Post-Peak Behavior of Cement-Based Materials in Compression

(ProQuest: ... denotes formulae omitted.)

INTRODUCTION

The existence of strain localization in the post-peak stage of cement-based materials in tension is generally accepted. Strains that exceed the value at tensile strength f^sub ct^ are localized in the so-called process zone, whereas unloading occurs in the remaining part of the specimen in tension. Under these conditions, the stress-strain (σ^sub ct^-ε^sub ct^) relationship is adopted for the ascending branch, whereas, according to the fictitious crack model introduced by Hillerborg et al.,1 the softening stage can be modeled by means of stress-crack opening displacement (σ^sub ct^-COD) relationships. The area under the entire σ^sub ct^-COD curve is defined as the fracture toughness G^sub F^, and represents the energy absorbed per unit area of crack. The values of f^sub ct^ and G^sub F^, together with the Young's modulus of concrete E^sub c^, define the characteristic length, which can be considered as a material property. This is true both for plain concrete and for fiber-reinforced concrete (FRC) with a fiber volume content V^sub f^ ≤ 2%. For this type of cement-based composite, however, fracture toughness can be even 10 times higher than that of plain concrete.2 This is due to the pullout of fibers, bridging the parts of a specimen in tension separated by cracks.3 In the case of high-performance fiber-reinforced cementitious composite (HPFRCC), bridging actions are increased as much as possible to have strain hardening in the post-cracking stage.2

From the physical point of view, crushing of compressed cement-based composites is similar to cracking failure observed in tension. In both the cases, strain localization appears when compressive and tensile strengths are reached, respectively. Thus, the post-peak response in compression has to be defined by means of a suitable stress-inelastic displacement (σ^sub c^-w) relationship. It should be related to the strain localization zone, where sliding planes appear during failure.4 As observed in different test campaigns,4,5 the socalled size effect is no longer evident in the σ^sub c^-w curves of compressed prisms and cylinders. The stress-inelastic displacement relationships can, however, be affected by the presence of fiber. According to Fanella and Naaman,6 the fracture toughness in compression seems to increase with the increase of reinforcing index RI (RI is the product of V^sub f^, in percent, and the fiber aspect ratio l/D).

From a practical point of view, the bearing capacity as well as the ductility of cement-based structures depend on the dimensions of the zone where strains localize.7 This is clearly evident in the moment-curvature (M-μ) relationships of beams in bending, which can show a remarkable dependence on the effective depth of cross sections.8-10 Therefore, to correctly predict the mechanical response of different cement-based structures, the definition of σ^sub c^-w, to which this paper is devoted, is of primary importance.

RESEARCH SIGNIFICANCE

Although crushing of cement-based materials is commonly considered as a strain localization phenomenon, stress-inelastic displacement relationships, which should reproduce the post-peak stage in compression, cannot be found in the existing literature. The present study aims to establish possible σ^sub c^-w relationships for different concretes. They are empirically defined from stress-strain relationships experimentally measured in ordinary concrete, FRC, and HPFRCC cylinders in compression. Because the reliability of structural analysis is strictly connected to the correct material characterization, numerical procedures, able to evaluate the structural response of simply-supported beams up to the failure, are also provided. By means of these algorithms, the effectiveness of the proposed relationships is checked through their capability of predicting the structural response of tested beams.

IDEALIZED BEHAVIOR OF CEMENT-BASED COMPOSITES IN COMPRESSION

The stress-strain relationships of cement-based materials in compression (Fig. 1(a)) can be divided into two parts (Fig. 1(b)). In the first part, when the stress is lower than the strength f^sub c^' (and ε^sub c^ < ε^sub c1^), the specimen can be considered undamaged. In the case of plain concrete, the ascending branch of σ^sub c^-ε^sub c^ can be defined by the Sargin's relationship proposed by the CEB-FIP Model Code.11

As soon as the peak stress is reached, localized damage develops and strain softening begins. In this stage, the progressive sliding of two blocks of cement-based material is evident. In Fig. 1(a), the angle between the vertical axe of the specimen and the sliding surfaces is assumed to be α = 18 degrees, according to the experimental observations of Fujita et al.12 In the case of compressed concrete, a similar value of α is obtained through the Mohr-Coulomb failure criterion, if the tensile strength is assumed to be 1/10 of that in compression (f^sub ct^ = 0.1f^sub c^'). For this reason, the value of α only depends on the strength of materials and, therefore, can change only in the presence of efficient confinement (that is, the presence of stirrups in reinforced concrete beams). As shown by Fantilli et al.,9 however, ordinary values of confinement do not significantly change the value of α.

The inelastic displacement w of the specimen and the consequent sliding s of the blocks along the sliding surface rule the average post-peak compressive strain ε^sub c^ of the specimen (Fig. 1). Referring to the specimen depicted in Fig. 1(a), post peak strains can be defined by the following equation4 (Fig. 1(b))

... (1)

where ε^sub c1^ is the strain at compressive strength f^sub c^', Δσ^sub c^ is the stress decrement after the peak, and H is the height of the specimen.

According to test measurements,4-5 the post-peak slope of σ^sub c^-ε^sub c^ increases in longer specimens (Fig. 1(b)), due to the ratio w/H involved in the evaluation of ε^sub c^ (Eq. (1)).

The stress decrement Δσ^sub c^ can be defined as

Δσ^sub c^ = f^sub c^' - σ^sub c^ = f^sub c^'[1 - F(w)] (2)

where F(w) is the nondimensional function that connects the inelastic displacement w and the relative stress σ^sub c^/f^sub c^' during softening (Fig. 1(c)), and f^sub c^' is the compressive strength (assumed to be positive).

Substituting Eq. (2) into Eq. (1), it is possible to obtain a new equation for ε^sub c^

... (3)

Equation (3), adopted for the post-peak stage of a generic cement-based material in compression, is based on the definition of F(w), which has to be considered as a material property. In all the cement-based composites, this function should be evaluated experimentally on cylindrical specimens, as performed by Jansen and Shah5 for plain concrete (Fig. 1(c)). It must be remarked, however, that only when some specified conditions are satisfied, such as those prescribed by ASTM standards at the end of specimens, can the failure mode depicted in Fig. 1(a) be obtained and F(w) be independent from the height H.4,5 Thus, the range defined by Jansen and Shah5 (depicted in Fig. 2(c)) can also be considered as a parameter to measure the consistency of a campaign test on fiber-reinforced concrete specimens in compression. In the case of fiberreinforced composites, test campaigns for defining F(w) cannot be found in the literature. Only the influence of the fibers on σ^sub c^-ε^sub c^ relationships has been measured in the existing experimental data.6 In particular, test results clearly show the dependence of the adsorbed energy on the type of fiber, on the aspect ratio l/D (where l is the length of the fiber and D is the diameter of the fiber), and on the fiber volume content V^sub f^ . For these reasons, starting from uniaxial compression tests on cylindrical specimens, F(w) relationships are herein defined for different cement-based composites.

UNIAXIAL COMPRESSION TESTS ON CYLINDRICAL SPECIMENS

In what follows, the function F(w) is defined by using test results of different authors.6,13-16 The properties of the 27 cylindrical specimens are summarized in Table 1, where they are arranged in four groups.

In the first group, named PL, were included five plain concrete specimens, whose heights were 152.4 to 300 mm (6 to 11.8 in.). They were made of different cement-based materials: mortar6 (f^sub c^' [congruent with] 60 MPa [8700 psi]), ordinary concrete13 (f^sub c^' [congruent with] 40 MPa [5800 psi]), lightweight concrete with pumice stones and expanded clay aggregates15 (f^sub c^' [congruent with] 25 MPa [3630 psi]), and high-strength concrete14 (f^sub c^' [congruent with] 80 MPa [11,600 psi]).

The second group, named SS, contained nine cementbased specimens reinforced with straight steel fibers, whose aspect ratios were l/D = 47 to 100. The specimens had different heights (152.4 to 300 mm [6 to 11.8 in.]) and their reinforcing indexes were 82 to 249%. The cement-based matrixes consisted of mortar6 (f^sub c^' [congruent with] 60 MPa [8700 psi]), to which different amounts of fibers (V^sub f^ = 1 to 3%) were added, and of ordinary concrete13 (f^sub c^' [congruent with] 40 MPa [5800 psi]) containing 2% in volume of straight steel fibers.

The third group, named SH, consisted of eight cementbased cylinders reinforced with hooked steel fibers, whose aspect ratio was l/D = 60. The specimens had different heights (152.4 to 200 mm [6.0 to 7.87 in.]) and their reinforcing indexes were 30 to 120%. The cement matrixes consisted of lightweight concrete with pumice stones and expanded clay aggregates15 (f^sub c^' [congruent with] 25 MPa [3630 psi]), and high-strength concrete14 (f^sub c^' [congruent with] 80 MPa [11,600 psi]), to which different amounts of hooked steel fibers (V^sub f^ = 0.5 to 2%) were added.

Finally, the fourth group, named SC, was composed of five cylinders (100 to 200 mm [3.94 to 7.87 in.]) made of the HPFRCC tailored by Mihashi et al.16-18 It was a hybrid composite, in which steel cords (V^sub f^ = 0 to 1%), having an aspect ratio of l/D = 76.2, were added to a cement-based matrix already reinforced with polyethylene fibers (1% in volume). In such a HPFRCC (f^sub c^' = 40 to 50 MPa [5800 to 7250 psi]), reinforcing indexes, only referred to steel cords, are 0 to 76.2%. It is quite unusual to investigate HPFRCC elements in compression. In fact, they were mainly tailored to show high performances only in tension. The tests of the SC group, however, made by the authors and described for brevity in another work,16 are herein considered. In this way, a wider panorama of all the possible cement-based materials, and the corresponding behavior under compression, can be analyzed.

An acronym has been assigned to each specimen reported in Table 1. It is composed of two capital letters (PL = plain concrete; SS = FRC made of steel straight fibers; SH = FRC made of steel hooked fibers; and SC = HPFRCC made of steel cord), followed by two numbers (that is, the value of RI and the number of the specimen having that reinforcing index). The post-peak curves of each group are reported in Fig. 3 in terms of F = σ^sub c^/f^sub c^' versus w (that is, in Fig. 3(a), the PL group; in Fig. 3(b), the SS group; in Fig. 3(c), the SH group; and in Fig. 3(d), the SC group). They are obtained from the stress-strain relationships measured in the tests,6,13-15 respectively, referred to the whole height H of each specimen. In particular, for a given ε^sub c^ > ε^sub c1^, the value of compressive stress σ^sub c^ (of Δσ^sub c^ = f^sub c^' - σ^sub c^, and of F = σ^sub c^/f^sub c^') can be obtained through the σ^sub c^-ε^sub c^ diagrams experimentally evaluated, whereas the corresponding w (Fig. 1(a)) can be obtained from Eq. (1) (if f^sub c^', E^sub c^, and H are known).

EMPIRICAL FORMULATION OF F(w)

For the specimens reported in Table 1, a general definition of F(w) seems not to be possible for all the inelastic displacements, because tests are generally concluded when w 3 mm (0.118 in.) (Fig. 3). At this level of inelastic displacement, compressive stresses σ^sub c^ stabilize to a nonzero residual value, which increases with the increase of RI (Fig. 3). To model the post-peak response of reinforced concrete structures, it is necessary to define F(w) also for a higher value of w. Because no experimental data are available for modeling the tail of F(w), it can be theoretically defined by considering that the curve should vanish with the increase of w and should have an order of continuity with the previous part. For the sake of simplicity, F(w) is assumed to be composed, both in the first and the second part, by bivariate polynomial functions. In other words, the function F(w) can be modeled by means of the two parabolas depicted in Fig. 2(a)

... (4a)

... (4b)

... (4c)

As can be easily observed in Fig. 2(a), the parabolas (Eq. (4a) and (4b)) are defined by the same coefficients a and b and have the same extreme point at w = -0.5b/a. While w = -b/a (that is, twice the value at the extreme point) is considered the maximum value of the inelastic displacement for which F(w) is assumed to be higher than zero. Both the coefficients, however, can be defined only through the first parabola (Eq. (4a)) because of the reduced range of experimental observation (0 < w < -0.5b/a).

Values of a and b for plain concrete (PL group)

In the case of the plain concrete specimens (Fig. 2(a)), the values a = 0.320 mm^sup -2^ (206 in.^sup -2^) and b = -1.12 mm^sup -1^ (-28.4 in.^sup -1^) are obtained by means of the least square approximation of Eq. (4a). As the curves defined by Eq. (4a) to (4c) fall within the range of test data measured by Jansen and Shah,5 these values can be considered consistent for plain concrete. As shown in Fig. 2(b), however, this range only covers the first part of the theoretical F(w) curve.

Values of a and b for FRC made of straight steel fibers (SS group)

In the case of straight steel fibers in a cementitious matrix, the specimens reported in the SS group of Table 1 are considered. Tested in compression, they give the F(w) relationships depicted in Fig. 3(b). The coefficients a and b of Eq. (4a) to (4c) are computed separately in the specimens6,13 having the same RI. By means of the least square approximation algorithm, five values of a (Fig. 4(a)) and b (Fig. 4(b)) can be obtained. All the evaluated coefficients are reported in Table 1 and Fig. 4 as functions of RI. In the same figure, the coefficients a = 0.320 mm^sup -2^ (206 in.^sup -2^) and b = -1.12 mm^sup -1^ (-28.4 in.^sup -1^), regarding the PL group, are also depicted, as they have to be equal to those of the SS group when RI = 0. With a good approximation, the coefficients a and b can be written as linear functions of RI (Fig. 4)

a = (-0.00129 . RI + 0.320) . mm^sup -2^ = (-0.832 . RI + 206) . in.^sup -2^ (5a)

b = (0.00424 . RI - 1.12) . mm^sup -1^ = (0.108 . RI - 28.4) . in.^sup -1^ (5b)

Values of a and b for FRC made of hooked steel fibers (SH group)

In the case of hooked steel fibers in a cementitious matrix, the considered specimens are reported in the SH group of Table 1, whereas the corresponding F(w) relationships are depicted in Fig. 3(c). The coefficients a and b are computed separately in specimens14,15 having the same RI. By means of least square approximation algorithm, three values can be obtained, respectively, for the coefficients a (Fig. 5(a)) and b (Fig. 5(b)). They are reported in Table 1 and Fig. 5 as functions of RI. In the same figure, the coefficients a = 0.320 mm^sup -2^ (206 in.^sup -2^) and b = -1.12 mm^sup -1^ (-28.4 in.^sup -1^), regarding the PL group, are also depicted. In fact, they have to be equal to those of the SH group when RI = 0. As Fig. 5(a) and (b) show, both the coefficients can be written as linear functions of RI

a = (-0.00217 . RI + 0.320) . mm^sup -2^ = (-1.4 . RI + 206) . in.^sup -2^ (6a)

b = (0.00645 . RI - 1.12) . mm^sup -1^ = (0.164 . RI - 28.4) . in.^sup -1^ (6b)

The consistency of the values defined in the Eq. (5) and Eq. (6) is indirectly proven by the fact that when RI = 0, the curve F(w) falls within the range defined by Jansen and Shah4 (Fig. 2(b)).

Values of a and b for HPFRCC made of steel cords (SC group)

The specimens made of steel cords in a fiber-reinforced cementitious matrix are reported in the SC group of Table 1, whereas the corresponding F(w) relationships are depicted in Fig. 3(d). The coefficients a and b are computed separately in specimens16 having the same RI. By means of least square approximation algorithm, three values can be obtained, respectively, for the coefficients a (Fig. 6(a)) and b (Fig. 6(b)). As Fig. 6(a) and (b) show, the coefficients a and b can be written as linear functions of RI

a = (-0.00761 . RI + 0.704) . mm^sup -2^ = (4.91 . RI + 454) . in.^sup -2^ (7a)

b = (0.00130 . RI - 1.53) . mm^sup -1^ = (0.033 . RI - 38.9) . in.^sup -1^ (7b)

The values of a and b obtained by means of Eq. (5) through (7) clearly show the different post-peak responses obtained by changing the mixture of cement-based composites. In other words, the evaluation of a and b cannot be of general validity, but should be experimentally evaluated for each kind of reinforcing fiber (for example, steel and carbon) and matrix.

STRUCTURAL ANALYSIS OF SIMPLY-SUPPORTED BEAMS

Only if strain localization phenomena are taken into account can the post-peak stage of beams in bending be correctly predicted by structural analysis. Although structures can be affected by shear and tensile cracks, only beams in bending that fail via crushing of compressed concrete are considered herein. In particular, beams in four-point bending and in three-point bending with a span-height ratio higher than 10 are analyzed. According to Hillerborg,7 due to this crushing, moment-curvature relationships (M-μ) cannot be evaluated from the stress-strain response of materials.19 On the contrary, stress-inelastic displacement relationships have to be taken into consideration in a more effective approach.

Moment-curvature relationship

For the beam depicted in Fig. 7, assuming plane section compatibility, the strain profile ε(y) of a generic cross section (Fig. 7(a)) can be computed through the following equation

ε = μ . y (8)

In the compressed cement-based materials, strains that exceed ε^sub c1^, which rule the evolution of sliding planes, can be split up into several rectangles of base Δε^sub ci^ (Fig. 7(b)). Their values are computed in accordance with Eq. (3)

... (9)

where Δ^sub wi^ is the inelastic displacement increment that affects a volume of concrete in compression of width y^sub c,max^ - y^sub ci^ and length 2H^sub i^ (Fig. 7(d)).

The corresponding sliding increment Δs^sub i^ (Fig. 7(c)) is evaluated by means of the following equation

... (10)

It must be clarified that Δε^sub ci^ is related to the stripe of height Δy^sub ci^ located within the crushing zone (of length equal to y^sub c,max^ - y^sub c1^) of a cross section. On the contrary, both Δw^sub i^ and Δs^sub i^ affect the distance y^sub c,max^ - y^sub ci^ and, therefore, all the stripes from i to n.

According to Fig. 7, the length Hi and the strain increment y^sub c,max^ - y^sub ci^ can be respectively written as

...

and consequently Eq. (10) becomes

... (11)

In the i-th strip, the total amount of the inelastic displacement w^sub i^ can be evaluated by summing all the contributions Δw^sub j^ of the strips in the following

... (12)

When ε^sub c,max^ > ε^sub c1^ (Fig. 7(b)), the bending moment M corresponding to a given curvature μ can be computed by means of the following iterative procedure:

1. Select a trial value for ε^sub c,max^ > ε^sub c1^;

2. The crushed zone y^sub c,max^ - y^sub c1^ is divided into n strips with a height of Δy^sub ci^; constant strain increment Δε^sub ci^ is assumed within the i-th strip;

3. By means of Eq. (11) (Fig. 7(b)), the sliding displacements Δs^sub i^ (and Δw^sub i^ = Δs^sub i^cosα) are computed in each strip ([dF/dw]^sub i^ can be easily obtained from Eq. (4));

4. The complete inelastic shortening w^sub i^ of the i-th strip can be evaluated with Eq. (12), whereas the corresponding stress can be obtained by means of Eq. (4a) to (4c);

5. In uncrushed zones, where ε^sub c^ < ε^sub c1^, a simple stressstrain relationship11 can be used to evaluate the state of stress (Fig. 1(b));

6. For the whole cross section of Fig. 7(b), if the equilibrium of the horizontal forces is not satisfied, change ε^sub cmax^ and go back to Step 2; and

7. Compute the bending moment M by imposing the equilibrium to rotation.

It is possible to observe how the state of stress, related to the crushed zone of a beam, depends on the extension of the compression zone y^sub c,max^ (Eq. (11) and Fig. 7), according to the size effect model proposed by Hillerborg.7 In the present approach (Eq. (11)), however, the mechanical response of crushed concrete is also a function of the cross-sectional curvature.8-10

Load-midspan deflection curves of statically determinate beams

Instead of a single cross section, the softening branch of the moment-curvature diagram should model the behavior of a wide block of the beam. As observed experimentally,8 the crushing failure of reinforced concrete beams is characterized by the expulsion of V-shaped blocks,8 whose surfaces are inclined of α. Precisely, when crushing in compression occurs, cross-sectional curvatures should be extended to the length Lc (Fig. 7(d)). This length, which is proportional to the width (y^sub c,max^ - y^sub c1^) where ε^sub c^ > ε^sub c1^, can be computed by means of the following equation (Fig. 7(d))

L^sub c^(z) = (y^sub c,max^ - y^sub c1^)/tanα (13)

For all cement-based materials considered in this paper, the angle α between the sliding plane and the horizontal line (Fig. 7(d)) is assumed to be equal to that of ordinary concrete (α = 18 degrees).

A complete M-μ-Lc relationship is depicted in Fig. 8(a). The length L^sub c^, usually called the softening region,20 must be larger than zero to avoid nonobjective results from structural analyses. When L^sub c^ &rarr 0, the energy dissipation during softening reduces with the decrease of the mesh size (mesh dependence or mesh sensitivity), which is not admissible (refer to Reference 20 for a review).

To overcome this problem, during crushing of compressed concrete (μ > μ^sub 1^ in Fig. 8(a)), curvatures are assumed to be constant within the softening region. In other words, μ is shifted from the cross section where ε^sub c^ > ε^sub c1^, to the nearest cross sections within the length L^sub c^. As this length increases with the increase of beam deflection, M(z) and μ(z) distributions can be evaluated in statically determinate beams by means of the following procedure (Fig. 8):

1. Select a curvature μ(z = 0) in the midspan cross section of a three-point bending beam (Fig. 8(b));

2. Compute the bending moment M(z = 0) in the same cross section (by means of the M-μ relationship of Fig. 8(a)) and in all the cross sections of the beam;

3. If μ(z = 0) ≤ μ^sub l^, then the curvature distribution μ(z) is obtained referring to the ascending branch of the M-μ relationship (Fig. 8(a));

4. If μl < μ(z = 0) ≤ μp, then compute μ(z) referring to the ascending branch of M-μ in the cross sections where μ > μl curvatures are shifted (and assumed to be constant) within L^sub c^;

5. If μ(z = 0) > μp (Fig. 8(b)) in the cross sections at distance z ≤ L^sub c^(z = 0), the condition μ(z) = μ(z = 0) is assumed; in the cross sections at distance z > L^sub c^(z = 0), curvature distribution μ(z) are obtained referring to the descending branch of M-μ (Fig. 8(a)) in the cross sections where μ^sub l^ < μ < μ^sub p^ curvatures are shifted within the length L^sub c^(z).

For a given value of applied load P, this procedure furnishes a more reliable distribution of curvature μ(z) and, consequently, the midspan deflection η can be correctly computed in the three-point bending beam of Fig. 8(b).

Theoretically, in four-point bending beams, the softening regions cannot be univocally defined because compressive strains may be localized in one of the cross sections where the bending moment is constant. Strain localization is usually observed around the midspan cross section of the beam10 because confinement effects produced by applied loads are lower in this region. Thus, the previous procedure can be effectively applied to evaluate μ(z) distributions also in four-point bending beams.

COMPARISON OF PREDICTIONS AND EXPERIMENTAL DATA

In accordance with the procedures previously described, when the constitutive relationships of a cement-based material are known, the mechanical response of a simplysupported beam can be adequately predicted. In beams that fail via the crushing in compression, the accuracy of the proposed F(w) can be estimated from the comparison between predicted results and experimental data. To this aim, load-midspan deflection curves of simply-supported beams (Fig. 9) made of three different cement-based materials (Table 2) and tested in four-point bending21 (Fig. 9(a)) and three point bending16 (Fig. 9(b)), are analyzed in the following. Both the three-point bending beams and the constant moment zone of the four-point bending beams do not have steel stirrups, thus their post-peak response is only affected by the presence of fibers.

Four-point bending tests of Mansur et al.21

To study the behavior of over-reinforced four-point bending beams (Fig. 9(a)), a wide test campaign was conducted by Mansur et al.21 In particular, the behavior of two beams, named B4-0.0C and B4-1.0F, and made, respectively, of plain high-strength concrete and FRC with hooked steel fibers, is herein taken into consideration. The geometrical dimensions of the beams are shown in Fig. 9(a), whereas the mechanical properties of the adopted materials are reported in Table 2. As they are over-reinforced, crushing failures in compression have been observed in both the beams before the yielding of the steel bars in tension.

To predict the structural response of both beams, the previous procedures can be used after defining the constitutive relationships of materials. The prepeak response of both plain concrete and FRC in compression can be reproduced by the Sargin's relationship proposed by the CEB-FIP Model Code11 (founded on the parameters f^sub c^' , E^sub c^, and ε^sub c1^ reported in Table 2). For Beam B4-0.0C, the postpeak stage in compression is described by the function F(w) in Eq. (4a) to (4c), and the coefficients a = 0.320 mm^sup -2^ (206 in.^sup -2^) and b = -1.12 mm^sup -1^ (-28.4 in.^sup -1^) obtained for the PL group (Table 1). For Beam B4-1.0F, the coefficients a and b are computed according to Eq. (6), assuming RI = 60%. For the sake of simplicity, tensile stresses of concrete are neglected because they give an irrelevant structural contribution during the failure stage of over-reinforced beams. At this stage in fact, due to the presence of high curvatures and, consequently, of high tensile stresses in the steel bars, cracks are completely developed and tension stiffening does not affect the ductility of concrete beams.9

The procedure for evaluating moment-curvature relationships, if applied to the cross sections without stirrups (Section A-A in Fig. 9(a)), furnishes the M-μ-L^sub c^ curves depicted in Fig. 10(a). The slight difference that exists between the two analyzed beams is only related to the coefficients a and b of their F(w). Despite this, in terms of M-μ-L^sub c^, Beam B4-1.0F appears more ductile than Beam B4-0.0C (Fig. 10(c)). This is further confirmed by the load-midspan deflection curves depicted in Fig. 10 (in Fig. 10(b) for Beam B4-0.0C, and in Fig. 10(c) for Beam B4-1.0F). In these figures, the diagrams theoretically computed and those experimentally measured seem to agree. Therefore, the effectiveness of the adopted functions F(w) to describe the crushing of plain concrete and FRC is definitely proven.

Three-point bending tests of Mihashi et al.16

To complete the investigation on the F(w) relationships, the behavior of three beams (Fig. 9(b)), named HSC 1, HSC 2, and HSC 3, is herein analyzed. The beams have been tested in three-point bending by Mihashi et al.16 to investigate the mechanical response of structures made of highperformance concrete (HPFRCC) and traditional steel reinforcing bars. The HPFRCC is obtained by a mixture of 1% in volume of steel cord in a cement-based mortar, already reinforced by polyethylene fibers (1% in volume). The mechanical properties of the material are summarized in Table 2. Although the considered beams are normally reinforced, crushing in the compressed zone is systematically observed after yielding of steel bars. As widely discussed by the authors,22,23 it is reasonable to expect such a behavior from reinforced HPFRCC members in bending. The capability of cement-based composites to sustain tensile stresses also at high strains can correspond to an increase of the steel reinforcement area in tension.23 Thus, such tests can be used herein to verify the effectiveness of the proposed model to predict crushing of steel reinforced HPFRCC in bending. Other tests of the same type are not reported in the existing literature.

Unlike the beams tested by Mansur et al.,21 the contribution of tension stiffening cannot be neglected in the evaluation of the whole structural response of reinforced HPFRCC beams in bending. Thus, it can be taken into account by means of a suitable stress-strain relationship for HPFRCC in tension, which has been experimentally evaluated from tensile tests16 (Fig. 11(a)) together with the stress-strain relationship of steel reinforcing bars (Fig. 11(b)). Similar to ordinary concrete and FRC, the pre-peak response of compressed HPFRCC can be reproduced by Sargin's relationship proposed by the CEB-FIP Model Code11 (defined by the parameters f^sub c^', E^sub c^, and ε^sub c1^ reported in Table 2). For the three beams, the post-peak stage in compression is described by the function F(w) of Eq. (4a) to (4c), in which the coefficients a and b are computed by means of Eq. (7) (RI = 76.2%). As the spanheight ratio of the beam is approximately 10, shear stresses are not considered in the structural analysis. Conversely, they significantly affect the response of reinforced HPFRCC beams with a lower span-height ratio.24

The M-μ-L^sub c^ relationship and the load-midpan deflection curves, depicted, respectively, in Fig. 12(a) and (b), are computed by applying the procedures described previously. A good agreement between numerical and experimental results is evidenced in Fig. 12(b) if the lengths of softening regions are defined in accordance with Eq. (13). Conversely, if Lc = 0 is assumed, only the midspan cross section follows the post-peak branch of M-μ (Fig. 12(a)) and the predicted behavior underestimates the ductility of the beam. For this beam, due to instrument limitations, the measured midspan deflection was lower than 20 mm (0.787 in.). At this level of deflection, however, the steel bar reaches a tensile strain higher than 10%.

In conclusion, also in the case of HPFRCC, the functions of F(w) as well as the procedures introduced for the structural analysis are effective in describing the bending failure of beams. In the proposed model, the mechanical response of reinforced cement-based beams in bending is based on the experimental evaluation of the compressive strength f^sub c^'. Thus, in the case of large beams, such tests should be able to furnish f^sub c^' through a size effect law20 whose parameters could be different in case of plain concrete and fiber reinforced concrete, respectively. In fact, highly ductile materials appear less sensitive to the size effect of strength.25

CONCLUSIONS

With the goal of modeling the failure stage in beams reinforced with steel bars, a characterization of different cement-based materials has been presented. In particular, the function F(w), able to reproduce crushing of compressed cement-based materials, has been introduced for different composites. In this way, the ultimate stage of RC, reinforced FRC, and reinforced HPFRCC beams in bending, which fail via crushing in compression, has been analyzed. Based on the results of this investigation, the following conclusions can be drawn:

1. The post-peak response of ordinary and highperformance concrete in compression, with and without fibers, can be defined by means of stress-inelastic displacement relationships σ^sub c^-w (or F(w));

2. In the case of fiber-reinforced mixtures, there is not a unique relationship for F(w), as it should be related to the type of cementitious matrix (having low or high ductility), to the type of steel fibers (straight, hooked, or cord), and to the value of RI;

3. Moment-curvature relationships of beams reinforced with steel bars cannot be independent of the length Lc of the softening region, because compressive strains mainly localize there; and

4. If the M-μ-L^sub c^ relationships are computed according to the proposed procedure, based on the definition of F(w), the mechanical response of cement-based beams can be correctly predicted.

In the present case, the good agreement between the numerical predictions and the experimental data, in terms of load midspan-deflection of simply supported beams, validate the approaches introduced to characterize crushing of cement-based composites.

ACKNOWLEDGMENTS

The authors wish to express their gratitude to the Italian Ministry of Education, University and Research, for financing this research work (PRIN 2004-2005).

NOTATION

A^sub s^ = area of steel reinforcing bars in cross section

a, b = coefficients of F(w) (Eq. (4))

B = width of cross section

COD = crack opening displacement of cement-based material in tension

c = concrete cover in cross section

d = effective depth of cross section

E^sub c^ = Young's modulus of cement-based material

E^sub s^ = Young's modulus of steel reinforcing bars

F(w) = σ^sub c^/f^sub c^' = relative stress in post-peak stage of cement-based material in compression

f^sub c^' = compressive strength of cement-based material

f^sub ct^ = tensile strength of cement-based material

f^sub y^ = yielding stress of steel reinforcing bars

G^sub F^ = fracture toughness of cement-based material in tension

H, ΔH = height of cement-based specimen under compression (and its increment)

L^sub c^ = length of softening region produced by crushing in compression

l/D = fiber aspect ratio (l = length of fiber; D = diameter of fiber)

M = bending moment in cross section

M^sub 1^ = bending moment in cross section at beginning of crushing

M^sub max^ = maximum bending moment in cross section

P = applied load on simply-supported beam

RI = reinforcing index (RI = V^sub f^ . l/D)

s, Δs = sliding displacement during crushing (and its increment)

V^sub f^ = fiber volume content in cement-based materials, %

w, Δw = inelastic displacement of cement-based material in compression (and its increment)

y = vertical coordinate

y^sub c^ = distance of point in compression and neutral axis of cross section

y^sub c1^ = distance of points where ε^sub c^ = ε^sub c1^ and neutral axis of cross section

y^sub c,max^ = depth of neutral axis in cross section

z = horizontal coordinate

α = angle between sliding plane and direction of applied loads in cement-based material in compression

ε^sub c^, Δε^sub c^ = compressive strain in cement-based material (and its increment)

ε^sub c1^ = compressive strain at maximum stress of cement-based material

ε^sub c,max^ = compressive strain in top fiber of cross section

ε^sub s^ = tensile strain in steel reinforcing bars

η = midspan deflection of simply-supported beam

μ = curvature in cross section

μ^sub 1^ = curvature in cross section at beginning of crushing

μ^sub p^ = curvature in cross section at maximum bending moment

σ^sub c^, Δσ^sub c^= compressive stress in cement-based material (and its increment)

σ^sub c^t = tensile stress in cement-based material

σ^sub s^ = tensile stress in steel reinforcing bars