IMPACT AND BLAST PROTECTION WITH FIBER REINFORCED...

IMPACT AND BLAST PROTECTION WITH FIBER REINFORCED CONCRETE

N. Banthia, V. Bindiganavile and S. Mindess The University of British Columbia, Canada

AbstractFiber reinforcement is undoubtedly one of the most effective means of enhancing the impact and blast resistance of concrete. Unfortunately, the dynamic properties of fiber reinforced concrete also remain some of its least understood. Since 9/11, there has been an increased interest in developing a better understanding of the properties of fiber reinforced concrete under impact and blast loading and enhancing such resistance. This paper provides a historical perspective of our efforts aimed at understanding the impact resistance of fiber reinforced concrete and highlights some of the issues and challenges encountered. A synopsis of our current state of the art with respect to experimental and modeling attempts is presented. Finally, the paper attempts to chart our future course and identify areas where further research will be highly valuable.

1. Introduction

Concrete in most instances is loaded at the strain rates approximated in our standardized tests. However, there are dynamic events in which the strain rates may significantly exceed those obtained in standardized tests such as in fast moving traffic (e = 10-6 10-

4s-1), gas explosions (e = 5x10-5 5x10-4 s-1), earthquakes (e = 5x10-3 5x10-1 s-1), pile driving (e = 10-2 100 s-1) and aircraft landing (e = 5x10-2 2x100s-1). Inappropriately, though, the properties of materials used in the design of these structures are derived from standardized tests that are run at low quasi-static strain-rates (generally around 10-6s-1). Since concrete is known to be highly strain-rate sensitive in compression (2), tension (3,4) and flexure (5,6,7), one cannot use its quasi-statically obtained properties to design structures that undergo high rates of loading; proper impact tests are needed. Loads during impact and blast are generally so large that the structure will often respond beyond its elastic state with cracking in concrete and plastic yielding in steel. An

6th RILEM Symposium on Fibre-Reinforced Concretes (FRC) - BEFIB 200420-22 September 2004, Varenna, Italy

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understanding of the complete constitutive response of the material including its post-elastic stress transfer capability is therefore critical. A large amount of energy is often imparted to the structure during an impact event. If the structure is not capable of absorbing the incoming energy, a failure during the event itself may ensue. Concrete, unfortunately, is a very brittle material with a poor energy absorption capacity. Any attempt at improving its energy absorption capacity by enhancing its post-fracture stress-transfer capability will therefore be an effective way of improving its resistance to impact loads. Research in the last thirty years has indicated that the ideal way to enhance the post-fracture stress transfer capability in concrete is by reinforcing it with fibers. Fibers bridge matrix cracks and undergo pull-out processes resulting in a significant enhancement in the fracture toughness of the material. While we understand very well the quasi-static properties of fiber reinforced concrete (FRC), its impact and blast properties largely remain poorly understood. For a better understanding of such properties, one needs to scrutinize and analyze the responses at the fiber-matrix interface level, at the level of micro-mechanical crack growth and finally at the level of the engineering properties useful in design. It is only through such a multi-pronged approach that we can arrive at FRCs that can be classified as truly ‘high’ performance.

2. Test Techniques

A primary reason behind our lack of understanding of the impact resistance of FRC is the complete absence of a standardized test setup. Several techniques have been developed, including the Drop Weight Test (5) in which a mass is raised to a predetermined height, and then allowed to drop directly on a concrete specimen. Similar in principle are the Swinging Pendulum Machines (6,8)--as in the conventional Charpy or Izod impact machines used by the metallurgists--where a swinging pendulum is allowed to strike a specimen in its path thereby transferring momentum and causing high stress-rates. Other significant impact tests include the Split Hopkinson Pressure Bar Test (2-4) in which the specimen is sandwiched between two elastic bars and high stress-rates are generated by propagating a pulse through one of the elastic bars using a drop weight or similar device. In most modern impact test systems, sufficient instrumentation is provided such that along with loads and deformations, additional specimen responses such as accelerations, velocities, etc., are also measured; these are needed for a proper analysis of the data. One major issue that needs to be dealt with is that of inertial loading. With the exception of the Split Hopkinson Pressure Bar, all other impact tests result in high accelerations in the specimen that manifest as inertial forces in the system. In the case of ductile metals, one can ignore accelerations since failure occurs long after the specimen has accelerated and the inertial oscillations have died down. For a 3-point bend specimen, the period of apparent specimen oscillation (t) is given by (9):

2/1

0

)(36.3 sEBCSW

=t (1)

where, W and B are specimen width and depth, respectively, Cs is the specimen compliance, So is the speed of sound in the material and E is its Young’s modulus. It is


32

recommended that only when the time to failure is at least 3 times greater than t can the inertial forces be ignored. Unfortunately, in the case of brittle materials like concrete, the failure is sudden and this requirement cannot be met; inertial forces must be accounted for to derive any meaningful information. One commonly adopted technique is to carry out direct measurements of accelerations, and then use the principle of virtual work to derive expressions for generalized inertial loads. For a simply supported plate specimen impacted in the center, the generalized inertial load (Pi(t)) is given by (10):

l.xt).cosecl/2,(x,u

4hl(t)P

2

ipr

= (2)

where, l is width (also length) of the plate, r is the mass density, h is the thickness of the plate and ü(x,y,t) = acceleration at any point (x,y) on the plate at time t. Once the generalized inertial load is obtained, the plate can be modelled as a Single Degree of Freedom (SDOF) system and the generalized bending load can be obtained from the Equation of dynamic equilibrium,

P (t) P (t) P (t)b t i= - (3) Similar expressions have been developed for beam specimens (11) and other geometries (12). Equally important in impact tests is to provide a proper energy balance. It has been shown that at the point of peak load only 20% of the hammer energy is transmitted to the specimen and at failure this percentage rises only to about 50%; the rest of the energy remains within the machine as elastic and vibrational energy. This has significant implications, and one can not simply equate the hammer energy loss to that absorbed by the specimen. Clearly, a number of these issues could be resolved if there existed a standardized test for concrete under impact loading. Data reported in the literature can not be compared as they are obtained using different impact machines with different specimen support techniques, different energy loss mechanisms and different ways of generating high stress-rates, all of which have a strong influence on the results. For example, drop weight machines that vary in size and capacity can lead to very different conclusions. In a recent study, Bindiganavile and Banthia (13) demonstrated (Figure 1) that very different flexural toughness values can be obtained for the same FRC materials by using machines of different capacities. Data from different labs therefore should only be compared for identical machine capacities.


33

Figure 1. Influence of Machine Capacity on Measured Impact Response of a Given FRC Material

2. Bond–Slip Mechanisms Under Impact

The interaction between a fiber and its surrounding matrix is the fundamental mechanism of fiber reinforced concrete (FRC) performance, and it is only at this level that its performance can be optimized. The routine method of understanding the interaction between a fiber and its surrounding matrix is by conducting a pull-out test on a bonded fiber. While significant research has been conducted to understand the quasi-static bond-slip response of various types of fibers, very little is known of bond-slip responses at high rates of loading, and there is a lack of general agreement in the literature. Gokoz and Naaman (8) carried out bond-slip tests on aligned steel, glass and fibrillated polypropylene fibers at crack opening displacement (COD) rates of 4.23x10-2

to 3000 mm/s and reported negligible sensitivity for steel and glass fibers but an extreme sensitivity in the case of the fibrillated polypropylene fiber. Banthia and Trottier (14) performed pull-out tests on aligned steel fibers at a COD rate of 1500 mm/s and reported that the deformed steel fibers showed a greater sensitivity to rate than smooth fibers and produced smaller slips at higher rates of COD. This was in sharp contrast to the findings of Pacios et al (15) who showed that for straight steel fibers in an aligned group, the slips under impact were higher than those under quasi-static conditions, and the overall sensitivity to rate was only minor. One major difference between the various studies cited above is how the inertial part of the applied load was dealt with. While a proper dynamic analysis was performed in Ref. (14), in References (8) and (15), a rubber pad was introduced in the contact zone and inertial forces were ignored. As discussed above, it is now well recognized that inertial loading must be explicitly considered in all impact testing. Some high performance polymeric macro-fibers (also called polymeric structural fibers) have recently been developed (16). While the quasi-static response of these high performance fibers is well understood, their response to impact and dynamic loads remains unknown. Preliminary testing (17) has, however, uncovered an interesting trend (Figure 2): slip at peak load was seen to decrease as the rate of crack opening increased, but this decrease was far more pronounced in the case of polymeric macro-fibers than for

-1-0.5

00.5

11.5

22.5

33.5

4

-5 0 5 10 15

ln (stress rate) MPa/s

ln (f

lexu

ral t

ough

ness

) N

m


34

steel macro-fibers. This pronounced bond stiffening in polymeric macro-fibers is very encouraging and can be very useful in developing high performance composites for impact. Clearly, significant further research is needed to fully understand the bond-slip mechanisms of various fibers under impact. In particular, the following variables need attention: fiber type and geometry, fiber inclination, fiber embedment length, fiber grouping, fiber coatings and chemical modification of fiber surfaces, temperature of the environment, concrete strength and the presence of a secondary fiber in the matrix.

0

1

2

3

4

5

6

-2 -1 0 1 2 3 4Log (Crack opening rate) mm/s

Slip

@ P

eak

(mm

) Structural Polypropylene Fiber (PP); 30 mm

Flat-End Steel Fiber; 30 mm

Structural Polypropylene Fiber (PP); 50 mm

Figure 2. Bond Stiffening in PP Fiber under Impact

Limited efforts have been made to model the bond-slip process as a function of the strain-rate. From the traditional R-curve approach in fracture mechanics, Pacios et al.[15] assumed that the point of intersection of the R and G curves was a function of the slip-rate. That is (Figure 3), the ratio of the crack length at ‘m’ to that at ‘n’ depends upon the slip-rate as:

( )Bn

c

ASaa

-= exp*

*

(4)

Figure 3. R-Curves at Different Slip-Rates (15)

R (or) G

Crack Length

G-curve at peak load at new slip rate

G-curve at peak load at reference slip rate

R-curve at new slip rate

R-curve at reference slip rate

m'

n m


35

where, ac* and a* are crack lengths at points m and n, respectively. A and B are

constants to be determined and Sn is given by:

ö÷õ

æçå= *log

ssSn (5)

where, s = given slip-rate

and s*= reference slip-rate.

Peak pull-out load, P, is then obtained from:

( )[ ]( )

21

20

32

coth4

îý

îüû

îí

îìë

-

-=

aLwaarE

Pnd

nf bp (6)

where, Ef = modulus of elasticity of the fiber material a = debonded length at the fiber-matrix interface r = radius of the fiber a0 = initial flaw size at the interface L = fiber embedded length

w = a parameter quantifying the shear stiffness of the interface

na and nb are constants to be determined by using a reference specimen and

21

211

411

21

îý

îüû

îí

îìë

öö÷

õææç

å --

-+-

-+=

n

n

n

n

n

nnd

aa

aa

aa

(7)

The existing models apply only to straight undeformed fibers and assume that the interface properties including fracture toughness are not affected by the applied strain-rate. Need exists to develop sophisticated models that can apply to different fiber geometries and can explicitly include the strain-rate dependent properties of the interface. 3. Dynamic Crack Growth Mechanisms

Analogous to the plastic zone in metals, at the tip of a critical crack in concrete, a microcracked ‘process zone’ develops and the crack essentially propagates in a stable manner before acquiring unstable conditions of failure. Solutions to the problem include the ‘effective crack model’ (18) and the ‘cohesive crack model’ (19), but these have not been applied under dynamic conditions. In fiber reinforced concrete, in addition to crack closing pressures due to aggregate interlocking, fiber bridging occurs behind the tip of a propagating crack where fibers undergo bond-slip processes and provide additional closing pressures. The fracture processes in fiber reinforced cement composites are


36

therefore even more complex and need advanced models to simulate them. Attempts have been made to model fracture in FRC using the cohesive crack model (20) or by using the J-integral (21). However, these are strictly speaking only crack initiation criteria and fail to define conditions for continued crack growth. To define both crack initiation and growth, there is now general agreement that a continuous curve of fracture conditions at the crack tip is needed, as in an R-curve (22). An R-curve is a much more suitable representation of fracture in FRC, as one can monitor variations in the stress intensity as the crack grows and produce a multi-parameter fracture criterion. Further, an R-Curve is particularly suited for dynamic loading, as one can incorporate dynamic stress intensity factors in the analysis and key-in their reduction as the crack speed approaches the Rayleigh wave velocity (23). Unfortunately, the study of dynamic fracture mechanics of FRC is in its infancy, and significant doubt exists even on the exact velocity that cracks acquire during impact. Mindess et al. (24) observed crack velocities to be only one-tenth (~115 m/s) of the theoretically predicted Rayleigh wave velocity and found a further reduction to ~75 m/s in the presence of steel fibers. Similar observations were reported by others (25) for plain concrete. Some preliminary dynamic R-Curves were recently obtained (12) using a contoured double cantilever beam specimen. The clarity and effectiveness of such a representation of dynamic fracture was clearly demonstrated (Figure 4). Unfortunately, however, much remains to be done. In particular, the following variables need to be studied: influence of loading rate, fiber volume fraction, fiber type and geometry, fiber coatings and fiber surface modifications, initial crack size, displacement rate, aggregate characteristics, temperature of the environment, strength of concrete matrix, and presence of a secondary fiber in the matrix.

0

100

200

300

400

500

600

700

47 97 147 197 247a- ef f ( mm)

PFRC

SFRC

Plain Concrete

Figure 4. A dynamic crack growth test (left) and some dynamic R-Curves (drop height = 1 m)


37

4. Engineering Properties Under Impact

The micromechanical bond-slip tests and crack growth studies are a great tool to understand and optimize material characteristics. However, for designing structures, global responses of the materials, as determined by standardized tests for engineering properties, are required. Figure 5 shows a plot of some engineering properties under variable strain-rates. Note that concrete is most rate sensitive under tensile load and least sensitive under compressive load. The sensitivity in flexure is in-between that under compressive and tensile modes of loading.

Tension Flexure Compression Birkimer (26) Zielinsky (27)

Mindess et al. (28) Suaris et al. (6) Banthia (29 )

Watstein (30) Evans (31) Hughes et al. (32)

Figure 5. Strain-rate sensitivity of concrete in compression, tension and flexure

In spite of continued efforts in this area, our understanding of the engineering properties of concrete and particularly fiber reinforced concrete under impact loading remains severely limited, and contradictory data exist in the literature. For example, while Ross (33) reported a reduced sensitivity to stress-rate with an increase in the static strength of concrete, Bischoff and Perry (34) reported otherwise. Also, while Banthia et al. (5) reported an increase in the failure strains under impact John and Shah (35) reported a reduction in the ultimate strains. There is also a ‘knee’ reported in strength (sf) vs. stress-rate (s ) plots, and the value of n in equation: sf =ks (1/1+n) (8)

appears to decrease with stress-rate. Origins of these anomalies are not known. In the case of fiber reinforced concrete, while an improvement in impact properties is widely reported, the exact magnitude of these improvements is uncertain. On a worrisome note, steel fibers are reported (14) to fracture across cracks at high rates of loading and thus produce a brittle response at very high strain-rates. As seen in Figure 6, SFRC may


38

depict inferior impact toughness than PFRC which is in agreement with Figure 2. Shear properties of FRC under impact have not been studied at all.

Figure 6. Toughness of steel and polypropylene FRC under impact loading. Note a continual drop in the toughness of SFRC and an increase in the toughness of PFRC

On the modeling side, only sporadic attempts have been made to predict the engineering properties of fiber reinforced concrete at high strain rates. Kaadi (36) expressed the rate effects by the following expression:

( )05.0

öö÷

õææç

å=

stst e

eatat (9)

where a is the efficiency factor for discontinuous fibers and t is the average interfacial bond strength. He then introduced this rate-dependent term into a model proposed by Visalvanich and Naaman (37) to predict the rate dependent post-cracking stress-displacement relationship as follows:

ùú

øéê

èö÷õ

æçå-ù

ú

øéê

è-ö

÷õ

æçå=

ùú

øéê

èö÷õ

æçå l

allV f

dd

fat

s 21122

(10)

where a = ùú

øéê

èöö÷

õææç

å-

st

xee51078.1exp15.1

and s = post-cracking stress at a strain rate of ed = displacement corresponding to s

ste = static strain rate equal to 1.57 x 10-5/s.

0102030405060708090

200 500 750 1000Drop Height (mm)

Toug

hnes

s (N

m)

Steel FiberPolypropylene Fiber


39

In a recent study, Bindiganavile and Banthia (38) adopted a model proposed by Armelin and Banthia (39) to predict the load-displacement response of a beam tested as per ASTM C1018. (Fig. 7). The compressive strain, eo, at the top-most fiber of the specimen leads to an axial shortening, Do, as shown. This in turn leads to stress, sc, in the uncracked concrete. On the other hand, it results in fiber slippage, wi, below the neutral axis and corresponding forces, f1, as the fibers pull-out. Thus, the flexural load carried during the post-crack phase is obtained by satisfying the equilibrium of moments:

lM

P e2= (11)

Figure 7. Schematic view of forces and stresses acting on the cracked section of an SFRC beam

The equilibrating moment, Me, may be calculated by summing the moments generated by concrete stresses and the individual moments generated by the N individual fibers bridging the crack below the neutral axis. It follows from Figure 7, that

( )ñ ä =+'

0 10.

c N

ic fdybs (equilibrating forces) (12)

( ) ( )äñ +=N

ii

c

ce yfydybM10

.'

s (equilibrating moments) (13)

where b is the width of the beam, c’ is the depth of the uncracked section and y is the distance from the neutral axis.

CMOD

D0

e0

disp

lace

men

ts

f i..f 3

f 2f 1

ƒi = f(wi , ai , li )

sc

uncr

acke

d se

ctio

n

wi

12

3i

stra

ins


40

In the model, the pull-out force in each fiber (fi) is expressed as a function of the crack width, wi, according to the average pull-out force versus slip (or crack width) relationship obtained experimentally at the full embedment length, le=l/2. To enable this, single fibers must be pulled out from a concrete matrix at various inclinations with respect to the pull-out load. The bond-slip response is then represented using the Ramberg-Osgood formulation so that the force carried by each fiber may be expressed in terms of its orientation, ai and the slip, wi, as follows:

( )( )[ ] îý

îü

û

îí

îì

ë

+

-+=

CCi

ipiii

Bw

AAwEwf 1

1

1,a (14)

where the constants A, B, C and Ep, are obtained for each orientation through the Ramberg-Osgood formulation. Recognizing that the average force in the fibers at a layer which is at a distance ‘y’ from the neutral axis is averaged over the entire range of embedment and inclination that is possible, the value of ‘fi’ in Equations 12 and 13 may be computed as follows:

( ) ( ) ( ) ( ) ( ) ( )ýüû

íìë

+ùúø

éêè ++++= wf

wfwfwfwf

wff geometryi 4

1222

1 905.67455.22

0 (15)

or

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )ýüû

íìë

+ùúø

éêè ++++++= wf

wfwfwfwfwfwf

wff geometryi 6

1222

1 907560453015

0 (16)

depending upon the number of different inclinations (4 or 6, respectively) tested.

Note that a change in the rate of fiber slip will result in a new pull-out load, fi in Equations 15 and 16. This rate dependence of fi is reflected in a change in the constants for the Ramberg-Osgood fitted curve (Figure 8).

Flattened end steel fibers were pulled out individually from a concrete matrix under dynamic loading at four orientations in increments of 22.5º. The resulting bond-slip responses is shown in Figure 8. Clearly, under impact, the bond-slip response is not a monotonic response unlike under quasi-static loading (39). Therefore, one may question the wisdom of choosing a Ramberg-Osgood formulation to represent the bond-slip response of these fibers under impact. Nonetheless, as a first approximation, in this study


41

the bond-slip curves were fitted using the Ramberg-Osgood expressions as shown in Figure 8 for an aligned fiber. The constants for the fitted curves are also shown.

Figure 8. Impact response of single steel fibers during pull-out [with the corresponding Ramberg-Osgood formulation for the aligned case]

Concrete is a stress-rate sensitive material as witnessed by the apparent increase in the flexural strength under impact (Figure 5). Accordingly, in the model the compressive strength has been adjusted to include this apparent increase.

0

5

10

15

20

25

30

35

40

45

0 0.5 1 1.5 2 2.5 3deflection (mm)

load

(kN)

ExperimentalAnalytical

Experimental Response Analytical Response Peak Load

(kN) 38 34

JSCEToughness

Factor (MPa) 6.30 7.05

Figure 9. Impact response of an SFRC beam in flexure [notice the monotonic nature of the fitted curve in Figure 8 and reproduced by the model for the flexural response above.]


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The resulting model prediction is plotted together with the experimental flexural response under impact in Figure 9. Not surprisingly, the analytical response is also monotonic; nevertheless it is able to approximate the experimental response to a large degree. Notice that the model captures the flexural performance quite satisfactorily, as shown by the peak load and toughness measurements.

5. Concluding Remarks

The paper provides a brief description of our current understanding of the response of fiber reinforced concrete to impact and impulsive loads. While fiber reinforcement is the most effective way of enhancing concrete’s resistance to impact and other dynamic loads, significant issues remain unresolved. There is a critical need to develop a standardized technique of testing FRC under impact loading. Without a standardized technique, progress will remain slow. At very high strain-rates, some fiber reinforced concrete materials may exhibit brittle responses due to changes in fiber failure modes. Crack growth processes and crack velocities under impact remain poorly understood. Finally, more sophisticated models are needed that can capture the responses of a variety of fiber reinforced concrete materials to a wide range of applied strain-rates.

6. References

1. Comite Euro-International du Beton (CEB), Bulletin 187 (1988) 3.6. 2. Grote, D. L. et al., Int. J. Impact Engineering, 25 (2001) 869-910. 3. Malvar, L. J. and Ross, C. A., ACI Materials Journal, 95 (6) 735-773. 4. Zielinski, A.J. and Reinhardt, H.W., Cem. & Conc. Res. 12 (1982) 309-319. 5. Banthia, N., Mindess, S., Bentur, A. and Pigeon, M., Expt. Mech. 29 (2) (1989) 63-69. 6. Suaris, W. and Shah, S.P., Composites, 13 (1982) 153-159. 7. Gopalaratnam, V.S. et al., Experimental Mechanics, 24 (1984) 102-111. 8. Gokoz, U. and Naaman, A. E., Int. J. Cement Composites, 3 (3) (1981) 187-202. 9. Server, W. L., Journal of Testing & Evaluation, ASTM, 6 (1) (1978) 28-34. 10. Gupta, P. et al., Journal of Materials in Civil Engrg., ASCE, 12 (1) (2000) 81-90. 11. Banthia et al., SEM/RILEM Int. Conf. on Fract. of Concrete & Rock, (1987) 26-36. 12. Bindiganavile, V. and Banthia, N., ACI-SP “FRC: Innovations for Value”, in press. 13. Bindiganavile, V. and Banthia, N., Proceedings, CONSEC ’01, 1 (2001) 589-596. 14. Banthia, N. and Trottier, J-F., Cement and Concrete Res, 21 (1991) 158-168. 15. Pacios, A., Ouyang, C. and Shah, S. P., Mater. & Struct., RILEM, 28 (1995) 83-91. 16. Trottier, J.-F. and Mahoney, M., Concrete International, 23 (6) (2001) 23-28. 17. Bindiganavile, V. and Banthia, N., ACI, Materials Journal, 98 (1) (2001) 10-16. 18. Hillerborg, A. et al, Cement and Concrete Research, 6 (1976) 773-82. 19. Jeng, Y.S. and Shah, S.P., ASCE J of Eng. Mech., 111 (10) (1985) 1227-41. 20. Hillerborg, A., Cement and Concrete Composites, 2 (1980) 177-84. 21. Mindess, S. et al, Cement and Concrete Research, 7 (1977) 731-742. 22. Mobasher, B., Ouyang, C. and Shah, S.P., Int. J. of Fracture, 50 (1991) 199-219.


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23. Reinhardt, H. W., Materials Research Society, Proc. 64 (1985) 1-14. 24. Mindess, S. et al, Materials Research Society, Proc. 64 (1985) 217-224. 25. Yon, J-H, Hawkins, et al., ACI Mat. J., 88 (5) (1991) 470-479. 26. Birkimer, D. L., Proc.: 12th Symp. on Pock Mechanic, (1971) 573-589. 27. Zielinsky, A.J., Ph.D Thesis, TU Delft, (1982). 28. Mindess, S. and Nadeau, J.S., Am. Cer. Soc. Bull., 56 (1977) 429-430. 29. Banthia et al., Mat. And Sturc. (RILEM), 20 (1987) 293-302. 30. Watstein, D., J.of ACI, 49 (1953) 729-756. 31. Evans, R. H., J. of Inst. Of Civil Engrs, 18 (1942) 296. 32. Hughes et al., Magazine of Concrete Res., 24 (1972) 25-36. 33. Ross, C. A., ASME, Proc: Pressure Vessels & Piping Conf., (1997) 255-262 34. Bischoff, P. H. and Perry, S. H., ASCE J. of Eng. Mech, 121 (6) (1995) 685-693 35. Shah, S.P. and John, R., Materials Research Society, Proc. 64 (1985) 21-37. 36. Kaadi, G. W., MS Thesis, The University of Illinois, Chicago, (1983). 37. Visalvanich, K. and Naaman, A. E., ACI Journal, 80 (2), (1983), pp. 128-138. 38. Bindiganavile, V. and Banthia, N., Predicting Impact Response of Fiber Reinforced

Concrete Beams, in preparation, (2004) 39. Armelin, H. and Banthia, N., ACI Mat. J., 94 (1), (1997), pp. 18-31.


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