Toughness of steel fiber reinforced silica fume concrete under compression.

Ramadoss, P. ; Prabakaran, V. ; Nagamani, K. 等

Introduction

Steel fiber reinforced concrete (SFRC) has gained acceptance for a variety of applications, namely industrial floors, bridge decks, pavements, hydraulic and marine structures, precast elements, nuclear vessels, repair and rehabilitation works, blast resistance structures [1, 2, 3]. Balaguru and Shah [1] and ACI Committee 544-1989 [3] have reported that the addition of steel fibers into concrete improves all engineering properties of concrete such as tensile strength, compressive strength, impact strength, ductility and toughness. The improved toughness in compression imparted by fibers is useful in preventing sudden and explosive failure under static loading and in absorption of energy under dynamic loading [3].

High-performance concrete (HPC) is achieved by using super-plasticizer to reduce water-binder ratio and by using supplementary cementing materials such as silica fume, which usually combines high-strength with high durability [1]. Addition of fibers has shown to improve ductility of normal and HSC/ HPC, particularly concrete containing silica fume [4].

The demand for HSC/ HPC has been growing at an ever-increasing rate over the past years, which lead to the design of smaller sections. Reduction in mass is also important for the economical design of earthquake resistant structures [10, 11]. ACI Committee 363-1992 [11] reported that as the concrete strength increases for HSC, the post peak portion of the stress-strain diagram almost vanishes or descents steeply. The application of high-strength or high-performance concrete in practice is severely limited by its more brittle behavior. However, the brittleness of HSC/ HPC can be eliminated by the addition of discrete fibers of small diameter in the concrete matrix [5, 6]. To incorporate such improvement in structural design, it is necessary to establish the complete stress-strain response of the resulting fiber reinforced concrete. While the compressive strength is used for the assessment of the structural components, the stress-strain curve is needed to evaluate the toughness of the material.

Nataraja et al. [14] have generated the complete stress-strain curve experimentally and proposed an analytical expression similar to Ezeldin and Balaguru [15]), for SFRC using crimped fibers for compressive strength ranging from 30 to 50 MPa. Equations were proposed to quantify the effects of fibers on compressive strength, strain at peak stress and toughness of concrete in terms of fiber reinforcing index (RI). Mansur et al. [16] proposed an analytical model to generate the complete stress-strain curve of high-strength fiber concrete with strength ranges from 70 to 120 MPa derived from cylinders and horizontally cast prisms. In their study, the toughness index is determined as the ratio of the area under stress-strain curve up to a strain of 3[[epsilon].sub.o] to the area up to a strain of [[epsilon].sub.o]. Ramadoss and Nagamani [17] have generated the complete stress-strain curve experimentally for high performance fiber reinforced concrete in compression.

In the present study, an experimental work has been carried out to study the complete stress-strain response and toughness of steel fiber reinforced silica fume concrete (SFRSFC) with compressive strength ranging from 52 to 75 MPa. Crimped fibers having an aspect ratio of 80, with four fiber volume fractions of 0%, 0.5%, 1.0% and 1.5% (0, 39, 78 and 117.5 kg/[m.sup.3]) were used in this investigation. The variation in concrete strength was achieved by varying the w/cm ratio with 10% silica fume replacement. The influence of fiber content in terms of fiber reinforcing index (RI) on compressive strength and toughness of HSFRC, were investigated. Based on the test data, complete stress-strain ([sigma]-[epsilon]) curves have been drawn and toughness ratios evaluated for steel fiber reinforced silica fume concrete.

Experimental Programme

Materials, Mixture Proportioning, and Preparation of Specimens

Ordinary Portland cement--53 grade, having 28-day compressive strength = 54.5 MPa and fineness by Blaine's fineness by specific surface = 245 [m.sup.2]/ kg, conforming to IS: 12269-1987 and silica fume having fineness by specific surface area = 23000 [m.sup.2]/kg, specific gravity = 2.25 were used. The chemical analysis of silica fume (Grade 920-D) is: silicon dioxide = 88.7%, LOI at 975[degrees]C = 1.8% and carbon = 1.8%, are conforming to ASTM C1240-1999, AASHTO M307--1990 and Canadian Standard Association 1986.

Fine aggregate of locally available river sand passing through 4.75 mm IS sieve, conforming to grading zone-II of IS: 383-1970 was used. Sand has fineness modulus of 2.65, a specific gravity of 2.63 and water absorption of 0.98 % @ 24 hrs. Coarse aggregate of crushed granite stones with maximum size of 12.5 mm, conforming to IS: 383-1978 was used. The characteristics of coarse aggregate are: Specific gravity = 2.70, Fineness modulus = 6.0, Dry rodded unit weight = 1600 kg/[m.sup.3] and Water absorption = 0.65 % @ 24 hrs.

Superplasticizer of sulphonated naphthalene formaldehyde (SNF) condensate as HRWR admixture conforming to ASTM Type F (ASTM C494) and IS: 9103-1999, which has a specific gravity of 1.20, was used.

Crimped steel fiber conforming to ASTM A820-2001 has been used in this investigation. Properties of crimped fibers (undulated) are: length = 36 mm, diameter = 0.45 mm, aspect ratio = 80, ultimate tensile strength, [f.sub.u] = 910 MPa and elastic modulus, Esteel = 2.1 x [10.sup.5] MPa.

Mixtures were proportioned using guidelines and specifications given in ACI 211.4R-1993 [7], and recommended guidelines of ACI 544--1995 [8]. Mixture proportions used in this test programme are summarized in Table 1. This aspect of work has been carried out elsewhere [13]. For each water--cementitious materials ratio (w/cm), one plain concrete (silica fume concrete) mix, and three fibrous concrete mixes having fiber volume fractions ([V.sub.f]) of 0.5, 1.0 and 1.5 percent by volume of concrete (39, 78 and 117.5 kg/[m.sup.3]) were prepared. Superplasticizer with dosage range of 1.75 to 2.5% by weight of cementitious materials (Cm = OPC + SF) has been used to maintain the adequate workability of silica fume concrete and fiber reinforced concrete mixes. Slump value obtained was 75 [+ or -] 25 mm for silica fume concrete and Vebe value of 12 [+ or -] 3 sec. for fibrous concrete mixes.

Eight series of fiber reinforced silica fume concrete mixes were used in this investigation. Concrete was mixed using a tilting type mixer and specimens were cast using steel moulds, and compacted by using table vibrator. For each mix at least three 150 mm diameter cylinders were prepared. Specimens were demoulded 24 hours after casting, and water cured at 27 [+ or -] 2[degrees] C until the age of testing at 28 days. For maintaining uniform curing all the specimens were cured in the same curing tank.

In mix designation FC1 * to FC4 *, silica fume replacement is 10 percent by weight of cementitious materials, after hyphen denotes fiber volume fraction in percent.

Water present in Super plasticizer is excluded in calculating the water to cementitious materials ratio (w/cm).

[V.sub.f] (%) denotes Steel fiber volume fraction in percent in total volume of concrete.

Compressive strength test

Compressive strength tests were performed according to ASTM C 39-1992 [12] standards using 150 mm diameter cylinders loaded uniaxially. Before testing, the cylinders were capped with a hard plaster on the cast faces to ensure parallel loading faces of the test specimens and constant height for all the cylinders. A compressometer equipped with dial gauges available in the laboratory was used to record the deformation of the cylinder. Efforts were made to take as many readings as possible, to get considerable length of post-peak portion of the stress-strain curve. In the descending portion readings were taken at random intervals. Stresses and corresponding strains were evaluated and average values are reported with the compressive strength ranges from 52 to 75 MPa, as given in Table 4.

Results and Discussion

The shape of the stress-strain curve in uniaxial compression is strongly affected by the testing conditions and concrete characteristics. To minimize the testing condition effects, careful attention was exercised to avoid variations in the testing setup and specimen's instrumentation. Ultrasonic pulse velocity measurements show the SFRC mixes having uniform mixing; compaction and fiber distribution (Table 2). The stress-strain relationship of concrete essentially consists of two distinct branches--an ascending branch up to the peak stress followed by a descending branch until the concrete crushes.

Compressive Strength ([f.sub.0])

Concrete strength was achieved in the range of 55-75 MPa, by varying the water-cementitious materials ratio (w/cm) and fiber volume fraction ([V.sub.f]) with 10% silica fume replacement, is presented in Table 2. Figure 1 shows the effect of fiber reinforcing index (RI) on compressive strength at 10% silica fume replacement. The peak compressive strength [f.sub.o] or [f.sup.'.sub.cf] and the corresponding strain [[epsilon].sub.0] depend on the response of the specimen at ultimate load. At this stage, cracks will form in the specimen due to lateral expansion of the concrete. Fibers aligned normal to the loading direction will therefore, be intercepted by these cracks and offer some resistance to their growth. The effect of fibers on the compressive strength of concrete may be evaluated from the stress-strain curves for fiber reinforced concretes. It may be seen from Table 2 that the inclusion of fibers in silica fume concrete (plain concrete) results in moderate increase in compressive strength.

[FIGURE 1 OMITTED]

A least square regression analysis was performed using the experimental results to establish a possible relationship between the peak compressive stress and the fiber-reinforcing index. The proposed expression for the steel fiber reinforced silica fume concrete for compressive strength ranging from 52 to 70 MPa is given as:

Compressive strength ([f.sub.o] or [f.sup.'.sub.cf])

[f.sup.'.sub.cf] = [f.sup.'.sub.c] + 1.397 (RI) (R= 0.95) (1)

where [f.sup.,.sub.c] and [f.sup.,.sub.cf] are the compressive strength of silica fume concrete (HPC) and SFRSFC, respectively in MPa.

RI = fiber reinforcing index.

The percentage variation in absolute value has been obtained as 3.35.

Stress-Strain Curves

The stress-strain response of silica fume and SFRC with compressive strength ranging from 52 to 70 MPa has been investigated. Typical stress-strain ([sigma]-[epsilon]) curve for silica fume concrete and steel fiber reinforced concrete is shown in Fig. 2. Figures 3, 4 and 5 show the stress-strain ([sigma]-[epsilon]) curves for fiber reinforced silica fume concrete in compression with different fiber volume fractions ([V.sub.f]). It clearly shows (Figs. 3, 4, and 5) that the post-peak segment of the [sigma]-[epsilon] curve is affected by the addition of steel fibers. From the stress-strain curves generated in this study, it can be observed that an increase in concrete strength increases the extent of curved portion in ascending branch and renders the drop in the descending part more steeper for non-fibrous concrete and gradually flatter for SFRC. An increasing in the slope of the descending part of the stress-strain curve is also observed by increasing the fiber volume fraction. The gradual change in shape with an increase in strength have, however, been reported on by many investigators in the past. Previous researchers noticed that crimped and hook end fibers are effective in improving the mechanical properties, and energy absorption at post peak load capacity and ductility.

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

Modulus of Elasticity ([E.sub.c])

Modulus of elasticity (secant modulus) was defined according to ACI Building code (ACI 318-1995) as the slope of the line drawn from a stress of zero to a compressive stress of 0.45 [f.sup.,.sub.c].

The static modulus of elasticity evaluated from the stress-strain curves are in the range of 29.68 x [10.sup.3] Mpa--41.06 x [10.sup.3] MPa (Table 2). Results of modulus of elasticity obtained for various concrete mixes show that modulus of elasticity increases with increase in fiber volume fraction or fiber reinforcing index.

Based on the experimental results, using least square regression analysis, the expression obtained for the elastic modulus ([E.sub.c]) as a function of compressive stress is given as:

[E.sub.c] = 1515 [f.sub.o.sup.0.75] (R = 0.99) (2)

where secant modulus, [E.sub.c] and compressive strength, [f.sub.o] are all expressed in megapascals.

The equation (2) gives the lower bound values for the steel fiber reinforced concretes to that of ACI 318-1995, ACI 363-1992 and IS: 456-2000 recommended equations. On comparing the CEB-FIP model code (1990) and BS code (BS: 8110) formulae for modulus of elasticity, the proposed formula for modulus of elasticity gives the upper bound values.

Compressive Toughness

Toughness is a measure of the capability of the material to absorb energy during deformation when subjected to compressive load, estimated using the area under stress-strain curve. The energy absorption per unit volume under compression is expressed mathematically as, Toughness = [[integral].sup.[epsilon]] [sigma]. d[epsilon]

The convenient way to quantify ductility is to use toughness ratio, TR (Fig. 6).

TR = Area (OABC)/ [f.sup.'.sub.cf] x 0.015 (3)

where Area(OABC) = area of OABC in stress-strain diagram (Fig. 6) and [f.sup.,.sub.cf] = peak compressive strength of concrete.

Fanella and Naaman [5], Hsu and Hsu [4] have defined the toughness index of FRC as the ratio of the toughness of FRC matrix to that of unreinforced control matrix. Ezheldin and Balaguru [15] have proposed a rigid plastic approach to define the toughness ratio. In the results presented in this paper, the toughness is measured as the total area under stress-strain curve up to a strain of 0.015 mm/mm, which is five times the ultimate concrete strain of 0.003mm/mm as adopted in the ACI building code 318-95 [18]. Fanella and Naaman [5], and Ezheldin and Balaguru [15] have also used an ultimate strain of 0.015 for computing the toughness as it is sufficient to represent the trend of post peak behavior of SFRC. This toughness is compared to the toughness of a rigid plastic material in the form of toughness ratio (TR) as indicated in Fig. 4. It is observed in the present investigation that the area under stress-strain curve increases with the increase in fiber content, and fiber type and geometry (crimped steel fiber) compared with other fibers used by previous researchers. To combine the effect of both fiber volume fraction and their aspect ratio, the fiber reinforcing index, (RI = [w.sub.f] * (l/d)) can be used as the fiber reinforcing parameter for a given type of fiber. Weight fraction ([w.sub.f]) is approximately equal to 3.27 times the volume fraction of fibers. The increase in fiber reinforcing index, RI would yield a large area under the stress-strain curve making a flatter descending part and a higher toughness ratio as shown in Fig.2. Table 2 presents the experimental values of toughness ratios for various SFRSFC. Nataraja et al. [14] have obtained maximum toughness ratio of 0.77 for SFRC with w/c = 0.38 and RI = 2.67, which is comparable with the experimentally calculated maximum value of 0.685 for SFRC with w/cm = 0.25 and RI= 3.88.

[FIGURE 6 OMITTED]

A least square regression analysis was performed using the experimental results to establish a possible relationship between the toughness ratio of the concrete based on the stress-strain behavior and fiber-reinforcing index as shown in Fig. 7. Results of this equation are presented in Table 3 and the predicted values match with the values computed from the experimental results. The proposed equations (linear and nonlinear models) from the complete stress-strain behavior of the steel fiber reinforced silica fume concrete for the compressive strength ranges from 52 to 70 MPa are given as:

Toughness ratio of concrete ([TR.sub.f])

[TR.sub.f] = [TR.sub.c]+ 0.142 (RI) (R = 0.78) (4)

[TR.sub.f] = [TR.sub.c] + 0.03241 (RI) - 0.0548 [(RI).sup.2] (R = 0.98) (5)

where [TR.sub.c] and [TR.sub.f] are the toughness ratio of HPC and SFRSFC, respectively.

The percentage variation in absolute value has been obtained as 2.56. The above equations can be used for reinforcing index (RI) up to 3.9 for the crimped steel fibers. The experimentally determined peak stress-RI and toughness ratio-RI relationships are given in Figures 8 and 9 for SFRSFC with w/cm ratio = 0.40. It is observed from Figures 8 and 9 that toughness ratio as a function of RI increases as peak stress increases.

[FIGURE 7 OMITTED]

[FIGURE 8 OMITTED]

[FIGURE 9 OMITTED]

Conclusions

Based on the experimental study, following conclusions can be drawn from the compression response of steel fiber reinforced silica fume concrete.

(1) Addition of crimped steel fibers to silica fume concrete (HPC) chances the basic characteristics of its stress-strain response. The slope of the descending branch increases with increasing the fiber reinforcing index (RI).

(2) Compressive toughness and ductility are increased considerably for steel fiber reinforced silica fume concrete. The increase in toughness is directly proportional to the reinforcing index.

(3) A moderate increase in compressive strength, strain at peak stress is also observed, which is proportional to the reinforcing index. The expression proposed is valid for steel fiber reinforcing index ranging from 0 to 3.9.

(4) The toughness ratio predicted based on the empirical equation arrived based on the stress-strain curves, matches with values calculated from the experimental results. The proposed expression is giving good correlation with the experimental values.

Acknowledgement

The authors would like to thank the Structural Engineering Division of Anna University, India for extending the facilities for the above research work.

Notations

HPC = high-performance concrete

SFRSFC = steel fiber reinforced silica fume concrete

[f.sup.'.sub.cf] or [f.sub.o] = cylinder compressive strength of SFRSFC, MPa

[f.sup.'.sub.c] = compressive strength of silica fume concrete (HPC), MPa

[TR.sub.f] = toughness ratio of SFRSFC

[TR.sub.c] = toughness ratio of HPC

[V.sub.f] = volume fraction of fiber, percent

[w.sub.f] = weight fraction of fiber

l/d = aspect ratio of fiber

RI= fiber reinforcing index.

References

[1] Balaguru, N., and Shah, S.P. (1992), "Fiber reinforced concrete composites", McGraw Hill International edition, New York.

[2] ACI Committee 544 (2006), "State-of-the-art report on fiber reinforced concrete", ACI 544.1R-82, American Concrete Institute, Detroit.

[3] ACI Committee 544 (1989), "Design considerations for steel fiber reinforced concrete", ACI 544.4R-89, American Concrete Institute, Detroit.

[4] Hsu, L.S., and Hsu, C.T.T. (1994), "Stress-strain behavior of steel fiber reinforced high-strength concrete under compression", ACI Structural Journal, 91(4), pp. 448-457.

[5] Fanella, D.A. and Naaman, A.E. (1985), "Stress-strain properties of fiber reinforced mortar in compression",. ACI Journal, 82(4), pp. 475-583.

[6] Ezeldin, A.S., and Balaguru, P.N. (1989), "Bond behavior of Normal and high strength fiber reinforced concrete", ACI Materials Journal, 86(5), pp. 515-523.

[7] ACI Committee 211 (1999), "Guide for selecting proportions for High strength concrete with Portland cement and Fly ash", ACI 211.4R-93, ACI Manual of concrete practice.

[8] ACI Committee 544 (2006), "Guide for specifying, mixing, placing and finishing steel fiber reinforced concrete", ACI 544.3R-93, American Concrete Institute, Detroit.

[9] ACI Committee 544 (2006), "Measurement of properties of fiber reinforced concrete", ACI 544.2R-89, American Concrete Institute, Detroit.

[10] Chin, M.S., Mansur, M.A., and Wee, Y.H. (1999), "Effects of shape, size and casting direction of Specimens on Stress-strain curves of high strength concrete", ACI Materials Journal, 94(3), pp. 209-219.

[11] ACI Committee 363 (1992), "state-of-the-art report on high strength concrete", ACI 363-1992, American Concrete Institute, Detroit.

[12] ASTM C39--1992, "Standard test method for compressive strength of fiber reinforced concrete", Annual book of ASTM standards. American Society for Testing and Materials.

[13] Ramadoss, P., and Nagamani, K. (2008), "A new strength model for high-performance fiber reinforced concrete", Computers and Concrete--an International Journal, 5(1), pp. 21-36.

[14] Nataraja, M.C., Dhang, N., and Gupta, A.P. (1999), "Stress-strain curve for steel fiber reinforced concrete in compression", Cement and Concrete Composites, 21(5/6), pp. 383-390.

[15] Ezeldin, A.S., and Balaguru, P.N. (1992), "Normal and high strength fiber reinforced concrete under compression", ASCE, Journal of Mate. in Civil Eng., 4(4), pp. 415-429.

[16] Mansur, M.A.., Chin, M.S., and Wee, Y.H. (1999). Stress-strain relationship of high strength fiber concrete in compression. ASCE Journal of Mate. in Civil. Eng, 13(1), pp. 21-29.

[17] Ramadoss, P., and Nagamani, K. (2008), "Stress-strain curves for high-performance fiber reinforced concrete under compression", Journal of Civil Engineering Research and Practice, 5(1), pp. 1-14.

[18] ACI Building code 318 (1995), "Building code requirements for structural concrete", ACI 318-1995, American Concrete Institute, Detroit.

P. Ramadoss (1), *, V. Prabakaran (2) and K. Nagamani (3)

(1,2) Department of Civil Engineering, Pondicherry Engineering College, Pondicherry-605014, India.

(3) Department of Civil Engineering, Anna University, Chennai-600 025, India.

* Corresponding author: Email: dosspr@gmail.com

Table 1: Mix proportions for SFRSFC (data for 1 [m.sup.3]).

    Mix       W/Cm   C, kg   FA, kg   CA, kg   SF, kg   W, kg
Designation

FC1 * - 0     0.4    394.2    691      1088     43.8     175
FC1 * - 0.5   0.4    394.2    687      1079     43.8     175
FC1 * - 1     0.4    394.2    682      1071     43.8     175
FC1 * - 1.5   0.4    394.2    678      1062     43.8     175
FC2 * - 0     0.35   437.4    664      1088     48.6     170
FC2 * - 0.5   0.35   437.4    660      1079     48.6     170
FC2 * - 1     0.35   437.4    655      1071     48.6     170
FC2 * - 1.5   0.35   437.4    651      1062     48.6     170
FC3 * - 0     0.3     495     624      1088      55      165
RC3 * - 0.5   0.3     495     620      1079      55      165
FC3 * - 1     0.3     495     615      1071      55      165
FC3 * - 1.5   0.3     495     611      1062      55      165
FC4 * - 0     0.25    576     562      1088      64      160
FC4 * - 0.5   0.25    576     558      1079      64      160
FC4 * - 1     0.25    576     553      1071      64      160
FC4 * - 1.5   0.25    576     549      1062      64      160

    Mix         Steel     SP, kg
Designation    fiber,
              [V.sub.f]
                 (%)

FC1 * - 0         0        7.66
FC1 * - 0.5      0.5       7.66
FC1 * - 1        1.0       7.66
FC1 * - 1.5      1.5       7.66
FC2 * - 0         0        9.72
FC2 * - 0.5      0.5       9.72
FC2 * - 1        1.0       9.72
FC2 * - 1.5      1.5       9.72
FC3 * - 0         0       13.75
RC3 * - 0.5      0.5      13.75
FC3 * - 1        1.0      13.75
FC3 * - 1.5      1.5      13.75
FC4 * - 0         0       17.60
FC4 * - 0.5      0.5      17.60
FC4 * - 1        1.0      17.60
FC4 * - 1.5      1.5      17.60

Table 2: Experimental results for steel fiber reinforced concrete
and SF concrete.

    Mix        RI      UPV     [f.sup.'.sub.cf]   [[member of].sub.o]
Designation          (m/sec)      (MPa)            (mm/mm)

FC1 * -0       0      4398        52.56            0.00260
FC1 * -0.5    1.29    4377        54.77            0.00305
FC1 * -1      2.58    4367        56.01            0.00320
FC1 * -1.5    3.88    4524        57.40            0.00330
FC2 * -0       0      4318        55.85            0.00305
FC2 * -0.5    1.29    4382        59.65            0.00325
FC2 * -1      2.58    4435        61.05            0.00338
FC2 * -1.5    3.88    4559        61.44            0.00345
FC3 * -0       0      4516        63.86            0.00335
FC3 * -0.5    1.29    4603        67.12            0.00335
FC3 * -1      2.58    4667        68.91            0.00365
FC3 * -1.5    3.88    4790        69.67            0.00370
FC4 * -0       0      4615        74.87            0.00360
FC4 * -0.5    1.29    4716        77.42            0.00376
FC4 * -1      2.58    4889        79.96            0.00388
FC4 * -1.5    3.88    5004        80.41            0.00395

    Mix       [E.sub.c]     TR
Designation     (GPa)

FC1 * -0        29.68     0.2038
FC1 * -0.5      30.14     0.6155
FC1 * -1        30.92     0.6507
FC1 * -1.5      31.78     0.6715
FC2 * -0        30.87     0.2161
FC2 * -0.5      32.32     0.6061
FC2 * -1        33.26     0.6343
FC2 * -1.5      33.97     0.6647
FC3 * -0        34.14     0.2252
FC3 * -0.5      35.96     0.6340
FC3 * -1        36.52     0.6592
FC3 * -1.5      36.98     0.6851
FC4 * -0        37.65     0.2454
FC4 * -0.5      39.47     0.6313
FC4 * -1        40.52     0.6469
FC4 * -1.5      41.06     0.6789

UPV = Ultrasonic pulse velocity (m/sec.)

[f.sup.'.sub.cf] = 150 x 300 mm cylinder compressive strength of
SFRSFC (MPa)

[E.sub.c] = secant modulus (GPa), [[member of].sub.o] = strain at
peak stress and TR = toughness ratio.

Fiber reinforcing index (RI) = [w.sub.f] * (l/d) and average density
of SFRSFC = 2415 kg/[m.sup.3]

Weight fraction ([w.sub.f]) = (density of fiber/density of fibrous
concrete) * [V.sub.f]

Aspect ratio (l/d) = length of fiber/diameter of fiber.

Table 3: Calculating results for fiber reinforced concrete and
SF concrete.

    Mix        RI               Calculated values
Designation          [f'.sub.cf]  % error   [E.sub.c]   % error
                       (MPa)                  (GPa)

FC1 * -0       0       52.56        0.00     30.293     -2.065
FC1 * -0.5    1.29     54.36       -0.75     31.115     -3.235
FC1 * -1      2.58     56.16        0.27     31.571     -2.105
FC1 * -1.5    3.88     57.98        0.98     32.078     -0.938
FC2 * -0       0       55.85        0.00     31.512     -2.080
FC2 * -0.5    1.29     57.65       -3.35     32.890     -1.764
FC2 * -1      2.58     59.45       -2.62     33.390     -0.391
FC2 * -1.5    3.88     61.27       -0.28     33.528      1.301
FC3 * -0       0       63.86        0.00     34.381     -0.706
FC3 * -0.5    1.29     65.66       -2.18     35.512      1.246
FC3 * -1      2.58     67.46       -2.10     36.124      1.084
FC3 * -1.5    3.88     69.28       -0.56     36.383      1.614
FC4 * -0       0       74.87        0.00     38.126     -1.264
FC4 * -0.5    1.29     76.67       -0.97     38.965      1.279
FC4 * -1      2.58     78.47       -1.86     39.791      1.799
FC4 * -1.5    3.88     80.29       -0.15     39.936      2.737

              Calculated Values
    Mix         TR     % error
Designation

FC1 * -0       0.204     0.00
FC1 * -0.5     0.622    -0.982
FC1 * -1       0.648     0.397
FC1 * -1.5     0.675    -0.518
FC2 * -0       0.216     0.00
FC2 * -0.5     0.622    -2.555
FC2 * -1       0.648    -2.178
FC2 * -1.5     0.675    -1.537
FC3 * -0       0.225     0.00
FC3 * -0.5     0.622     1.957
FC3 * -1       0.648     1.676
FC3 * -1.5     0.675     1.488
FC4 * -0       0.245     0.00
FC4 * -0.5     0.622     1.546
FC4 * -1       0.648    -0.187
FC4 * -1.5     0.675     0.589