Cracks and Crack Control in Concrete Structures, F. Leonhardt. July-August 1988

Special Report

Cracks and Crack Control in Concrete Structures

Fritz Leonhardt Professor Emeritus Dr.-Ing. Dr.-Ing. h.c. mull. Consulting Engineer Stuttgart, FRG

he material presented in this paper is ing in concrete structures. In this paper, T based on more than 30 years of re- causes of concrete cracking are dis- search, observations and experience cussed, including tensile strength of concerning causes, control, and conse- concrete, temperature, shrinkage and quences of cracking in concrete struc- creep effects. Recommended crack tures. This extensive background was widths are presented along with design helpful in the preparation of this paper methods for sizing reinforcement to which deals with questions of concrete control crack widths. cracking. The presence of cracking does not necessarily indicate deficiency in CAUSES OF CRACKING strength or serviceability of concrete Concrete can crack due to a number of structures. While currently available de- causes. Some of the most significant sign code provisions lead to reasonable causes are discussed in detail. control of cracking, additional control can be achieved by understanding the basic causes and mechanisms of crack- Tensile Strength of Concrete The tensile strength of concrete is a widely scattering quantity. Cracking oc- Note: This paper is a revised and updated version of curs when tensile stresses exceed the an article originally published in the Proceedings of tensile strength of concrete. Therefore, the International Association for Bridge and Struc- tural Engineering (1ABSE), Zurich, Switzerland, to control concrete cracking, the tensile 1987, p. 109. strength of concrete is of primary im-

124 portance. Laboratory test data con- ducted by H. Busch were analyzed statistically. As presented in Ref. 1, this Synopsis analysis furnished the following re- Simple design rules are presented lationships for the mean direct tensile to control cracking in concrete struc- strength, f tm , related to the 28-day tures. Causes of cracking and its ef- compressive cylinder strength f,' of con- fect on serviceability and durability are crete: discussed. The paper is primarily ap-

/3 plicable to large structures such as fcm = 2.1 (fc)z (psi) bridges. However, general concepts

fim = 0.34 (ff)2"(N/mm2) presented are applicable to any concrete structure. Prestressing forces The statistical analysis indicated that are considered. A numerical example the coefficient in this equation can be showing application of the method modified to 1.4 (0.22) and 2.7 (0.45) to and use of simple design charts is in- obtain the 5 and the 95 percentiles, re- cluded. spectively, of the tensile strength, fI. The tensile strength of concrete is slightly higher in flexure. However, it is recommended that values for direct tension be used in practice. Concrete cement during concrete hardening. This cracks when the tensile strain, € °t, ex- effect is usually neglected except in ceeds 0.010 to 0.012 percent. This massive structures as indicated in Ref. 2. limiting tensile strain is essentially in- However, depending on cement content dependent of concrete strength. and type of cement, the temperature The 5 percentile of the tensile within concrete members with dimen- strength, f, should be used in design to sions of 12 to 36 in. (30 to 91 cm) can locate areas in the structure that are increase approximately 36°F to 108°F likely to crack by comparing calculated (20°C to 60°C) during the first 2 days stresses with the expected concrete after casting. strength. The 95 percentile, f 5, should If concrete members are allowed to be used to obtain conservative values for cool quickly, tensile stresses may reach restraint forces that might occur before values higher than the developing ten- the concrete cracks. These restraint sile strength of the concrete. Even if this forces are used to calculate the amount process results only in microcracking, of reinforcement needed for crack width the effective tensile strength of the control. hardened concrete is reduced. How- ever, very often wide cracks appear even when reinforcement is provided. Causes of Cracking During In addition, the reinforcement may not Concrete Hardening be fully effective since bond strength is Concrete cracking can develop during also developing and is yet too low. It is the first days after placing and before necessary to minimize such early cracks any loads are applied to the structure. by keeping temperature differentials Stresses develop due to differential within the concrete as low as possible. temperatures within the concrete. This can be done by one or more of the Cracking occurs when these stresses ex- following measures: ceed the developing tensile strength, f;, 1. Choice of cement — A cement with of the concrete as indicated in Figs. 1 low initial heat of hydration should be and 2. Differential temperatures are selected. Table 1 shows that there is a mainly due to the heat of hydration of significant variation in heat develop-

PCI JOURNAUJuly-August 1988 125 b concrete

compression

+ +G = OT a T Ec GT tension

Internal stresses in equilibrium

Fig. 1. Temperature distribution due to heat of hydration and internal stresses caused by outside cooling in a free standing concrete block.

ment among different types of cements. concrete, as it was done years ago, is not The cement content of concrete should recommended. be kept as low as possible by good 4. Precooling — This is a necessity for grading of the aggregates. Heat de- large massive concrete structures such velopment can also be reduced by ad- as dams. For more usual structures, in ding fly ash or using slag furnace ce- which shortening after cooling can take ment. place without creating significant re- 2. Curing — Evaporation of water straint forces, precooling is expensive must be prevented by using curing and unnecessary. In this case, thermal compounds or by covering the concrete insulation is preferable and it also has with a membrane. Rapid evaporation the benefit of accelerating concrete can lead to plastic shrinkage cracking. strength development. An exception 3. Curing by thermal insulation - may be made in very hot climates since Rapid cooling of the surface must be precooling can keep concrete workable prevented. The degree of thermal insu- for a longer period of time. lation depends not only on the climate, Often shrinkage is considered as a but also on the thickness of the concrete cause of early cracking. However, this is member and on the type of cement used. not true under normal climatic condi- Spraying cold water on warm young tions. Shrinkage needs time to produce a

126 Table 1: Heat of hydration of various types of cements.

Heat of hydration (Btu/1b)

Type of cementt 1 day 3 days 7 days 28 days I 92 144 157 167

II 76 115 135 148 III 139 184 194 205

IV 50 81 94 117 V 58 88 101 124

Data obtained from Concrete Manual, U.S. Bureau of Reclamation, 1975, pp. 45-46. t Federal Specifications SS-C-192G, including Interim Amendment 2, classified the live types according to usage as follows: Type I for use in general concrete construction when Types 11, 111, 1V, and V are not required; Type 11 for use in construction exposed to moderate sulfate attack; Type III for use when high early strength is required; Type IV for use when low heat of hydration is required; and Type V for use when high sulfate resistance is required. Note: 1.0 Btu/Ib = 2.32 J/g.

shortening as high as the tensile rupture Causes of Cracking After strain. Only in very hot and dry air Concrete Hardening shrinkage can cause early cracks in young concrete, if measures against Tensile stresses due to dead and live evaporation are not applied. loads cause cracking. Normal rein-

cracking due to restraint

tensile strength fit

Internal Stress

5 10 15 20 h Concrete hardening time, hours

Fig. 2. Development of the tensile strength and stresses due to nonlinear temperature distribution within the concrete.

PCI JOURNAL/July-August 1988 127 Deformed Shape considering Upper Face of Beam Warmer Than Bottom Face and assuming beam freed from interior Supports^

I I I

I MDT Moment Diagram

VAT

Shear Diagram

Fig. 3. Forces in a concrete beam due to a temperature rise OT at the upper face of the beam and external restraint provided by interior supports.

forcement or prestressing should be de- treme temperatures that occur at 20 to signed to provide required strength and 50-year intervals must be considered. As keep crack widths within permissible indicated in Refs. 3, 4, 5 and 6, temper- limits. Tensile stresses due to service atures in bridge structures were mea- loads can be controlled by prestressing. sured in several countries. Recently, the The degree of prestressing can be cho- U.S. Transportation Research Board sen based on structural or economic published in Ref. 7 temperature data for considerations. Normally, partial pre- bridge design. stressing leads to better serviceability Temperature differentials should be than full prestressing. considered along with recommended Cracks can also be initiated by ten- mean temperatures, Tm, used for cal- sile stresses due to restrained defor- culating maximum and minimum mations from temperature variations or changes in the lengths of structural from shrinkage and creep of concrete. members. In Central Europe values for Imposed deformations such as differ- T. are specified for concrete bridges as ential settlement between foundations varying from +68°F to –22° (+20°C to can also cause cracks. –30°C). There are two types of restraint which The temperature distribution over a cause stress in concrete members, beam cross section can be subdivided namely, internal restraint as shown in into three parts as shown in Fig. 4. The Fig. 1, and external restraint in indeter- constant part, 0 T,, causes axial forces if minate structures, as shown in Fig. 3. overall length changes are restrained. Restrained deformations caused crack- The linear part, AT,, causes restraint ing in concrete bridges and it was forces, M AT and V AT, in indeterminate primarily due to temperature differ- structures as shown in Fig. 3 for a three ences produced by heating under the span continuous beam. The nonlinear sun and cooling during the night. Ex- part, AT3i causes stresses, which are in

128 Table 2. Recommended cross section temperature differentials for bridge design in Europe.

Box girder T-beams Type of cross section Mari- Conti- Mari- Conti- and exposure time nental time nental Top of cross section warmer than bottom (°F) 18 27 14.4 21.6 Bottom of cross section warmer than top(F) 9 14.4 7.2 10.8

Note: 1.0 A N _ (9/5) A°C.

equilibrium over the cross section and pansion for concrete. Cooling causes produce no action forces. These tensile stresses in areas near extremities stresses, which also exist in statically of the section. determinate structures, can be calcu- For bridges in Europe, the AT values lated by imposing equilibrium condi- given in Table 2 are recommended. In tions and considering that: addition to temperature, restrained concrete creep and shrinkage can cause JcT = AT3aTEc stresses. Shrinkage often leads to cracks between connected members of signifi- where a T is equal to 6 x 10 -6 /0 F cantly different sizes. Stress due to re- (10-5/° C), the coefficient of thermal ex- strained creep and shrinkage can be cal-

rig. 4. uivision of temperature aiagram into its constant, unear ana noniinear parts.

PCI JOURNAL/July-August 1988 129 G i i h i i

compressive G H due to DL+w LL+ P plan A-A —_–-- cracks due to AT, AS and ACr

Fig. 5. Transverse cracks in thin bottom slab of box girder due to differential temperature, creep and shrinkage despite prestressing.

culated in the same way as stresses due the linear theory of elasticity, consider- to temperature. ing the structure initially uncracked. In Transverse cracks due to temperature, these calculations, f,s should be taken as creep and shrinkage effects are fre- the tensile strength of the concrete. In quently found in the relatively thin the tension side of a beam, cracking will bottom slabs of box girders despite the occur in areas where bending moments fact that calculations show considerable due to service loads and restraint cause longitudinal compressive stresses due to stresses in the extreme tensile fiber prestressing. Compressive stresses tend above f15 . As bending increases, the to shift towards the thick webs which depth of cracking can be calculated by undergo less creep and shrinkage strains considering a maximum concrete tensile as illustrated in Fig. 5. strain of 0.015 percent as shown in Fig. Box sections are indeterminate struc- 7. tures. Therefore, restraint moments are Calculation of possible maximum developed when the section is heated bending moments due to restraint on one side by the sun. This leads to should be based on f 5 . As shown in Fig. vertical cracking in bridge piers and 8, consideration of such moments in- tower shafts as shown in Fig. 6. Ref. 8 creases the areas in which cracking may shows examples of temperature cracks be expected to occur. in prestressed concrete structures. Bending moments due to restraint define only the location and quantity of reinforcement or prestressing necessary Determination of Areas to limit the crack width for serviceability Likely to Crack purposes. As proven long ago by Cracking occurs whenever the princi- Priestley, and illustrated in Fig. 9, pal stresses due to service loads or due these moments do not decrease the ul- to restraint forces or due to a combina- timate strength of the structure because tion of service loads and restraints ex- they are reduced and finally disappear ceed the tensile strength of concrete. due to cracking and plastic deformation These stresses can be calculated using as service loads are increased until the

130 deflection line

vertical cracks

-MAT II

+ +MOT

+ MAT

Fig. 6. Bridge pier cross section and moments due to temperature rise on one side of the pier.

-EG XII

-O,015°/. cracked Zone of web

7,5db rLE +ill flange Zone S

Fig. 7. Cracked zones in webs of beams under combined moment due to dead load, live load and restraint.

limit state is reached. However, the ments due to service loads in sizing of structure must be checked for possible main reinforcement. It must, however, brittle failure of the compression zone if be observed that restraint due to pre- a relatively high degree of prestressing stressing does not decrease on the way is used, especially for continuous T- up to limit state. beains. Therefore, to satisfy strength re- Restraint forces decrease beginning quirements, bending moments due to with the first crack since the stiffness of restraint should not be added to mo- the structure is progressively reduced

PCI JOURNAL/July-August 1988 131

Fig. 8. Increased cracked area due to restraint moment.

with each crack that occurs. Steel Evaluation of Cracks stresses due to restraint are highest As indicated in Fig. 10, crack widths when the first crack occurs and decrease are greater at the surface and decrease with each further crack. This tends to re- towards the reinforcement. Long years duce crack widths. Fig. 9 shows the effect of research reported in Refs. 9 and 10, on moment due to reduction of restraint. and experience indicate that crack

service loads ultimate lim. state M due to load and restraint forces brittle failure by too high prestress

1,75MDL+LL ductile failure

MAT 1,75 MDL effect of AT effect of 1,5T M DL+ LL MDL

0 Curvature 4 D Curvature OT OT

Fig. 9. Illustration of reduction of restraint forces as the limit state is approached.

132 Ag. 10. Crack width at the surface is used as a measure of the effect of cracking on concrete members.

Table 3. Allowable crack widths.

Ambient condition of w90t Maximum w Crack exposuret (in.) permitted (in.) appearance Mild 0.012 0.020 Easily visible

Moderate 0.008 0.016 Difficult to see with the naked Severe 0.004 0.0012 eye

w90 denotes the 90 percentile of the crack width, w. t Defined as indicated in the CEB-FIP Model Code: Mild exposure — The interiors of buildings for normal habitation or offices. — Conditions where a high level of relative humidity is reached for a short period only in any one year (for example 60 percent relative humidity for less than 3 months per year). Moderate exposure — The interior of buildings where the humidity is high and where there is a risk for the temporary presence of corrosive vapors. — Running water. — Inclement weather in rural or urban atmospheric conditions, without heavy condensation of aggressive gases. — Ordinary soils. Severe exposure — Liquids containing slight amounts of acids, saline or strongly oxygenated waters. — Corrosive gases or particularly corrosive soils. — Corrosive industrial or maritime atmospheric conditions. Note: 1 in. = 2.54 cm.

PCI JOURNALJJuly-August 1988 133 widths up to 0.016 in. (0.4 mm) do not sensitivity to corrosion of various types significantly harm the corrosion protec- of reinforcement led to different re- tion of the reinforcement furnished by quirements for concrete cover. It is the concrete, provided the cover is suffi- prudent to vary crack width limita- ciently thick and dense. However, to tions depending on environmental con- avoid undue concern by casual observ- ditions. ers, crack widths should be limited to For the environmental criteria of CEB 0.008 in. (0.2 mm) at surfaces which are and Eurocode No. 2, crack widths can be often seen from a short distance. defined as presented in Table 3. These Polluted air containing CO 2 (which values are valid for a concrete cover, c, causes carbonation), and SO 2 (which of 1.18 in. (30 mm) and for bar diame- forms acids), or chlorides from deicing ters, db, smaller than c/1.2 but not greater salts, can cause damage to concrete than 1 in. (25 mm). For larger cover, the structures. Having cracks or not, con- allowable crack width should be in- crete structures must be protected creased to c130 (c in mm). For cover against such attacks. greater than 2 3/s in. (60 mm) and bar di- Despite the evidence that crack ameters of main reinforcement greater widths up to 0.016 in. (0.4 mm) do not or equal to No. 10 (32 mm), small diam- significantly affect the corrosion protec- eter and closely spaced reinforcement tion of reinforcement, different levels of should be provided within the cover to environmental exposure and different control crack widths.

Fig. 11. Stress-strain diagram of a reintorced concrete member under direct tension.

134 DESIGN OF tension between cracks, referred to as tension stiffening REINFORCEMENT effect (see Fig. 11)

Reinforcement can be designed to As indicated in Ref. 9, AE S can be ex- control crack widths using information pressed as: presented in the following sections. AE 8 = (1/E 3) ( o2scr / a ) (3)

Basic Analysis where The following presentation follows (T ,. = reinforcement stress immedi- the 1978 CEB-FIP Model Code and the ately after cracking 1983 CEB Manual. The material is mss = steel stress in cracked state based on theoretical considerations and E s = Youngs modulus for the rein- experimental results. forcement Fig. 11 shows a plot of steel stress ver- sus longitudinal strain over a given The strains E° and A E S are significantly length, 1, of a reinforced concrete ele- affected by concrete strength and rein- ment in direct tension. As the load inforcement ratio. creases, cracks are assumed to occur The mean crack spacing can be ex- within this length. The crack spacing pressed as: and the longitudinal mean strain define s c,.m =2(c+s/10) +k,kzdb /pe(4) the mean crack width: where Wm = Scan E m (1) c = concrete cover in mm s = bar spacing in mm where = 0.4 for deformed reinforcement, wm = mean crack width kl = mean crack spacing considering bond strength E m = mean strain = Al/l k2 = 0.125 for bending members, considering shape of E diagram As the load increases, reinforcement = 0.25 for members under direct stress at a potential crack location varies tension, linearly. When the crack occurs, rein- < 0.125 for combined bending and forcement stress at the crack, o-$, in- compression creases suddenly without a significant db = bar diameter change in the mean strain. As the load Pe = effective reinforcement ratio, continues to increase and more cracks As /ACei where A, is the effective appear, the relationship between the concrete area around the bar mean strain, E m , and reinforcement defined as indicated in Fig. 13. stress at the crack, a-8 approaches that of , Using the equations presented, the the reinforcement alone, as indicated in mean crack width, can be calcu- Fig. 11. Conditions before cracking will Wm, lated. The 90 percentile of w can be as- be referred to as State I and conditions sumed to be: assuming the reinforcement working in a cracked section will be referred to as wso = k4 W m (5) State II: where k4 is given in the Eurocode as 1.3 E,n= ES — AEg (2) and 1.7 for restraint forces and service loads, respectively. The author recom- where mends k4 = 1.5 for all cases. The effect of E, = steel strain in State II repeated loads can be considered by re- AEs = strain reduction by concrete in ducing the value of A Eg in Eq. (2):

PCI JOURNAL/July-August 1988 135

b= 13ocm (5(.2 in.) (0.6in,) (#4) c=1,5cm db=l4mm

L i E v o °1 r ( ^ l I c=2 cm (0.8 in.)

i ) E c / O M 18cm 3 (a2)db=4.2mm db 6mm db=26mm J1 ( (7 y / 1 \ ` f I s= 15 cm (it8) J ` l 1 I1 `l (6in.)

Fig. 12. Crack pattern on T-beam with large bars near extreme tension fiber and light reinforcement in the stem.

Des = k5 (1/E3 ) (crgcr/(Tg) Sizing Reinforcement for Direct Tension where k5 varies from 0.4 to 0.8 depending on frequency of load repetition as The diagram in Fig. 14 can be used to indicated in Ref. 11. obtain the required reinfor( ement area If the direction of the reinforcement is for a given bar diameter, specified crack not normal to the crack, as in the case of width limit, and given concrete shear or torsion, the crack width can be strength. The diagram applies for the case of axial tension under service loads multiplied by a factor k. which can be taken as 1.0 for angles up to 15 degrees or restraint forces. The full lines in Fig. from the normal with the crack direction 14 refer to a characteristic strength of the and 2.0 for a 45 degree angle. This factor concrete C 20, the dotted lines to C 40. can be interpolated for angles between For other strengths, the factor k, must be 15 and 45 degrees. used. Fig. 12 shows that the reinforcement For crack control one should always controls crack width only within a small choose the concrete class above the one area around the bars. This. area is de- specified for ultimate strength of the fined in the CEB-FIP Model Code as structure. Bar diameter should be cho- the effective area, A Ce, shown in Fig. 13. sen for obtaining small bar spacings as indicated in Section 5.5. Fig. 14 also shows steel stress at first crack and Sizing Reinforcement for minimum reinforcement percentage re- Crack Control quired to avoid reinforcement yielding when first crack occurs. Steel stress at For practical design, the use of simple first crack is given by: charts is recommended to obtain the necessary crack control reinforcement 08CT = .f rm/Pe = 2.1 (.{x) 2 3/Pe area. Design charts are presented in Section 2.42 of the 1983 CEB Manual. For reinforcement with a yield Use of these charts will be explained in strength of 60,000 psi (413 N/mm 2 ) and the following sections. 3000 and 5000 psi (21 and 34 N/mml)

136 ii

16 in. (40cm) Direct 7.5 d bi5ldp dA Tension

S 16 in.

Fig. 13. Definition of the effective concrete area according to CEB-FIP 1978 Model Code.

concrete strength, the minimum rein- the reinforcement area obtained to forcement percentage can be calculated satisfy strength requirements will be as 0.57 and 0.81 percent, respectively. sufficient for crack control requirements Steel stresses at cracking can be in the effective area. higher than allowable stress under If the load is high, then the steel working stress provisions of design stresses can rise above 0 cr and cause an codes. This is acceptable for restraint additional crack width Aw. This A w forces because additional cracking will can be estimated using Eqs. (1) and (2) decrease stresses. For loads however, and obtaining the mean crack spacing such high stresses will not occur if the for given db and Pe from Fig. 15. The A w structure meets the strength require- must then be subtracted from the ments of applicable codes. Normally, specified w 90 to read the higher Pe from

PCI JOURNAL/July-August 1988 137 II 32 10 9 E 28 ^8 a

N 7 . 6 -0 20 y .0 5 16

4 a` 12 3 a8 v 4 U,4 1,2 1,4 1,6 1,8 2.0 percentage of reinforcement pe_As (ova) for C 20 (2900psi 1 (6Oksi)(50) I (40) I (30) I I I)2Oksii ACe 420 300 2ó01 150 125 Gs cr min 9 i{ for C40 (5800psi) ( ksi) (50) (40) (3Oksi) 420 300 200 min p -2,3 N/mm 2at C 20 Steel stress at first crack Gs cr [N/mm 2 ] for fctm 3,4 N/mm 2 at C 40 (334 psi of 2900psi ) (493psi at 5800psi)

Fig. 14. Design chart ( Ps — db diagram) for determining direct tension.

138 Gs,cr small p 0 largep AGscr l } I_-nGc Q)

N Ncr,Mcr N,M Fig. 16. Illustration of jump in steel applied load stress at cracking for different reinforcement ratios.

500 (70)- +N MandM,-N

(60)- ^^ M 400 N \ASI E I I As E (50)- 1 _ As z \ p bd e/d =M/Nd 1 \ 300 1 axial (40), tension N,

Ln a, pure cS (30) bending 20C a, \ s;

° (20) ^o C2p a a^^ 900PS!) E 00 . 10 C M ^^50 (^2s ^-- 20 (2

0 0,5 1,0 1,5 2,0 percentage of reinforcement p = bd /°

Fig. 17. Diagram to obtain increase of steel stress at cracking under tension, bending or bending with compression for varying eccentricity elh of resultant forf;,n = 1.2 (f,1 )2/3.

PCI JOURNAL/July-August 1988 139

Fig. 14 along a line for (w 90 – Ow). Rough estimates are sufficient in practice.

_ X T Sizing Reinforcement for Bending LJ L and Combined Bending and T Compression L a, Members subjected to bending plus compression require much less rein- p O forcement for crack control than mem-

C bers under direct tension. This can be O understood considering the sharp in- U N crease in steel stress that occurs at cracking. As indicated in Figs. 16 and L V Z 17, the increase in steel stress depends 3 W o on concrete tensile strength, f,, rein-

x AA forcement percentage, p, and type of J) a, stress distribution such as produced by direct tension, or combined bending (D C^ X and compression. It should be noted x I a that in Fig. 17 the reinforcement ratio refers to the concrete area A, = bh. L W V As can be seen from Fig. 17, there is a significant difference between the stress 0 increase for tension compared to the in- Ii) crease for bending. For prestressed concrete structures, the stress increase is w significantly reduced depending on the }f L degree of prestressing. In Fig. 17 the values of e/h = –1.0 and –0.4 corre- spond to moderate and limited prestressing levels, respectively, e/h = –0.17 to full prestressing. Even moderate prestressing leads to low steel stresses at cracking and, therefore, small per- centages are sufficient for crack control. For bending, and combined bending

Cn and compression, the diagram given in Fig. 14 can be used to obtain Pe applying the correction factor:

k5 = (h – x ")lh c where x" is the depth of the neutral axis U for State II and under the cracking moment considering axial loads from restraint and prestressing, and including reinforcement to satisfy strength re- Fig. 18. Illustration of effective areas and quirements. associated strain distributions under For sizing longitudinal crack control cracking moment. web reinforcement, the correction factor

140 stresses before cracking crack pattern due to loads to A T Gc

neutr. axis state II-

to i i 5

F 3tcr Gct

crack width Ect _ 0,4 to f e w90 ~ 1000

Fig. 19. Cracks and stresses in members without reinforcement.

becomes: pends on the possible depth to of the crack and can be calculated from the kB = (ha — x°)Ih,, maximum tensile strain of concrete ect,,, 0.012 percent with k4 = 1.6. Thus: where h,, is the depth to the reinforcement as shown in Fig. 18. w90 = 1.6 2 tcr' Ect,u 0.4 (10 -3 ) to For crack control of members sub- In a dry climate, shrinkage of the jected to shearing or torsional stress, the cracked zone should be taken as A E8h equations given in Ref. 11 should be 0.01 percent. Thus: used. w90 = 1.6•2tcr(Ect,u+ A Esh) 0.6 (10 -3 ) Crack Control Without to Reinforcement Forw90 = 0.004 in. (0.1 mm), the depth In massive concrete structures or in of the cracks can be as large as 10 in. (25 moderately prestressed concrete struc- cm). For restrained bending (e.g., un- tures subjected to tensile stresses due to reinforced but moderately prestressed temperature, as shown in Figs. 1 and 4, slabs or beams), the depth should re- cracks that occur often remain as fine main below t cr -- h/5. cracks having widths below w 90 even without reinforcement. Larger cracks do Recommendations for Size and not occur because the tensile strain in Spacing of Reinforcement the concrete, Ect, is restrained by the adjoining zone under compression as Optimal crack control is obtained by shown in Fig. 19. choosing small bar diameters and a The width, w90, of such cracks de- small spacing. Table 4 shows recom-

PCI JOURNAL/July-August 1988 141 Table 4. Recommended upper limits for reinforcing bar spacing.

Allowable crack width, w 90 (in.) 0.004 0 .008 0.012

Direct tension 4 6 8

Bending (o = 36,000 psi) 4 6 8 Bending (&j= 18,000 psi) 6 8 12

Shear (v0 = 290 psi) Stirrups rectangular to axis of member 4 6 8

Shear (v0 = 435 psi) Stirrups rectangular to axis of member 2 4 6

Shear (v0 = 435 psi) Stirrups at 45 to 60 deg with axis of member 4 8 10 Torsion (t0>290 psi) Ties rectangular to axis of member 2 3 5

Torsion (t0 > 290 psi) Ties at 45 deg with axis of member 4 8 10

*The steel stress Q'; and the nominal vertical shear and nominal torsional stresses v 0 and t o , respectively, refer to the load specified for the serviceability limit state of crack control. Note: 1.0 in. = 2.54 cm; 1.0 psi = 6.89 x 10 N/mm2.

mended upper limits for the spacing of a. Direct tension: p,,,i„ = fps / ff = 2.7 reinforcing bars. (f) 2 3/f5 b. Bending:

Minimum Reinforcement I',,i iii 0.5 J [95 /J Minimum reinforcement should meet where Pmin refers to the full cross-sec- the following conditions: tional area, A, = bh and f, is the yield (a) Satisfy strength requirements — It strength of the reinforcement. If crack- is noted in passing that in most Euro- ing is caused by restraint forces due to pean codes not only a maximum con- temperature, creep, shrinkage or differ- crete strain but also a maximum steel ential settlement, then the A, can be strain is specified for strength design. limited to two to three times the effec- (b) Prevent sudden failure when tive area as indicated in Fig. 13. Table 5 cracking occurs — This can occur when presents minimum reinforcement re- the force transferred from the concrete quirements. to the reinforcement is greater than the (c) Satisfy serviceability requirements strength of the reinforcement. Cracking by controlling crack widths — Minimum can be due to applied loads or restraint reinforcement should be used in all forces. The minimum amount of rein- areas where in the cracked state con- forcement can be calculated as follows: crete tensile stresses due to loads or restraint forces exceed ft5 . In these areas, Table 5. Minimum reinforcement the minimum reinforcement can be ob- percentage for f,, = 60,000 psi. tained from Fig. 14 using the kB factor according to Fig. 18. Concrete strength Related f, (psi) 3000 5000 area ACKNOWLEDGMENT Direct tension 0.76 1.08 A, = b h The author wishes to express his ap- preciation to Dr. Walter Dilger and Bending 0.15 0.22 A,= b h Eduardo A. B. Salse for reviewing the Note: 1.0 psi = 6.8 x 10 -3 N/mm2. paper and for their helpful suggestions.

142 REFERENCES

1. Riisch, H., "Die Ableitung der charak- Transportation Research Board, Report teristischen Werte der Beton-zugfestig- 276, September 1985. keit," Beton, Heft, February 1975. 8. Leonhardt, F., "Ri/3schaden an Beton- 2. Widmann, R., "Massenbetonprobleme brucken, Ursachen and Abhilfe," beim Ban der Gewolbemauer Ko1n- Beton-und Stahlbetonbau, Heft, Feb- brein," Lectures of Betontag, 1977, ruary 1979. Deutscher Beton-Verein. 9. Schiessl, P., "Admissible Crack Width in 3. Kehlheck, F., "Einfluss der Son- Reinforced Concrete Structures," Pre- nenstrahlung bei Bruckenbauwerken," liminary Report, V. 2 of IABSE-FIP- Werner Verlag, Dusseldorf, 1975. CEB-RILEM-IASS, Colloquium, Liege, 4. Priestley, M. J. N., "Design of Concrete June 1975. Bridges for Temperature Gradients," 10. Beeby, A. W., "Design Considerations, ACI Journal, Proceedings, V. 75, May In Debate: Crack Width Cover and Cor- 1978, pp. 209-217. rosion," Concrete International, V. 7, 5. Thurston, S. J., Priestley, M. J. N., and No. 5, May 1985, pp. 24-25. Cooke, N., "Influence of Cracking for 11. Leonhardt, F., Vorlesungen fiber Mas- Thermal Response of Reinforced Con- sivbau, 4, Teil, Springer Verlag, 1977. crete Bridges," Concrete International, 12. Falkner, H., "Zur Frage der Ri/3bildung V. 6, No. 8, August 1984, pp. 36-48. durch Eigen-und Zwangs-spannungen 6. Dilger, W., and Ghali, A., "Temperature infolge Temperatur in Stahlbeton- Stresses in Composite Box Girder bauteilen," Deutscher Ausschu,6 fur Bridges," Journal of Structural En- Stahlbeton, Heft, 208. gineering, ASCE, V. 109, ST6, June 1983, 13. Schlaich, J., and Dieterle, H., "Versuche p. 1460. mit Ferrozement," 15, Forschungskol- 7. Imbsen, et al., "Thermal Effects in Con- loquium des DAfStB in Stuttgart, April crete Bridge Superstructures," U.S. 1984.

PCI JOURNAL/July-August 1988 143 APPENDIX A - NOTATION

Ae = gross area of concrete cross h = total depth of member section k,...k5 = coefficients in crack width Ace = effective area of concrete ten- equations sile zone surrounding rein- M = bending moment forcing bar Mc, = cracking moment AP, = area of prestressed reinforce- MDL = moment due to dead load ment MLL = moment due to live load A, = area of nonprestressed rein- N = normal force (positive if ten- forcement sion) b = width of member n = modular ratio = E8/Ee C = thickness of concrete cover s = bar spacing d = effective depth of member SCr = crack spacing dp = bar diameter Berm = mean crack spacing Ec = Youngs modulus of concrete T = temperature ES = Youngs modulus of steel AT = temperature change e = eccentricity w = crack width = tensile stress in concrete Wm = mean crack width ft, = percentile, p, of direct tensile w90 = 90 percentile of crack width strength of concrete (per- x ° = depth of neutral axis in cracked centile is that value of the sections quantity below or equal to aT = coefficient of thermal expansion which p percent of all mea- eet = tensile strain in concrete surements may be expected to em= mean strain over an average fall) length of cracked concrete .f; = strength of concrete in direct e° = steel strain in cracked stage tension (neglecting tension stiffening) lm = mean strength of concrete in A eg = reduction of steel strain by direct tension tension stiffening 08 = steel stress in cracked state p = reinforcement ratio 0scr = steel stress at first crack im- Pc = A8/ACe = inforcement ratio in mediately after cracking effective .2nsion zone of con- f. = yield strength of steel rein- crete forcement P„ to = minimum reinforcement ratio

NOTE: Discussion of this article is invited. Please submit your comments to PCI Headquarters by April 1, 1989.

144 APPENDIX B - DESIGN EXAMPLE

Given: A 12 in. (30.5 cm) square rein- 12 forced concrete section in direct tension. 2.5" f = 5800 psi (40 N/mmz) 14 60,000 psi (413 N/mmz) f,= 12° N = 100 kips (445 N) 7° Cover, c = 2 in. (5 cm) db = 0.875 in. (22 mm)

Required: Check the bar diameter to limit crack width at crack formation to 0.012 in. (0.305 mm).

Solution: Fig. 14 provides a chart relat- (0.178 mm) is obtained. Since this value ing bar diameter and effective rein- is less than the 0.012 in. (0.305 mm) re- forcement ratio for various limiting quired crack width limit in this example, crack widths and concrete strengths. the No. 7 bars provided are considered Calculate effective area, Ace: adequate for crack width control in this ACe = 6 x 6 = 36 in. 2 (232 cm2) case. Use of the chart is illustrated c= 0.6/36 = 0.0167 or 1.67 percent below. Entering the chart in Fig. 14 with A= From Fig. 14, a designer can also deter- 1.67 percent and No. 7 bar size, a crack mine that the steel stress at cracking is width, w90 , of approximately 0.007 in. approximately 30 ksi (207 N/mm2).

6 in. 3.5" <7.5db=7.5x7/8=6.6in.(ok) <8 in. (ok) 2.5" —J- Qy. 6 00

Obtained by interpolation

v m

Pe = 1.6 7 % .Qe

PCI JOURNAL/July-August 1988 145