Effect of the concrete compressive strength and tensile reinforcement ratio on the flexural behavior of fibrous concrete beams

Engineering Structures 22 (2000) 1145–1158 www.elsevier.com/locate/engstruct

Effect of the concrete compressive strength and tensile reinforcement ratio on the flexural behavior of fibrous concrete beams Samir A. Ashour *, Faisal F. Wafa, Mohmd I. Kamal Civil Engineering Department King Abdulaziz University, Jeddah, Saudi Arabia Received 18 January 1999; received in revised form 22 April 1999; accepted 4 June 1999

Abstract Twenty seven reinforced concrete beams were tested to study the effects of steel fibers, longitudinal tensile reinforcement ratio and concrete compressive strength on the flexural behavior of reinforced concrete beams. Concrete compressive strengths of 49, 79 and 102 MPa and tensile reinforcement ratios of 1.18, 1.77 and 2.37% were used. The fiber contents were 0.0, 0.5 and 1.0% by volume. The results show that the additional moment strength provided by fibers was not affected by the amount of tensile reinforcement ratio. However, the concrete compressive strength influenced the fiber contribution significantly. The flexural rigidity increases as the concrete compressive strength and steel fiber content increases. The transition of effective moment of inertia from uncracked to fully cracked sections depends strongly on the considered variables. A previously proposed formula in the literature for the estimation of the effective moment of inertia is modified to consider the effect of reinforcement ratio and concrete compressive strength as well as fiber content.  2000 Published by Elsevier Science Ltd. All rights reserved. Keywords: Beams (supports); Compressive strength; Cracking; Deflection; Flexural strength; Flexural rigidity; High-strength concrete; Moment of inertia; Reinforced concrete; Steel fibers; Tensile reinforcement ratio

1. Introduction The maximum potentiality of high-strength concrete (HSC) cannot be realized fully in structures due to the brittleness of the material and the serviceability problems associated with the resulting reduced cross-sectional dimension. Addition of fibers to high-strength concrete converts its brittleness into a more ductile behavior. When concrete cracks, the randomly oriented fibers arrest both microcracking and its propagation, thus improving strength and ductility. Addition of fibers only slightly influences the ascending portion of the stressstrain curve but leads to a noticeable increase in the peak strain (strain at peak stress) and a significant increase in ductility [1,2]. Researches conducted on the flexural behavior of fiber reinforced concrete (FRC) beams have been concen-

* Corresponding author. Tel.: +966-2-695-2488; fax: +966-2-6952179. E-mail address: [email protected] (S.A. Ashour).

trated on the prediction of the ultimate flexural strength and the load deformation behavior in terms of various material parameters [3–15]. Less attention was given to the flexural rigidity of FRC beams in the service load range. Several methods have been proposed for calculating the deflections of reinforced concrete flexural members subjected to short and long-term loadings [16–20], however, those methods deal mainly with nonfibrous concrete, and differences may exist for FRC beams. The determination of short-term deflection requires the estimation of the moment of inertia, I, of the beam which depends on the degree of cracking that has taken place in the member. For loads below the cracking load, computation of deflection may be based on the gross concrete section, Ig. However, as the load increases above the cracking load, the member will crack at discrete intervals because the tensile strength of the concrete has been exceeded, and all tensile stress is carried by the steel reinforcement. The neutral axis will fluctuate between cracks causing variation of the curvatures along the member length and reducing the flexural rigidity of the section. The value of I changes along the beam span

0141-0296/00/$ - see front matter  2000 Published by Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 1 - 0 2 9 6 ( 9 9 ) 0 0 0 5 2 - 8

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Nomenclature a Shear span Area of longitudinal tension reinforcement As b Width of beam section c Neutral axis depth from the top surface of the beam d Effective depth h Thickness of beam section Ec Modulus of elasticity of concrete Modulus of elasticity of concrete (with steel fibers) Ecf Modulus of elasticity of steel reinforcement Es (Ec.Icr)exp Experimental flexural rigidity of the cracked section Compressive strength of concrete (at 28 days) f’c Modulus of ruputre of concrete fr f’sp Splitting tensile strength of concrete Icr(exp) Experimental moment of inertia of cracked transformed section Effective moment of inertia Ie Iexp Experimental moment of inertia Moment of inertia of gross concrete section ignoring reinforcement Ig Iut Moment of inertia of untracked transformed section l Clear span of beam m Power in Branson’s equation Ma Maximum bending moment in the span Mcr(exp) Experimental cracking moment My(exp) Experimental yield moment Mu(exp) Experimental ultimate moment carried by the section Steel fiber content Vf Distance from tensile fibers to neutral axis yt r Tension reinforcement ratio=As/bd ⌬ Deflection at mid span

from a maximum value of Ig for the untracked (gross) section to a minimum value of Icr for the fully cracked (transformed) section. This variation of I along the span length makes the deflection calculation lengthy and tedious and makes the accurate determination of deformation from the moment-curvaturerelationships in the elastic range difficult. Hence, in a cracked member, it is desirable to consider an effective moment of inertia, Ie, that will have a value between those derived by cracked and untracked sections. To provide a smooth transition between the moments of inertia Ig and Icr, the ACI Building Code [21] has adopted since 1971, the expression developed by Branson [18] for the computation of the effective moment of inertia Ie over the entire length of a simply supported beam. ACI 318-95 [22] recommends the use of the following expression for the effective moment of inertia Ie⫽

冉 冊 冋 冉 冊册 Mcr 3 Mcr Ig⫹ 1⫺ Ma Ma

3

Icr

(1)

where Ma=maximum moment in member at stage deflection is computed, kN.m; Mcr=cracking moment of beam=frIg/yt, kN.m; fr=modulus of rupture; yt=neutral

axis depth from the bottom tension side of the beam. The effective moment of inertia Ie is estimated using Eq. (1) when Ma⬎Mcr; otherwise Ie=Ig. The effect of concrete compressive strength, f’c, and tensile reinforcement ratio, r on the flexural behaviour of reinforced concrete beams has been investigated by Ashour [23]. It has been found that flexural rigidity increases as f’c increase. The exponent in Branson’s equation Eq. (1) also increases as f’c increases. This yields a faster decay rate of the effective moment of inertia, Ie, from the untracked transformed moment of inertia, Iut, to the fully cracked section, Icr, as Ma/Mcr increases above one. The inclusion of steel fibers in high-strength concrete beams enhances the arresting mechanism of crack propagation and consequently enhances the effective moment of inertia to be used in the deflection calculation. The effect of steel fibers on the serviceability and ultimate strength of reinforced high-strength concrete beam has been reported by Ashour et al. [24]. The exponent in Branson’s equation was related to the amount of fiber content, and the exponent decreases as Vf increases. The objective of this research is to investigate the

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Fig. 1.

Details of test beams and testing arrangement.

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Table 1 Concrete mix proportioning Mix Mix Designation Proportion C:FA:CA (1) (2)

W/C Ratio

N M H

1:1:2 1:1.2:1.8 1:1:2

Silica Fume (%) (5)

f’c (MPa)

(3)

Superplasticizer (%) (4)

0.37 0.24 0.23

2 6 6

0 0 20

49 79 102

(6)

Table 2 Experimental modulus of elasticity, Ec (MPa) Vf(%)

Fig. 3.

Mix Designation (1)

0.0 (2)

0.5 (3)

1.0 (4)

N M H

24612 35443 38423

26823 37169 40241

30131 38247 41889

Experimental secant modulus of concrete.

2. Experimental program

effect of concrete compressive strength, tensile reinforcement ratio and steel fiber content on the deflection and strength of reinforced concrete beams. Modifications to a previously proposed formula for the effective moment of inertia are presented.

Fig. 2.

2.1. Test specimen Twenty seven fiber reinforced concrete beams were tested in this investigation. All beams were singly reinforced and provided with shear reinforcement except at the constant moment zone. The variables were the concrete compressive strength, f’c, the steel fiber content, Vf, and the longitudinal tensile reinforcement ratio, r. The compressive strengths used were 49, 79 and 102 MPa, the fiber contents were 0.0, 0.5 and 1.0% by vol-

Compressive stress-strain diagram of concrete cylinders.

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Table 3 Mechanical properties of test beams Beam (1)

As (2)

r (%) (3)

Vf (%) (4)

f’c (MPa) (5)

fr (MPa) (6)

f’sp (MPa) (7)

B-0.0-N2 B-0.5-N2 B-1.0-N2 B-0.0-N3 B-0.5-N3 B-1.0-N3 B-0.0-N4 B-0.5-N4 B-1.0-N4 B-0.0-M2 B-0.5-M2 B-1.0-M2 B-0.0-M3 B-0.5-M3 B-1.0-M3 B-0.0-M4 B-0.5-M4 B-1.0-M4 B-0.0-H2 B-0.5-H2 B-1.0-H2 B-0.0-H3 B-0.5-H3 B-1.0-H3 B-0.0-H4 B-0.5-H4 B-1.0-H4

2φ18 2φ18 2φ18 3φ18 3φ18 3φ18 4φ18 4φ18 4φ18 2φ18 2φ18 2φ18 3φ18 3φ18 3φ18 4φ18 4φ18 4φ18 2φ18 2φ18 2φ18 3φ18 3φ18 3φ18 4φ18 4φ18 4φ18

1.18 1.18 1.18 1.77 1.77 1.77 2.37 2.37 2.37 1.18 1.18 1.18 1.77 1.77 1.77 2.37 2.37 2.37 1.18 1.18 1.18 1.77 1.77 1.77 2.37 2.37 2.37

0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0

48.61 55.82 65.16 48.61 55.82 65.16 48.61 55.82 65.16 78.50 81.99 87.37 78.50 81.99 87.37 78.50 81.99 87.37 102.40 106.93 111.44 102.40 106.91 111.44 102.40 106.93 111.44

5.64 5.88 7.95 5.64 5.88 7.95 5.64 5.88 7.95 7.04 7.24 9.75 7.04 7.24 9.75 7.04 7.24 9.75 9.36 10.13 11.23 9.36 10.13 11.23 9.36 10.13 11.23

3.69 4.67 6.72 3.69 4.67 6.72 3.69 4.67 6.72 5.05 6.01 7.69 5.05 6.01 7.69 5.05 6.01 7.69 5.59 6.53 8.13 5.59 6.53 8.13 5.59 6.53 8.13

ume, and the longitudinal tensile reinforcement ratios used were 1.18, 1.77 and 2.37%. Fig. 1 presents the detailed testing program. Each beam is designated to indicate the fiber content, compressive strength level and amount of longitudinal reinforcement. Thus, Beam B-1, 0-M3 represents a beam with 1.0% fiber content, medium compressive strength of approximately 79 MPa, and three 18 mm diameter steel bars that provide a reinforcement ratio of 1.77%. 2.2. Materials Deformed steel bars having yield strength of 530 MPa (76 800 psi) were used as flexural reinforcements. Three concrete mix proportions were used to provide the required compressive strengths as presented in Table 1 Ordinary Portland cement (Type-I), desert sand with a fineness modules of 3.1, and coarse aggregate (crushed basalt) of 10 mm (3/8 in.) maximum size were used. Light gray densified microsilica (20% by weight of cement) with a specific gravity of 2.2, a bulk density of 6.0 kN/m3 (37.4 lb/ft3) and a specific surface of 2.3 m2/g was used for the high-strength concrete mix (H). Hooked-ends mild carbon steel fibers with average length of 60 mm (2.36 in.), nominal diameter of 0.8 mm (0.03 in.), aspect ratio of 75 and yield strength of 1100 MPa (159 500 psi) were used. A superplasticizer was

used, and enough mixing time was allowed to produce uniform mixing of concrete without any segregation. Six 150×300 mm (6×12 in.) cylinders were cast to determine the concrete compressive and splitting tensile strengths. Additionally, three 150×150×530 mm (6×6×21 in.) prisms were cast to determine the modulus of rupture of the concrete used. The concrete was placed in three layers and was vibrated internally and externally immediately afterward. All beams and control specimens were cast and cured under similar conditions. The beams and specimens were kept covered with polyethylene sheets for 28 days until 24 hours before testing. 2.3. Test procedure The test beams were simply supported and were subjected to two-point loads as shown in Fig. 1. The distance between the two loading points was kept constant at 500 mm (20 in.). The beam midspan deflection and the end rotation were measured with the help of transducers. Strains in the tensile steel were measured by electrical foil-type strain gages. Compressive strains at the center of the top surface of the concrete at three locations were measured with electrical resistance wire-type strain gages. These gages were located in the constant moment zone at midspan. The load was applied in 25 to 35 increments up to failure by means of a 400 kN (90 kips)

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hydraulic testing machine. At the end of each load increment, observations and measurements were recorded for the midspan deflection, end rotation, strain gage readings, and crack development and propagation on the beam surfaces.

3. Experimental results 3.1. Compressive stress-strain diagram Fig. 2 shows the stress-strain diagram of 150×300 concrete cylinder tested in compression. The effect of steel fibers is obvious on the stress-strain diagram especially for the lower strength concrete (f’c=49 MPa). As the fiber content increases the maximum compressive strength increases slightly, and the slope of the ascending portion increases accordingly. The ascending part of the high-strength concrete (f’c=102 MPa) is almost a straight line up to the maximum compressive strength. The concrete secant modulus, Ec, was evaluated at a stress level of 0.5 f’c and is given in Table 2 and Fig. 3. The secant modulus of concrete, Ec, is increased as Vf and f’c increase. The increase of the fiber contents from 0.0 to 1.0% increases Ec by 22.4, 7.9 and 9.0% for concrete with f’c of 49, 79 and 102 MPa, respectively. Table 3 and Fig. 4 present the mechanical properties of the FRC used in the test beams. The increase of the fiber contents from 0.0 to 1.0% increases the compressive strength by 34.0, 11.3 and 8.8%, increases the modulus of rupture by 41.0, 38.5 and 20.0%, and increases the splitting tensile strength by 82.1, 52.3 and 45.4% for concrete with 49, 79 and 102 MPa compressive strengths, respectively. 3.2. Flexural behavior

Fig. 4. Mechanical properties of test beams (a) Concrete compressive strength, f’c; (b) Modulus of rupture, fr; (c) Splitting tensile strength, f’sp.

The test beams were designed to fail in flexure. All beams exhibited vertical flexural cracks in the constantmoment region before final failure of the beams due to crushing of concrete. The presence of fibers reduced the crack width, increased the number of cracks, increased the ductility, and delayed the final crushing of concrete. The effectiveness of steel fibers in arresting cracks is related to the average spacing of fibers inside the matrix. The spatial distribution and orientation of fibers in FRC beams are random, however, boundary conditions such as edges constrain the fiber orientation in a uni-directional alignment. Fig. 5 shows the load versus deflection relationships for all test beams. The test results clearly show the fibers contribution on the stiffnesses and strengths of all beams. The fibers have a clear enhancement of the post cracking stiffness and ductility (area under P-⌬ curve) for all beams. Table 4 presents the experimental cracking moment, Mcr(exp), the moment at first yielding of the flexural reinforcement, My(exp), and the ultimate moment, Mu(exp), of the test beams. Test results show that the increase of Vf increase Mcr(exp), My(exp), and Mu(exp), for all test beams irrespective to the f’c and r values. However, the increase due to the presence of fibers is reduced as r increases.

S.A. Ashour et al. / Engineering Structures 22 (2000) 1145–1158

Fig. 5.

Load deflection curves for beams.

The additional moment enhancement at ultimate stage due to the presence of steel fiber, Mf, can be estimated as the difference between the ultimate moments of beams with Vf of 0.0 and 1.0%, and are shown in Table 5 and Fig. 6. The figure shows that the increase of f’c increases Mf, however, a lower rate of increase is noticed when f’c increases from 79 to 102 MPa. Fig. 6 also shows that the value of r has no effect on Mf. The enhancement of the flexural capacities varies between 7.52 and 26.43%. 3.3. Cracking moment The analytical evaluation of deflection depends greatly on the cracking moment of the beam. The theoretical cracking moment Mcr(th) is estimated as: fr·Ig Mcr(th)⫽ yt

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(2)

The use of the untracked transformed moment of inertia, Iut, instead of Ig in Eq. (2) will give a better prediction of Mcr(th), Fig. 7 shows the variation of Mcr(exp)/Mcr(th) ratio as a function of fiber content and concrete com-

pressive strength for the different reinforcement ratios. The figure shows that the experimental cracking moments are about 55 to 85% of the theoretical cracking moments calculated using the modulus of rupture values. Thus, the use of fr, to calculate the cracking moment Eq. (2) overestimated the experimental cracking moments, and this overestimation increases as the concrete compressive strength increases. This is attributed to the size effect phenomena. For normal and medium concrete strengths, the value of Mcr(exp)/Mcr(th) ratio increases as Vf increases from 0.0 to 0.5%, and thereafter decreases as Vf increases to 1.0%. 3.4. Neutral axis depth The experimental neutral axis depth, c, of the test beams is obtained from the experimentally measured strain values in the concrete and the tensile reinforcement. The variation of the ratio of c to the effective depth of the section, d, in the constant moment zone is shown in Fig. 8. For loading levels below the cracking load, Ma/Mcr=1, the c/d ratio is about 0.6. When cracks occurred, the neutral axis shifted upward and the c/d value drops to a value of about 0.4 and remains constant

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Table 4 Experimental and theoretical results of test beams Beam (1)

Mcr(exp) (kN- My(exp) (kN- Mu(exp) (kN- Icr(exp) m) (2) m) (3) m) (4) (mm4×103) (5)

B-0.0-N2 B-0.5-N2 B-1.0-N2 B-0.0-N3 B-0.5-N3 B-1.0-N3 B-0.0-N4 B-0.5-N4 B-1.0-N4 B-0.0-M2 B-0.5-M2 B-1.0-M2 B-0.0-M3 B-0.5-M3 B-1.0-M3 B-0.0-M4 B-0.5-M4 B-1.0-M4 B-0.0-H2 B-0.5-H2 B-1.0-H2 B-0.0-H3 B-0.5-H3 B-1.0-H3 B-0.0-H4 B-0.5-H4 B-1.0-H4

8.02 9.39 9.61 8.64 9.92 11.51 9.82 11.29 11.51 8.97 9.82 11.51 9.81 10.97 11.82 10.56 12.56 13.67 9.18 10.77 11.82 10.35 11.54 13.20 11.82 12.77 14.78

50.29 54.47 60.27 74.40 74.85 86.35 94.06 101.22 105.04 49.51 56.58 65.13 75.16 80.97 86.77 97.44 109.36 113.48 48.56 58.27 68.93 77.48 84.35 91.31 100.91 107.78 113.38

58.17 60.17 64.50 77.08 83.8 87.72 98.37 103.98 105.77 55.27 63.34 69.88 80.86 89.62 92.05 103.77 113.59 115.70 55.89 62.60 69.25 82.76 89.84 95.64 108.10 114.96 120.61

103.14 94.84 98.22 126.78 117.67 119.38 153.08 135.30 96.48 77.09 76.38 78.90 86.91 98.85 105.22 108.81 114.80 104.79 75.21 70.75 73.49 82.71 91.00 85.67 100.31 108.84 107.81

Icr(th) (mm4×103) (6)

(Ec.Icr)exp (109×Nmm2) (7)

m (8)

c/d(th) (9)

c/d(exp) (10)

110.81 104.14 95.34 147.51 139.07 128.25 178.28 168.67 135.16 84.23 81.09 79.40 114.15 110.30 108.73 138.94 135.54 132.74 82.48 77.08 73.96 109.12 104.01 100.96 132.31 128.07 124.48

2.54 2.54 2.96 3.12 3.16 3.60 3.77 3.63 2.91 2.73 2.84 3.02 3.08 3.67 4.02 3.86 4.27 4.01 2.89 2.85 3.08 3.18 3.66 3.59 3.85 4.38 4.52

1.96 1.97 1.49 1.55 1.56 1.10 1.19 1.34 0.88 2.36 2.37 1.61 2.14 2.03 1.34 1.72 1.52 1.19 3.23 2.5 1.96 2.13 2.43 1.84 1.85 1.52 1.91

0.355 0.343 0.328 0.415 0.402 0.385 0.460 0.447 0.428 0.307 0.301 0.298 0.361 0.355 0.351 0.403 0.396 0.392 0.297 0.292 0.287 0.350 0.344 0.338 0.391 0.385 0.379

0.290 0.331 0.326 0.359 0.392 0.390 0.384 0.423 0.441 0.246 0.247 0.355 0.312 0.348 0.380 0.362 0.385 0.402 0.226 0.259 0.292 0.308 0.348 0.333 0.344 0.356 0.338

Table 5 Moment enhancement due to fibers addition (%) Concrete compression strength r (%) 1.18 1.77 2.37

N 10.88 13.8 7.52

M

H

26.43 13.84 11.50

23.90 15.56 11.57

upto the yielding of the reinforcement. Some fluctuations of the c/d values took place at low level of loading due to the sensitivity of the strain gage readings specially before cracking. It is noticed that the value of c does not vary between the cracking and yielding levels. For a specific level of loading, Ma/Mcr, the neutral axis depth is larger for the lower compressive strength, irrespective of the amount of flexural reinforcement. The c value increases as Vf increases, and this is attributed to the decrease in curvature of the beam, and also due to that the fibers bridge the cracks and reduce crack width which in turn reduce the strain in the tension zone. The theoretical depth of the neutral axis can be obtained from the statistical moment given by:

Fig. 6.

Fiber contribution in moment enhancement.

bc2 ⫺nAs(d⫺c)2⫽0 2

(3)

Table 4 gives the theoretical and experimental c/d values for the tested beams. The theoretical values generally underestimated the experimental values, however, for beams with 1.0% fiber content the theoretical c/d values overestimate slightly the experimental values.

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as a function of load level, therefore it is more appropriate to consider the effect of the f’c, r and Vf on the flexural rigidity, (Ec.I)exp, of the beam rather than considering the experimental moment of inertia, Iexp, alone. Fig. 9 shows the variation of (Ec.I)exp obtained using Eq. (4) as a function of level of loading, Ma/Mcr. In general, the flexural rigidity increases with the increase of the fiber content. The effect of concrete strength on the experimental flexural rigidity is shown in Fig. 10. For beams with no fibers, f’c has very little effect on (Ec.I)exp, however, for beams with fibers, f’c has a significant influence especially for beams with high r, The test results show that the higher the flexural reinforcement ratio, the higher the flexural rigidity and the lesser the rate of transition of the flexural rigidity from the uncracked to fully cracked section values. This suggests that the exponent in Branson’s equation Eq. (1) is inversely proportional to r, which supports the conclusion by Al-Sheikh et al. [26] that the exponent of 3 in Branson’s equation should be reduced as r increases. 3.6. Cracked moment of inertia The value of Iexp is assumed to approach Icr(exp) when the applied moment approaches My, which is a realistic assumption [25]. At that level of loading, the Mcr/Ma ratio is quite small and the contribution of Ig in Eq. (1) is negligible. The calculation of deflection during the service stage of a structure depends mainly on the cracked moment of inertia, Icr. The experimental cracked moment of inertia is obtained by considering: Pya(3l2−4a2) Icr(exp)⫽ 48Ec⌬exp

(5)

where Py=the load that causes yielding in the steel reinforcement. The values of Icr(exp) and (EcIcr)exp, are calculated using Eq. (5) and are presented in Table 4. The value of Icr(exp) decreases as Vf increases, however, the value of (EcIcr)exp increases as Vf increases.

Fig. 7. Ratio of experimental to theoretical cracking moment.

3.7. Modification of Branson’s equation 3.5. Experimental moment of inertia Based on the elastic deformation theory, the experimental moment of inertia of a simply supported beam subjected to a two-points load is obtained as: Iexp⫽

Pa(3l2−4a2) 48Ec⌬exp

(4)

where; P=applied load; a=shear arm; l=clear span of the beam; ⌬exp=experimental midspan deflection; Ec=experimental secant modulus of elasticity of concrete. However, as shown in Fig. 2 the value of Ec varies

In the evaluation of the deflection of the test beams, the determination of Mcr, Ec, and Icr, are the required parameters in calculating Eq. (1). These parameters control the serviceability and deflection calculation. Al-Sheikh et al. [26] proposed the following formula to include the effect of reinforcement ratio in Branson’s equation [18]:

冉 冊 冋 冉 冊册

Ie⫽

Mcr m Mcr I ⫹ 1⫺ Ma g Ma

where m=3⫺0.8r.

m

Icr

(6)

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Fig. 8.

Behavior of neutral axis depth.

Eq. (6) was based on beams with 33 MPa concrete compressive strength. Ashour [23] showed that the transition rate of Ie from Ig to Icr drops quicker as f’c increases, and proposed a modification on Branson’s equation to consider the effect of f’c as:

冪f⬘

m⫽3⫺0.8r

33

(7)

c

where f’c⬎33 MPa. The experimental variation of the exponent m in Eq. (6) as a function of the level of loading, Ma/Mcr, can be evaluated by replacing Ie and Ig by the values of Iexp and Iut respectively:

冋

册

Iexp−Icr(exp) log Iut−Icr(exp) m⫽ Mcr(exp) log Ma

冋

册

(8)

The variation of m obtained from Eq. (8) as a function of Ma/Mcr, is shown in Fig. 11 for all test beams. In general, the value of m increases as f’c increases and Vf and r decrease.

Fig. 11 shows that the value of m for each test beam has an almost constant value for level of loadings Ma/Mcr between 1.5 to 6.0. The experimental average value of m obtained for each beam within this range of level of loading is given in Table 4. Based on the test results, a regression analysis is performed and an empirical formula that incorporates the considered variables (r, f’c and Vf) in the expression of m is proposed as: 3−0.8r m⫽

冪f⬘

33 c

1+0.4Vf

(9)

For beams with no fibers and with 33 MPa compressive strength, Eq. (9) is reduced to Eq. (6). The deflection calculation requires the determination of other factrors such as Mcr, Ec and Icr. The variations of the secant modulus of concrete in terms of f’c and Vf are given in Fig. 12. Based on regression analysis, the secant modulus of FRC, Ecf, in terms of the that of plain concrete, Eco is given as: Vf Ecf⫽Eco(1⫹600( )2) (10) f⬘c

S.A. Ashour et al. / Engineering Structures 22 (2000) 1145–1158

Fig. 9.

Experimental flexural rigidity as a function of Vf.

where Eco for high-strength concrete is given as [27]:

冑

Eco⫽3200 f⬘c⫹6900

(11)

Eqs. (10) and (11) are presented in Fig. 12. The equations give good estimation of the experimental values, however, the equations overestimate the value for normal strenth concrete. As discussed earlier and shown in Fig. 7, the ratio of Mcr(exp)/Mcr(th) are about 55 to 85%. Thus the use of fr to calculate the cracking moment Eq. (2) overestimated the experimental values. Thus a reduced value of fr should be used to predict with reasonable accuracy the beam deflection: 0.6frIg Mcr(th)⫽ Yt

(12)

Ashour [23] proposed an equation to predict the theoretical cracked moment of inertia and is given as: Icr(th)⫽Icr[1.129⫺0.0011f⬘c⫺0.0133r]

(13)

where Icr is the cracked moment of inertia and is given by: bc3 Icr⫽ ⫹nAs(d⫺c)2 3

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(14)

where n=Es/Ecf and c is the neutral axis depth. The predicted deflections of the test beams are evaluated in terms of m (Eq. (9)), Mcr (Eq. (12)) and Icr (Eq. (13)), and the values are presented in Fig. 13. The figure shows that the predicted deflections give good estimation of the experimental values. 4. Conclusions Based on the test results of twenty seven reinforced concrete beams tested in flexure, the following conclusions are drawn: 1. The presence of steel fibers reduces the crack propagation in the tested beams. 2. The flexural rigidity increases as fc⬘ and Vf increases. 3. The increase of the fiber content increase the cracking, the yielding and the ultimate moments. 4. The predicted cracking moments estimated in terms of the modulus of rupture overestimated the experimental values. 5. Additional moment strength due to the presence of fibers is almost independent of the amount of reinforcement, r. However, this additional moment is proportional to concrete compressive strength, f’c.

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Fig. 10.

Experimental flexural rigidity as a function of f’c.

Fig. 12.

Theoretical secant modulus of concrete.

6. The rate of decay of the beam effective moment of inertia from the untracked transformed to a fully cracked section is lower for beams with fibers than that of beams with no fibers.

7. The exponent, m, in Branson’s equation decreases as Vf increases, however, it increases as f’c increases.

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Fig. 11. Variation of exponent, m, in Eq. (7).

Fig. 13.

Prediction of deflection for some test beams.

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