Experimental and numerical investigation on postcracking behavior of steel fiber reinforced concrete

Engineering Fracture Mechanics 98 (2013) 326–349

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Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech

Experimental and numerical investigation on postcracking behavior of steel fiber reinforced concrete Julien Michels a,⇑, Rouven Christen a, Danièle Waldmann b a b

Swiss Federal Laboratories for Materials Science and Technology (Empa), Überlandstrasse 129, CH-8600 Dübendorf, Switzerland University of Luxembourg, Rue Coudenhove-Kalergi, L-1359 Luxembourg City, Luxembourg

a r t i c l e

i n f o

Article history: Received 14 May 2012 Received in revised form 30 August 2012 Accepted 10 November 2012

Keywords: Steel fiber reinforced concrete Experimental investigation Numerical modeling Fracture energy Size effect

a b s t r a c t This paper presents an experimental and numerical investigation on the postcracking strength, energy absorption and fracture energy of steel fiber reinforced concrete (SFRC). The aim of the conducted research was a study on the effect of fiber type, fiber dosage and specimen size on the postcracking behavior of steel fiber reinforced concrete. Another objective was the development of a numerical fitting procedure being able to deliver stress–strain relations in tension for a given experimental test. For this purpose, a failure pattern based on plastic hinge and yield line theory is considered and subsequently the experimental force–deflection curves are fitted through an numerical optimization procedure. In a first step, experimental investigation on 4-point bending beam specimens following an SFRC design recommendation as well as on large scale plates has been conducted. These results give first impressions on the evolution of bearing forces and energy absorptions under different material and geometry conditions. It could be demonstrated that both maximal force and total energy absorption increase when fiber dosage increase. Furthermore, a higher fiber aspect ratio has a positive effect on the mentioned characteristics, too. In a second step, a parametric optimization procedure has been performed with the simulation code on two postcracking constitutive laws in tension, assuming either an exponential or a tangent hyperbolic decreasing stress evolution with growing tensile strain. It is shown that a hyperbolic relation offers a more accurate overall approximation of the experimental curves, although the exponential law offers higher precision at small deflection levels. For both, evaluation of fracture energy reveals similar values and trends, and the comparison between different specimen sizes reveals a size effect resulting in lower tensile strength and fracture energy values when dealing with large scale specimens. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction Fiber reinforcement for cementitious materials have gained increasing popularity since a few decades already. Naaman [1] presents an extensive overview on the evolution of such additional discontinuous reinforcements. Nowadays, conventional steel fiber reinforced concrete (SFRC) is mostly applied in the field of industrial floor [2] and tunnel constructions (e.g. precast tubbings [3]) with typical dosages Cf ranging from 20 to 60 kg/m3, corresponding to approximately 0.26– 0.78% in volume (Vf). Other suggestions use fibers as partial stirrup or punching reinforcement in RC beams and plates [4], respectively, or even a complete substitution of conventional steel reinforcements in flat slab construction [5–9].

⇑ Corresponding author. Tel.: +41 58765 4339. E-mail address: [email protected] (J. Michels). 0013-7944/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.engfracmech.2012.11.004

J. Michels et al. / Engineering Fracture Mechanics 98 (2013) 326–349

Nomenclature parameter for exponential SFRC tensile-softening geometry factor DFi difference between experimental and numerical force at deflection level i d midspan (beam) or border (plate) deflection di deflection level din initial deflection level du ultimate deflection level  SFRC strain in compression 0 concrete compressive strain at ultimate strength t SFRC strain in tension u ultimate SFRC strain in tension u⁄ transformed ultimate strain for large-scale elements fiber aspect ratio kf r stress h hinge rotation a column diameter b beam width c, d parameters for hyperbolic SFRC tensile-softening c.o.v. coefficient of variation fiber mass content Cf D plate diameter df fiber cross section diameter e column side length Ec concrete elastic modulus err average relative error between the experimental and fitted force–deflection curve errCS difference between compressive and tensile forces in the cross section F force f1 immediate postcracking tensile strength f2 postcracking SFRC strength parameter Fc compression force in the cross section concrete compression strength on cylinder fc Ft tension force in the cross section fcm,cube,28 concrete compression strength on cylinder after 28 days fct uniaxial concrete tensile strength maximum force during 4-point bending test Fmax fres,1,plate first transformed postcracking strength for large-scale plates fres,1 first postcracking strength obtained from small-scale laboratory test fres,2,plate second transformed postcracking strength for large-scale plates second postcracking strength obtained from small-scale laboratory test fres,2 fres residual postcracking tensile strength at ultimate strain u ftu fiber tensile strength Gf fracture energy Gf,150,w=4 mm fracture energy at crack width w = 4 mm for 4-point bending beam with b = 150 mm and h = 150 mm Gf,h,w=4 mm fracture energy at crack width w = 4 mm for a large scale SFRC element Gf,w=4 mm fracture energy until crack width w of 4 mm h specimen height L beam length lf fiber length lc characteristic length m moment in the yield line n number of yield lines ni substep ntot total number of substeps r overhang s.d. standard deviation v fiber wave depth Vf fiber volume content W SFRC energy absorption in tension w crack width b

vf

327

328

wu Wtot xc y

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ultimate crack width total energy absorption height of the compression zone fiber wave length

Strength classification of a certain SFRC mixture is generally performed according to a specific standard or recommendation. Although suggestions for uniaxial tensile tests exist [10] (RILEM recommendations [11,12]), execution is considered rather difficult. For practical reasons, it is generally preferred to use the indirect strength determination by means of bending tests. The most common test procedures are 3-point bending tests on notched beams (e.g. RILEM recommendations, presented in Barr et al. [13–15]) or 4-point bending tests on non-notched beam specimens (DAfStb-guideline [16] and DBV-recommendations [17]). Additionally, the ASTM recommendation C1550-10a [18] suggests a round-panel test on a round plate supported on three points, whilst the Swiss SIA 162-6 [19] presents three different possibilities to evaluate the material’s ultimate and postcracking tensile strengths. Despite a certain number of differences, all testing methods have in common the evaluation of postcracking strength(s) at defined deflection levels for specimens with constant geometry. In general, it is known that a higher fiber dosage involves higher postcracking strength (see, among many others, Barros et al. [20], Rossi [21], Soulioti et al. [22,23] and Bencardino et al. [24]), whereas maximum uniaxial tensile strength is only marginally influenced [25]. Furthermore, fiber geometry (for instance fiber aspect ratio kf, corresponding to the ratio of the fiber length lf to its diameter df), is a crucial factor when evaluating postcracking resistance. Similar to the fiber content, a higher fiber aspect ratio generally evokes larger postcracking tensile strength. Usually, upper limits for both fiber content and aspect ratio are given by workability issues (see [26–29]). A third group of crucial impact factors are the fibers’ dispersion in the cement matrix and orientation in direction of the applied stress [30]. The first parameter describes the numerical dispersion of the fibers through the concrete volume, the second one the fiber alignment in the matrix. Both values are governed by several influence factors during production, such as concrete consistency at fresh state, formwork dimensions, casting directions, compacting technique and concrete composition [31]. A perfect fiber alignment in stress direction offers the highest fiber efficiency resulting in higher postcracking strength than for a random orientation, whereas a perpendicular orientation would theoretically not offer any strength contribution by the fiber at all. Results on four-point bending tests showing higher bending strength with better fiber alignment are given by Stähli et al. [32]. Generally larger dimensions (for instance plates) implicate a more three-dimensional (random) fiber orientation compared to small-scale specimens, resulting in lower residual strength values. An extensive experimental study confirming the aforementioned results was performed by Lin [33] on beam elements with different cross-sections. Additionally, the so-called wall effect implicates fiber orientation along the formwork wall as described in Stähli and van Mier [34], presenting higher bending stresses with moulded specimens that for geometrically identical samples previously cut out from a bigger cast block (see also [35]). Furthermore, fibers tend to orient perpendicularly to the casting direction. For instance, Barragán [36] presents uniaxial tensile tests on drill cones taken horizontally and vertically from a cast plate and beam. In both cases, it is shown that the specimens cut parallel to the casting direction exhibited considerably lower postcracking tensile strength. Recent developments in the field are presented by Laranjeira et al. [37]. Even though certain studies have presented correlations between postcracking strength and specimen geometry and casting directions, experimental data on structural level are considered few. Regarding numerical approaches in fiber reinforced cementitious materials, a large number of publications are available. The authors want to draw the attention to several recent developments in the field. Zhang et al. [38] present a fracture mechanics based approach to analyse crack propagation under direct tension. The cited researchers calculate the stress intensity factor at the crack tip used for evaluating several parameters on the material’s behavior in tension. Interface models are presented by Caggiano et al. [39,40] and Etse et al. [41]. Discrete embedded elements for simulating the fibers in finite element modeling are used by Cunha et al. [42] and Radtke et al. [43]. The present manuscript aims at confirming a certain number of known effects regarding fiber dosage and aspect ratio on 4-point bending specimens. Additionally, four large-scale SFRC plates are evaluated regarding postcracking strength. A numerical inverse analysis with parameter fitting of two different tensile-softening laws (see also [44]) is used to derive fracture energy. Through the respective fracture energies, laboratory-scale values can eventually be compared to the results obtained on a structural level. 2. Experimental investigation 2.1. Materials 2.1.1. Steel fibers Two different undulated steel fiber types (Table 1) are used during the experimental campaign. On the one hand, Tabix+ 1/60 (type A) have a fiber length lf of 60 mm, and diameter df of 1 mm, fiber aspect ratio kf is 60. Average tensile strength ft,u is 1450 MPa. Tabix 1.3/50 (type B) on the other hand have a fiber length lf of 50 mm, a diameter df of 1.3 mm. Fiber aspect ratio in this case is 39, whilst maximum tensile strength averages 900 MPa. Further characteristics are given in Table 1 and the

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J. Michels et al. / Engineering Fracture Mechanics 98 (2013) 326–349 Table 1 Technical data for the two fiber types Tabix 1.3/50 and Tabix+ 1/60 (also refer to [45]).

Fiber diameter df (mm) Fiber length lf (mm) Wave depth v (mm) Wave length y (mm) Aspect ratio kf (–) Tensile strength ftu (MPa)

Type A

Type B

1 60 0.4–0.65 8 60 1450

1.3 50 0.4–0.65 8 39 900

technical data sheet by the producer [45]. Type B is only used in a dosage Vf of 1.3%, type A in four ratios of 0.2%, 0.52%, 0.65% and 0.92%. 2.1.2. Concrete composition Concrete composition is given in Table 2. For the presented fiber type A, Mixture 1 is used, whereas Mixture 2 was put in place with type B. Average compression strength after 28 days on cube fcm,cube,28 for Mixture 2 averages 60.4 MPa, whilst Mixture 1 showed values of 60.5, 58.7, 62.3 and 63.5 MPa for a dosage of 0.2%, 0.52%, 0.65% and 0.92%, respectively. 2.2. Test setups 2.2.1. 4-point bending tests 4-point bending specimens at an age of 28 days were tested according to the recommendations of the German Concrete Association (DBV) [17]. The beams with a cross-section (b  h) of 150  150 mm2 and a total length L of 700 mm were cast along the longitudinal side. For testing, specimens were rotated by 90° along their horizontal axis, so that the former casting surface became a side surface. Span length was 600 mm. The tests were run under controlled displacement at a velocity of 0.2 mm/min up to a total midspan deflection d of 4 mm with reference to the central beam axis. The setup is shown in Fig. 1. Specimens with fiber type B were tested at University of Luxembourg (L), beams with type A at the laboratory of ArcelorMittal Bissen (L). 2.2.2. Large-scale plate tests Large-scale tests were conducted with octogonal plate specimens symmetrically loaded around the column. Testing diameters D were 2.34 and 1.9 m, respectively, and plate thickness h varied from 20 to 40 cm. The square column support cross section was either 350  350 mm2 or 250  250 mm2 (see Table 3). Casting was performed from the top, no change in plate orientation was performed prior to testing. In total, eight hydraulic jacks applied constant displacement at a velocity of 0.1–0.4 mm/min. Vertical displacement was measured by means of LVDTs at the plate border at a radial distance of 150 mm from the loading point. All plate specimens were casted with Fiber Type B at a dosage of Cf = 100 kg/m3, corresponding to a fiber volume Vf = 1.3% and tested at University of Luxembourg. The test setup is presented in Fig. 2. 3. Numerical modeling 3.1. Background and algorithm 3.1.1. Definitions The basis of the numerical modeling is the plastic hinge or yield line theory [46], assuming a crack pattern for a specific static system (see Section 3.2). This calculation technique is assumed realistic as all specimens exhibited a clear bending failure. Once uniaxial tensile strength fct of the SFRC reached, all deformation is supposed to occur in the plastic hinge or yield lines. A strain-dependant stress relation r(t) in tension of the SFRC is calculated by inverse analysis from the experimental force–deflection curves.

Table 2 Concrete composition for Mixtures 1 and 2 (*plasticizer mass/cement mass). Mixture 1 Sand Aggregate 2/8 Aggregate 8/16 Cement CEM II 32.5 R Water/cement Plasticizer

Mixture 2 640 kg/m3 790 kg/m3 370 kg/m3 350 kg/m3 0.56 0.80%

Sand Aggregate 4/8 Aggregate 8/16 Cement CEM I 42.5 R Fly ash Water/cement Plasticizer

700 kg/m3 500 kg/m3 550 kg/m3 350 kg/m3 120 kg/m3 0.5 1.26%

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Fig. 1. Test setup for 4-point bending tests [7,17].

Table 3 Dimensions of the large scale plates. Plate no.

h (mm)

e  e (mm mm)

D (mm)

1 2 3 4

200 250 300 400

350  350 350  350 350  350 250  250

2340 2340 2340 1900

Fig. 2. Test setup for large scale plate tests.

The hinge rotation h (see Fig. 3) can be linked to each deflection level di by the Equations presented in Section 3.2. By using the geometric relations presented in Fig. 3, one can write:

tanðhÞ ffi h ffi

w h  xc

ð1Þ

As presented in Eq. (2), crack width w can be linked to the tensile strain t through the characteristic length lc over which the concrete undergoes plastic deformation. The concept was initially introduced by Bazˇant and co-workers [47,48] in the crack band model. The characteristic length can be used as a transition between smeared (stress–strain relationship) and discrete (stress-crack width relationship, see [49]) crack models.

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Strain

Stress

c

c

Fc Compression zone Tension zone

m

xc

z

m

ft Ft

t

w Fig. 3. Hinge rotation.

w ¼ t  lc

ð2Þ

The evaluation of the indicated characteristic length is a strongly discussed topic in the related scientific literature. In general, the characteristic length is influenced by a certain number of parameters, such as specimen geometry, fiber type and fiber content (see [50]). This reference gives a good overview on different approaches on how to treat characteristic length in numerical simulation and design. The presented studies suggest fixed values ranging from h/4 to 2h (h being the specimen height) or variable values depending on the respective load level. Di Prisco et al. [51] suggest to define the characteristic length as the minimum value of the crack spacing and the height of the cracked zone if conventional steel rebars are present. In case of a purely steel fiber reinforcement, due to a limited extension of the compression zone, the cited researchers propose a simplified evaluation in which the characteristic length can be set equal to the element height h. In the present investigation, the recommendations from the DBV-Merkblatt [17], also presented in Falkner et al. [52], are adopted. With this approach, the characteristic length is defined as the height of the cracked tensile zone h  xc at the respective load level (see Eq. (3)). The height of the compression zone xc is derived from the equilibrium between the compression and tension forces in the cross-section (see Section 3.1.2).

lc ¼ h  xc

ð3Þ

Since h  xc corresponds to lc and by combining Eqs. (1)–(3), one obtains:

h ¼ t

ð4Þ

A cross-section analysis in the yield line, requesting an equilibrium between the concrete compressive and tensile forces Fc and Ft, is performed (see Fig. 3) for each deflection level di. Once an equilibrium is found, the code moves to the next deflection level. The obtained bearing moment mR at each step is transformed into an outer force F by the yield line theory relations presented in Section 3.2. The different material parameters, introduced in Section 3.3, are iteratively upgraded by the developed code based on the fminsearch function (see [53]).

Fig. 4. Experimental and numerical force–deflection curves and relative error err (nf,i = total number of substeps).

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Fig. 5. Flow chart of the numerical calculation procedure.

Hence, for each experimental specimen, two best-fit curves are obtained and can be compared with the test values. Eventually the best-fitting stress–strain curves in tension for each specimen will be used for deriving fracture energy Gf. Further details about this aspect are given in Section 3.4. A similar, yet polylinear ’step-by-step’ inverse analysis is applied by Kurihara et al. [54] based on suggestions by Kitsutaka [55,56]. 3.1.2. Structure of the algorithm The developed algorithm is structured according to the flow chart in Fig. 5. Initially, material parameters such as compression strength, tensile strength and elastic modulus have to be introduced. Afterwards, the crack pattern has to be chosen a priori according to the considered test setup. Experimental data for each test is loaded and used as reference for curve fitting. The two loops (inner and outer) are finally used for curve fitting. 3.1.2.1. External loop. The external loop defines and evaluates material parameters for a specific tensile-softening relation. For one fixed parameter group, the complete force–deflection curve is calculated and compared with the original data from the experimental investigation. As mentioned, the algorithm aims at minimizing the average relative error err (see Eq. (5)) between the experimental and numerical values as presented in Fig. 4.

Pdu err ¼

din jDF i j

ntot  F max

ð5Þ

An initial target value (in our case for err) is defined and serves as guidance for the optimization procedure. Subsequently, the algorithm tries to optimize the stress–strain law parameters in order to reach the defined relative error. In the present case err is set to 0.05 (5%). Additionally, the maximum number of iterations is set to 500. In case that number is reached, the optimization procedure is stopped. Furthermore, the algorithm disposes of a self-check tool which controls its relative

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333

Fig. 6. Crack pattern of the 4-point bending beam tests.

improvement over the optimization span. The code hereby discovers whether further improvement is possible or not. With this tool, the initially defined target error does not represent an absolute value, but can even be underran in case the maximum number of iterations is not reached and further improvement is possible. In the opposite case, in order to save calculation time, the calculation procedure can be stopped before completing the maximum iteration number if the self-check tool indicates that the target value will not be reached. Details about the exact algorithm structure can be found in Lagarias et al. [53]. 3.1.2.2. Internal loop. The internal loop calculates an equilibrium between tensile and compressive forces in the hinge or yield line (see Fig. 3) for a given rotation (tensile strain) under constant constitutive parameters. The height of the compression zone xc is iteratively determined by minimizing the square of the relative error errCS in the cross-section between the two forces (see Eq. (6)). The calculation is also based on the fminsearch algorithm (see Section 3.1.2.1).

errCS ¼

4  ðF c  F t Þ2

ð6Þ

ðF c þ F t Þ2

2. Π n

D. cos ( Πn ) _ a 2 Fig. 7. Crack pattern of the large scale plate tests.

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Fig. 8. Idealized stress–strain curve for concrete in compression [58].

Once one equilibrium state is reached, the next measured displacement level is considered and the equilibrium procedure repeated. After one complete cycle up to the maximum measured displacement, constitutive parameters are evaluated and adapted in the external loop before restarting the equilibrium phase. 3.2. Crack pattern 3.2.1. 4-point bending beams For the material testing on small scale beams under 4-point bending, hinge rotation h at midspan and the corresponding bending moment m in the hinge can be obtained with the following expressions (Eqs. (7) and (8), see Fig. 6). It is mentioned that the hinge location ad midspan represents a simplified approach, as specimens without a notch do not necessarily exhibit crack location exactly at this specific location.

4d L F L m¼ 6

h¼

ð7Þ ð8Þ

3.2.2. Large scale plates In case of the large scale plates, the following Eqs. (9) and (10) (see Fig. 7) express the yield line rotation h for n yield lines and the corresponding moment m in the yield line [57]:

  4  d  sin Pn P  D  cos n  a     F  D  cos Pn  a   m¼ 2  n  ðD þ 2  rÞ  sin Pn

h¼

(a)

ð9Þ ð10Þ

(b)

Fig. 9. Exponentially decreasing and hyperbolic tangent equations for experimental curve fitting.

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335

Fig. 10. Force against midspan deflection for 4-point-bending beams with different fiber volumes for Fiber type A.

It is worth mentioning that the crack pattern in the present investigation is assumed regular, whereas in many cases yield lines develop irregularly. By adopting this procedure, an overestimation of the bending moment in the yield line is generally the consequence. This eventually leads to an overestimation of the fracture energy, depending on how much the real crack pattern differs from the idealized one. The main goal of the presented investigation is to give a first impression on possible differences between small- and large-scale SFRC structural elements. In this sense, the chosen calculation procedure is judged sufficient for a first analysis. Higher precision could be obtained by exactly assessing the yield line orientation in each case and redefining the relation between the moment in the yield line and the applied load. 3.3. Constitutive laws 3.3.1. Compression In compression, a parabolic Hognestad stress–strain relation [58] is implemented and kept constant throughout the whole numerical analysis. The following Eqs. (11) and (12) as well as Fig. 8 define the SFRC behavior in compression. If  < 0:

"

rðÞ ¼ fc 

2

0

 2 #



 0

ð11Þ

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Fig. 11. Force against midspan deflection for 4-point-bending beams with fiber volume Vf of 1.3% for Fiber Type B.

Fig. 12. Crack pattern on top side of the large scale plate tests [8].

J. Michels et al. / Engineering Fracture Mechanics 98 (2013) 326–349

If

 6 0:



rðÞ ¼ fc  1 

0:15  ð   0 Þ 0:004  0

337

ð12Þ

For all simulations, a value of 30 GPa was considered for the Young’s modulus Ec. Average compression strength on cylinder fc was estimated 80% of the compression strength on cube. For each simulation series, the respective value was implemented according to the performed cube tests (with a side length of 150 mm). 3.3.2. Tension In tension, the material behavior is assumed linear elastic until uniaxial tensile strength fct is reached, followed by a decreasing stress–strain branch. The principal focus of the investigation lies in the evaluation and comparison of two different constitutive (r, t) material laws for SFRC in tensile-softening. The first one assumes an exponentially decreasing behavior after having reached uniaxial tensile strength, whilst the second one is based on a hyperbolic tangent function. Both relations, presented in Fig. 9 and Eqs. (13) and (14), were chosen for fitting the experimental force–deflection curves. The fitting code optimizes the given parameters f1, f2, b, c and d for the chosen tensile-softening relation in the outer loops. For an exponential decreasing, the following equation is implemented:

rðÞ ¼ ðf1  f2 Þ  ebðt ct Þ þ f2

ð13Þ

For a hyperbolic decreasing, the stress–strain dependency is the following:



rðÞ ¼ f1  f2 þ

1  f2  ½1  tanhðc  ðt  dÞÞ 2

 ð14Þ

After completion of the optimization procedure, the stress–strain curve is transformed into a stress-crack opening relation by means of Eqs. (2) and (3). Eventually, as presented in the following section, fracture energy can be calculated. 3.4. Fracture mechanics In the present paper, the two main fracture mechanics parameters are the energy absorption W and the fracture energy Gf defined in Eqs. (15) and (16). The total energy absorption is defined as the area under the force deflection curve:

W¼

Z

dmax

F  df

ð15Þ

0

The fracture energy, necessary to obtain complete crack opening with zero stress transfer, is obtained by deriving the area under the stress-crack width curve:

Gf ¼

Z

wu

rðwÞ  dw

ð16Þ

0

The assessment of total fracture energy Gf up to zero stress transfer would require much larger crack opening than the ones evoked by the small beam test procedures according to design recommendations. Hence, it is necessary to limit the evaluation of the fracture energy to a common crack width value. As the 4-point bending tests were all performed up to

Fig. 13. Force–deflection curves of the large scale plate tests with different specimen height h including self-weight [7,8].

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a midspan deflection dmax of 4 mm, equivalent to a crack width w of 4 mm, this value is set as reference. For the large scale plate tests, the comparison to the small beams is preformed by evaluating fracture energy up to the same crack width (Gf,w=4 mm). Regarding extrapolation techniques up to zero postcracking stress, the Reader is referred to Denneman et al. [59].

4. Results and discussion 4.1. Experimental force–deflection Force–deflection curves of the 4-point bending tests are given in Figs. 10 and 11 for Fiber Types A and B, respectively. All beam specimens exhibited bending failure with creation of one hinge in which rotation was concentrated with ongoing midspan deflection. Crack pattern on top side of the large scale plates is given in Fig. 12. Similar to the 4-point-bending specimens, all plates failed in bending with creation of yield lines. Force–deflection curves of the four tests are presented in Fig. 13.

Fig. 14. Experimental against numerical force–deflection curves for different fiber volumes Vf with % Fiber type A under 4-point bending.

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339

4.2. Numerical results and evaluation of the tensile-softening laws Average numerical curves (obtained from the individually fitted curves with the respective tensile-softening relation) compared to the average experimental curve are given in Figs. 14 and 15. All fitted parameters f1, f2, b, c and d are presented in Table 4. For both numerical approaches it is apparent that the postcracking tensile strength f1 immediately after cracking is increasing with growing fiber volume Vf in case of Fiber Type A. In case of an exponential tensile-softening, higher values for b can be observed for a low fiber content of Vf = 0.2% in the same fiber type category. The reason is a faster stress drop in uniaxial tension compared to higher fiber contents. Hence, a higher exponential decrease factor b is necessary for the curve fitting. A similar effect can be observed for the hyperbolic softening. Whereas no clear trend in the parameter c can be read out, it is evident that the position d of the inflection point is increased when dealing with higher fiber content. This is again due to the fact that postcracking tensile strength decreases much faster when dealing with low fiber-volume ratios. Due to the different mathematical expressions used for curve fitting, higher values for f1 in case of an exponential tensile softening result from the calculations. For Fiber Type B, a clear strength drop in f1 is noticed when moving from the small-scale beams to the plates, indicating lower residual tensile strength for large scale elements. Relative errors err for all tested beams and plates are given in Figs. 16 and 17 as well as fracture energies Gf,w=4 mm in Tables 5–7.

Fig. 15. Experimental against numerical force–deflection curves for Vf = 1.3% Fiber Type B for (a) 4-point bending tests and (b) large scale plate tests.

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Table 4 Fitted parameters f1, f2, b, c and d for an exponential and hyperbolic decreasing stress–strain law. Vf (%)-no. (fiber type)

Exponential f1

f2

b

f1

f2

c

d

0.2-1 0.2-2 0.2-3 0.2-4 0.2-5 0.2-6 0.2-7 0.2-8 0.2-9

2.04 2.62 1.40 1.55 1.94 1.78 1.81 1.74 1.95

0.33 0.27 0.27 0.14 0.25 0.44 0.27 0.26 0.22

381.0 367.1 362.2 87.4 342.2 326.2 307.0 332.0 375.4

0.98 1.20 0.56 1.38 1.85 2.25 0.97 1.96 0.78

0.21 0.03 0.11 0.18 0.99 0.11 0.15 0.18 0.07

563.2 739.9 307.2 239.9 35.1 89.1 491.1 1141.7 407.4

0.009 0.009 0.015 0.007 0.018 0.001 0.008 0.002 0.010

0.65-1 0.65-2 0.65-3 0.65-4 0.65-5 0.65-6 0.65-7 0.65-8

2.65 2.39 2.58 2.15 2.17 2.33 2.34 2.11 2.25

0.62 2.16 1.45 1.10 1.09 0.44 1.23 1.88 0.04

56.9 23.0 140.6 28.2 89.6 30.2 311.1 20.3 59.7

2.25 2.02 2.20 1.77 1.93 2.08 1.78 1.85 1.93

0.42 0.35 0.47 0.29 0.61 0.54 0.56 0.45 0.38

238.6 316.8 258.9 396.4 238.6 262.0 272.2 334.7 357.6

0.013 0.011 0.013 0.013 0.011 0.011 0.011 0.011 0.009

0.92-1 0.92-2 0.92-3 0.92-4 0.92-5 0.92-6 0.92-7 0.92-8 0.92-9

2.88 2.78 2.82 2.75 3.40 2.49 2.31

0.97 0.82 1.13 0.28 1.31 0.10 0.25

302.5 175.1 288.6 106.0 37.5 65.7 84.6

2.95 2.06 1.90 2.41 3.01 2.10 1.91

0.37 0.37 0.46 0.33 0.35 0.36 0.39

427.6 290.0 361.5 446.0 264.6 441.1 376.2

0.002 0.009 0.010 0.006 0.009 0.008 0.008

PL1 PL2 PL3 PL4

(A) (A) (A) (A) (A) (A) (A) (A) (A)

0.52-1 0.52-2 0.52-3 0.52-4 0.52-5 0.52-6 0.52-7 0.52-8 0.52-9 1.3-1 1.3-2 1.3-3 1.3-4 1.3-5 1.3-6 1.3-7

(A) (A) (A) (A) (A) (A) (A) (A) (A)

(B) (B) (B) (B) (B) (B) (B)

Hyperbolic

Vf (%)-no. (fiber type)

Exponential

Hyperbolic

f1

f2

b

f1

f2

c

d

(A) (A) (A) (A) (A) (A) (A) (A)

2.25 2.29 2.39 2.45 2.64 2.37 2.74 2.77

0.04 5.38 6.14 1.66 1.42 1.50 1.03 1.61

59.7 11.7 10.6 264.7 296.6 29.7 315.6 261.0

2.05 2.03 2.09 2.13 2.07 1.98 2.02 2.82

0.23 0.36 0.29 0.52 0.21 0.23 0.26 0.25

369.2 328.6 236.3 197.0 110.7 351.8 161.7 124.6

0.011 0.011 0.013 0.014 0.016 0.012 0.010 0.013

(A) (A) (A) (A) (A) (A) (A) (A) (A)

3.26 2.78 3.11 2.51 2.78 2.88 3.14 3.03 3.02

1.23 0.82 7.27 0.84 0.52 1.00 1.38 1.40 5.23

283.8 175.1 13.2 29.8 53.2 25.2 56.3 113.2 12.7

2.46 2.36 2.56 2.10 2.42 2.55 2.96 3.02 2.82

0.42 0.50 0.18 0.30 0.43 0.35 0.62 0.55 0.25

281.6 278.4 293.9 334.7 247.0 264.4 201.7 249.3 124.6

0.008 0.010 0.013 0.014 0.012 0.014 0.010 0.006 0.013

1.00 1.04 0.98 0.88

0.46 0.89 0.66 1.32

32.0 177.5 18.2 10.7

0.93 0.98 0.91 0.79

0.70 0.76 0.53 0.44

178.4 404.0 253.0 277.2

0.014 0.016 0.011 0.014

(B) (B) (B) (B)

The results of the numerical investigation demonstrate that a hyperbolic tangential decreasing law seems to be the most appropriate law of both to simulate the SFRC behavior due to lower relative errors err between the experimental and numerical curves. Fig. 14 also shows that an exponential law is able to better approximate the experimental results at low deflection values (<1 mm), whereas a hyperbolic decreasing is more accurate with growing deformation. This is due to the more sophisticated formulation form Eq. (14), allowing the user to adjust one more parameter in order to minimize relative errors. However, fracture energy values do not considerably differ from one tensile-softening relation to the other. This is due to the fact that for both numerical approaches, over- and underestimations of the experimental values cancel each other out. Hence, as for the evaluation of the fracture energy only the total area under the stress-crack opening curve is taken into account, both numerical simulations give similar results. Additionally, as presented in Tables 5–7, relative differences between the fracture energies for one tensile-softening relation are almost identical.

Fig. 16. Relative errors err for 4-point bending specimens with Fiber Types A and B for both exponential and hyperbolic curve fitting.

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Fig. 17. Relative errors err for plate tests with Fiber Type B for both exponential and hyperbolic curve fitting.

Table 5 Average relative error err and fracture energy Gf,w=4 mm for both tensile-softening relations in case of 4-point bending tests with Fiber Type A (s.d. = standard variation, c.o.v. = coefficient of variation (%)). Vf = 0.2 %

Vf = 0.52%

Exponential

Hyp. tangent

Exponential

Hyp. tangent

err (%)

Gf,w=4 mm (N mm)

err (%)

Gf,w=4 mm (N mm)

err (%)

Gf,w=4 mm (N mm)

err (%)

Gf,w=4 mm (N mm)

Average S.d. C.o.v.

5.27 2.13 38

1.78 0.26 14

5.35 0.96 18

1.53 0.38 25

4.85 1.33 27

5.15 0.79 15

3.97 0.96 24

5.04 0.65 13

Average S.d. C.o.v.

Vf = 0.65% 6.84 1.61 24

5.14 1.01 20

4.12 1.08 26

4.89 0.85 17

Vf = 0.91% 4.66 1.23 26

6.34 0.95 15

3.39 0.72 21

6.23 0.97 16

Table 6 Average relative error err and fracture energy Gf,w=4 mm for both tensile-softening relations in case of 4-point bending tests with Fiber Type B (s.d. = standard variation, c.o.v. = coefficient of variation (%)). Vf = 1.3% Exponential

Average S.d. C.o.v.

Hyperbolic tangent

err (%)

Gf,w=4 mm (N mm)

err (%)

Gf,w=4 mm (N mm)

3.5 2.28 65

4.66 0.74 16

3.75 1.12 30

4.72 0.74 16

Table 7 Relative error err and fracture energy Gf,w=4 mm for both tensile-softening relations for the large scale plate tests. h (mm)

Vf = 1.3% Exponential

200 250 300 400

Hyperbolic tangent

err (%)

Gf,w=4 mm (N mm)

err

Gf,w=4 mm (N mm)

1.71 4.67 3.32 4.63

3.37 3.72 3.12 2.93

0.93 2.28 1.90 3.96

3.35 3.83 3.13 3.09

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(a)

(b) Fig. 18. Influence of fiber volume Vf on the (a) maximum force and (b) total energy absorption with fiber type A under 4-point bending.

Table 8 Average maximum force and total energy absorption values for beam specimens with Fiber Type A (s.d. = standard deviation, c.o.v. = coefficient of variation (%)). Vf = 0.2%

Fmax (kN)

Wtot (J)

Vf = 0.52%

Fmax (kN)

Average S.d. C.o.v.

26.9 2.0 7.4

38.2 5.6 14.7

Average s.d. c.o.v.

30.6 2.9 9.5

Wtot (J)

Vf = 0.65%

Fmax (kN)

Wtot (J)

Vf = 0.91%

Fmax (kN)

Wtot (J)

Average S.d. C.o.v.

31.4 1.4 4.5

98.9 11.0 11.1

Average s.d. c.o.v.

39.1 3.7 9.5

121.9 14.4 11.8

97.0 9.3 9.6

4.3. Influence of fiber volume Vf For the experimental 4-point bending test series with fiber type A, force F is plotted against midspan deflection d in Fig. 10a–d. The general tendencies of the dependence of Fmax and Wtot on the fiber volume Vf are shown in Fig. 18a and b,

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J. Michels et al. / Engineering Fracture Mechanics 98 (2013) 326–349 Table 9 Average maximum force and total energy absorption values for beam specimens with Fiber Type B (s.d. = standard deviation, c.o.v. = coefficient of variation (%)). Vf = 1.3%

Fmax (kN)

Wtot (J)

Average S.d. C.o.v.

34.5 4.3 12.5

96.5 15.4 16

respectively. Three regression curves (linear, exponential and logarithmic) are indicated as a qualitative trend for both the maximum force and the energy absorption. In case of the maximum force, the respective coefficients of determination R2 indicate that a linear and a logarithmic expressions are of similar quality, whilst an exponential fit is of relatively poor quality. Summing up, it is apparent that the three regressions approximate the experimental data to a limited extent. In case of the energy absorption, the overall fitting is significantly better. Among the three methods, a logarithmic approximation shows the best agreement. Average values are given in Tables 8 and 9. It is mentioned that the authors refrain from an often applied procedure consisting in reconverting measured forces into equivalent flexural stresses, as there is no change in geometry for the 4-point bending tests. Hence, stress results would follow the exact tendencies as for measured forces. The presented results indicate that a growing fiber content has an increasing effect on both the maximum force and the energy absorption. This effect has been observed in a very large number of experimental investigations with different fiber type and content. The presented results show a force increase of 13.8%, 16.7% and 45.4% for Vf = 0.52%, 0.65% and 0.91%, respectively, compared to a low fiber content of Vf = 0.2%. Furthermore, it can be observed that a low fiber dosage of 0.2% maximum force occurs at a clearly lower deflection value than with higher fiber content. For instance, a clear force drop to approximately half the force level after the peak at 0.15 mm midspan deflection is visible in Fig. 10 a), whereas higher fiber dosages offer the possibility to obtain a (quasi-) hardening behavior under bending. Dosages of 0.52% and 0.65% (Fig. 10b and c) induce a supplementary force increase after a short force drop when leaving the linear domain followed by a force plateau up to approximately 1 mm deflection at midspan. However, almost no increase in the maximum force is noticed when using 0.65 instead of 0.51% of steel fiber type A. Another of the fiber content to 0.91% involves further force increase (Fig. 10 d)) with a pronounced hardening behavior. Regarding the scatter in about the maximal forces listed in Table 8, no tendency about the evolution of the coefficient of variation depending on the fiber volume can be pointed out. More important than the maximum force in SFRC strength analysis is the postcracking behavior up to high strain levels. Similar to the presented comments about force evolution is the enhancement of the energy absorption Wtot until maximum midspan deflection. From both Table 8 and Fig. 18b), it gets obvious that a higher fiber dosage induces higher energy absorption. On average, relative enhancements of 154%, 158% and 219% are observed when increasing dosage from 0.2% to 0.52%, 0.65% and 0.91%, respectively. Again no significant difference between 0.52% and 0.65% fiber volume can be observed. However, the general tendency is in accordance to the previously mentioned sudden force drop after reaching the peak value at low fiber dosages, whereas a higher fiber content allows transferring higher stresses at large crack openings and thus higher residual load carrying capacity at large deflections. As shown in Table 8, scatter of the energy absorption is slightly reduced when moving from a fiber volume fraction of about 0.2% to higher values.

Fig. 19. Influence of fiber volume Vf on the fracture energy Gf,w=4 mm with both exponential and hyperbolic tangent tensile-softening relations (4-point bending).

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Fig. 19 as well as Tables 5 and 6 present the numerical results from the best-fitting calculation procedure. Regarding fracture energy, the same tendency, i.e. higher values with an increasing fiber dosage, is noticed. This observations is proved by the same aspect of the presented regression curves. For each optimization procedure, three regression curves are shown. In both cases, an exponential regression offers moderate agreement with regard to the experimental values (R2 = 0.64 in both cases). Linear regression offers better results, optima being reached with a logarithmic approximation with the highest coefficients of determination (R2 = 0.82 and 0.83). It is repeated that, when referring to Tables 5 and 6, it becomes apparent that the hyperbolic tangent stress–strain curves allow a slightly better curve fitting with the experimental results due to lower relative error values than the corresponding exponential decreasing. Such a result, as stated earlier, can be explained by a more sophisticated tensile-softening model, having at disposal one more parameter to be fitted. 4.4. Influence of fiber aspect ratio kf The influence of fiber aspect ratio on the maximum force Fmax as well as the total energy absorption Wtot and fracture energy Gf,w=4 mm until midspan deflection d of 4 mm is presented in Figs. 20 and 21. The first two plots present experimental data, whereas the last two show numerical results from the fitting procedure. Fig. 20 shows average maximum forces and total energy absorption resulting from the experimental investigation on 4-point bending beams with both Fiber Types A (kf = 60) and B (kf = 39). It gets obvious that a fiber with kf = 60 in a content of Vf = 0.91% induces a clearly higher maximum force than a more compact fiber with kf = 39 in a higher content of Vf = 1.3%. A difference of 13% in force can be noted be-

(a)

(b) Fig. 20. Comparison of (a) average maximum force and (b) average total energy absorption between 0.2%, 0.52%, 0.65% and 0.91% fiber type A and 1.3% Fiber Type B (4-point bending).

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345

tween the two mentioned configurations. Whereas a higher maximum force with 1.3% of Fiber Type B than with 0.52% and 0.65% of fiber type A can still be observed, total energy absorption until maximum midspan deflection is already slightly higher for the fiber type with higher aspect ratio. This indicates a higher efficiency of a longer and thinner fiber instead of a more compact type. The same tendency is observed for both fracture energy calculations shown in 21a and b. Fiber type A with 0.52%, 0.65% and 0.91% in volume implicates clearly higher energy results than the more compact geometry with a fiber content of 1.3%.

4.5. Influence of specimen thickness h The influence of specimen thickness h is described by the plots in Fig. 22 and by Table 7. For both tensile-softening relations, lower fracture energies with large scale elements compared to laboratory specimens are observed. As postcracking strength is for instance also governed by fiber orientation in stress direction, the latter seems to be less pronounced for large scale plates than for small beams. The results are in accordance to several experimental findings, such as for Michels [7] and Lin [33], showing a reduced number of fibers per unit area and thus a reduced fiber orientation [30] in the cracked section with growing element (formwork) thickness and width. The results indicate a fracture energy drop when switching from the 4-point bending beams to the larger plates, most likely due to a less pronounced fiber orientation in consequence of no walleffect (see Section 1). The differences between the large scale elements themselves are rather small.

(a)

(b) Fig. 21. Comparison between fracture energies Gf,w=4 mm with (a) exponential and (b) hyperbolic tangent decreasing relations with different fiber types and volumes (4-point bending).

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Special attention for the current results has to be paid to the observed fiber segregation towards the formwork bottom. Especially the first two plates with a thickness of 20 and 25 cm, respectively, exhibited a more or less strong fiber ’drowning’ during and right after the casting process. Results of manual counting of the fiber dispersion in the yield lines are given in Michels et al. [8]. The consequence is a lower fiber presence and thus lower residual tensile strength in the upper negative bending area. With further crack opening however, tensile zone moves down and hence more fiber participate again in the tension force transfer. For the plate with a thickness of 400 mm, fiber segregation was almost null, whereas some fiber balling was noticed during the casting, possibly having a reduced total number of fibers over the complete concrete volume as a consequence [7]. In order to verify the presented results and obtain more detailed information on scatter, further experimental research regarding fiber orientation and dispersion as well as fracture energy is necessary. With regard to the differences in fracture energies between small and large scale specimens, a geometry factor can be defined for different thickness as the ratio between the large-scale fracture energy to its reference from the 4-point bending beams (see Eq. (17)). The calculated geometry factors with both tensile-softening relations are presented in Fig. 23. All the suggested trendlines give similar tendencies with a low general agreement though. However, as experimental data from plate tests are few in the present case, the indicated values should be treated in a qualitative matter. In order to obtain more reliable results, further experimental investigation on large scale elements is mandatory.

(a)

(b) Fig. 22. Comparison of fracture energies Gf,w=4 mm with (a) exponential and (b) hyperbolic tangent decreasing relations between 4-point bending beams and large-scale plates.

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Fig. 23. Geometry factor vf for different specimen heights resulting from the fitting procedure of the experimental results.

Fig. 24. Example of a possible integration of the geometry factor v into a design procedure for large scale SFRC plate elements for a bilinear stress–strain law in tension.

vf ¼

Gf ;h;w¼4mm Gf ;150;w¼4mm

ð17Þ

Fig. 24 gives a suggestion on how a geometry factor could be integrated into a design concept. Usually strength values are obtained by performing small-scale laboratory tests. In case of SFRC, the presented investigations revealed clearly lower strength values when dealing with large-scale elements. In case of a particular design problem, it is suggested to multiply the obtained laboratory strength values (obtained by following a specific testing recommendation) by the geometry factor in order to respect the differences in postcracking behavior. It is important to seize the fact that the geometry factor has been derived from fracture energies up to equal crack openings w. For a stress–strain transformation, the geometric differences between small laboratory beams and large plates have to be respected. Hence, an equal crack opening is obtained at a by a lower strain value for a higher thickness. The newly obtained large-scale strengths can subsequently be lowered by an adequate safety factor and used for ultimate limit state verifications.

5. Conclusions and outlook Several conclusions can be drawn from the presented results. It was shown that the developed algorithm allowed a numerical recalculation of the experimental beam and plate tests and thus a comparison of fracture energies of SFRC elements with different size and geometry. As known from earlier research in the field, experiments on laboratory-scale beam specimens confirmed a higher postcracking strength when increasing fiber content and/or fiber aspect ratio. A fiber with a higher aspect ratio exhibited higher

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efficiency than a similar, but more compact type in a higher content. If workability restrictions are respected, economic advantages are obtained when preferring fibers with a higher aspect ratio to reach a specific strength class. The numerical application showed that the two tensile-softening relations are able to well recalculate the experimental force–deflection curves after the adequate parameter fitting. Average error between experiment and numerical model over the total deflection range was below 10% in the worst case, intensifying the aforementioned aptitude of the calculation technique. A hyperbolic decreasing seems slightly more precise in approximating the experiments than an exponential stress– strain relation. However, as the difference of the absolute fracture energy values between the two laws is small and the evolution tendency identical for both relations, the simpler exponential decreasing with one parameter less to calibrate is judged sufficient for fracture energy evaluation. An important finding of the investigation is a size effect in postcracking strength when increasing dimensions from small 4-point bending beam specimens up to plate elements with real dimensions. The observed decrease in fracture energy for large scale elements indicates the necessity to include a realistic geometry factor in the design procedure of large scale elements when using strength values from laboratory-scale investigations. For design purposes, it is recommended to apply such a geometry factor as a multiplier for postcracking strength obtained with small bending elements. 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