10
Fracture Mechanics of Concrete and Concrete Structures - High Performance, Fiber Reinforced Concrete, Special Loadings and Structural Applications- B. H. Oh, et al. (eds) 2010 Korea Concrete Institute, ISBN 978-89-5708-182-2 Fiber reinforced concrete characterization through round panel test - part I: experimental study F. Minelli & G.A. Plizzari University of Brescia, Italy ABSTRACT: Standard test methods for determining the mechanical properties of Fiber Reinforced Concrete (FRC) are properly defined if they reproduce the actual structural behavior. Among many proposals, the round panel test seems to have all potentials to become an easy-to-use tool and, at the same time, a reliable procedure for the characterization of FRC, in terms of toughness and post-cracking constitutive cohesive law. A compari- son between different test typologies for characterizing FRC is reported and discussed in the present paper, whit special emphasis on the different scatter that each tests produces. Tests are performed on beams as well as on panels. All specimens herein compared have the same concrete mechanical properties and fiber content. Aim of the experimental investigation is to critically discuss advantages and disadvantages of each testing proce- dure, focusing on the applicability of the method and on the reliability of results toward a consistent characteri- zation of the structural behavior. A new geometry for the panel test is herein proposed and discussed in order to make the panel easier to place, handle and test, therefore avoiding one of the major drawbacks which limit an extensive utilization of the panel tests. Suitable correlations among the different fracture and energy parame- ters defined in the assumed standards are finally reported, resulting very useful for a harmonization of the available standards. 1 INTRODUCTION Fiber Reinforced Concrete (FRC) is gaining an in- creasing interest among the concrete community for the reduced construction time and labor costs. For this reason, many structural elements are now rein- forced with steel fibers as partial or total substitution of conventional reinforcement (rebars or welded mesh; di Prisco et al.2004b). Besides cost issues, quality matters are of paramount importance for a construction and FRC also fulfills these require- ments since fibers allow for more distributed cracks with a smaller opening that enhances dura- bility (Schumacher, 2006). Construction materials require Standards for mea- suring their mechanical properties and for quality control. At the same time new material require new rules in building codes and special guidelines for structural design (Vandewalle, 2004; CEN, 2003). As far as FRC is concerned, design guidelines are al- ready available in some Countries (Rilem, 2003; di Prisco et al.2004; CNR, 2006) and work is in pro- gress for including them in the coming new fib Model Code (2010). In these guidelines, structural design is usually based on design values of the mate- rial parameters that are normally determined by di- viding the characteristic values by a partial safety fac- tor (γ M ). Mechanical properties of FRC are traditionally de- termined from beam tests that are usually based on a three (CEN, 2003) or four point bending schemes (UNI, 2003). Early experiences with the low volume fractions of fibers that are nowadays mostly used in practice (V f <0.8-1.0%), evidence that the character- istic values determined from beam tests (CEN, 2003; UNI, 2003) are quite low because of the high scatter in beam test results. It should be observed that this scatter is not related to the material itself by is mainly due to the small fracture areas (ranging from 160 to 180 cm 2 ). Such a scatter becomes particularly high when low contents (25-50 kg/m 3 ) of macro steel fi- bers (length ranging between 30 and 60 mm) are used (Sorelli et al.2005). It is commonly accepted that FRCs with a low volume fraction of fibers are particularly suitable for structures with a high degree of redundancy where stress redistribution may occur. Because of this redis- tribution, large fracture areas are involved (with a high number of fibers crossing them) and, therefore, structural behavior is mainly governed by the mean value of the material properties. Furthermore, be- cause of the large fracture areas, the scatter of ex- perimental results from structural tests is remarkably lower than that obtained from beam tests. A typical example is shown in Figure 1, which exhibits a set of curves obtained from a standard (bending) test on notched specimens (Fig. 1a) and from structural tests on full scale slabs on grade made of the same mate-

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Fracture Mechanics of Concrete and Concrete Structures -High Performance, Fiber Reinforced Concrete, Special Loadings and Structural Applications- B. H. Oh, et al. (eds)

ⓒ 2010 Korea Concrete Institute, ISBN 978-89-5708-182-2

Fiber reinforced concrete characterization through round panel test - part I: experimental study

F. Minelli & G.A. Plizzari University of Brescia, Italy ABSTRACT: Standard test methods for determining the mechanical properties of Fiber Reinforced Concrete (FRC) are properly defined if they reproduce the actual structural behavior. Among many proposals, the round panel test seems to have all potentials to become an easy-to-use tool and, at the same time, a reliable procedure for the characterization of FRC, in terms of toughness and post-cracking constitutive cohesive law. A compari-son between different test typologies for characterizing FRC is reported and discussed in the present paper, whit special emphasis on the different scatter that each tests produces. Tests are performed on beams as well as on panels. All specimens herein compared have the same concrete mechanical properties and fiber content. Aim of the experimental investigation is to critically discuss advantages and disadvantages of each testing proce-dure, focusing on the applicability of the method and on the reliability of results toward a consistent characteri-zation of the structural behavior. A new geometry for the panel test is herein proposed and discussed in order to make the panel easier to place, handle and test, therefore avoiding one of the major drawbacks which limit an extensive utilization of the panel tests. Suitable correlations among the different fracture and energy parame-ters defined in the assumed standards are finally reported, resulting very useful for a harmonization of the available standards.

1 INTRODUCTION

Fiber Reinforced Concrete (FRC) is gaining an in-creasing interest among the concrete community for the reduced construction time and labor costs. For this reason, many structural elements are now rein-forced with steel fibers as partial or total substitution of conventional reinforcement (rebars or welded mesh; di Prisco et al.2004b). Besides cost issues, quality matters are of paramount importance for a construction and FRC also fulfills these require-ments since fibers allow for more distributed cracks with a smaller opening that enhances dura-bility (Schumacher, 2006).

Construction materials require Standards for mea-suring their mechanical properties and for quality control. At the same time new material require new rules in building codes and special guidelines for structural design (Vandewalle, 2004; CEN, 2003). As far as FRC is concerned, design guidelines are al-ready available in some Countries (Rilem, 2003; di Prisco et al.2004; CNR, 2006) and work is in pro-gress for including them in the coming new fib Model Code (2010). In these guidelines, structural design is usually based on design values of the mate-rial parameters that are normally determined by di-viding the characteristic values by a partial safety fac-tor (γM).

Mechanical properties of FRC are traditionally de-

termined from beam tests that are usually based on a three (CEN, 2003) or four point bending schemes (UNI, 2003). Early experiences with the low volume fractions of fibers that are nowadays mostly used in practice (Vf<0.8-1.0%), evidence that the character-istic values determined from beam tests (CEN, 2003; UNI, 2003) are quite low because of the high scatter in beam test results. It should be observed that this scatter is not related to the material itself by is mainly due to the small fracture areas (ranging from 160 to 180 cm2). Such a scatter becomes particularly high when low contents (25-50 kg/m3) of macro steel fi-bers (length ranging between 30 and 60 mm) are used (Sorelli et al.2005).

It is commonly accepted that FRCs with a low volume fraction of fibers are particularly suitable for structures with a high degree of redundancy where stress redistribution may occur. Because of this redis-tribution, large fracture areas are involved (with a high number of fibers crossing them) and, therefore, structural behavior is mainly governed by the mean value of the material properties. Furthermore, be-cause of the large fracture areas, the scatter of ex-perimental results from structural tests is remarkably lower than that obtained from beam tests. A typical example is shown in Figure 1, which exhibits a set of curves obtained from a standard (bending) test on notched specimens (Fig. 1a) and from structural tests on full scale slabs on grade made of the same mate-

rial (Fig. 1b); the different scatter between material and structural tests is clearly evident, as a confirma-tion of the above presented discussion.

In order to obtain a more realistic value of the scatter of FRC material tests, specimens with larger fracture areas are needed; this suggests the use of larger beams or different specimens like slabs, where stress redistribution may also occur.

A square panel was proposed to simulate a por-tion of sprayed concrete in tunnel lining applications (EN 1488-5, 2004). However, since it is simply sup-ported along the whole border, any geometrical ir-regularity involves that the real support may vary in different specimens; in fact, although the support can lie in a perfect (and controlled) plane, specimens are normally deformed because of shrinkage effect. The crack pattern is therefore hardly predictable (the ac-tual three points on which any rigid body takes sup-port are generally randomly located along the four edges) and the determination of the constitutive laws for cracked concrete becomes very difficult.

A Round Determinate Panel (RDP) test was pro-posed by ASTM (2004); it is a statically determinate test (a round slab having a diameter φ=800 mm and a thickness of 75 mm, with three supports at 120 de-grees) where the crack pattern is predictable and the post-cracking material properties can be adequately determined. However, handling and placing such a specimen is quite complicated due to the large size and, consequently, high weight (91 kg). In addition, standard servo-controlled loading machines may not fit with the geometry of the panel, which is too big for many of them. The need of having a specimen easier to handle brought the Authors to come up with a proposal of a smaller round panel having a di-ameter of 600 mm and a depth of 60 mm, with a weight of only 40 kg.

0

1

2

3

4

5

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5CTODm [mm]

Nom

inal

Str

ess

sN [M

Pa]

Figure 1. Experimental results from bending tests on notched beams (di Prisco et al. (1), 2004).

The present paper focuses on the comparison of

different tests for FRC materials tested during the last few years at the University of Brescia. A com-parison between beam and panel tests, a discussion on the smaller round panel herein proposed as well as

the correlations between several fracture properties obtained from different Standards are also presented.

Load vs. Central deflection

0

50

100

150

200

250

300

350

0 1 2 3 4 5 6

Central deflection [mm]

Load

[kN

]

Slab P22 50/0,75 Vf=0,38 %

Slab P21 50/0,75 Vf=0,38 %

Slab P20 50/0,75 Vf=0,38 %

Figure 2. Experimental results from full-scale slabs on grade made of the same FRC as for Figure 1 (di Prisco et al. 2004).

2 MATERIALS, SET UP, EXPERIMENTAL RESULTS AND DISCUSSION

Among more than 100 comparative experiments car-ried out on beams and panels (60 tests on small round panels and 52 on large round panels), the fol-lowing discussion will deal with a number of tests performed on members containing either 20 or 30 kg/m3 of hooked-end steel fibers having a length of 50 mm and a diameter of 1 mm (the aspect ratio L/φ is 50). Fibers have a circular cross section and a ten-sile strength of 1100 MPa. Besides the FRC speci-mens, plain concrete beams and panels were also made.

Table 1 reports the main geometrical characteris-tics of the five testing typologies studied herein; the weight of each specimen, assuming a density of 24 kN/m3, is also outlined.

In order to study the behavior of all specimens up to failure, including any possible unstable branch af-ter cracking, a displacement controlled testing me-thod was adopted. The equipment shown in Figure 3 was utilized for all beam tests, whereas the dimen-sions of the round panel specimens, according to ASTM, as already mentioned, required utilizing a dif-ferent equipment, exhibited in Figure 4. In the first case, an INSTRON 1274 machine was used (having a closed loop and a maximum load of 300 kN). In the second case, the displacement was imposed by adopting an electro-mechanical screw jack (having a maximum load of 500 kN and a stroke of 300 mm) placed into a steel frame (Fig. 4); no closed loop was provided in this case. More recently, a steel support-ing and loading system was designed for performing tests on small round panels using the INSTRON ma-chine. Figure 5 and Figure 6 show a small round panel ready to test and the steel supporting system to be used in the servo-controlled machine.

Proceedings of FraMCoS-7, May 23-28, 2010

hThD ∇−= ),(J (1)

The proportionality coefficient D(h,T) is called moisture permeability and it is a nonlinear function of the relative humidity h and temperature T (Bažant & Najjar 1972). The moisture mass balance requires that the variation in time of the water mass per unit volume of concrete (water content w) be equal to the divergence of the moisture flux J

J•∇=∂

∂−

t

w (2)

The water content w can be expressed as the sum

of the evaporable water we (capillary water, water vapor, and adsorbed water) and the non-evaporable (chemically bound) water wn (Mills 1966, Pantazopoulo & Mills 1995). It is reasonable to assume that the evaporable water is a function of relative humidity, h, degree of hydration, αc, and degree of silica fume reaction, αs, i.e. we=we(h,αc,αs) = age-dependent sorption/desorption isotherm (Norling Mjonell 1997). Under this assumption and by substituting Equation 1 into Equation 2 one obtains

nscw

s

ew

c

ew

hh

Dt

h

h

ew

&&& ++∂

=∇•∇+∂

− αα

αα

)(

(3)

where ∂we/∂h is the slope of the sorption/desorption isotherm (also called moisture capacity). The governing equation (Equation 3) must be completed by appropriate boundary and initial conditions.

The relation between the amount of evaporable water and relative humidity is called ‘‘adsorption isotherm” if measured with increasing relativity humidity and ‘‘desorption isotherm” in the opposite case. Neglecting their difference (Xi et al. 1994), in the following, ‘‘sorption isotherm” will be used with reference to both sorption and desorption conditions. By the way, if the hysteresis of the moisture isotherm would be taken into account, two different relation, evaporable water vs relative humidity, must be used according to the sign of the variation of the relativity humidity. The shape of the sorption isotherm for HPC is influenced by many parameters, especially those that influence extent and rate of the chemical reactions and, in turn, determine pore structure and pore size distribution (water-to-cement ratio, cement chemical composition, SF content, curing time and method, temperature, mix additives, etc.). In the literature various formulations can be found to describe the sorption isotherm of normal concrete (Xi et al. 1994). However, in the present paper the semi-empirical expression proposed by Norling Mjornell (1997) is adopted because it

explicitly accounts for the evolution of hydration reaction and SF content. This sorption isotherm reads

( ) ( )( )

( ) ( )⎥⎥

⎢⎢

⎥⎥⎥

⎢⎢⎢

−∞

+

−∞

−=

1110

,1

110

11,

1,,

hcc

ge

scK

hcc

ge

scG

sch

ew

αα

αα

αα

αααα

(4)

where the first term (gel isotherm) represents the physically bound (adsorbed) water and the second term (capillary isotherm) represents the capillary water. This expression is valid only for low content of SF. The coefficient G1 represents the amount of water per unit volume held in the gel pores at 100% relative humidity, and it can be expressed (Norling Mjornell 1997) as

( ) ss

s

vgkc

c

c

vgk

scG αααα +=,1

(5)

where k

cvg and k

svg are material parameters. From the

maximum amount of water per unit volume that can fill all pores (both capillary pores and gel pores), one can calculate K1 as one obtains

( )1

110

110

11

22.0188.00

,1

−⎟⎠

⎞⎜⎝

⎛−∞

⎥⎥⎥

⎢⎢⎢

⎡⎟⎠

⎞⎜⎝

⎛−∞

−−+−

=

hcc

ge

hcc

geGs

ssc

w

scK

αα

αα

αα

αα

(6)

The material parameters k

cvg and k

svg and g1 can

be calibrated by fitting experimental data relevant to free (evaporable) water content in concrete at various ages (Di Luzio & Cusatis 2009b).

2.2 Temperature evolution

Note that, at early age, since the chemical reactions associated with cement hydration and SF reaction are exothermic, the temperature field is not uniform for non-adiabatic systems even if the environmental temperature is constant. Heat conduction can be described in concrete, at least for temperature not exceeding 100°C (Bažant & Kaplan 1996), by Fourier’s law, which reads

T∇−= λq (7)

where q is the heat flux, T is the absolute temperature, and λ is the heat conductivity; in this

A CMOD (Crack Mouth Opening Displacement)-controlled procedure was adopted for the notched beams (UNI and CEN prescribe a notch with a dif-ferent depth) whereas a displacement-controlled test (screw control of the electro-mechanical jack) was performed with the big round panel tests, resulting in an instability of the response immediately after the peak of the concrete matrix. For more details con-cerning the geometry, tests set-up and instrumenta-tion, one can refer to Marinoni et al. (2007).

Table 1. Main geometric characteristics of specimens. *Large Round Panel refers to the standard test of ASTM (2004).

Figure 3. Set up for performing beam tests.

In the latest small round panels, tested with the

servo-controlled machine available in the laboratory, a fictitious CMOD was set in a region close to the midpoint in the bottom panel surface. By imposing a very small opening of this point, a much more refined and stable control of the test was possible, without the sudden load drop experienced in all previous panels (both large and small) tested with a simple screw jack. This recent improvement in the test set-up guaranteed a helpful refinement of the test, especially concerning the crack monitoring and its use for material charac-

terization, as shown in the second part of this re-search report.

After the peak load, once the load path was stabi-lized, the test was conducted under a stroke-control procedure, as for beam tests.

Figure 4. Set up for performing Large Panel tests.

Figure 5. Set up for performing Large Panel tests.

Several LVDTs (Linear Variable Displacement

Transducers) were used in each test to measure the vertical displacements (under the load points and in other locations) and the crack openings (including the Crack Tip Opening Displacements in notched beams). In the beam tests, instruments were placed for measuring the point load displacement/s in the front and rear face, the crack width (CTOD) both in the front and back face, and the CMOD gauge.

Concerning the round panels, besides the central

Specimen Length [mm]

Height [mm]

Notch [mm]

Weight [kg]

Loading Scheme

Beam UNI 600 150 25 32.4 3 points Beam CEN 550 150 45 29.7 4 points

Dimensions [mm]

Thickness [mm]

Weight [kg]

Loading Scheme

Large Round Panel* (RPL) 800 (radius) 75 90.5 Central

point load Small Round Panel (RPS)

600 (radius) 60 40.7 Central point load

Square Panel (SP)

600 (side) 100 86.4 Central point load

Proceedings of FraMCoS-7, May 23-28, 2010

hThD ∇−= ),(J (1)

The proportionality coefficient D(h,T) is called moisture permeability and it is a nonlinear function of the relative humidity h and temperature T (Bažant & Najjar 1972). The moisture mass balance requires that the variation in time of the water mass per unit volume of concrete (water content w) be equal to the divergence of the moisture flux J

J•∇=∂

∂−

t

w (2)

The water content w can be expressed as the sum

of the evaporable water we (capillary water, water vapor, and adsorbed water) and the non-evaporable (chemically bound) water wn (Mills 1966, Pantazopoulo & Mills 1995). It is reasonable to assume that the evaporable water is a function of relative humidity, h, degree of hydration, αc, and degree of silica fume reaction, αs, i.e. we=we(h,αc,αs) = age-dependent sorption/desorption isotherm (Norling Mjonell 1997). Under this assumption and by substituting Equation 1 into Equation 2 one obtains

nscw

s

ew

c

ew

hh

Dt

h

h

ew

&&& ++∂

=∇•∇+∂

− αα

αα

)(

(3)

where ∂we/∂h is the slope of the sorption/desorption isotherm (also called moisture capacity). The governing equation (Equation 3) must be completed by appropriate boundary and initial conditions.

The relation between the amount of evaporable water and relative humidity is called ‘‘adsorption isotherm” if measured with increasing relativity humidity and ‘‘desorption isotherm” in the opposite case. Neglecting their difference (Xi et al. 1994), in the following, ‘‘sorption isotherm” will be used with reference to both sorption and desorption conditions. By the way, if the hysteresis of the moisture isotherm would be taken into account, two different relation, evaporable water vs relative humidity, must be used according to the sign of the variation of the relativity humidity. The shape of the sorption isotherm for HPC is influenced by many parameters, especially those that influence extent and rate of the chemical reactions and, in turn, determine pore structure and pore size distribution (water-to-cement ratio, cement chemical composition, SF content, curing time and method, temperature, mix additives, etc.). In the literature various formulations can be found to describe the sorption isotherm of normal concrete (Xi et al. 1994). However, in the present paper the semi-empirical expression proposed by Norling Mjornell (1997) is adopted because it

explicitly accounts for the evolution of hydration reaction and SF content. This sorption isotherm reads

( ) ( )( )

( ) ( )⎥⎥

⎢⎢

⎥⎥⎥

⎢⎢⎢

−∞

+

−∞

−=

1110

,1

110

11,

1,,

hcc

ge

scK

hcc

ge

scG

sch

ew

αα

αα

αα

αααα

(4)

where the first term (gel isotherm) represents the physically bound (adsorbed) water and the second term (capillary isotherm) represents the capillary water. This expression is valid only for low content of SF. The coefficient G1 represents the amount of water per unit volume held in the gel pores at 100% relative humidity, and it can be expressed (Norling Mjornell 1997) as

( ) ss

s

vgkc

c

c

vgk

scG αααα +=,1

(5)

where k

cvg and k

svg are material parameters. From the

maximum amount of water per unit volume that can fill all pores (both capillary pores and gel pores), one can calculate K1 as one obtains

( )1

110

110

11

22.0188.00

,1

−⎟⎠

⎞⎜⎝

⎛−∞

⎥⎥⎥

⎢⎢⎢

⎡⎟⎠

⎞⎜⎝

⎛−∞

−−+−

=

hcc

ge

hcc

geGs

ssc

w

scK

αα

αα

αα

αα

(6)

The material parameters k

cvg and k

svg and g1 can

be calibrated by fitting experimental data relevant to free (evaporable) water content in concrete at various ages (Di Luzio & Cusatis 2009b).

2.2 Temperature evolution

Note that, at early age, since the chemical reactions associated with cement hydration and SF reaction are exothermic, the temperature field is not uniform for non-adiabatic systems even if the environmental temperature is constant. Heat conduction can be described in concrete, at least for temperature not exceeding 100°C (Bažant & Kaplan 1996), by Fourier’s law, which reads

T∇−= λq (7)

where q is the heat flux, T is the absolute temperature, and λ is the heat conductivity; in this

Figure 6. Steel supporting frame for performing Small Panel tests.

displacement of the bottom side and the CMOD, as already reported, 3 LVDTs were also disposed for the measurements of the three cracks. The three in-struments were placed at a distance of 120 mm from the midpoint, to make their placement easy. The measurement length was 150 mm, which generally al-lowed intercepting the crack. Figure 7 shows a pic-ture with the instrumentation on a round panel small prior testing.

Figure 7. Instrumentation, bottom side of a small round panel.

Figure 8-Figure 11 show the experimental curves

of a series of experiments made of different speci-mens from the same concrete batch. Figure 8 and Figure 9 report the Nominal Stress (according to a linear stress distribution in the cracked section) ver-sus the CTOD for both CEN and UNI beam tests whereas Figure 10 and Figure 11 exhibit the load vs. the central vertical displacement of both large and small round panels made of the identical material. As expected, the post-peak behavior is similar for the two types of beam tests, which are characterized by a rather large scatter. A smaller dispersion can be seen in the panel plots: this is further confirmed from Figure 12, which shows the load-displacement curve

of small round panels cast with High Strength Con-crete and with two fiber contents: 30 and 60 kg/m3.

Figure 14 and Figure 15 exhibit the coefficient of variation from the dispersion of previous figures, it results that the coefficient is definitely much smaller in panels than in the corresponding beam tests, both for large and small round panels.

0

1

2

3

4

5

6

7

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5N

omin

al S

tres

s s

N[M

Pa]

CTODm [mm]

Comparison UNI Beams Steel fibres FF1

20 kg/m³

30 kg/m³

Figure 8. Nominal stress-CTOD of Beam tests according to UNI, for FRC with 20 and 30 kg/m3.

0

1

2

3

4

5

6

7

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Nom

inal

Str

ess

sN

[MPa

]

CTODm [mm]

Comparison CEN BeamsSteel fibres FF1

20 kg/m³

30 kg/m³

Figure 9. Nominal stress-CTOD of Beam tests according to CEN, for FRC with 20 and 30 kg/m3.

With the aforementioned three LVDTs, crack

widths greater than 20 mm were measured with a post-cracking load higher than one third of the peak load. Such an instrumentation is not required by the ASTM Standard, which states that one should only calculate the energy absorption that is defined by the vertical load and the vertical displacement. By moni-toring the crack widths, whose location is predictable because of the statically determinate support system, it was also possible to come up with nominal stress (from elastic analysis) vs. crack width plots, as a suitable tool for the mechanical characterization of

Proceedings of FraMCoS-7, May 23-28, 2010

hThD ∇−= ),(J (1)

The proportionality coefficient D(h,T) is called moisture permeability and it is a nonlinear function of the relative humidity h and temperature T (Bažant & Najjar 1972). The moisture mass balance requires that the variation in time of the water mass per unit volume of concrete (water content w) be equal to the divergence of the moisture flux J

J•∇=∂

∂−

t

w (2)

The water content w can be expressed as the sum

of the evaporable water we (capillary water, water vapor, and adsorbed water) and the non-evaporable (chemically bound) water wn (Mills 1966, Pantazopoulo & Mills 1995). It is reasonable to assume that the evaporable water is a function of relative humidity, h, degree of hydration, αc, and degree of silica fume reaction, αs, i.e. we=we(h,αc,αs) = age-dependent sorption/desorption isotherm (Norling Mjonell 1997). Under this assumption and by substituting Equation 1 into Equation 2 one obtains

nscw

s

ew

c

ew

hh

Dt

h

h

ew

&&& ++∂

=∇•∇+∂

− αα

αα

)(

(3)

where ∂we/∂h is the slope of the sorption/desorption isotherm (also called moisture capacity). The governing equation (Equation 3) must be completed by appropriate boundary and initial conditions.

The relation between the amount of evaporable water and relative humidity is called ‘‘adsorption isotherm” if measured with increasing relativity humidity and ‘‘desorption isotherm” in the opposite case. Neglecting their difference (Xi et al. 1994), in the following, ‘‘sorption isotherm” will be used with reference to both sorption and desorption conditions. By the way, if the hysteresis of the moisture isotherm would be taken into account, two different relation, evaporable water vs relative humidity, must be used according to the sign of the variation of the relativity humidity. The shape of the sorption isotherm for HPC is influenced by many parameters, especially those that influence extent and rate of the chemical reactions and, in turn, determine pore structure and pore size distribution (water-to-cement ratio, cement chemical composition, SF content, curing time and method, temperature, mix additives, etc.). In the literature various formulations can be found to describe the sorption isotherm of normal concrete (Xi et al. 1994). However, in the present paper the semi-empirical expression proposed by Norling Mjornell (1997) is adopted because it

explicitly accounts for the evolution of hydration reaction and SF content. This sorption isotherm reads

( ) ( )( )

( ) ( )⎥⎥

⎢⎢

⎥⎥⎥

⎢⎢⎢

−∞

+

−∞

−=

1110

,1

110

11,

1,,

hcc

ge

scK

hcc

ge

scG

sch

ew

αα

αα

αα

αααα

(4)

where the first term (gel isotherm) represents the physically bound (adsorbed) water and the second term (capillary isotherm) represents the capillary water. This expression is valid only for low content of SF. The coefficient G1 represents the amount of water per unit volume held in the gel pores at 100% relative humidity, and it can be expressed (Norling Mjornell 1997) as

( ) ss

s

vgkc

c

c

vgk

scG αααα +=,1

(5)

where k

cvg and k

svg are material parameters. From the

maximum amount of water per unit volume that can fill all pores (both capillary pores and gel pores), one can calculate K1 as one obtains

( )1

110

110

11

22.0188.00

,1

−⎟⎠

⎞⎜⎝

⎛−∞

⎥⎥⎥

⎢⎢⎢

⎡⎟⎠

⎞⎜⎝

⎛−∞

−−+−

=

hcc

ge

hcc

geGs

ssc

w

scK

αα

αα

αα

αα

(6)

The material parameters k

cvg and k

svg and g1 can

be calibrated by fitting experimental data relevant to free (evaporable) water content in concrete at various ages (Di Luzio & Cusatis 2009b).

2.2 Temperature evolution

Note that, at early age, since the chemical reactions associated with cement hydration and SF reaction are exothermic, the temperature field is not uniform for non-adiabatic systems even if the environmental temperature is constant. Heat conduction can be described in concrete, at least for temperature not exceeding 100°C (Bažant & Kaplan 1996), by Fourier’s law, which reads

T∇−= λq (7)

where q is the heat flux, T is the absolute temperature, and λ is the heat conductivity; in this

FRC materials and for the definition of simplified stress-crack width cohesive constitutive laws.

Figure 13 reports a typical experimental curve of the crack width vs. load. The three crack locations are in most of cases in good agreement with the ex-pectations (120°) and their values are rather similar one to each other.

0

5

10

15

20

25

30

35

0 10 20 30 40 50

Load

[KN

]

Displacement [mm]

Comparison LARGE ROUND PANELSteel fibres FF1

20 kg/m³

30 kg/m³

Figure 10. Load-Displacement curve of Large Round Panel Tests according to ASTM, for FRC with 20 and 30 kg/m3.

0

5

10

15

20

25

0 10 20 30 40

Load

[KN

]

Displacement [mm]

Comparison SMALL ROUND PANELSteel fibres

20 kg/m³

30 kg/m³

Figure 11. Load-Displacement curve of Small Round Panel Tests, for FRC with 20 and 30 kg/m3.

The coefficient of variation, as an indicator of the

test-result scatter, was calculated for all properties and indexes defined in the different standards, both for quantities which refer to the serviceability limit states (SLS, Fig. 14) and ultimate limit states (ULS, Fig. 15). Once again, a significant lower coefficient of variation can be outlined for panel tests in com-parison with beam tests.

It seems worth using the low scatter of panel tests for the determination of more suitable fracture prop-erties consistent with the ones determined from the

standard beam tests (i.e. finding residual stresses or post-cracking strengths as stated by beam standard but using round panel tests). To this aim, the follow-ing procedure was undertaken: 1. Find correlation between the different parameters required by different standards using experimental re-sults; 2. Given the correlations, calculate average value of the equivalent (UNI) or local (CEN) post-cracking strength (beam tests) from the experimental values of energy absorptions from panels; 3. Calculate the characteristic values of the post-cracking strength from panels accounting for a lower experimental scatter; 4. Compare the values of equivalent post-cracking strength determined from beam tests (direct method) with those from large and small panel tests.

Figure 12. Load-Displacement curve of Small Round Panel Tests, HSC panels, FRC with 30 and 60 kg/m3.

0

5

10

15

0 5 10 15 20 25

Aver

age

Cra

ck W

idth

[mm

]

Load [KN]

Comparison SMALL ROUND PANEL Steel fibres FF1

20 kg/m³

30 kg/m³

Figure 13. Crack width-Load curve of Small Round Panel Tests, HSC panels, FRC with 30 and 60 kg/m3.

Figure 16, as an example, describes the correlation

between the equivalent post-cracking strength of the UNI standard and the energy absorption of the

Proceedings of FraMCoS-7, May 23-28, 2010

hThD ∇−= ),(J (1)

The proportionality coefficient D(h,T) is called moisture permeability and it is a nonlinear function of the relative humidity h and temperature T (Bažant & Najjar 1972). The moisture mass balance requires that the variation in time of the water mass per unit volume of concrete (water content w) be equal to the divergence of the moisture flux J

J•∇=∂

∂−

t

w (2)

The water content w can be expressed as the sum

of the evaporable water we (capillary water, water vapor, and adsorbed water) and the non-evaporable (chemically bound) water wn (Mills 1966, Pantazopoulo & Mills 1995). It is reasonable to assume that the evaporable water is a function of relative humidity, h, degree of hydration, αc, and degree of silica fume reaction, αs, i.e. we=we(h,αc,αs) = age-dependent sorption/desorption isotherm (Norling Mjonell 1997). Under this assumption and by substituting Equation 1 into Equation 2 one obtains

nscw

s

ew

c

ew

hh

Dt

h

h

ew

&&& ++∂

=∇•∇+∂

− αα

αα

)(

(3)

where ∂we/∂h is the slope of the sorption/desorption isotherm (also called moisture capacity). The governing equation (Equation 3) must be completed by appropriate boundary and initial conditions.

The relation between the amount of evaporable water and relative humidity is called ‘‘adsorption isotherm” if measured with increasing relativity humidity and ‘‘desorption isotherm” in the opposite case. Neglecting their difference (Xi et al. 1994), in the following, ‘‘sorption isotherm” will be used with reference to both sorption and desorption conditions. By the way, if the hysteresis of the moisture isotherm would be taken into account, two different relation, evaporable water vs relative humidity, must be used according to the sign of the variation of the relativity humidity. The shape of the sorption isotherm for HPC is influenced by many parameters, especially those that influence extent and rate of the chemical reactions and, in turn, determine pore structure and pore size distribution (water-to-cement ratio, cement chemical composition, SF content, curing time and method, temperature, mix additives, etc.). In the literature various formulations can be found to describe the sorption isotherm of normal concrete (Xi et al. 1994). However, in the present paper the semi-empirical expression proposed by Norling Mjornell (1997) is adopted because it

explicitly accounts for the evolution of hydration reaction and SF content. This sorption isotherm reads

( ) ( )( )

( ) ( )⎥⎥

⎢⎢

⎥⎥⎥

⎢⎢⎢

−∞

+

−∞

−=

1110

,1

110

11,

1,,

hcc

ge

scK

hcc

ge

scG

sch

ew

αα

αα

αα

αααα

(4)

where the first term (gel isotherm) represents the physically bound (adsorbed) water and the second term (capillary isotherm) represents the capillary water. This expression is valid only for low content of SF. The coefficient G1 represents the amount of water per unit volume held in the gel pores at 100% relative humidity, and it can be expressed (Norling Mjornell 1997) as

( ) ss

s

vgkc

c

c

vgk

scG αααα +=,1

(5)

where k

cvg and k

svg are material parameters. From the

maximum amount of water per unit volume that can fill all pores (both capillary pores and gel pores), one can calculate K1 as one obtains

( )1

110

110

11

22.0188.00

,1

−⎟⎠

⎞⎜⎝

⎛−∞

⎥⎥⎥

⎢⎢⎢

⎡⎟⎠

⎞⎜⎝

⎛−∞

−−+−

=

hcc

ge

hcc

geGs

ssc

w

scK

αα

αα

αα

αα

(6)

The material parameters k

cvg and k

svg and g1 can

be calibrated by fitting experimental data relevant to free (evaporable) water content in concrete at various ages (Di Luzio & Cusatis 2009b).

2.2 Temperature evolution

Note that, at early age, since the chemical reactions associated with cement hydration and SF reaction are exothermic, the temperature field is not uniform for non-adiabatic systems even if the environmental temperature is constant. Heat conduction can be described in concrete, at least for temperature not exceeding 100°C (Bažant & Kaplan 1996), by Fourier’s law, which reads

T∇−= λq (7)

where q is the heat flux, T is the absolute temperature, and λ is the heat conductivity; in this

ASTM panels at SLS. One should note that the en-ergy absorption relevant for the SLS was conven-tionally defined by the authors as the energy meas-ured at 5 mm displacement for the classical round panels, whereas at 3.75 mm (just by scaling the di-mensions) for the small panels. These two values re-fer to crack stages which are significant for general situation related to serviceability limit states (com-pare, as an example, Fig. 11 and Fig. 13).

0.24

0.05

0.09

0.14

0.23

0.41

0.16

0.08

0.0

0.1

0.2

0.3

0.4

0.5

UNI CEN RPL RPS

Comparison coefficient of variation Steel Fibers

Properties determined at SLS

20 kg/m³

30 kg/m³

Figure 14. Coefficient of variation calculated for all parame-ters required by the standards considered in the present ex-perimental campaign.

0.32

0.19

0.37

0.15

0.23

0.46

0.23

0.08

0.0

0.1

0.2

0.3

0.4

0.5

UNI CEN RPL RPS

Comparison coefficient of variation Steel Fibers FF1

Properties determined at ULS

20 kg/m³

30 kg/m³

Figure 15. Coefficient of variation calculated for all parame-ters required by the standards considered in the present ex-perimental campaign.

Thirteen comparative studies are reported with

different points in the plots, each one referring to the average of at least three round panels per series. Dif-ferent fiber contents, fiber materials, fiber “cock-tails”, fiber geometry and concrete classes are con-sidered in these plots. Figure 17 shows the identical correlation calculated with the parameters at ULS. From these two plots, one can notice that a linear re-gression between the two quantities represents well

the trend and that the coefficient of variation R2 is quite small.

EL,5RPL = 26.84 feq (0-0,6)UNIR2 = 0.85

0

20

40

60

80

100

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

EL

,5R

PL

[J]

feq (0-0,6) UNI Beams [MPa]

Relation Large Round Panels - UNI Beams ; SLS

Figure 16. Correlations between energy absorption and equiva-lent post cracking strength: Classical Large Round Panel tests ASTM vs. Beam Tests UNI, serviceability limit states.

EL,40 RPL = 127.23 feq (0.6-3)UNIR2 = 0.86

0

100

200

300

400

500

600

0.0 1.0 2.0 3.0 4.0

EL,

40R

PL [

J]

feq (0.6-3) UNI Beams [MPa]

Relation Large Round Panels - UNI Beams ; ULS

Figure 17. Correlations between energy absorption and equiva-lent post cracking strength: Classical Large Round Panel tests ASTM vs. Beam Tests UNI, ultimate limit states.

The two relationships found are as follows: ,5 (0 0.6)

,40 (0.6 3)

26.84

127.23L eq

L eq

E f

E f−

=

= (1)

where:

EL,5 is the energy absorption for the large round panel up to a vertical displacement of 5 mm (set by the authors);

EL,40 is the energy absorption for the large round panel up to a vertical displacement of 40 mm (de-fined in the ASTM Standard);

Proceedings of FraMCoS-7, May 23-28, 2010

hThD ∇−= ),(J (1)

The proportionality coefficient D(h,T) is called moisture permeability and it is a nonlinear function of the relative humidity h and temperature T (Bažant & Najjar 1972). The moisture mass balance requires that the variation in time of the water mass per unit volume of concrete (water content w) be equal to the divergence of the moisture flux J

J•∇=∂

∂−

t

w (2)

The water content w can be expressed as the sum

of the evaporable water we (capillary water, water vapor, and adsorbed water) and the non-evaporable (chemically bound) water wn (Mills 1966, Pantazopoulo & Mills 1995). It is reasonable to assume that the evaporable water is a function of relative humidity, h, degree of hydration, αc, and degree of silica fume reaction, αs, i.e. we=we(h,αc,αs) = age-dependent sorption/desorption isotherm (Norling Mjonell 1997). Under this assumption and by substituting Equation 1 into Equation 2 one obtains

nscw

s

ew

c

ew

hh

Dt

h

h

ew

&&& ++∂

=∇•∇+∂

− αα

αα

)(

(3)

where ∂we/∂h is the slope of the sorption/desorption isotherm (also called moisture capacity). The governing equation (Equation 3) must be completed by appropriate boundary and initial conditions.

The relation between the amount of evaporable water and relative humidity is called ‘‘adsorption isotherm” if measured with increasing relativity humidity and ‘‘desorption isotherm” in the opposite case. Neglecting their difference (Xi et al. 1994), in the following, ‘‘sorption isotherm” will be used with reference to both sorption and desorption conditions. By the way, if the hysteresis of the moisture isotherm would be taken into account, two different relation, evaporable water vs relative humidity, must be used according to the sign of the variation of the relativity humidity. The shape of the sorption isotherm for HPC is influenced by many parameters, especially those that influence extent and rate of the chemical reactions and, in turn, determine pore structure and pore size distribution (water-to-cement ratio, cement chemical composition, SF content, curing time and method, temperature, mix additives, etc.). In the literature various formulations can be found to describe the sorption isotherm of normal concrete (Xi et al. 1994). However, in the present paper the semi-empirical expression proposed by Norling Mjornell (1997) is adopted because it

explicitly accounts for the evolution of hydration reaction and SF content. This sorption isotherm reads

( ) ( )( )

( ) ( )⎥⎥

⎢⎢

⎥⎥⎥

⎢⎢⎢

−∞

+

−∞

−=

1110

,1

110

11,

1,,

hcc

ge

scK

hcc

ge

scG

sch

ew

αα

αα

αα

αααα

(4)

where the first term (gel isotherm) represents the physically bound (adsorbed) water and the second term (capillary isotherm) represents the capillary water. This expression is valid only for low content of SF. The coefficient G1 represents the amount of water per unit volume held in the gel pores at 100% relative humidity, and it can be expressed (Norling Mjornell 1997) as

( ) ss

s

vgkc

c

c

vgk

scG αααα +=,1

(5)

where k

cvg and k

svg are material parameters. From the

maximum amount of water per unit volume that can fill all pores (both capillary pores and gel pores), one can calculate K1 as one obtains

( )1

110

110

11

22.0188.00

,1

−⎟⎠

⎞⎜⎝

⎛−∞

⎥⎥⎥

⎢⎢⎢

⎡⎟⎠

⎞⎜⎝

⎛−∞

−−+−

=

hcc

ge

hcc

geGs

ssc

w

scK

αα

αα

αα

αα

(6)

The material parameters k

cvg and k

svg and g1 can

be calibrated by fitting experimental data relevant to free (evaporable) water content in concrete at various ages (Di Luzio & Cusatis 2009b).

2.2 Temperature evolution

Note that, at early age, since the chemical reactions associated with cement hydration and SF reaction are exothermic, the temperature field is not uniform for non-adiabatic systems even if the environmental temperature is constant. Heat conduction can be described in concrete, at least for temperature not exceeding 100°C (Bažant & Kaplan 1996), by Fourier’s law, which reads

T∇−= λq (7)

where q is the heat flux, T is the absolute temperature, and λ is the heat conductivity; in this

feq(0-0.6) is the equivalent post-cracking strength calculated for a CTOD range varying from 0 to 0.6 mm (SLS) included in the UNI Standard.

feq(0.6-3.6) is the equivalent post-cracking strength calculated for a CTOD range varying from 0.6 to 3 mm (ULS) included in the UNI Standard.

Other correlations were also determined between panels (small and large) and beams (CEN and UNI).

Figure 18 and Figure 19 reports the correlations between UNI beam test and Small Round Panel, ob-tained using 12 experimental points. The coefficient of variation is again considerably good and the linear approximation is also consistent, leading to the fol-lowing relations:

)36.0(30,

)6.00(75.3,

84.7000.15

=

=

eqS

eqS

fEfE

(2)

ES,3.75RPS = 15.00 feq (0-0.6)UNIR2 = 0.97

0

10

20

30

40

50

60

70

0.0 1.0 2.0 3.0 4.0 5.0

E S,3

.75

RPS

[J]

feq (0-0,6) UNI Beams [MPa]

Relation Small Round Panels - UNI Beams; SLS

( Only Strain-Softening Materials Under Flexure)

Figure 18. Correlations between energy absorption and equiva-lent post cracking strength: Small Round Panel tests vs. Beam Tests UNI, serviceability limit states.

ES,30RPS = 70.84 feq (0.6-3)UNIR2 = 090

0

50

100

150

200

250

300

0.0 1.0 2.0 3.0 4.0

E S,3

0R

PS [

J]

feq (0.6-3) UNI Beams [MPa]

Relation Small Round Panels - UNI Beams; ULS

( Only Strain-Softening Materials Under Flexure)

Figure 19. Correlations between energy absorption and equiva-lent post cracking strength: Small Round Panel tests vs. Beam Tests UNI, ultimate limit states.

Figure 20, Figure 21 and Figure 22 exhibit the

correlation between the residual strengths and equiv-

alent post-cracking stresses defined in the CEN and UNI standard beam tests. Even though a smaller da-tabase is available (only 7 experimental points are plotted, each one referring to the average values of at least three beam tests per series), the coefficient of variation is quite good and the relation already con-sistent. Further comparative studies or collection of experimental comparative experimental data world-wide would be useful for a refinement of these rela-tionships. Note that all correlations were set as lines and force to cross the axes origin. This in fact seems to be in good agreement with the experimental re-sults.

fr1 CEN = 0.98 feq(0-0.6) UNI R2 = 0.83

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0

f r1 C

EN B

eam

s [M

Pa]

feq (0-0.6) UNI Beams [MPa]

Relation CEN Beams - UNI Beams; SLS

Figure 20. Correlations between residual stress and equivalent post cracking strength: Beam Test CEN vs. Beam Tests UNI, serviceability limit states.

fr4 CEN = 0.78 feq(0.6-3) UNIR2 = 0.75

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 1.0 2.0 3.0 4.0 5.0

f r4CE

N Be

ams

[MPa

]

feq (0.6-3) UNI Beams [MPa]

Relation CEN Beams - UNI Beams; ULS

Figure 21. Correlations between residual stress and equivalent post cracking strength: Beam Test CEN vs. Beam Tests UNI, ultimate limit states.

Once the correlations are found, it is possible to

determine, for example, beam toughness properties from panel tests, using the corresponding lower dis-persion. As an example, the characteristics values of the equivalent post-cracking strength, suitable for de-sign purposes, were calculated by using the corre-sponding experimental scatter respectively of UNI

Proceedings of FraMCoS-7, May 23-28, 2010

hThD ∇−= ),(J (1)

The proportionality coefficient D(h,T) is called moisture permeability and it is a nonlinear function of the relative humidity h and temperature T (Bažant & Najjar 1972). The moisture mass balance requires that the variation in time of the water mass per unit volume of concrete (water content w) be equal to the divergence of the moisture flux J

J•∇=∂

∂−

t

w (2)

The water content w can be expressed as the sum

of the evaporable water we (capillary water, water vapor, and adsorbed water) and the non-evaporable (chemically bound) water wn (Mills 1966, Pantazopoulo & Mills 1995). It is reasonable to assume that the evaporable water is a function of relative humidity, h, degree of hydration, αc, and degree of silica fume reaction, αs, i.e. we=we(h,αc,αs) = age-dependent sorption/desorption isotherm (Norling Mjonell 1997). Under this assumption and by substituting Equation 1 into Equation 2 one obtains

nscw

s

ew

c

ew

hh

Dt

h

h

ew

&&& ++∂

=∇•∇+∂

− αα

αα

)(

(3)

where ∂we/∂h is the slope of the sorption/desorption isotherm (also called moisture capacity). The governing equation (Equation 3) must be completed by appropriate boundary and initial conditions.

The relation between the amount of evaporable water and relative humidity is called ‘‘adsorption isotherm” if measured with increasing relativity humidity and ‘‘desorption isotherm” in the opposite case. Neglecting their difference (Xi et al. 1994), in the following, ‘‘sorption isotherm” will be used with reference to both sorption and desorption conditions. By the way, if the hysteresis of the moisture isotherm would be taken into account, two different relation, evaporable water vs relative humidity, must be used according to the sign of the variation of the relativity humidity. The shape of the sorption isotherm for HPC is influenced by many parameters, especially those that influence extent and rate of the chemical reactions and, in turn, determine pore structure and pore size distribution (water-to-cement ratio, cement chemical composition, SF content, curing time and method, temperature, mix additives, etc.). In the literature various formulations can be found to describe the sorption isotherm of normal concrete (Xi et al. 1994). However, in the present paper the semi-empirical expression proposed by Norling Mjornell (1997) is adopted because it

explicitly accounts for the evolution of hydration reaction and SF content. This sorption isotherm reads

( ) ( )( )

( ) ( )⎥⎥

⎢⎢

⎥⎥⎥

⎢⎢⎢

−∞

+

−∞

−=

1110

,1

110

11,

1,,

hcc

ge

scK

hcc

ge

scG

sch

ew

αα

αα

αα

αααα

(4)

where the first term (gel isotherm) represents the physically bound (adsorbed) water and the second term (capillary isotherm) represents the capillary water. This expression is valid only for low content of SF. The coefficient G1 represents the amount of water per unit volume held in the gel pores at 100% relative humidity, and it can be expressed (Norling Mjornell 1997) as

( ) ss

s

vgkc

c

c

vgk

scG αααα +=,1

(5)

where k

cvg and k

svg are material parameters. From the

maximum amount of water per unit volume that can fill all pores (both capillary pores and gel pores), one can calculate K1 as one obtains

( )1

110

110

11

22.0188.00

,1

−⎟⎠

⎞⎜⎝

⎛−∞

⎥⎥⎥

⎢⎢⎢

⎡⎟⎠

⎞⎜⎝

⎛−∞

−−+−

=

hcc

ge

hcc

geGs

ssc

w

scK

αα

αα

αα

αα

(6)

The material parameters k

cvg and k

svg and g1 can

be calibrated by fitting experimental data relevant to free (evaporable) water content in concrete at various ages (Di Luzio & Cusatis 2009b).

2.2 Temperature evolution

Note that, at early age, since the chemical reactions associated with cement hydration and SF reaction are exothermic, the temperature field is not uniform for non-adiabatic systems even if the environmental temperature is constant. Heat conduction can be described in concrete, at least for temperature not exceeding 100°C (Bažant & Kaplan 1996), by Fourier’s law, which reads

T∇−= λq (7)

where q is the heat flux, T is the absolute temperature, and λ is the heat conductivity; in this

beam, large (ASTM) and small (our proposal) round panels. In doing so, Figure 23 and Figure 24 exhibit a comparison between the average and the characteris-tic values of the fracture parameter feq(0-0.6), for FRC with 20 (Fig. 23) and 30 kg/m3 of steel fibers (Fig. 24). Due to a rather high scatter, using the beam test the characteristic values turn out to be at least 40% lower than the average one whereas, in the case of panel test, the reduction varies from 15 to 30%; this lower difference represents a beneficial effect on the values adopted in the design process and, conse-quently, on the structural dimensions. The design ad-vantages are, therefore, doubtless.

fr3 CEN = 0.90 feq(0.6-3) UNIR2 = 0.69

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

0.0 1.0 2.0 3.0 4.0 5.0

f r3C

EN B

eam

s [M

Pa]

feq (0.6-3) UNI Beams [MPa]

Relation CEN Beams - UNI Beams; ULS

Figure 22. Correlations between residual stress and equivalent post cracking strength: Beam Test CEN vs. Beam Tests UNI, ultimate limit states.

2.53

1.39 -45%

2.75

2.26-18%

2.65

1.96-26%

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Comparison feq(0-0.6),m - feq(0-0.6),kSteel fibres Vf=0.255%

Properties determined at SLS

UNI RPSRPL RPLUNI RPS

feq(0-0.6),m feq(0-0.6),k

[MP

a]

Figure 23. Average vs. characteristic feq(0-0.6) using the scatter of different standards: Round Panel tests vs. Beam Tests UNI.

One should notice that the three average values

are almost identical, as a further proof of the strength and consistency of the aforementioned correlations.

As a further confirmation of the abovementioned trend, one should also look at the results plotted in Figure 25 and Figure 26, showing the calculation of

the characteristic value of the equivalent post-cracking strength feq(0.6-3) for the two fiber contents considered. The scatter of round panels, small in these cases, is even lower than that of classical panels ASTM, even though, as a general observation, based on the total experimental program (more than 100 panel tests) carried out during the last 4 years in Bre-scia, the dispersion of results in rather similar be-tween the two panels.

3.08

1.78-42%

3.27

2.28-30%

3.162.70-15%

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Comparison feq(0-0.6),m - feq(0-0.6),kSteel fibres Vf=0.38%Properties determined at SLS

UNI RPL RPS RPLUNI RPS

feq(0-0.6),m feq(0-0.6),k

[MP

a]

Figure 24. Average vs. characteristic feq(0-0.6) using the scatter of different standards: Round Panel tests vs. Beam Tests UNI.

2.50

1.00-60%

2.45

1.74-29%

2.76

1.72-38%

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Comparison feq(0.6-3),m - feq(0.6-3),kSteel fibres FF1 Vf=0.255%

Properties determined at ULS

UNI RPL RPS RPLUNI RPS

feq(0.6-3),m feq(0.6-3),k

[MP

a]

Figure 25. Average vs. characteristic feq(0.6-3) using the scatter of different standards: Round Panel tests vs. Beam Tests UNI.

3 CONCLUDING REMARKS

A comparative study between beam and panel tests was discussed in the present paper. Results show that the high experimental scatter generally present in beam tests is definitely caused by the small geometry and fracture area involved in the tests. It does not represent, in general, the actual structural behavior where much larger fracture areas are involved and, consequently, a lower dispersion occurs.

Proceedings of FraMCoS-7, May 23-28, 2010

hThD ∇−= ),(J (1)

The proportionality coefficient D(h,T) is called moisture permeability and it is a nonlinear function of the relative humidity h and temperature T (Bažant & Najjar 1972). The moisture mass balance requires that the variation in time of the water mass per unit volume of concrete (water content w) be equal to the divergence of the moisture flux J

J•∇=∂

∂−

t

w (2)

The water content w can be expressed as the sum

of the evaporable water we (capillary water, water vapor, and adsorbed water) and the non-evaporable (chemically bound) water wn (Mills 1966, Pantazopoulo & Mills 1995). It is reasonable to assume that the evaporable water is a function of relative humidity, h, degree of hydration, αc, and degree of silica fume reaction, αs, i.e. we=we(h,αc,αs) = age-dependent sorption/desorption isotherm (Norling Mjonell 1997). Under this assumption and by substituting Equation 1 into Equation 2 one obtains

nscw

s

ew

c

ew

hh

Dt

h

h

ew

&&& ++∂

=∇•∇+∂

− αα

αα

)(

(3)

where ∂we/∂h is the slope of the sorption/desorption isotherm (also called moisture capacity). The governing equation (Equation 3) must be completed by appropriate boundary and initial conditions.

The relation between the amount of evaporable water and relative humidity is called ‘‘adsorption isotherm” if measured with increasing relativity humidity and ‘‘desorption isotherm” in the opposite case. Neglecting their difference (Xi et al. 1994), in the following, ‘‘sorption isotherm” will be used with reference to both sorption and desorption conditions. By the way, if the hysteresis of the moisture isotherm would be taken into account, two different relation, evaporable water vs relative humidity, must be used according to the sign of the variation of the relativity humidity. The shape of the sorption isotherm for HPC is influenced by many parameters, especially those that influence extent and rate of the chemical reactions and, in turn, determine pore structure and pore size distribution (water-to-cement ratio, cement chemical composition, SF content, curing time and method, temperature, mix additives, etc.). In the literature various formulations can be found to describe the sorption isotherm of normal concrete (Xi et al. 1994). However, in the present paper the semi-empirical expression proposed by Norling Mjornell (1997) is adopted because it

explicitly accounts for the evolution of hydration reaction and SF content. This sorption isotherm reads

( ) ( )( )

( ) ( )⎥⎥

⎢⎢

⎥⎥⎥

⎢⎢⎢

−∞

+

−∞

−=

1110

,1

110

11,

1,,

hcc

ge

scK

hcc

ge

scG

sch

ew

αα

αα

αα

αααα

(4)

where the first term (gel isotherm) represents the physically bound (adsorbed) water and the second term (capillary isotherm) represents the capillary water. This expression is valid only for low content of SF. The coefficient G1 represents the amount of water per unit volume held in the gel pores at 100% relative humidity, and it can be expressed (Norling Mjornell 1997) as

( ) ss

s

vgkc

c

c

vgk

scG αααα +=,1

(5)

where k

cvg and k

svg are material parameters. From the

maximum amount of water per unit volume that can fill all pores (both capillary pores and gel pores), one can calculate K1 as one obtains

( )1

110

110

11

22.0188.00

,1

−⎟⎠

⎞⎜⎝

⎛−∞

⎥⎥⎥

⎢⎢⎢

⎡⎟⎠

⎞⎜⎝

⎛−∞

−−+−

=

hcc

ge

hcc

geGs

ssc

w

scK

αα

αα

αα

αα

(6)

The material parameters k

cvg and k

svg and g1 can

be calibrated by fitting experimental data relevant to free (evaporable) water content in concrete at various ages (Di Luzio & Cusatis 2009b).

2.2 Temperature evolution

Note that, at early age, since the chemical reactions associated with cement hydration and SF reaction are exothermic, the temperature field is not uniform for non-adiabatic systems even if the environmental temperature is constant. Heat conduction can be described in concrete, at least for temperature not exceeding 100°C (Bažant & Kaplan 1996), by Fourier’s law, which reads

T∇−= λq (7)

where q is the heat flux, T is the absolute temperature, and λ is the heat conductivity; in this

Panel tests are therefore more suitable for repre-senting the actual behavior of FRC materials. Pro-vided that the test is performed under a close-loop control with crack width measurements, the round panel test can be adopted for the characterization of FRC as it implies much lower dispersion of experi-mental results.

3.45

1.98-43%

3.55

2.05-42%

3.37

2.86-15%

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Comparison feq(0.6-3),m - feq(0.6-3),kSteel fibres FF1 Vf=0.38%

Properties determined at ULS

UNI RPL RPS RPLUNI RPS

feq(0.6-3),m feq(0.6-3),k

[MP

a]

Figure 26. Average vs. characteristic feq(0.6-3) using the scatter of different standards: Round Panel tests vs. Beam Tests UNI.

The proposed smaller specimens for panel tests

does not affect the low scatter of the standard ASTM panel; moreover, it allows for an easier placing and handling (lower weight and smaller geometry that fits with many servo-controlled testing machines). In fact, results from round panels with a diameter of 600 mm and a thickness of 60 mm, are consistent, re-liable and provide a repeatable and predictable crack pattern with a consistently lower dispersion than classical beam tests.

The correlation among fracture parameters found in the experimental program are in general very promising and allow engineers to analytically deter-mine the fracture parameters of different tests from performing only one or few test typologies. These correlations, that were somehow expected since standard tests always refer to fracture properties of the same material, may be useful in practice since they can give immediate indications of the fracture properties without performing expensive tests. In ad-dition, they are useful for exchanging results from a broad database of beam and panel test available worldwide form different laboratories and universi-ties.

Finally, the round panel test could be considered as a complete test for the characterization of FRC once suitable range of crack widths will be defined. From these ranges, the corresponding equivalent (or residual) post-cracking strengths can be defined (from σ-w plots) following the same procedure as done for beam tests.

The second part of the paper will focus on the definition of a simplified cohesive constitutive law.

4 ACKNOWLEDGMENTS

The research was partially co-funded either by Mag- netti Building (Carvico, Italy) and Officine Maccaf-erri (Bologna, Italy) whose support is gratefully ac-knowledged.

The Authors would like to thank also Engineers Moira Conti, Giovanni Flelli, Yuri Marinoni, Raoul Martin Milla and Nicola Vezzoli for their assistance in performing the experiments and in data processing.

REFERENCES

ASTM C1550-04, “Standard test method for flexural tough-ness of fiber reinforced concrete (using centrally loaded round panel), pp. 9, 2004.

CEN/TC 229 - PrEN 14651-5, “Precast Concrete Products – Test method for metallic fibre concrete – Measuring the flexural tensile strength”, Europ. Standard, 2003, pp.15.

CNR DT 204/2006. 2006. "Guidelines for the Design, Con-struction and Production Control of Fibre Reinforced Con-crete Structures", National Research Council of Italy.

di Prisco, M., Failla, C., Plizzari, G.A. and Toniolo, G. (2004), “Italian guidelines on SFRC”, in Fiber Reinforced Concrete: from theory to practice, di Prisco, M. and Pliz-zari, G.A. Eds., Proceedings of the International Workshop on advances in Fibre Reinforced Concretes, Bergamo (It-aly), September 24-25, pp. 39-72.

di Prisco, M., Felicetti, R. and Plizzari, G.A. Eds. (2004b), Proceedings of the 6th RILEM Symposium on Fibre Rein-forced Concretes, BEFIB 2004, RILEM PRO 39, 1514 pp.

Eurocode 2. 2003. Design of Concrete Structures, prEN 1992 Ver. December 2003.

fib SAG 5. 2010. Draft proposal the new fib Model Code, in preparation.

Lambrechts, A.N., “The variation of steel fibre concrete char-acteristics. Study on toughness results 2002-2003”, Pro-ceeding of the International Workshop “Fiber Reinforced Concrete. From theory to practice”, Bergamo, September 24-25, 2004, pp. 135-148.

Marinoni, Y., Minelli, F., Plizzari, G.A., and Vezzoli, N.: “Prove di frattura su calcestruzzi fibrorinforzati in accordo con le principali normative internazionali”, Technical Report, University of Brescia, Department DICATA, 2007.

Meda, A., Minelli, F., Plizzari, G.A. and Riva, P. 2005. Shear Behavior of Steel Fibre Reinforced Concrete Beams. Mate-rials and Structures, 38, 343-351.

Minelli, F. 2005. Plain and Fiber Reinforced Concrete Beams under Shear Loading: Structural Behavior and Design As-pects. Ph.D. Thesis, Department of Civil Engineering, University of Brescia, Starrylink Editor, pp. 430.

Minelli, F. & Plizzari, G.A. 2006. Steel Fibers as Shear Rein-forcement for Beams. Proceedings of The Second Fib Congress, Naples, Italy, 5-8 June, 2006, abstract on page 282-283, full length paper available on accompanied CD, pp.12.

Minelli, F. & Vecchio, F.J. 2006. Compression Field Model-ing of Fiber Reinforced Concrete Members under Shear Loading. ACI Structural Journal, 103(2), 244-252.

Minelli, F., Cominoli, L., Meda, A., Plizzari, G.A., and Riva, P. 2005. Full Scale Tests on HPSFRC Prestressed Roof Elements Subjected to Longitudinal Flexure. Proceedings of the International Workshop on High Performance Fiber Reinforced Cementitious Composites in Structural Appli-

Proceedings of FraMCoS-7, May 23-28, 2010

hThD ∇−= ),(J (1)

The proportionality coefficient D(h,T) is called moisture permeability and it is a nonlinear function of the relative humidity h and temperature T (Bažant & Najjar 1972). The moisture mass balance requires that the variation in time of the water mass per unit volume of concrete (water content w) be equal to the divergence of the moisture flux J

J•∇=∂

∂−

t

w (2)

The water content w can be expressed as the sum

of the evaporable water we (capillary water, water vapor, and adsorbed water) and the non-evaporable (chemically bound) water wn (Mills 1966, Pantazopoulo & Mills 1995). It is reasonable to assume that the evaporable water is a function of relative humidity, h, degree of hydration, αc, and degree of silica fume reaction, αs, i.e. we=we(h,αc,αs) = age-dependent sorption/desorption isotherm (Norling Mjonell 1997). Under this assumption and by substituting Equation 1 into Equation 2 one obtains

nscw

s

ew

c

ew

hh

Dt

h

h

ew

&&& ++∂

=∇•∇+∂

− αα

αα

)(

(3)

where ∂we/∂h is the slope of the sorption/desorption isotherm (also called moisture capacity). The governing equation (Equation 3) must be completed by appropriate boundary and initial conditions.

The relation between the amount of evaporable water and relative humidity is called ‘‘adsorption isotherm” if measured with increasing relativity humidity and ‘‘desorption isotherm” in the opposite case. Neglecting their difference (Xi et al. 1994), in the following, ‘‘sorption isotherm” will be used with reference to both sorption and desorption conditions. By the way, if the hysteresis of the moisture isotherm would be taken into account, two different relation, evaporable water vs relative humidity, must be used according to the sign of the variation of the relativity humidity. The shape of the sorption isotherm for HPC is influenced by many parameters, especially those that influence extent and rate of the chemical reactions and, in turn, determine pore structure and pore size distribution (water-to-cement ratio, cement chemical composition, SF content, curing time and method, temperature, mix additives, etc.). In the literature various formulations can be found to describe the sorption isotherm of normal concrete (Xi et al. 1994). However, in the present paper the semi-empirical expression proposed by Norling Mjornell (1997) is adopted because it

explicitly accounts for the evolution of hydration reaction and SF content. This sorption isotherm reads

( ) ( )( )

( ) ( )⎥⎥

⎢⎢

⎥⎥⎥

⎢⎢⎢

−∞

+

−∞

−=

1110

,1

110

11,

1,,

hcc

ge

scK

hcc

ge

scG

sch

ew

αα

αα

αα

αααα

(4)

where the first term (gel isotherm) represents the physically bound (adsorbed) water and the second term (capillary isotherm) represents the capillary water. This expression is valid only for low content of SF. The coefficient G1 represents the amount of water per unit volume held in the gel pores at 100% relative humidity, and it can be expressed (Norling Mjornell 1997) as

( ) ss

s

vgkc

c

c

vgk

scG αααα +=,1

(5)

where k

cvg and k

svg are material parameters. From the

maximum amount of water per unit volume that can fill all pores (both capillary pores and gel pores), one can calculate K1 as one obtains

( )1

110

110

11

22.0188.00

,1

−⎟⎠

⎞⎜⎝

⎛−∞

⎥⎥⎥

⎢⎢⎢

⎡⎟⎠

⎞⎜⎝

⎛−∞

−−+−

=

hcc

ge

hcc

geGs

ssc

w

scK

αα

αα

αα

αα

(6)

The material parameters k

cvg and k

svg and g1 can

be calibrated by fitting experimental data relevant to free (evaporable) water content in concrete at various ages (Di Luzio & Cusatis 2009b).

2.2 Temperature evolution

Note that, at early age, since the chemical reactions associated with cement hydration and SF reaction are exothermic, the temperature field is not uniform for non-adiabatic systems even if the environmental temperature is constant. Heat conduction can be described in concrete, at least for temperature not exceeding 100°C (Bažant & Kaplan 1996), by Fourier’s law, which reads

T∇−= λq (7)

where q is the heat flux, T is the absolute temperature, and λ is the heat conductivity; in this

cations, Honolulu, Hawaii, USA, May 23-26, 2005, 323-332.

PrEN 14488-5-CEN/TC 104, “Testing sprayed concrete – Part 5: Determination of energy absoprtion capacity of fibre re-inforced slab specimens”, Europ. Standard, 2004, pp.1-7.

RILEM Final Recommendation TC-162-TDF, “Test and De-sign Methods for Steel Fibre Reinforced Concrete; σ−ε De-sign Method”, Materials and Structures, Vol 36, October 2003, pp. 560-567.

RILEM TC162-TDF: “Test and Design Methods for Steel Fi-bre Reinforced Concrete: Recommendations. Bending test”, Materials and Structures, 33, (225), (2000) 3-5.

Schumacher, P., “Rotation capacity of self-compacting steel fiber reinforced concrete”, Ph.D Thesis, Delft University of Technology, The Netherlands, 2006

Sorelli, L., Meda, A. and Plizzari, G.A., “Bending and uni-axial tensile tests on concrete reinforced with hybrid steel fibers”, ASCE Materials Journal, Vol.15(5), 2005, pp. 519-527.

UNI 11039, “Steel Fiber Reinforced Concrete - Part I: Defini-tions, Classification Specification and Conformity - Part II: Test Method for Measuring First Crack Strength and Duc-tility Indexes”, Italian Board for Standardization, 2003.

Vandewalle, L., “Test and design methods for steel fibre rein-forced concrete proposed by RILEM TC 162-TDF”, Pro-ceeding of the International Workshop “Fiber Reinforced Concrete. From theory to practice”, Bergamo, September 24-25, 2004, pp. 3-12.

Proceedings of FraMCoS-7, May 23-28, 2010

hThD ∇−= ),(J (1)

The proportionality coefficient D(h,T) is called moisture permeability and it is a nonlinear function of the relative humidity h and temperature T (Bažant & Najjar 1972). The moisture mass balance requires that the variation in time of the water mass per unit volume of concrete (water content w) be equal to the divergence of the moisture flux J

J•∇=∂

∂−

t

w (2)

The water content w can be expressed as the sum

of the evaporable water we (capillary water, water vapor, and adsorbed water) and the non-evaporable (chemically bound) water wn (Mills 1966, Pantazopoulo & Mills 1995). It is reasonable to assume that the evaporable water is a function of relative humidity, h, degree of hydration, αc, and degree of silica fume reaction, αs, i.e. we=we(h,αc,αs) = age-dependent sorption/desorption isotherm (Norling Mjonell 1997). Under this assumption and by substituting Equation 1 into Equation 2 one obtains

nscw

s

ew

c

ew

hh

Dt

h

h

ew

&&& ++∂

=∇•∇+∂

− αα

αα

)(

(3)

where ∂we/∂h is the slope of the sorption/desorption isotherm (also called moisture capacity). The governing equation (Equation 3) must be completed by appropriate boundary and initial conditions.

The relation between the amount of evaporable water and relative humidity is called ‘‘adsorption isotherm” if measured with increasing relativity humidity and ‘‘desorption isotherm” in the opposite case. Neglecting their difference (Xi et al. 1994), in the following, ‘‘sorption isotherm” will be used with reference to both sorption and desorption conditions. By the way, if the hysteresis of the moisture isotherm would be taken into account, two different relation, evaporable water vs relative humidity, must be used according to the sign of the variation of the relativity humidity. The shape of the sorption isotherm for HPC is influenced by many parameters, especially those that influence extent and rate of the chemical reactions and, in turn, determine pore structure and pore size distribution (water-to-cement ratio, cement chemical composition, SF content, curing time and method, temperature, mix additives, etc.). In the literature various formulations can be found to describe the sorption isotherm of normal concrete (Xi et al. 1994). However, in the present paper the semi-empirical expression proposed by Norling Mjornell (1997) is adopted because it

explicitly accounts for the evolution of hydration reaction and SF content. This sorption isotherm reads

( ) ( )( )

( ) ( )⎥⎥

⎢⎢

⎥⎥⎥

⎢⎢⎢

−∞

+

−∞

−=

1110

,1

110

11,

1,,

hcc

ge

scK

hcc

ge

scG

sch

ew

αα

αα

αα

αααα

(4)

where the first term (gel isotherm) represents the physically bound (adsorbed) water and the second term (capillary isotherm) represents the capillary water. This expression is valid only for low content of SF. The coefficient G1 represents the amount of water per unit volume held in the gel pores at 100% relative humidity, and it can be expressed (Norling Mjornell 1997) as

( ) ss

s

vgkc

c

c

vgk

scG αααα +=,1

(5)

where k

cvg and k

svg are material parameters. From the

maximum amount of water per unit volume that can fill all pores (both capillary pores and gel pores), one can calculate K1 as one obtains

( )1

110

110

11

22.0188.00

,1

−⎟⎠

⎞⎜⎝

⎛−∞

⎥⎥⎥

⎢⎢⎢

⎡⎟⎠

⎞⎜⎝

⎛−∞

−−+−

=

hcc

ge

hcc

geGs

ssc

w

scK

αα

αα

αα

αα

(6)

The material parameters k

cvg and k

svg and g1 can

be calibrated by fitting experimental data relevant to free (evaporable) water content in concrete at various ages (Di Luzio & Cusatis 2009b).

2.2 Temperature evolution

Note that, at early age, since the chemical reactions associated with cement hydration and SF reaction are exothermic, the temperature field is not uniform for non-adiabatic systems even if the environmental temperature is constant. Heat conduction can be described in concrete, at least for temperature not exceeding 100°C (Bažant & Kaplan 1996), by Fourier’s law, which reads

T∇−= λq (7)

where q is the heat flux, T is the absolute temperature, and λ is the heat conductivity; in this