14
FRACTURE MECHANICS ANALYSIS OF HIGH-STRENGTH CONCRETE By Reji John' and Surendra P. Shah, 2 Members, ASCE ABSTRACT: Concrete with compressive strength up to 140 MPa (20,000 psi) can now be economically produced. For these high-strength concretes it has been shown that one cannot use the same empirical relationships between compressive strength and other properties such as splitting tensile strength, flexural strength, shear strength, and bond strength as those currently being used, and different relationships for high-strength concrete have been proposed. In this paper, a fracture-mechanics- based theoretical model is used to predict various experimentally observed trends for high-strength concrete. The size-independent fracture parameters needed for this model can be derived from a single test. The proposed fracture-mechanics- based model satisfactorily predicts the variation of uniaxial tensile strength, split- cylinder strength, and modulus of rupture with compressive strength up to 110 MPa (16,000 psi). The relatively linear behavior of high-strength concrete is also predicted by the model. INTRODUCTION Due to its cost-effectiveness, the commercial use of high-strength concrete has been increasing rapidly during the past decade (ACI Committee 1984; Ahmad and Shah 1985; Shah 1979). ACI Committee 363 (1984) has defined high-strength concrete as that having compressive strength higher than or equal to 41 MPa (6,000 psi). Studying the differences in mechanical prop- erties and behavior between high-strength concrete and normal-strength con- crete, Carrasquillo et al. (1981) observed that cracking in high-strength con- crete is more localized and approaches homogeneous material behavior as compared to normal concrete, based on quantitative microcracking studies. Although it has been recognized that the response of high-strength concrete is relatively more linearly elastic and comparatively more brittle than nor- mal-strength concrete, no quantitative measure is available to express this observation. If a linear-elastic-fracture-mechanics-based single parameter such as K, c (critical stress intensity factor) is used to represent fracture toughness, then it is observed that the value of K lc increases with increasing compressive strength, implying an increased ductility with an increase in compressive strength. This is contrary to the observation in metals that with increasing strength, K lc decreases, thus indicating reduced toughness and hence in- creased brittleness (Barsom and Rolfe 1987). The fracture energy of con- crete, G f , has also been found to increase with increasing compressive strength (Nallathambi and Karihaloo 1986), thus making it not very useful as a single parameter. In this paper, applicability of the two parameter fracture model (TPFM) developed by Jenq and Shah (1985a, 1985b) for predicting the var- ious experimentally observed trends for high-strength concrete is shown. 'Assoc. Res. Engr., Univ. of Dayton, Res. Inst., Struct. Integrity Div., 300 Col- lege Park, Dayton, OH 45469-0001. 2 Prof. and Dir., NSF Sci. and Tech. Ctr. for Advanced Cement-Based Materials, Dept. of Civ. Engrg., Northwestern Univ., Evanston, IL 60208. Note. Discussion open until April 1, 1990. To extend the closing date one month, a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on January 24, 1989. This paper is part of the Journal of Materials in Civil Engineering, Vol. 1, No. 4, November, 1989. ©ASCE, ISSN 0899-1561/89/0004-0185/S1.00 + $.15 per page. Paper No. 24026. 185 J. Mater. Civ. Eng. 1989.1:185-198. Downloaded from ascelibrary.org by Mississippi State Univ Lib on 12/18/14. Copyright ASCE. For personal use only; all rights reserved.

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Page 1: Fracture Mechanics Analysis of High‐Strength Concrete

FRACTURE MECHANICS ANALYSIS

OF HIGH-STRENGTH CONCRETE

By Reji John' and Surendra P. Shah,2 Members, ASCE

ABSTRACT: Concrete with compressive strength up to 140 MPa (20,000 psi) can now be economically produced. For these high-strength concretes it has been shown that one cannot use the same empirical relationships between compressive strength and other properties such as splitting tensile strength, flexural strength, shear strength, and bond strength as those currently being used, and different relationships for high-strength concrete have been proposed. In this paper, a fracture-mechanics-based theoretical model is used to predict various experimentally observed trends for high-strength concrete. The size-independent fracture parameters needed for this model can be derived from a single test. The proposed fracture-mechanics-based model satisfactorily predicts the variation of uniaxial tensile strength, split-cylinder strength, and modulus of rupture with compressive strength up to 110 MPa (16,000 psi). The relatively linear behavior of high-strength concrete is also predicted by the model.

INTRODUCTION

Due to its cost-effectiveness, the commercial use of high-strength concrete has been increasing rapidly during the past decade (ACI Committee 1984; Ahmad and Shah 1985; Shah 1979). ACI Committee 363 (1984) has defined high-strength concrete as that having compressive strength higher than or equal to 41 MPa (6,000 psi). Studying the differences in mechanical prop­erties and behavior between high-strength concrete and normal-strength con­crete, Carrasquillo et al. (1981) observed that cracking in high-strength con­crete is more localized and approaches homogeneous material behavior as compared to normal concrete, based on quantitative microcracking studies. Although it has been recognized that the response of high-strength concrete is relatively more linearly elastic and comparatively more brittle than nor­mal-strength concrete, no quantitative measure is available to express this observation. If a linear-elastic-fracture-mechanics-based single parameter such as K,c (critical stress intensity factor) is used to represent fracture toughness, then it is observed that the value of Klc increases with increasing compressive strength, implying an increased ductility with an increase in compressive strength. This is contrary to the observation in metals that with increasing strength, Klc decreases, thus indicating reduced toughness and hence in­creased brittleness (Barsom and Rolfe 1987). The fracture energy of con­crete, Gf, has also been found to increase with increasing compressive strength (Nallathambi and Karihaloo 1986), thus making it not very useful as a single parameter. In this paper, applicability of the two parameter fracture model (TPFM) developed by Jenq and Shah (1985a, 1985b) for predicting the var­ious experimentally observed trends for high-strength concrete is shown.

'Assoc. Res. Engr., Univ. of Dayton, Res. Inst., Struct. Integrity Div., 300 Col­lege Park, Dayton, OH 45469-0001.

2Prof. and Dir., NSF Sci. and Tech. Ctr. for Advanced Cement-Based Materials, Dept. of Civ. Engrg., Northwestern Univ., Evanston, IL 60208.

Note. Discussion open until April 1, 1990. To extend the closing date one month, a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on January 24, 1989. This paper is part of the Journal of Materials in Civil Engineering, Vol. 1, No. 4, November, 1989. ©ASCE, ISSN 0899-1561/89/0004-0185/S1.00 + $.15 per page. Paper No. 24026.

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Page 2: Fracture Mechanics Analysis of High‐Strength Concrete

NONLINEAR FRACTURE MECHANICS MODEL

The tensile fracture of concrete is usually localized and dominated by a single macrocrack (Gopalaratnam and Shah 1985b; Jenq and Shah 1985a; John and Shah 1986; Maji and Shah 1986). For conditions where LEFM (linear elastic fracture mechanics) is applicable, one can calculate the critical stress intensity factor Kic from the notched beam tests using observed values of the maximum load of Pmax, and initial notch length, aa. For cement-based composites, since the precritical (prepeak) crack extension can be substan­tial, this stable crack growth (also often called fracture process zone or fic­titious crack) should be included in determining the fracture toughness value (Jenq and Shah 1985a; Maji and Shah 1986; John and Shah, unpublished 1989). It has been recognized that the exclusion of this precritical crack growth will yield values of Klc that will depend on specimen size and ge­ometry (Bazant 1984; Hillerborg et al. 1976; Hilsdorf and Brameshuber 1985; Jenq and Shah 1985a). The extent of the precritical crack growth cannot be accurately determined from the surface crack growth measurement or from the conventional compliance technique, since the crack front is tortuous and discontinuous (Diamond and Bentur 1985; Mindess and Diamond 1980; Swartz and Go 1984).

To overcome this difficulty, Jenq and Shah (1985a, 1985b) proposed an effective elastic crack length approach to obtain a valid fracture toughness value. The effective crack length a was defined such that the measured (elas­tic) crack-tip opening displacement (CTODc) was the same as that calculated using LEFM (Fig. 1). They observed that K,c and the corresponding CTODc values determined using the effective crack length were essentially indepen­dent of the size and geometry of the beams they tested (Jenq and Shah 1985a, 1985b). Recently Alvarado et al. (1989) validated these results using center-cracked plate specimens. The effective pre-critical stable crack growth as defined by Jenq and Shah (1985a, 1985b) is dependent upon the value of

(a)

'(2<rfX)V2

—»- x

Effect ive Gr i f f i t h Crack

/ Z1

/ ft"

pre-crit ical crack growth " | cr i t ica l point:

• ~ / * ~ \ KrKfc, CTOD= CTODc

\ post-critical \ y K[= K : C

(b) CMOD

Typical Plot of Load vs. Crack Mouth Opening Displacement

FIG. 1. Two-Parameter Fracture Model (Jenq and Shah 1985a, 1985b)

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Page 3: Fracture Mechanics Analysis of High‐Strength Concrete

a0, geometry of the specimen, measured Proax, and elastic crack mouth open­ing displacement at Pmax. Using this two-parameter fracture toughness model, Jenq and Shah (1985b) were able to explain many observed size-effect re­lated phenomena for cement composites. Note that the Klc defined by them is termed Ks

lc. It is possible to calculate the maximum load of a structure or a specimen

of a given geometry using the two parameters Klc and CTODc. In general, LEFM yields the following equations:

K, = CT- V^a-Fiia) (1)

4-u-a CMOD = VM) (2)

E

CTOD = CMOD -Z(a,p) (3)

where cr = applied stress; E = Young's modulus; D = depth of the speci­men; CMOD = crack mouth opening displacement; CTOD = crack tip open­ing displacement; a = a/D = notch-depth ratio; a = a0 4- Aap; a0 = initial notch length; Aap = pre-peak crack extension; p = a0/a; and Fit Vlt and Z = geometry correction factors. Combining Eqs. 1,2, and 3, we obtain

CTOD-E'VTT K, = Y(a) (4)

4-Va where

Y(a) = — (5) V,(a)-Z(a,P)

Knowing Kslc and CTODc, the peak stress a can be evaluated for any structure

using Eqs. 1 and 4. Note that for an unnotched specimen CTOD = CMOD (since a0 = 0, Z(a, P) = 1). Jenq and Shah (1985b) showed that the fracture parameters K,c and CTODc can be combined with E into a single parameter, Q, as shown in

/E-CTODC\2

The typical values of Q, which characterize a material length, are approx­imately 100 mm (4 in.) and 200 mm (8 in.) for mortar and concrete, re­spectively (Jenq and Shah 1985b).

For uniaxial tension and split cylinder tests, using the two-parameter frac­ture model, Jenq and Shah (1985a, 1985b) derived the following expression for the uniaxial tensile strength, /,' and split cylinder strength, f'sp:

f' or/,' = 7 • ——^-— (7) P J y ECTODc

where

7 = 1.47 for b, width of specimen > (0.54)2

(uniaxial tension test) (8) 187

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Page 4: Fracture Mechanics Analysis of High‐Strength Concrete

and

4 7 = — for d, diameter of specimen > 3<2 (split cylinder test) (9)

It should be noted that in the case of split-cylinder test, typically the diameter of the specimen is equal to 150 mm (6 in.) and hence 7 = 1.00 for concrete (Eq. 9) (Jenq and Shah 1986). It is difficult to perform purely uniaxial tests and some eccentricity is usually unavoidable during testing. The value of 7 in Eqs. 7 and 8 for different eccentricities was derived by Jenq and Shah (1986).

EFFECT OF HIGH-STRENGTH CONCRETE ON FRACTURE PARAMETERS, K)c AND CTODc

Experimental Program To study the effect of high-strength concrete on the fracture parameters

(Ks,c and CTODc), high-strength concrete specimens were tested. The com­pressive strength (f'c) of high-strength concrete was equal to 110 MPa (15,950 psi). The mix proportions and the experimental program are shown in Tables 1 and 2, respectively.

The beam specimens (see Fig. 2) were subjected to three point-bending tests—midspan deflection- and CMOD-controlled for unnotched and notched specimens. Static (slow-rate) tests were conducted using a 89-kN (20-kip) MTS closed-loop testing machine (John 1988; John and Shah, unpublished 1989). Stable post-peak responses were obtained for all static tests. A typical load-CMOD plot obtained at the static rate of loading is shown in Fig. 2. Unloading compliance was obtained very close to the peak load. Some of the experimental results are summarized in Table 2.

Test Results Young's modulus, E, was determined from the measured initial compli­

ance using Eq. 2 assuming linear elasticity. Jenq and Shah (1985a, 1985b) reported that E measured in this manner is the same as that obtained from compression tests in contrast to the low values of E calculated from load-

TABLE 1. Mix Proportions of High Strength Concrete"

Constituent (1)

Low alkali sulphate resistant cement Aggregate:

Quartz sand (1 /4-1 ram) Quartz sand (1-2 mm) Crushed diabas (2-8 mm)

Superplasticizer Water

Amount (kg/m3) (2)

511.0

330.5 447.1

1,166.4 12.8

110.0

"Compressive strength,/, = 110 MPa (15,950 psi). Note: Density = 2,323 kg/m3 (159 lb/cu ft); 1 lb/cu ft = 16.0 kg/m3; and 1 in. =

25.4 mm.

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Page 5: Fracture Mechanics Analysis of High‐Strength Concrete

TABLE 2. Experimental Program and Test Results

Type of specimen3

(1)

Unnotched Unnotched Unnotched Notched" Notched"

Depth of Specimen

mm (2)

79 51 25 76 76

in. (3)

3.1 2.0 1.0 3.0 3.0

Number of specimens

tested (4)

1 1 1 2 2

Peak Load

kN (5)

7.13 2.94 0.76 2.74 2.54

lb (6)

1,600 660 170 615 570

Modulus of Rupture,

MOR

MPa (7)

13.78 13.66 14.07

psi (8)

1,998 1,980 2,040

Elastic CMOD at Peak Load

mm (9)

0.0366 0.0310

x10~3 in. (10)

1.44 1.22

"Beam specimens: span = 203 mm (8 in.), thickness = 25 mm (1 in.); strain rate at critical section, k = 10 -6 sec"'; tests were conducted using 20-kip (89-kN) MTS closed-loop testing system.

"Notch at midspan, initial notch depth = 25 mm (1 in.).

midspan deflections plots obtained from bending tests. Gopalaratnam and Shah (1985a), and John and Shah (unpublished 1989) made similar obser­vations based on direct measurement of flexural strains using strain gages.

To obtain the fracture parameters Ks,c from the load-CMOD plot, Eqs. 1-3 are required for the present testing configuration. The beams tested in this study had a span-to-depth ratio (L/D) = 2.67. The geometry-correction fac­tors Fi(a), Vi(a), and Z(a,(3) in Eqs. 1-3 are available in handbooks (Tada et al. 1976) only for L/D = 4 and L/D = 8. Hence, an elastic finite element analysis was done for this purpose. Quarter-point singular elements were used to calculate the stress intensity factor. Based on the analysis, it was concluded that the geometry-correction factors for L/D = 2.67 were essen­tially the same as that of L/D = 4 (John 1988; John and Shah 1987).

Knowing the unloading compliance at (or very close to) the peak load, the fracture parameters (Ks

lc and CTODc) were evaluated using Eqs. 1-3. The effective elastic crack length was calculated such that the compliance calculated using Eq. 2 was equal to the experimentally measured unloading compliance. Then, substituting this value of crack length in Eqs. 1 and 3,

6 0 0

500

- , 4 0 0 xi

" 300 TJ o 3 200

100

0 1 2 3 4 Crack mouth opening displacement (CMODxIO in.)

FIG. 2. Experimental Load versus Crack Mouth Opening Displacement Plot for High-Strength Concrete Beam

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Page 6: Fracture Mechanics Analysis of High‐Strength Concrete

TABLE 3. Effect of Compressive Strength on Fracture Parameters of Concrete [Data Obtained from Present Study, and Jenq and Shah (1985b)]

Compressive Strength, / ;

MPa (1)

27.2 39.4 54.8

110.0

psi (2)

3,942 5,718 7,950

15,950

Critical .Stress Intensity Factor,

K'lc

MPaVm (3)

0.71 0.91 1.06 2.13

psiVirT (4)

644.1 816.3 963.5

1,936.5

Critical Crack Tip Opening Displacement,

CTODc

x10^3 mm (5)

9.1 9.7

10.0 11.2

x10~3 in. (6)

0.36 0.38 0.39 0.44

Young's Modulus E

GPa (7)

25.38 32.48 37.31 56.55

x10e psi (8)

3.68 4.71 5.41 8.20

KsiC and CTODc were calculated. These values, along with those of Jenq and Shah (1985a, 1985b), are reported in Table 3. The specimens tested in Jenq and Shah (1985a) had heterogeneity (in terms of aggregate size distribution) somewhat similar to those in the present study. It can be seen from Table 3 that Ks,c and CTODc increase with an increase mf'c, but the increase in CTODc is not as significant as that of Ks,c. Jenq and Shah (1985b) observed that CTODc varied from 0.00686 mm (0.00027 in.) for cement paste (with 5% sand) to 0.016 mm (0.00063 in.) for concrete (max. aggregate size = 13 mm, 0.5 in.). This implies that CTODc is predominantly dependent on the heterogeneity of the mix; i.e., the lesser the heterogeneity, the lower is the value of CTODc. Note that CTODc = 0 implies that LEFM is applicable to such materials.

The dependency of Kslc, CTODc, and E on the compressive strength, f'c,

is shown in Figs. 3-5. The corresponding relationships are given by the following empirical equations:

K'c = 1.3(/;f75 (10a)

CTODc = 1.24 X 10~4(/^)013 (10b)

E = 33(145)L5\/^ (10c)

with Kslc in psi Vin., CTODc in inches, E in psi, andf'c in psi, or

K'lc = 0.06(/;f75 (lla)

2500

2000

!•£ 1500

5-1000

"* 500

0 2 4 6 8 10 12 14 16 Compressive strength, f ' (x103psi)

FIG. 3. Effect of Compressive Strength on Critical Stress Intensity Factor, Kjc

@ " •

-

John Jenq

' 1 i i

and Shah and Shah

in. = 25.4 mm

1.3(fc'f75

1MPa 145

3--'

psi

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Page 7: Fracture Mechanics Analysis of High‐Strength Concrete

( j 0.30

0.25

-CTODc = 0.000124 ( f c ) a 1 3

1 in. = 2 5 . 4 m m , 1MPa = 145ps i _i i i i i i i L

0 2 4 6 8 10 12 14 16 Compressive s t rength , fc (x103 psi)

FIG. 4. Effect of Compressive Strength on Critical Crack Tip Opening Displace­ment, CTODc

'0 2 4 6 8 10 12 14 16 Compressive st rength, fc (x 103 psi)

FIG. 5. Effect of Compressive Strength on Young's Modulus, E

CTODc = 6.02 x 10"6(/;)0 1 3 (lift)

E = 4 ,785Vj^ (He)

with KS1C in MPaVm, CTODc in meters, E in MPa, a n d / ; in MPa.

Note that these relations are for normal-weight concrete. Based on the results obtained in this investigation, it was concluded that the ACI equation for determining E (Eqs. 10c and l i e ) is adequate for compressive strengths up tof'c — 110 MPa (= 16,000 psi)—Fig. 5. A similar observation has been made by Ahmad and Shah (1985).

Analysis of Effect of High-Strength Concrete Ahmad and Shah (1985) discussed in detail the behavior of high-strength

concrete and its application in reinforced concrete and precast prestressed concrete construction. They concluded that the ACI code equations for eval­uating split-cylinder strength (f'sp) and modulus of rupture (f'r) from com­pressive strength have to be modified for high-strength concrete. Carras-quillo et al. (1982) and Raphael (1984) made similar observations.

Uniaxial Tensile and Split Cylinder Strength Carrasquillo et al. (1982) proposed the following equation for split-cyl­

inder strength of concretes having compressive strengths between 21 MPa (3,000 psi) and 83 MPa (12,000 psi):

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Page 8: Fracture Mechanics Analysis of High‐Strength Concrete

1 0 0 0

01 Q .

8 0 0

C

"O c

600

400

200

0

analytical (d = 6in.)

_L _L

Walker and Bloem (6 x12 cy l inders) Houk (6"x 6" square p r i sms ) Gr ieb and Werner (6"x12"cyl inders) C a r r a s q u i l l o (.4 x 8 " c y I inders ) Ahmad (6 "x12 "cy l i nders) 1 in. : 2 5 . 4 m m , 1MPa=145ps i

I I I i I I 0 2 4 6 8 10

Compressive strength. fc' (x103psi) 12

FIG. 6. Split-Cylinder Strength Variation with Compressive Strength: Compari­son of Analytical and Experimental Results

/ ; = i.Wfi psi r / ; = 0 . 5 9 ^ MPa (12)

Ahmad and Shah (1985) proposed a mean curve, Eq. 13, for split-cylinder strength based on available data

/ ; = 4.34(/;) a 5 5 psi o r / ; = 0.46(/;f55 MPa (13)

Eq. 12 proposed for high-strength concrete is compared to data from Ah­mad and Shah (1985), Carrasquillo et al. (1982), Grieb and Werner (1962), Houk (1965), and Walker and Bloem (1960) in Fig. 6. Also shown is an equation often used for normal-strength concrete (f'sp = 6\/f[). Combining Eqs. 7 - 1 1 , we can theoretically obtain the following expression for uniaxial or split-cylinder strength:

/, ' ( o r ) / ; = (0 .237) - 7 - ( / ; f 8 7 psi (14a)

or

/,' ( o r ) / ; = (0.125) • 7 • (f'c)M1 MPa (14ft)

Most of the available results on the splitting test are for cylinders with diameter d = 152 mm (6 in.). As discussed earlier, this corresponds to y = 1 in Eqs. \Aa~b. Thus using Eqs. \Aa-b, the predicted response clearly shows the transition from normal-strength concrete to high-strength concrete (Fig. 6). It should be noted that the measured split-cylinder strength may not be independent of testing method and specimen size. The value of y in

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Page 9: Fracture Mechanics Analysis of High‐Strength Concrete

2000

„ 1500

Q-

J l 1000 I_

o " ^ 500

" 0 2 4 6 8 10 12 14 16 Compressive strength, fc' ( x10 3 ps i )

FIG. 7. Comparison of Uniaxial Tensile Strength and Split-Cylinder Strength Variation with Compressive Strength

Eqs. 7 and \Aa-b is dependent on the diameter of the cylinder. The mea­sured load is also affected by the geometry of the bearing area between the load cell and the cylinder during testing (Jenq and Shah 1986).

The uniaxial tensile strength,/,' can be theoretically predicted by assuming that cracks initiate simultaneously on both sides of a plate specimen and propagate towards the center of the specimen (Jenq and Shah 1985b). Eqs. 7 and 8 were derived based on this assumption. The predicted curves for /, ' a n d / ^ variation w i t h / ; are shown in Fig. 7. It can be observed that/ , ' is always greater than f'sp. It is also interesting to note that the predicted variation of /, ' with f'c is quite close to the general assumption of /, ' ~ (0.1) -/c- For cement-based composites, it is very difficult to conduct stable tests under uniaxial tension. Gopalaratnam and Shah (1985b) developed a deflection-controlled testing scheme to obtain stable uniaxial tension-defor­mation of plain and fiber-reinforced concrete. The data obtained from Go­palaratnam and Shah (1985b) are also plotted in Fig. 7. The data lie in between the predicted split-cylinder and uniaxial tension curves. This could be attributed to the following: (1) The specimens tested did not satisfy the condition in Eq. 8; and (2) even a very small eccentricity in loading, for example 5-10%, could result in a load reduction of 25-50% (Jenq and Shah 1986). The predicted tensile strength of a uniaxial tensile specimen with an arbitrarily selected eccentricity, e equal to (0.05)&, is also shown in Fig. 7.

Flexural Strength or Modulus of Rupture For determining the flexural strength (modulus of rupture, f'r) of beams

based on the corresponding compressive strength, ACI code provides the following expression:

/ ; = 7 .5V/ I psi o r / ; = 0.62\/fl MPa (15)

Carrasquillo et al. (1982) modified Eq. 15 to include the effect of high-strength concrete as follows:

/ ; = U.iy/fi psi o r / ; = 0 . 9 4 ^ MPa (16)

Ahmad and Shah (1985) concluded that the equation proposed by Raphael (1984) is valid for concretes up t o / ; ~ 83 MPa (12,000 psi). This equation is

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Page 10: Fracture Mechanics Analysis of High‐Strength Concrete

1200[-

1000

l/l CL

-^ *> L-M—

flT I_ 3

Q. 3

cr o

3

3 X}

o 2

800

600

400

200

0

analytical.

fr = 11.7/fTN

%o

r = 7.5/fT(ACI)

- ^ 1

Gonnerman and Shuman (7"x10beams) a Wa lker and B loem ( 6 ' x 6 " b e a m s ) A HouK (6"x 6 " beams ) v Gr ieb and Werner (6"x 6"x 21" beams) o Car rasqu i l l o (4"x 4 " x 1 4 " beams) • Khqjoo (6"x 6"x 20" beams)

1 in.= 2 5 . 4 m m , 1MPa = 145ps t

J_ _L _L 8 10

Compressive strength. Vc ( x 1 0 3 p s i ) 12

FIG. 8. Effect of Compressive Strength on Modulus of Rupture: Comparison of Analytical and Experimental Results

/ ; = 2.3(/;)2/3 psi or/ ; = 0.44(/;)2/3 MPa (17)

For a given compressive strength, Kslc, CTODc, and E are given by Eqs.

lOfl-llc. Substituting these in Eqs. 1 and 4, MOR (modulus of rupture) can be calculated for beams of different depths and span-depth ratios. The pre­dicted results for beam depth = 6 in. and span-to-depth ratio equal to 4 are plotted along with data from Ahmad and Shah (1985), Carasquillo et al. (1982), Gonnerman and Shuman (1928), Grieb and Warner (1962), Houk (1965), Khaloo (1986), and Walker and Bloem (1960), in Fig. 8. The tran­sition from normal-strength concrete to high-strength concrete is predicted well.

The proposed model predicted that the modulus of rupture is always higher than the uniaxial tensile strength and that the relationship between MOR and f'c is not unique. There is significant size effect on MOR as predicted by Jenq and Shah (1985b). The value of MOR approaches/,' as the depth of the beam increases, as shown in Fig. 9. The predicted trend is also con­firmed by the CEB-FIP code equation, as shown in Fig. 9. The predicted influence of compressive strength on the size effect on MOR is also shown

194

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Page 11: Fracture Mechanics Analysis of High‐Strength Concrete

1.3

1.2

O 5 1.1

1.0 0 4 8 ' 12 16 20 24 28 32 36 40

Depth of beam (in.)

FIG. 9. Size Effect on Modulus of Rupture

- \ h«—L£_ ^ .U-UJ V all dimensions in inches

"1 inch= 25,4 mm •— -1 psi = 0.0069 MPa

i I i i i I i I i I _ _ 0 2 4 6 8 , 10 12

Compressive strength. f c (x10 psi)

FIG. 10. Effect of Compressive Strength on Prepeak Nonlinearity

by the solid ( / ; = 27.6 MPa, 4,000 psi) and dashed (f'c = 110 MPa, 16,000 psi) lines in Fig. 9. High-strength concrete has a lesser size effect than nor­mal-strength concrete, although the difference is not significant.

Prepeak Nonlinearity As discussed earlier, the prepeak nonlinearity has been attributed to crack

growth. The two-parameter fracture model (TPFM) was developed on that basis. It should be noted that TPFM reduces to LEFM if prepeak (precritical) crack growth, Aap (= a — a0) is equal to 0.0. For a particular geometry of the SEN beam specimen, variation of prepeak crack growth with/c is plotted in Fig. 10. Prepeak nonlinearity (Aap) decreases with increasing/^. Hence the proposed theory predicts that high-strength concrete behaves more lin­early as compared to normal-strength concrete, as has been experimentally observed by many investigators (ACI Committee 363 1984; Ahmad and Shah 1985; Carrasquillo et al. 1982; Shah 1979).

COMMENTS

All of the foregoing remarks were based on K]c and CTODc dependency on f[, as described by Eqs. lOa-b and lla-b. These equations were ob­tained from relatively few tests on mortar-type specimens. More experiments are required to establish these relationships for both mortar and concrete specimens. The important point is that it is possible to predict various ex­perimentally observed phenomena using these two parameters that can be obtained from a single test. These two parameters provide a rational method

A \ *\ \ ~ * \ \

1 1 1 1

Predicted Size

Effect

_____ - — -

(psi)

4000

16000

<t (psi)

4 7 4

t584

CEB-FIP

7 ~ — r-'2'5"='™=l^»B*__

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Page 12: Fracture Mechanics Analysis of High‐Strength Concrete

of predicting the influence of composition and microstructure on various fracture properties of concrete.

CONCLUSIONS

1. The two-parameter fracture model has been successfully used to study the tensile and flexural fracture of high-strength concrete. This model is based on the observation that to evaluate a size-independent fracture parameter or param­eters for concrete, one must include pre-critical (pre-peak) stable crack growth.

2. The size-independent fracture parameters, critical stress intensity factor at the tip of an effective crack (Kjc), and critical crack tip opening displacement {CTODc) have been determined to depend on the static uniaxial compressive strength of concrete. Empirical relations were derived for concretes having com­pressive strength up to 110 MPa (16,000 psi). The ACI equation for determining Young's modulus was found to be adequate for high-strength concrete.

3. Using the proposed fracture mechanics model, the variation of uniaxial tensile strength, split-cylinder strength, and modulus of rupture with compressive strength was predicted satisfactorily. The present ACI code equations are inad­equate for high-strength concrete in this regard, as also reported by other in­vestigators .

4. The proposed model predicts that true split-cylinder strength (f'sp) is always less than the true uniaxial tensile strength (/,'). The model also predicts that for certain combinations of the size of specimens tested and eccentricities involved in testing procedures, f'sp could be higher than/, ' .

5. Size effect should be considered while testing specimens to determine the mode-I fracture strength.

6. The relatively linear behavior of high-strength concrete as compared to nor­mal-strength concrete is also predicted by the proposed model.

ACKNOWLEDGMENTS

The research reported here was supported in part by grant number AFOSR-88-C-0118 from the Air Force Office of Scientific Research (Program Man­ager: Dr. Spencer T. Wu) and from NSF-DMR-8808432 (Program Manager: Dr. Lance Haworth) to Northwestern University. The first writer gratefully acknowledges the support provided by the University of Dayton Research Institute (Division Supervisor: Dr. Joseph P. Gallagher) in the preparation of the manuscript. Thanks are due to Mr. Thorkild Bach of Aalborg Port­land, Denmark, for providing the high-strength concrete specimens.

APPENDIX. REFERENCES

ACI Committee 363. (1984). "State-of-the-art report on high strength concrete." ACI J., 81(4), 364-411. '

Ahmad, S. H., and Shah, S. P. (1985). "Structural properties of high strength con­crete and its implications for precast prestressed concrete." PCI J., 30(6), 92-119.

Alvarado, M., Shah, S. P., and John, R. (1989). "Mode I fracture in concrete using center cracked plate specimens." J. Engrg. Mech., ASCE, 115(2), 366-383.

Barsom, J. M., and Rolfe, S. T. (1987). Fracture and Fatigue Control in Structures. Prentice Hall, Inc., Englewood Cliffs, N.J.

Bazant, Z. P. (1984). "Size effect in blunt fracture: Concrete, rock, metal." J. Engrg. Mech., ASCE, 110(12), 1666-1692.

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Carrasquiilo, R. L., Nilson, A. H., and Slate, F. O. (1982). "Properties of high strength concrete subjected to short-term loads." ACI J., 78(3), 171-178; Reply to discussion, 79(2), 162-163.

Carrasquiilo, R. L., Slate, F. O., and Nilson, A. H. (1981). "Microcracking and behavior of high strength concrete subjected to short term loading." ACI J., 78(3), 179-186.

Diamond, S., and Bentur, A. (1985). "On the cracking in concrete and fiber rein­forced cements." Application of Fracture Mechanics to Cementitious Composites, S. P. Shah, ed., Martinus Nijhoff Publishers, The Netherlands, 361-397.

Gonnerman, H. F., and Shuman, E. C. (1928). "Compression, flexural and tension tests of plain concrete." Proc, ASTM, 28(11), 527-564, 1928.

Gopalaratnam, V. S., and Shah, S. P. (1985a). "Properties of steel fiber reinforced concrete subjected to impact loading." ACI J., 83(1), 117-126.

Gopalaratnam, V. S., and Shah, S. P. (1985b). "Softening response of plain concrete in direct tension." ACI J., 82(3), 310-323.

Grieb, W. E., and Werner, G. (1962). "Comparison of the splitting tensile strength of concrete with flexural and compressive strengths." Public Roads, 32(5), 97-106.

Hillerborg, A., Modeer, M., and Petersson, P. E. (1976). "Analysis of crack for­mation and crack growth in concrete by means of fracture mechanics and finite elements." / . Cement and Concr. Res., 6(6), 773-782.

Hilsdorf, H. K., and Brameshuber, W. (1985). "Size effects in the experimental determination of fracture mechanics parameters." Application of Fracture Me­chanics to Cementitious Composites, S. P. Shah, ed., Martinus Nijhoff Publishers, The Netherlands, 361-397.

Houk, A. (1965). "Concrete aggregate and concrete properties investigations, Dwor-shak Dam and Reservoir." Design Memorandum No. 16, U.S. Army Engr. Dis­trict, Walla Walla, Wash., 203-212.

Jenq, Y. S., and Shah, S. P. (1985a). "A fracture toughness criterion for concrete." Engrg. Fracture Mech., 21(5), 1055-1069.

Jenq, Y. S., and Shah, S. P. (1985b), "A two parameter fracture model for con­crete." / . Engrg. Mech., ASCE, 111(10), 1227-1241.

Jenq, Y. S., and Shah, S. P. (1986). Discussion of "Application of the highly stressed volume approach to correlated results from different tensile results of concrete," by R. J. Torrent and J. J. Brooks," Magazine of Concr. Res., 38(136), 168-172.

John, R. (1988). "Mixed mode fracture of concrete subjected to impact loading." Thesis presented to Northwestern University, at Evanston, 111., in partial fulfill­ment of the requirements for the degree of Doctor of Philosophy.

John, R., and Shah, S. P. (1986). "Fracture of concrete subjected to impact loading." / . Cement, Concr. and Aggregates, 8(1), 24-32.

John, R., and Shah, S. P. (1987). "Effect of high strength concrete and rate of loading on fracture parameters of concrete." Presented at SEM-RILEM Int. Conf. on Fracture of Concr. and Rock, Houston, Tex., June.

Khaloo, A. R. (1986). "Behavior of concrete under triaxial compression and shear loading." Thesis presented to North Carolina State University, at Raleigh, N.C., in partial fulfillment of the requirements for the degree of Doctor of Philosophy.

Maji, A., and Shah, S. P. (1986). "A study of fracture process of concrete using acoustic emission." Proc. Soc. of Exp. Mech. Spring Conf., New Orleans, La., June.

Mindess, S., and Diamond, S. (1980). "A preliminary SEM study of crack propa­gation in mortar." Cement and Concr. Res., 10(July), 509-519.

Nallafhambi, P., and Karihaloo, B. L. (1986). "Influence of slow crack growth on the fracture toughness of plain concrete." Fracture Toughness and Fracture En­ergy of Concrete, F. H. Wittmann, ed., Elsevier Science Publishers, The Neth­erlands, 271-280.

Raphael, M. J. (1984). "Tensile strength of concrete." ACI J., 81(2), 158-165. Shah, S. P., ed. (1979). High strength concrete, Proc, Workshop held at Univ. of

Illinois, Chicago, 111., Dec.

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Swartz, S. E., and Go, C. G. (1984). "Validity of compliance calibration to cracked concrete beams in bending." Exp. Mech., 24(2), 129-134.

Tada, H., Paris, P. C , and Irwin, G. R. (1976). The stress analysis of cracks hand­book. Del Research Corp., Hellertown, Pa.

Walker, S., and Bloem, D. L. (1960). "Effects of aggregate size on properties of concrete." ACI J., 57(3), 283-298.

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