pressive and tensile strength of concrete. The reinforce­ment used in the segments consisted of hot rolled de-

formed bars graded 60 (400 MPa). For each specimen, samples of the steel bar were tested for tension to deter­mine yield stress, ultimate stress, and modulus of elas­ticity. All samples showed typical ductile behaviour and well defined yield point. After casting,the speci­mens were left to dry in open air for about 1 h, then placed in a curing room for 7 days. Following this 7-day curing time, the specimens were removed from the curing room and covered with wet burlap and remained

in Ix

(2 Wi

m. in ac 10. an aq pn Th acl Af me (C Th the SOl

1 Ove line Rei rec

570 CAN. J. CIV. ENG. VOL. 10, 1983

FIG. 3. Typical specimen during testing.

bars, as shown in Fig. 4, and they are influenced by the diameter (d) of the reinforcement bar. After the for­mation of these internal cracks, the axial tensile force is carried by the uncracked shell of the concrete section. The small concrete teeth will resist the interlocking forces by flexural resistance. As one or two of the concrete teeth lose their resisting strength, almost no bond length (Lo) will extend equally each way from the crack (Fig. 4).

Leonhardt (1977) showed that when the concrete stress due to the applied load reaches the tensile strength, the concrete starts to crack, causing a sudden jump in steel stress. If this sudden increase in the steel stress is large enough, some bond-slip can occur, and the crack formation is likely to be a combination of bond-slip and internal cracking, as suggested by Goto (1971).

Leonhardt (1977) assumed the minimum possible value for the average crack spacing at the final crack pattern (SL) as:

La = Region of Almost Lost Bond

I" 'I

FIG. 4. Internal cracks at bar deformations.

[1]

where L, is the transfer length, which represents the length of active bond stress, and Lo is the length of almost no bond stresses. Based on experimental results, Leonhardt estimated the length Lo as:

[2] L = !s2.er d o 6500 (!s2,er in psi)

[2a] L = i,2.cr d o 45 (!s2,cr in N/mm2)

where d is the diameter of the reinforcement and!s2.cr is the stress in the steel at the crack immediately after cracking. Leonhardt's expression for the transfer length (L,) is:

[3] L, = Kda, c) + O.ld/p

where p is the percentage of steel reinforcement, and KJ is a factor depending on the minimum concrete cover measured from the surface of the bars perpendicular to the direction of the applied load, c, and the spacing between the longitudinal bars, a. The complete deri­vation is given by Leonhardt (1977).

Based on this proposed cracking mechanism, the average crack width W mL can be computed as follows:

[4] WmL = Lo E,2 + L, Em

where E,2 is the steel strain at the cracked section, Em is the average gross strain measured over the cracks in­cluding concrete contribution within the transfer length and is defined as follows:

where!s2 is the steel stress at the crack. Similarly, Beeby (1972, 1979) suggested an expres­

sion for the average crack spacing given by:

RIZKALLA ET AL. 571

(KN) (KIPS) Q) CI) -en If)

(489.28 ) 110.00 -0 Z crt 0 I:: c:5 (444.80) 100.00 :0 Q) w

a:: (400.32 ) 90.00

;::

-t (355.84) 80.00 ...:1:: u.2 0-

(311.36) 70.00 ~o (\J U'" :::J

Z Cl)crt SPECIMEN NO: 2 :0;;:::: 0 Cl (266.881 60.00 01:: c:5 « _0

I.LJ 0 enu a:: ...J (222.401 50.00

4 ( 177.92) 40.00 First Crack

( 133.44) 30.00

ES2 'z 0 88.96) 20.00 Uncracked Section (!i w

44.48) 10.00 a::

1 0.00) 0.00 0 2 3 4 5 6 7 8 9 10

STRAIN X 10-3

FIG. 5. Load-average gross strain relationship for typical specimen No.2.

[6] Sb = l.33c + 0.08d/p

and the average crack width Wmb is given by

[7] W mb = SbEm

where the average gross strain Em defined by Beeby is:

[8] K/;f,2,cr

Em = E52 - E f, ,p 52

where K is a constant depending on the type of bar.!; is the tensile strength of concrete, and E, is the modulus of elasticity of steel.

Typical load-average gross strain response is shown in Fig. 5, where specimen 2 was used as an example. In the same figure, the response based on steel alone is shown as a straight line. The shaded area represents the concrete contribution to the specimen.

Comparison of computed and measured· crack spacing

It was observed that most of the surface cracks for all the tested specimens coincided approximately with the locations of the transverse reinforcing bars. The aver-

age crack spacing between through cracks decreased as more cracks were formed, and the final crack pattern was reached at an average gross strain Em of 0.0011 (Fig. 5) for a typical specimen, No.2. The maximum and minimum crack spacings, as defined in Fig. 6 for a typical specimen, were compared to the average crack spacing for each specimen at the final crack pattern. The ratios of Smin/Save and Smax/Save were found to be 0.7 and 1.34 respectively (Fig. 7). These values emphasize the nature of the crack spacing distribution and intro­duce the range within which the prediction of crack spacing could be acceptable for reinforced concrete members.

Using Leonhardt's expression [1], and Beeby's ex­pression [6], the average crack spacings were computed for all segments (Table 2), and then compared with the measured values from the test.

Figure 8a illustrates the comparison between the ratio of Beeby's computed values to the measured values Sb/Save, as a function of bar diameter divided by steel percentage (d/p). The comparison suggests an under­estimation of the average crack spacing by Beeby, since the average ratio between the computed and measured

572 CAN. J. CIV. ENG. VOL. 10, 1983

Smin

Smax L

N-NUMBER OF THROUGH CRACKS

Save = NL_I

FIG. 6. Maximum, minimum, and average crack spacings for typical specimen.

values is approximately 0.80. Figure 8b suggests an overestimation ofthe average

crack spacing by Leonhardt, since the average value of the ratio between predicted and measured values is 1.25. Based on these comparisons, it may be concluded that neither of the above expressions is accurate enough to predict the crack spacing in the presence of transverse reinforcement. However, both expressions were capa-

2.5

ble of predicting the crack spacing within the acceptable range suggested by Fig. 7.

A proposed modification for crack spacing prediction

In this study an attempt was made to refine the crack spacing expression using Beeby's expression with the addition of the almost no-bond length concept intro­duced by Leonhardt. The discrepancy between the crack spacing values predicted by Beeby and the mea­sured values was used to re-evaluate the almost no bond length (l:;~r Based on the first experimental results, the proposed expression is as follows:

[9] Lorn = lO(d - 0.28) (d in in.)

[9a] Lorn = lO(d - 7.2) (d in mm)

where Lorn is the modified almost no bond length equiv­alent to the almost no bond length (Lo) proposed by Leonhardt [2l

The above expression assumes that the 'almost no bond length is a function of the diameter of the steel reinforcement bars and is virtually zero for bars with diameters less than 0.28 in. (7.2 mm). Another feature of the new expression is that it is expressed in terms of the section properties, which makes it much simpler to use than the original formula [2], which is a function of the steel stress level. This simplified expression was based on tests results from specimens having a constant

Smox 1.5

• MEAN = 1.34 I C.O.V. = 15.9% •• ST.DEV. = 0.213] •

Save

Smin

Save

1.0

0.5

-------- --~ "r-.!--...£~-- •• --~- -- 134 . - ..' '\ · -; ... .

~--------------r---~~~e---w-~~~~--e------0.70

MEAN = 0.70 I ST.DEV. : O. 15; C.O.V. - 22.1 Yo

25 50 75 00 100

o

125 150 175 (mm) o ~ ____ -L ____ ~~ ____ ~ ____ ~ ______ ~ ____ ~ ____ ~

o 2 3 4 5 6 7 (IN)

Save

FIG. 7. Comparison between maximum, minimum, and average measured crack spacings.

tral Ho oth etel (p) tun cral

[10

wh( ave:

RIZKALLA ET AL. 573

TABLE 2. Measured and calculated crack spacings for all specimens

Measured crack spacings Calculated crack spacing

Specimen Smin* Smax * Save * Beeby Sb Leonhardt SL Modified Sm Proposed Sc number (in.) (in.) (in.) eq. [6] (in.) eq. [I] (in.) eq. [II] (in.) eq. [12] (in.)

First experimental program

TIA 3.40 2.71 3.93 3.18 3.08 TIB 1.84 4.41 3.13 3.04 4.81 3.51 3.42 T2A 2.78 3.60 3.10 2.39 4.11 3.21 3.12 T2B 1.13 4.66 3.91 2.72 4.32 3.54 3.39 T3A 2.88 5.21 3.31 2.03 3.60 3.13 3.00 T3B 2.16 3.66 3.00 2.36 3.66 3.46 3.31 T4A 3.25 5.94 4.37 3.09 4.99 '3.91 3.75 T4B 2.38 4.47 3.47 3.42 5.34 4.24 4.07 T5A 2.34 6.44 4.61 2.60 4.41 3.70 3.57 T5B 3.06 5.41 3.79 2.93 4.68 4.03 3.94 T6A 2.41 5.75 3.80 2.38 4.07 4.13 3.95 T6B 2.50 3.59 3.03 2.71 4.29 4.46 4.35 TIA 4.31 6.63 5.16 3.39 5.64 4.49 4.32 T7B 3.47 8.59 6.21 3.72 5.67 4.82 4.71 T8A 3.28 7.63 5.36 3.10 4.87 4.85 4.68 T8B 3.47 11.63 6.36 3.43 5.05 5.18 5.01 T9A 4.28 7.78 6.13 2.71 4.40 5.06 4.96 T9B 3.84 7.69 5.48 3.04 4.72 5.39 5.17

Second experimental program

2A 2.25 4.63 3.34 3.42 5.37 4.24 4.16 2B 2.00 4.19 3.68 3.42 5.42 4.24 4.16 2C 2.44 4.13 3.22 3.42 5.48 4.24 4.16 4A 3.63 4.88 4.10 3.42 5.26 4.24 4.10 4B 3.63 4.56 4.07 3.42 5.29 4.24 4.10 4C 3.19 4.75 3.96 3.42 5.20 4.24 4.10 6A 2.31 6.38 3.45 3.42 5.37 4.24 4.01 6B 1.94 4.44 3.19 3.42 5.42 4.24 4.01 6C 1.31 6.19 4.07 3.42 5.34 4.24 4.01 1 2.63 5.31 3.78 3.42 5.62 4.24 4.03 2 2.44 7.66 4.09 3.42 5.48 4.24 3.91 3 4.41 6.31 5.40 4.42 6.44 5.24 5.17 4 3.38 4.34 4.01 4.42 6.38 5.24 5.10 5 5.52 6.53 6.13 4.42 6.40 5.24 4.97 6 4.81 5.44 5.Il 4.42 6.36 5.24 4.98 7 3.53 5.66 4.87 4.42 6.38 5.24 4.85

NOTE: I in. = 25.4 mm. *See Fig. 6.

transverse bar spacing of 3 in. (76 mm) centre to centre. The final form for the proposed expression for pre-However, it should be general enough to be useful in diction of the average crack spacing in this investigation other cases, since it was derived using different pararn- can be written as follows: eters, such as concrete cover (c), reinforcement ratio (p), and concrete thickness (I). Thus this expression in [11] Sm =:: 5 (d - 0.28) + 1.33c + 0.08 dip tum leads to a new equation for predicting the average (d, c in in.) crack spacing at the final crack patterns, Sm, as follows:

Ella] Sm = 5 (d - 7.2) + 1.33c + 0.08 dip 1 [10] Sm =:: "2 Lom + Sb (d, c in mm)

where Sb is equal to the original expression for the Figure 9 shows the ratio between the predicted crack average crack spacing given by Beeby's equation [6]. spacing based on [11] and the measured values. The

574 CAN. J. CIV. ENG. VOL. 10, 1983

2.0

MEAN • 0.80 STD. DEV. " O. 170

1.5 C.O.V. • 21.3 %

Q) >

• I •

CJ (f)

1.0 ....... .0

0.80

(f) • • • • • • • I· 0.5 •

400 550 700 850 1000 (mm)

15 20 25 30 35

dIp

G) MEAN = 1.25 STD. DEV. ,,0.259

2.0

• C.O.V. -20.7%

• I • • I 1.25

1.5

Q)

• • • • • • > CJ

(f) ....... 1.0 • • • • • •

...J (f)

0.5

400 550 700 850 1000 (mm)

15 20 25 30 35 40 (lNJ

dIp FIG. 8. (a) Comparison between measured and computed crack spacings based on Beeby's expression [6]. (b) Comparison

between measured and computed crack spacings based on Leonhardt's expression [1].

average value of the predicted crack spacing, based on the new equation, was found to overestimate the aver­age measured values by 5 percent, which is highly acceptable predictive accuracy.

Comparison between computed and measured crack widths

The average crack widths at the final crack pattern for all specimens were calculated using Leonhardt's ex­pression [4J and compared to measured values as given in Table 3. Figure lOa suggests very clearly an over­estimation of the average crack width resulting from the use of this expression, since the average value of the

ratio between the calculated and experimental values is as large as 2.17.

The crack widths for all the specimens were also calculated at the final crack pattern based on Beeby's expression [7] and compared with the measured values as given in Table 3. Figure lOb indicates that Beeby's expression [7] can be used adequately for predicting the average crack width, since the average ratio between C

both values is 1. 09 . , Influence of transverse reinforcement on t

crack spacing ~ Beeby (1972, 1979), Nawy and Blair (1971), and

RIZKALLA ET AL. 575

2.0

MEAN = 1.05

1.5 STD. DEV. = 0.170

• I

C.O.V. = 16.2% Q)

> 0

en • • 1.05 • ......... 1.0 • I

E • C· • en • •

0.5

400 550 700 850 1000 (mm)

0 10 15 20 25 30 35 40 (lNJ

dip FIG. 9. Comparison between measured and computed crack spacings based on equation [II].

McGregor et ai. (1980) have observed some correlation between the spacing of reinforcement parallel to the cracks and the spacing of the cracks themselves. To date, there are no available data in the literature that could be used to examine the effect of transverse rein­forcement. The two large reinforced concrete speci­mens tested at the University of Alberta were loaded biaxially. Other specimens loaded uniaxially were pre­stressed and do not conform with the conditions consid­ered in this paper. The preliminary (1981) results of the work at the Polytechnic of Central London under Dr. P. E. Ragen, which is related to the design of concrete gravity platforms for oil extraction, indicate that the transverse steel had a dominant influence on the distri­bution and size of the cracks. However, the data are not available for use in examining the proposed methodology.

The second experimental program in the present study involved testing of sixteen reinforced concrete specimens, which were divided into two groups with two concrete covers of 0.75 in. (19 mm) and 1.5 in. (38 mm). Within each group, concrete segments were identical in longitudinal reinforced ratio (p = 0.0147), concrete cover (c = !in. or 1.5 in. (19 or 38 rom», and concrete thickness (t = 7 in. or 178 mm). The only variation introduced in this experimental program was the spacing between transverse reinforcing bars per­pendicular to the direction of the applied load. The five spacings considered were 2 in. (51 mm), 4 in.

(102 mm), 6 in. (152 mm), 8.5 in. (216 mm), or 10.5 in. (254 mm), and additional specimens without transverse reinforcement.

According to theoretical predictions, if p, c, and t were constant, the calculated average crack spacing based on Beeby (1979) should have the same magnitude as given for the second experimental program in Table 2.

This paper proposes a general expression, based on extensive experimental results to predict the average crack spacings in reinforced concrete members in the presence of transverse reinforcement. The expression utilizes a dimensionless parameter (R), which repre­sents the ratio of transverse reinforcement spacing (SR) to the average crack spacing (Sm) predicted from the proposed equation [11].

R = SR/Sm

This expression is also intended to take into account the random nature of crack spacing distribution, as clearly illustrated in Fig. 7, and refines the proposed methodology presented earlier by Rizkalla et al. (1982).

For specimens that had transverse reinforcement located 2 in. (51 mm) apart, which was less than the predicted values for both groups based on equation [11], it was observed that cracks were randomly located with some cracks at the locations of transverse rein­forcement. The measured average crack spacings were

576 CAN. J. CIV. ENG. VOL. 10. 1983

0 • •

3.0 • MEAN = 2.17 STD. DEV. = 0.551 C.O.V. = 25.4%

• • • 2.5 • • • • tit

CI)

> • • 2.17 0

3: , 2.0 • • -' E

3: • • • •

• 1.5 • • ,

400 550 700 850 1000 (mm)

1.010 15 20 25 30 35 40 (IN.)

dip

2.0 0 MEAN = 1.09 STD. DEV. = 0.261

• C.O.v. = 24.0% • 1.5 • •

CI) I > • • 0 • ~ • • 1.09

, 1.0 I .D • E • • I

3: 0.5

400 550 700 850 1000 (mm)

0 10 15 20 25 30 35 40 (IN)

dIp

FIG. 10. (a) Comparison between measured and average crack widths based on Leophardt's expression [41 at stable crad pattern. (b) Comparison between measured and computed average crack widths based on Beeby's expression [71 at stable cracl

pattern.

RIZKALLA ET AL. 577

c=0.75" (l9mm) c-I 50" (38mm)

t l' c= /.50" (38 mm)

t

c=0.75" (l9mm) , c= /.50" (38mm) , c=0.75" (l9mm) c=1.50" (38mm)

J J

FIG. 11. (a) Final crack patterns of specimens with 2 in. (51 mm) spacing between transverse reinforcement. (b) Final crack patterns of specimens with 4 in. (102 rnm) spacing between transverse reinforcement. (c) Final crack patterns of specimens with 6 in. (152 mm) spacing between transverse reinforcement. (d) Final crack patterns of specimens with 8.5 in. (216 mm) and 10.5 in. (267 mm) spacing between transverse reinforcement.

578 CAN. J. CIV. ENG. VOL. 10, 1983

c= 1.50" (38 mm) ,

FIG. 11. (e) Final crack patterns of specimens with no transverse reinforcement.

approximately equal to the predicted values from [11] (Fig. lIa).

For specimens having 4 in. (102 mm) spacing be­tween the transverse reinforcements, cracks were found at each location of the transverse reinforcements (Fig. 11 b). This phenomenon was identical and repeated for all of the specimens in both groups. For specimens with 0.75 in. (19 mm) concrete cover, the predicted crack spacing based on equation [11] is 4.24 in. (108 mm), which is very close to the spacing of transverse rein­forcements. For the specimens in the second group that had 1.50 in. (38 mm) of concrete cover, the predicted value is 5.25 in. (133 mm), which is slightly greater than the spacing between the transverse reinforcements. However, due to stress concentration effects around the transverse reinforcements, cracks were induced at their locations. It is obvious that a parameter should be intro­duced to take this behaviour into account.

For specimens with 6 in. (152 mm) spacing, which is approximately 1.5 times the predicted value from equation [11] in the first group, cracks were formed first at every location of transverse reinforcement, with a subsequent crack between the previous cracks at final crack pattern, as shown in Fig. 11 c. In the second group, where the transverse reinforcement spacing is approximately 1.1 times the predicted value from equa­tion [11], it was observed that cracks were located only

TABLE 3. Measured and calculated average crack widths at stable crack pattern (j,2 = 50 ksi)

Specimen number

TlA TlB T2A TIB T3A T3B T4A T4B T5A T5B T6A T6B T7A T7B T8A T8B T9A T9B

2A 2B 2C 4A 4B 4C 6A 6B 6C 1 2 3 4 5 6 7

Crack width X 10-3 in.

First experimental program 2.69 6.52 3.67 7.19 2.32 7.25 4.30 7.47 2.64 6.75 4.45 6.84 3.48 8.46 3.69 8.47 3.10 7.91 3.26 8.18 3.26 7.76 4.86 8.12 2.62 9.20 4.16 9.49 2.88 8.98 3.95 9.28 3.40 8.39 3.79 9.00

Second experimental program 3.75 8.42 2.77 8.32 3.68 8.21 4.58 8.58 4.93 8.54 5.88 8.63 3.56 8.42 4.39 8.32 5.80 8.47 3.06 7.84 3.68 8.21 6.13 9.53 5.78 9.69 6.53 9.65 7.24 9.76 7.29 9.69

NOTES: 1 in. = 25.4 mm; 1 ksi = 6.895 N/mm2.

3.35 3.34 3.37 3.73 3.12 3.73 4.34 4.16 3.81 4.05 3.78 4.24 4.32 3.11 5.03 5.52 4.47 4.88

4.09 3.90 3.68 4.48 4.39 4.65 4.10 3.90 4.20 3.03 3.62 5.11 5.37 5.32 5.49 5.37

at every transverse reinforcement, with no additional cracks in between, which suggests that the spacing be· tween transverse reinforcement is not large enough fOI an additional crack to be formed in this case.

For specimens that have the transverse reinforce· ments located 8.5 in. (216 mm) apart in the first groul and 10.5 in. (267 mm) in the second group (twice th( predicted values based on equation [11] for bod groups), cracks were formed first at the location 01 every transverse reinforcement and followed by addi tional cracks formed between each transverse bar (Fig 11 d). The average crack spacings measured were ver:

RIZKALLA ET AL. 579

8

7

6

5

Save 4 Save 0.96 RSR 1 RSR = R2.02

3

2

2 3 4 5 6 7 8

R FIG. 12. Relationship of Save/RSR and R.

2.0 MEAN = 1.01

STD.DEV. = 0.163

C.O.V. = 16.3 %

b-__ ~r-~ ________________________________ ~ __ ~I.OI

600 750 (mm)

30 (lNJ

FIG. 13. Comparison between measured and computed crack spacings based on equation [12].

580 CAN. J. ClV. ENG. VOL. 10. 1983

close to the predicted crack spacings based on equation [11].

Specimens without transverse reinforcement (Fig. 11 e ), were considered to be portions of structures iso­lated by pre-existing cracks 30 in. (762 mm) apart at the location of transverse reinforcement. The measured crack spacings for both groups were close to the pre­dicted values from equation [11], which suggests that this equation is also applicable for reinforced concrete members without transverse reinforcement.

Based on the above observations, the following ex­pression is proposed,

[12] Sc = (3Sm

where Sc = average crack spacing in the presence of transverse reinforcement, Sm = average crack spacing on equation [11], (3 = 0.96/Ro.02

•

Equation [12] is based on the relationship presented in Fig. 12 for all of the specimens in both programs.

Based on the proposed equation [12], the predicted crack spacings were calculated and compared with the measured values (Fig. B). The average value was found to be 1.01, which clearly indicates a very high predictive accuracy.

Summary and conclusions

The following observations and conclusions are based on the two experimental programs discussed in this study. 1. The spacings of the through cracks in reinforced concrete members subjected to pure uniaxial tension are affected primarily by the spacing of the transverse rein­forcement steel parallel to the direction of the cracks. 2. The number of through cracks increases as the strain increases. A fully developed pattern of cracks is reach­ed at a strain approximately equal to 0.0011. 3. The new expression proposed for the "almost no bond" length (equation [9] or [9a]) is equally reliable as the expression proposed by Leonhardt. However, the new expression is simpler to use because it is expressed in terms of the section properties instead of the steel stress level. The average value of the predicted crack spacing based on the modified expression [11) over­estimates the average crack spacing of the measured values by only 5 percent. 4. Leonhardt's prediction [4] for the average crack width at the stabilized crack pattern is highly over­estimated, as the ratio between the measured values and those calculated by Leonhardt suggests an average value of 2.17 . 5. Beeby's expression [7] can be confidently used for predicting the average crack width at the final crack pattern. 6. A new expression is presented for predicting aver­age crack spacing in reinforced concrete members sub-

jected to axial tension with known transverse bar spac­ing. The ratio between predicted crack spacings based on this expression to the measured crack spacings is found to be 1.01, which indicates an excellent degree of predictive accuracy. The proposed expression is based on testing specimens reinforced with a maximum bar diameter of 20 mm in the direction of the applied load. Further study is in progress to include the effect of larger diameter bars.

Acknowledgments

This study was carried out in the Department of Civil Engineering at the University of Manitoba with financial assistance from the Natural Sciences and Engineering Research Council of Canada, and the University of Manitoba Research Board.

BEEBY, A. W. 1972. A study of cracking in reinforced con­crete members subjected to pure tension. Technical Report 42.468, Cement and Concrete Association, London.

--. - 1979. The prediction of crack widths in hardened concrete. Structural Engineering. 57A, No.1, pp. 9-17.

BROMS, B. B. 1965. Crack width and crack spacing in rein­forced concrete members. Journal of the American Con­crete Institute, 62(10), pp. 1237-1255.

BROMS, B. B. 1965. Technique for investigation of internal cracks in reinforced concrete members. Joumal of the American Concrete Institute, 62(1), pp. 35-44.

EL-SHAHAWl, M. 1981. Cracking behaviour of reinforced concrete members subjected to tensile membrane forces. M.Sc. thesis, Department of Civil Engineering, University of Manitoba, Winnipeg, Manitoba, Canada.

HWANG, L. S. 1983. Behavior of reinforced concrete in ten­sion at postcracking range. M.Sc. thesis, Department of Civil Engineering, University of Manitoba, Winnipeg, Manitoba, Canada.

GOTO, Y. 1971. Cracks formed in concrete around deformed tension bars. Journal of the American Concrete Institute. Proceedings, 68(4), pp. 244-251.

LEONHARDT, F. 1977. Crack control in concrete structures. IABSE Surveys, S-4/77 , IABE Periodical 3/1977, Inter­national Association for Bridge and Structural Engineering, Zurich, Switzerland.

MACGREGOR, J. G., RIZKALLA, S. H., and SIMMONDS, S. H. 1980. Cracking of reinforced and prestressed concrete wall segments. Structural Engineering Report No. 82, Univer­sity of Alberta, Edmonton, Alberta, Canada.

NAWY, E. G., and BLAIR, K. W. 1971. Further studies on flexural crack control in structural slab systems. Cracking, Deflection and Ultimate Load of Concrete Slab Systems, SP-30, American Concrete Institute, Detroit, Michigan, pp. 1-42.

RIZKALLA, S. H., EL-SHAHAWI, M., and KWOK, C. K. 1982. Cracking behaviour of reinforced concrete members. Pro­ceedings of the Canadian Society of Civil Engineering An­nual Conference, Edmonton, Alberta, Canada, Vol. 1, pp. 517-534.

SALlGAR, R. 1936. High grade steel in reinforced concrete. Preliminary Publication, 2nd Congress, International Asso-

ciation for Bridge and -Structural Engineering, Berlin- Save Munich, pp. 293-315. Sb

List of symbols Sc a spacing between longitudinal bars c concrete cover to surface of bar (measured SL

perpendicular to surface) d diameter of reinforcing bar Sm E, modulus of elasticity of steel

!s2 stresS in reinforcement at a crack Smin

!s2.cr stresS in reinforcement at a crack immediate- Sma. ly after cracking SR

f; tensile strength of concrete K,K1 empirical constants Wave L distance between two outermost cracks on Wmb

concrete segment Lo almost no bond length of bar at a crack WmL Lom modified almost no bond length of bar at a

crack Em L, transfer length N number of through cracks E,2 P steel ratio 13 R SR/Sm

581

measured average crack spacing average crack spacing based on Beeby's expression average crack spacing in the presence of transverse reinforcement average crack spacing based on Leonhardt's expression average crack spacing, based on equation [I I]

::= measured minimum crack spacing measured maximum crack spacing spacing between transverse bars

::= concrete thickness measured average crack width

::= average crack width, based on Beeby's ex­pression (equation [7]) average crack width, based on Leonhardt's expression (equation [4])

::= average gross strain measured over the cracks

::= steel strain at a crack dimensionless parameter