Size Effect of Synthetic Fibre Reinforced Concrete – Investigation using a Semi-Discrete Analytical Beam Model

The size effect of quasibrittle materials, such as concrete, is an important, and well-investigated area of interest. Since the specimen size of laboratory tests is always much smaller than that of real structures, it is essential to understand and handle the phenomena of size effect in order to gain reliable designing parameters. In the case of plain and reinforced concrete a great deal of research can be found in the literature in connection to the size effect (e.g. [1,2,3]). Fig. 1 shows the most widely used plot to present special aspects of the size effect of quasibrittle materials based on Bazant and Planas [1].

Size effect on the strength of concrete (adapted from Bazant and Planas [1])

The most accepted and simplest size effect law for concrete (Eq. 1) based on nonlinear fracture mechanics is derived by Bazant [2]. Where σ_N is the nominal stress, D is the characteristic dimension of the specimen, f_t is the tensile strength, B is a dimensionless constant and D₀ is a constant with the dimension of length). (1) $σ_{N} = \frac{B f_{t}}{\sqrt{1 + D / D_{0}}}$ {\sigma _N} = {{B{f_t}} \over {\sqrt {1 + D/{D_0}} }}

Although the use of fibre-reinforced concrete (FRC) is more and more popular, there are exiguous investigations on the size effect of FRC [4,5,6,7,8,9]. In fact, most of the existing model codes and guidelines of FRC constitutive laws (e.g. the fib Model Code [10] and the CNR [11]) do not take into account the size effect of FRC at all. Only the German code [12] and the Rilem recommendation [13]) introduced a size-dependent safety factor for steel fibre reinforced concrete to handle the reduction of the peak strength and also the residual stresses of the higher structural sized elements [14]. The common feature of the size effect investigations is handling FRC as a homogeneous material during the evaluation and modelling of experimental results. With this method, some papers [5–6] concluded that a smaller size effect can be obtained in the case of FRC compared to plain concrete, and Eq. 1 of Bazant can be used also for FRC material. In the case of synthetic fibre reinforcement, where the effect of the fibres appears after the first crack, it was stated by Vlietstra [8] that in the residual loadbearing capacity there is no effect of the structural size.

PROBLEM STATEMENT

According to the introduction, contradictory statements of the size-effect of FRC can be found in the literature, whether it should be taken into account during the design process, or it can be neglected. Any of these statements are proved satisfactorily. The reason for this is that measuring experimentally the effect of the specimen size on the mechanical behaviour of FRC precisely is not simple, as the dispersion of the experimental results is notoriously huge. For an adequate laboratory series of measurements, more specimens would be required than in the case of plain concrete, which, moreover, differs from the standard formwork. Because of the huge dispersion and the relatively few specimens, the commonly used evaluating methods (comparing the experimental force-crack opening or displacement curves) can lead to different conclusions even in the case of a repeated experiment under the same conditions. That is the reason for the lack in the literature of a common position in the case of the size effect of FRC.

Based on the observations of Vlietstra [8] it is probable that the size effect is the characteristic only of the concrete material, and the effect of the fibres can be handled separately. However, there is no evidence of this assumption until now. The commonly used analytical models handle FRC as a homogeneous material during the evaluation of experimental results and during the designing process. So, the material parameters of these models (e.g., [15,16,17,18]) depend on the combined behaviour of the concrete matrix and the fibres, so all of these parameters are necessarily dependent on the size effect. That is the reason, why these parameters are not applicable for verifying the above-mentioned assumption and for determining the share of the concrete matrix and the fibres in the size effect. A deeper understanding of the material behaviour is needed. In the following, authors examine the size effect of FRC separating the effect of the concrete matrix and the fibre reinforcement with the help of the semi-discrete analytical (SDA) beam model of Tóth and Pluzsik [19]. The SDA beam model examines FRC material at the meso level: the concrete material is handled as a homogeneous continuum, approximating its behaviour on the basis of fracture mechanics, and then the effect of the embedded fibres are built “one by one” into the model, handling them as discrete elements. Consequently, the SDA model contains new material parameters, which handle the effect of the concrete matrix and the fibres in the inhomogeneous cross-section separately. The dispersion of these parameters is smaller than that of the commonly used parameters of analytical models, so with their help, a relevant conclusion can be drawn in case of a standard number of experimental samples of concrete. The aim of this research is to prove that in the case of softening material the effect of the specimen size influences only the behaviour of concrete material, and so the peak strength, while the residual loadbearing capacity (which is mainly affected by the bridging effect of fibres) and the crack propagation are independent of the structural size.

TESTING METHOD

Based on the RILEM recommendation for the experimental procedure [20] twenty one test specimens were prepared. Table 1 contains the comparison of the performed experiment and the recommendation of RILEM (where l is the span, b is the width and D is the depth of the beam, a₀ is the notch depth; d_a is the maximum aggregate size and D_max and D_min are the maximal and minimal beam depth of the performed experiment). The notch depth to beam depth (a₀/D) ratio was smaller than the recommended value, while the maximal notch depth which could be performed was 25 mm. The notch at the bottom of the middle cross-section of the beams is to control the failure mode of the FRC beams: bending failure with a localized crack, started from the bottom cut. Although the notch depth was smaller than the recommendation, there was no disruption with the planned failure mode.

Table 1.

Comparison of the performed experiment and the recommendation of RILEM [20]

Proportions	RILEM recommendation	Performed experiment
l/D ratio	>2.5	3.33
a₀/D ratio	0.15<a₀/D<0.5	0.083
notch width	<0.5d_a (8 mm)	0.125 d_a (2 mm)
b	>3d_a (48 mm)	9.4 d_a (150 mm)
D_min	<5d_a (80 mm)	4.7d_a (75 mm)
D_max	>10d_a (160 mm)	18.75d_a (300 mm)
D_max/D_min	≥4	4

The test specimens were cast in three different sizes according to Table 2 and Fig. 2. Two plain concrete (PC) and five fibre reinforced concrete (FRC) beams were prepared in each size. The beam width was uniformly 150 mm, and all the beams had a bottom notch in the middle cross-section, with a depth of a₀. The strength class of concrete was C25/30 in all three cases. Table 3 contains the proportions of the concrete matrix. The FRC beams had 4 kg/m³ fibre reinforcement with 48 mm long synthetic polypropylene fibres (with a diameter of 0.72 mm).

Specimen dimensions and experimental layout

Table 2.

Dimensions of the test specimens

Specimen ID	Beam depth (D) [mm]	Span (l) [mm]	Notch depth (a₀) [mm]
FRC_S	75	250	6.25
FRC_M	150	500	12.5
FRC_L	300	1000	25

Table 3.

Proportions of concrete matrix

Component	Dosage [kg/m³]
Cement	350
Water	192
Aggregate 0/4	631
Aggregate 4/8	685
Aggregate 8/16	487
Superplasticizer	2

The concrete beams were examined by a three-point bending test according to the recommendations of RILEM TC 162-TDF [21] (Fig. 2). The measured values were the loading force (F), the deflection of the middle of the beam and the Crack Mouth Opening Displacement (CMOD). The speed of the movement-controlled loading was 0.5 mm/min. In the case of the largest beams (FRC_L), the testing machine could not measure the CMOD, therefore an alternative measuring method was necessary. Special brass measuring discs with two measuring points were fixed on the side of the beams, in case of the largest beams at four different heights, in case of the medium beams two, and in case of the smallest beams at only one height (at the bottom). The crack openings were measured manually on these discs with the help of a mechanical dial gauge crack width monitor (Fig. 3). Since the maximal displacement of the monitor is only 2.8 mm, first the two external points were used, as the crack opened an outer and an inner point, and finally, the two inner points were used for taking the crack opening data.

To make the results comparable, the above detailed CMOD measuring method was performed in the case of all beam sizes. However, in the case of the mediumsized beams (FRC_M), the CMOD gauge of the testing machine also measured the crack opening, so the two different methods could be compared. For example, in the case of specimen FRC_M_03 (Fig. 4), the CMOD was measured by the CMOD meter of the testing machine and the mechanical dial gauge crack width monitor, too. Since for the manual measurement (green line), the testing machine was stopped at regular intervals, in the experimental F-CMOD curve (measured by the testing machine – blue dashed line) relapses of the loading force can be seen.

Comparison of manual and machine CMOD measurement in the case of FRC_M_03

Before the first crack occurs and in the descending part of the F-CMOD curve the two different measuring method shows relatively high difference. The reason for this diversion is the different basis lengths of the two measurements: the manual measurement points were 60 mm apart, while the CMOD gauge of the testing machine measured directly at the edge of the notch. Therefore, the manually measured results were corrected by taking into account the elastic deformation of the 60 mm section. With this correction, the two F-CMOD curves fit perfectly.

After fracturing the number of fibres crossing the cracked cross-section was manually determined, divided the cross-sections into five horizontal zones (the bottom cut is the sixth zone) (Fig. 5). Table 5 contains the fibre distributions of the beams distinguishing between the two different failure modes. The appearance of the broken and pulled-out fibres is significantly different, the end of the pulled-out fibres remains intact, while the broken fibres disinte-grate into bundles of fibres (Fig. 6).

Cracked cross-sections divided into five zones

Appearance of pulled-out and broken fibres

Fig. 7 shows the experimental F-CMOD results of the three different sizes in the case of PC and FRC beams.

Experimental F-CMOD results of PC (left) and FRC (right) beams

EVALUATION OF THE RESULTS

4.1.

Description of the SDA model

The evaluation of the experimental results is performed with the help of the semi-discrete analytical (SDA) beam model of Tóth and Pluzsik [19]. The SDA model is not handling FRC as a homogeneous material, but as an inhomogeneous one which consists of fibres, concrete matrix and the bond between them. The effect of the fibres is modelled separately from the concrete matrix and is neglected before the first crack. The real fibre distribution (counted manually after fracture) of the cracked cross-section is taken into account. Fibres are modelled zone by zone (Fig. 5), using the pull-out model of Tóth et al. [22] based on Eq. (2–7), shown in Fig. 8. (Where τ is the bond strength between the fibre and concrete matrix, s is the relative displacement, ξ is the local coordinate, P is the axial force, F is the loading force, L_f is the embedded length of the fibre, A_f, d_f and E_f are the cross-sectional area, the diameter and the modulus of elasticity of the fibre respectively and u is the displacement of the fibre.) The maximum bond strength between the fibres and the concrete matrix (τ_max[N/mm²]) depends on the embedding matrix and the connection between the matrix and the fibre. It was shown in Tóth and Pluzsik [19] that τ_maxcharacterizes not only the single pull-out phenomenon but also the behaviour of the FRC material strengthened with the given fibres. According to Tóth and Pluzsik [19] τ_max, as the connection parameter of the SDA model can be used to evaluate test results. (2) $d P (ξ) = π d_{f} τ (ξ) d ξ, \frac{P (ξ)}{A_{f} E_{f}} = \frac{d s (ξ)}{d ξ}$ {\rm{d}}P\left( \xi \right) = \pi {d_f}\tau \left( \xi \right){\rm{d}}\xi ,\,{{P\left( \xi \right)} \over {{A_f}{E_f}}} = {{{\rm{d}}s\left( \xi \right)} \over {{\rm{d}}\xi }} Where: (3) $P (ξ = L_{f}) = F and s (ξ = L_{f}) = u$ P\left( {\xi = {L_f}} \right) = F\,{\rm{and}}\,s\left( {\xi = {L_f}} \right) = u (4) $τ (s) = {\begin{matrix} Gs, if s \leq s_{0}; \\ τ_{max}, if s_{0} < s \leq s_{1} and \\ (1 + \frac{s_{1} - s}{L_{f}}) [τ_{max} - (s - s_{1}) a], if s > s_{1} \end{matrix}$ \tau \left( s \right) = \left\{ {\matrix{ {Gs,\,\,\,\,\,{\rm{if}}\,s\, \le {s_0};} \cr {{\tau _{\max }},\,\,\,\,\,{\rm{if}}\,{s_0} < s \le {s_1}\,{\rm{and}}} \cr {\left( {1 + {{{s_1} - s} \over {{L_f}}}} \right)\left[ {{\tau _{\max }} - \left( {s - {s_1}} \right)a} \right],\,\,\,\,\,{\rm{if}}\,s > {s_1}} \cr } } \right. Where: (5) $s_{0} = \frac{τ_{max}}{G}$ {s_0} = {{{\tau _{{\rm{max\;}}}}} \over G} (6) $G = \frac{E_{c}}{d_{f} (1 + v_{m}) ln (R / r)}$ G = {{{E_{\rm{c}}}} \over {{d_f}\left( {1 + {v_m}} \right){\rm{\;ln\;}}\left( {R/r} \right)}} (7) $s_{1} = s_{0} + \frac{π d_{f} τ_{max} L_{f}^{2}}{2 A_{f} E_{f}} + \frac{F_{0} L_{f}}{A_{f} E_{f}}$ {s_1} = {s_0} + {{\pi {d_{\rm{f}}}{\tau _{{\rm{max\;}}}}L_f^2} \over {2{A_f}{E_f}}} + {{{F_0}{L_f}} \over {{A_f}{E_f}}}

Experimental and theoretical τ-s and F-u curves of the pulled-out fibres (Reprinted from Tóth and Pluzsik [19])

To take rupture failure mode into account a rupture condition is determined. The disruption of the fibres is related to the critical force (F_crit) of the fibres calculated by Eq. (8). (Where L_f is the embedded length, f_f is the tensile strength of the fibre and F_crit is the critical force of the fibre belonging to the rupture limit). (8) $F_{crit} = A_{f} f_{f}$ {F_{crit}} = {A_f}{f_f}

The model calculates if the force of the fibres of a given displacement with a given embedment length, is within the boundary condition or not. If the value of the fibre force is higher than F_crit the fibres tear in the model. In this case, after reaching the critical force, the force of the fibres of the zone drops to zero.

The use of the two different failure modes in the SDA model was verified based on experimental results: the ratio of broken fibres determined by the model was adjusted to the manually counted experimental results.

All the data defining the fibre (A_f, d_f, f_f and E_f) are given by the producer and are the same in the case of different mixtures of FRC. The run of the pull-out curve is determined by Eq. (2–8), so the only free parameter which can characterize the fibre-concrete matrix connection is the maximal bond strength (τ_max).

The softening curve of plain concrete is used (the softening curve does not contain the effect of fibres, the fibres are handled separately). The SDA model uses a linear stress – crack opening curve of plain concrete (Fig. 9) based on Eq. (9–13). (Where w_i is the width of the crack opening (Eq. 11) [13] in a given height (x_x,i) of the cross-section, x_w is the location of the CMOD meter (where CMOD is the Crack Mouth Opening Displacement, the opening on the surface of the crack, measured as the difference between the original opening and the final opening distance), x_c and x_t are the compressed and linearly tensiled concrete zones (Fig. 10).) The slope of the softening curve (q [N/mm³]) is the material parameter of the SDA model. (9) $σ (w_{i}) = f_{t} - {qw}_{i}$ \sigma \left( {{w_i}} \right) = {f_t} - q{w_i} (10) $σ (x_{x, i}) = f_{t} - q (2 \frac{1}{r} x_{x, i} x_{w})$ \sigma \left( {{x_{x,i}}} \right) = {f_t} - q\left( {2{1 \over r}{x_{x,i}}{x_w}} \right) (11) $x_{x, i} = t_{i} - x_{c} - x_{t}$ {x_{x,i}} = {t_i} - {x_c} - {x_t} (12) $w_{i} = 2 \frac{1}{r} x_{x, i} x_{w}$ {w_i} = 2{1 \over r}{x_{x,i}}{x_w} (13) $x_{w} = h + 25 - x_{c} - x_{t} + 2$ {x_w} = h + 25 - {x_c} - {x_t} + 2

Softening curve of plain concrete (Reprinted from Tóth and Pluzsik [23])

The model first calculates the moment (M) – curvature (1/r) connection based on Eq. (14–16), shown in Fig. 10: while the deflections are small, the FRC beam behaves in a linearly elastic manner: the beam is uncracked, the effect of the fibres is neglected, and the moment is calculated by Eq. (14). When the first crack occurs and the middle crack extends more and more fibres are affected. In the calculation of the fibre forces the displacement (u) is admitted to be equal to the width of the crack opening (w_i) of the given zone, and the embedment length of the fibres is assumed to be evenly distributed under the value of 0.5L. Finally, the F-CMOD relationship is determined with the help of the plastic hinge model of Vandewalle et al. [18] based on Eq. (17–18). (14) $M = \frac{\frac{1}{r} E_{c} b h^{3}}{12}$ M = {{{1 \over r}{E_c}b{h^3}} \over {12}} (15) $\sum N = \frac{{bx}_{c}^{2} \frac{1}{r} E_{c}}{2} = \frac{{bx}_{t} f_{t}}{2} + \int_{0}^{h - x_{x} - x_{t}} σ (x_{x}) d x_{x} + \sum_{i = 1}^{5} F_{fiber, i} n_{i}$ \sum {N = {{bx_{\rm{c}}^2{1 \over r}{E_{\rm{c}}}} \over 2} = {{b{x_{\rm{t}}}{f_{\rm{t}}}} \over 2} + \int\limits_0^{h - {x_x} - {x_t}} {\sigma \left( {{x_x}} \right){\rm{d}}{x_x} + \sum\limits_{i = 1}^5 {{F_{fiber,i}}{n_i}} } } (16) $\begin{array}{l} M & = \frac{{bx}_{c}^{2} \frac{1}{r} E_{c}}{2} \frac{x_{c}}{3} - \frac{{bx}_{t} f_{t}}{2} (x_{c} + \frac{2 x_{t}}{3}) - \\ - \int_{0}^{h - x_{c} - x_{t}} σ (x_{x}) d x_{x} \cdot (\frac{\int_{0}^{h - x_{c} - x_{t}} x_{x} σ (x_{x}) d x_{x}}{\int_{0}^{h - x_{c} - x_{t}} σ (x_{x}) d x_{x}} + x_{c} \\ + x_{t}) - \sum_{i = 1}^{5} F_{fiber, i} t_{i} n_{i} \end{array}$ \matrix{ M \hfill & { = {{bx_{\rm{c}}^2{1 \over r}{E_{\rm{c}}}} \over 2}{{{x_c}} \over 3} - {{b{x_{\rm{t}}}{f_t}} \over 2}\left( {{x_{\rm{c}}} + {{2{x_{\rm{t}}}} \over 3}} \right) - } \hfill \cr {} \hfill & { - \int\limits_0^{h - {x_c} - {x_t}} {\sigma \left( {{x_x}} \right){\rm{d}}{x_x} \cdot ( {{{\mathop \smallint \nolimits_0^{h - {x_c} - {x_t}} {x_x}\sigma \left( {{x_x}} \right){\rm{d}}{x_x}} \over {\mathop \smallint \nolimits_0^{h - {x_c} - {x_t}} \sigma \left( {{x_x}} \right){\rm{d}}{x_x}}} + {x_c}} } } \hfill \cr {} \hfill & {\left. { +\, {x_t}} \right) - \sum\limits_{i = 1}^5 {{F_{fiber,i}}{t_i}{n_i}} } \hfill \cr } (17) $CMOD = 2 \frac{1}{r} x_{w}^{2}$ CMOD = 2{1 \over r}x_w^2 (18) $F = \frac{4 \times M}{l}$ F = {{4 \times M} \over l}

Stress distribution and distances along the cross-section (Reprinted from Tóth and Pluzsik [19])

A thorough description of the model can be found in Tóth and Pluzsik [19, 24]. As input parameters the model contains the geometrical and material properties of the fibre and the concrete. Besides these given parameters the model has two free parameters τ_max the maximum bond strength between the fibres and the concrete matrix and q the slope of the softening curve. These two parameters of the model (τ_max and q) are determined by inverse analysis from the experimental results and have moderate deviation since the dispersion arising from the diverse fibre distribution is eliminated by the SDA model (taking into account the real fibre distribution). The SDA model uses τ_max and q parameters to characterize the FRC material. Evaluating experimental results with the help of these two parameters could lead to stronger conclusions because during the evaluation parameters with smaller deviation are used than that of the current practice.

In the following, the size-dependent properties of FRC beams will be examined with the help of the SDA model. The SDA model separates the behaviour of the concrete material (characterized by q) and the fibres (characterized by τ_max), so the share of the concrete matrix and the fibre in the size-effect become separable.

4.2.

PC results

The experimental results of the plain concrete beams are shown in Fig. 6. It can be seen that the loadbearing capacity increases with the size of the beam. However, the nominal stress (σ_N) at the peak load (calculated by Eq. 19) decreases with the greater size (Table 4). (19) $σ_{N} = \frac{3 l F_{\max}}{2 b D^{2}}$ {\sigma _N} = {{3l{F_{max }}} \over {2b{D^2}}}

Table 4.

Nominal stresses and q values of PC specimens

Specimen ID	Beam depth (D) [mm]	Nominal stress (σ_N) [N/mm²]	Average of σ_N [N/mm²]	Deviation of σ_N [%]	Slope of the softening curve (q) [N/mm³]
PC_S_06	75	2.73	2.9	−5.9	15
PC_S_07	75	3.07	2.9	5.9	15
PC_M_06	150	2.84	2.81	1.2	15
PC_M_07	150	2.77	2.81	−1.2	15
PC_L_06	300	2.50	2.48	1	15
PC_L_07	300	2.45	2.48	−1	15

The resulting nominal forces were compared to Bazant's size effect law (Eq. 1) (Bazant 1984). Bf_t and D₀ were identified with the least squares method, receiving Bf_t as 3.13 N/mm² and D₀ as 513 mm. Fig. 11 shows the size effect curve of Bazant and the experimental stress (σ_N /Bf_t) – size (D/D₀) data using a logarithmic scale.

Size effect of Bazant compared to experimental results of PC

After verifying Bazant's size effect law for the PC results, the material parameter of the SDA model was determined for all the PC specimens. The slope of the softening curve (q in Fig. 9) was calculated by inverse analysis, making equal the area under the experimental and theoretical F-CMOD curves. For the calculation, the average nominal stress of each sample size was used, and the inaccuracy of the first part of the F–CMOD curves (which is unfortunately significant in the case of plain concrete) was taken into account. Table 4 contains the values of the q parameter in the case of the different beam sizes. Conspicuous, that the same q value can be used for all beam sizes, thereby proving, that the slope of the softening curve is independent of the structural size. Fig. 12 shows the experimental and theoretical F–CMOD curves of PC_M specimens.

Experimental and theoretical F-CMOD curves of PC_M (q = 15 N/mm³)

4.3.

FRC results

In the case of the test results of the FRC beams the same phenomenon can be observed as with the plain concrete: the nominal stress (σ_N) at the peak load decreases with the greater size (Table 5). The nominal forces of the FRC beams were also compared to Bazant's size effect law (Eq. 1). Bf_t and D₀ were identified with the least squares method, receiving Bf_t as 3.76 N/mm² and D₀ as 314 mm. Fig. 13 shows that the experimental results of the FRC beams fit perfectly to the size effect curve of Bazant for plain concrete, as the maximal loading force belonging to the first crack of the FRC beam (F_max) mainly depends on the behaviour of concrete.

Table 5.

Nominal stresses and τ_max values of FRC specimens

Specimen ID	Nominal stress (σ_N) [N/mm²]	Average of σN² [N/mm²]	τ_max [N/mm²]	Average of τ_max [N/mm²]	Deviation of τ_max [%]	Pulled-out fibres [pcs]	Broken fibres [pcs]	Rate of pulled-out fibres [%]
FRC_S_01	3.29	3.47	3.4	3.16	8%	47	7	87%
FRC_S_02	3.47		2.4		−24%	51	10	84%
FRC_S_03	3.34		3.0		−5%	52	10	84%
FRC_S_04	4.25		3.4		8%	30	22	58%
FRC_S_05	3		3.6		14%	39	13	75%
FRC_M_01	3.43	3.10	3.6	3.64	−1%	82	26	76%
FRC_M_02	2.67		3.6		−1%	87	26	77%
FRC_M_03	2.94		3.4		−7%	70	35	67%
FRC_M_04	3.18		3.6		−1%	83	42	66%
FRC_M_05	3.3		4.0		10%	108	32	77%
FRC_L_01	2.58	2.71	2.6	3.52	−26%	179	15	92%
FRC_L_02	2.83		3.6		−2%	179	20	90%
FRC_L_03	2.88		4.4		25%	119	35	77%
FRC_L_04	2.47		3.8		8%	160	36	82%
FRC_L_05	2.79		3.2		−9%	159	24	87%

Size effect of Bazant compared to experimental results of FRC

The FRC beams were also modelled by the SDA model, using the real fibre distribution of the specimens. Parameter q of the PC specimens of each sizes (given in Table 4) were used during the invers analysis of the FRC beams (q = 15 N/mm³). Table 5 contains the values of τ_max defined by making equal the theoretical and experimental F-CMOD curves of the FRC beams. In case of the S, M and L sized beams the averages of τ_max are 3.16; 3.64 and 3.52 N/mm², respectively. It can be seen that there is no declining trend in the average of τ_max with the increasing beam sizes, so it can be stated that the structural size does not influences the connection between the concrete matrix and the fibres.

The calculated and measured F-CMOD curves fitted well in all cases, as shown in Fig. 14, for example for FRC_S_04 and FRC_M_05.

Experimental and theoretical F-CMOD curves of FRC_S_04 (left: q = 15 N/mm³, τ_max = 3.4 N/mm²) and FRC_M_05 (right: q = 15 N/mm³, τ_max = 4 N/mm²)

Fig. 15 shows the residual nominal strength (σ_RN) depending on the beam depth (D). Since the curve is approximately horizontal, it can be concluded that the residual loadbearing capacity of FRC beams is independent of the structural size.

It has to be noted, that in the case of this experiment, the fibre length was smaller than the critical fibre length, so the fibres in the majority were pulled out (Column 8 of Table 5). L_crit is the theoretical limit of the fibre length where the fibres rather break instead of pulling out. Based on Eq. (20) the critical length of the polypropylene fibres with a 0.72 mm diameter, embedded in the given concrete matrix, is 85 mm, which is much higher than the actual length (48 mm) of the fibres. In the case of longer fibres (exceeding the critical fibre length), when the failure mode of the fibres is mainly the rupture, the independence of the fibre-matrix connection from the structural size is not confirmed by the above experimental investigation. However, as the failure of the fibres is a local effect (even in the case of the disruption of fibres), the size-independent behaviour seems to be justified in that case, too. (19) $L_{crit} = \frac{F_{\max} - F_{0}}{π d_{f} τ_{\max}}$ {L_{crit}} = {{{F_{{{max}}}}_{\rm{\;}} - {F_0}} \over {\pi {d_f}{\tau _{{{max}}}}}}

It has to be also noted, that in this investigation concrete reinforced with synthetic fibres was used which has softening material behaviour. In the case of hardening materials (e.g. steel fibre reinforced concrete) the peak load is affected by the fibres also, therefore further research is needed to understand the size effect behaviour of them.

Finally, it is worth noting, that all the beams were prepared under laboratory conditions, where the mixing of fibres into the concrete is carefully controlled. However, in the case of larger structures the clumping of the fibres (because of unequal mixing) can cause serious problems. Therefore, during the preparation of the fibre reinforced concrete structures special attention should be paid to the appropriate mixing of fibres, ensuring the even fibre distribution is taken into account in the design process.

CONCLUSION

The size effect of synthetic fibre-reinforced concrete beams was investigated with the help of the SDA model of Tóth and Pluzsik [19]. The SDA model handles separately the concrete material, the fibres and the bond between them. Investigation of the FRC beams by inverse analysis has shown, that the size effect significantly influences the nominal stress at the peak load due to the softening concrete material, but does not affect the residual loadbearing capacity which is influenced by the connection between the concrete matrix and the fibres (namely the pull-out behaviour of the fibres, represented by τ_max).

Before the FRC structure cracks the effect of synthetic fibres is negligible (for safety), so the size effect of concrete can be handled as in the case of plain concrete, which is a well-investigated field of interest [e.g. 1–3]. After the first crack occurs, the residual load-bearing capacity of the FRC beam is mainly given by the bridging effect of fibres, and the effect of the concrete can be negligible. Since the pullout behaviour of the fibres is a local phenomenon, it is not affected by the size effect (parameter τ_max is independent of the structural size). It was shown, that the residual load-bearing capacity of FRC beams is independent of the structural size, hereby confirming the assumption of [8] with the help of the SDA beam model.

According to the above statements, the fib Model Code [10] and the CNR [11] fail against safety neglecting the size effect of FRC, while the German code [12] and the Rilem recommendation [13] use unnecessarily the size-dependent safety factor also in the residual stresses of the higher structural sized FRC elements [14].

It was also proven by the evaluation of experimental results that the two parameters of the SDA model (τ_max and q) are independent of the structural size, so the model is safely applicable in the case of real-size beams.

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Size Effect of Synthetic Fibre Reinforced Concrete – Investigation using a Semi-Discrete Analytical Beam Model

Online veröffentlicht: 20. Okt. 2023

Seitenbereich: 117 - 129

Eingereicht: 12. Mai 2022

Akzeptiert: 17. Apr. 2023

DOI: https://doi.org/10.2478/acee-2023-0039

Schlüsselwörter
Analytical beam model, Fibre reinforced concrete, Size effect, Synthetic fibre reinforcement, Three-point bending test

© 2023 Mária Erdélyiné Tóth et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

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Size Effect of Synthetic Fibre Reinforced Concrete – Investigation using a Semi-Discrete Analytical Beam Model

Online veröffentlicht: 20. Okt. 2023

Seitenbereich: 117 - 129

Eingereicht: 12. Mai 2022

Akzeptiert: 17. Apr. 2023

DOI: https://doi.org/10.2478/acee-2023-0039

SchlüsselwörterAnalytical beam model, Fibre reinforced concrete, Size effect, Synthetic fibre reinforcement, Three-point bending test

© 2023 Mária Erdélyiné Tóth et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Figure 1.

Figure 2.

Figure 3.

Figure 4.

Figure 5.

Figure 6.

Figure 7.

Figure 8.

Figure 9.

Figure 10.

Figure 11.

Figure 12.

Figure 13.

Figure 14.

Figure 15.

Schlüsselwörter
Analytical beam model, Fibre reinforced concrete, Size effect, Synthetic fibre reinforcement, Three-point bending test