A flexural capacity model for fibre-reinforced concrete (FRC) beams reinforced with PVA and basalt fibres is suggested for the rectangular beam sections. The proposed models are based on the concept of equivalent stress block parameters for both compressive and tensile stresses, similar to Eurocode and ACI code. The parameters are defined by allowing the conversion of the stress-strain models into equivalent rectangular stress blocks, similar to Eurocode 2. The flexural model is suggested to determine the loading capacity of 21 FRC beams containing up to 3% volume fraction of PVA and basalt fibres without reinforcing bars. In order to investigate the accuracy of the proposed flexure models, finite element analysis (FEA) of the same beams was carried out using the compressive and tensile stress-strain curves. Furthermore, 21 FRC beams subjected to three-point bending were tested. The results of the flexural models showed good agreement with the load-carrying capacity of the tested FRC beams, and the results of FEA of all beams showed a good correlation with the experimental results in terms of the maximum load, load versus midspan deflection patterns, and the maximum tensile strains.
Fibre-reinforced concrete (FRC) is concrete containing fibres. FRC is a well-established material, which is commonly used in several civil engineering-related structures. Steel fibre is the most commonly studied, but due to some drawbacks of steel fibres such as cost [
The modelling of the compressive stress-strain behaviour of FRC containing PVA and basalt fibres is reported in [
In this paper, the analytical flexural models are proposed which are based on the compressive and tensile stress-strain laws derived from experimental results, similar to the methodology followed by Barros and Figueiras [
It was earlier stated that the proposed analytical flexural model is based on the compressive and tensile stress-strain laws derived from experimental results, similar to the methodology followed by Barros and Figueiras [
FRC mixes were made up of three different mix binders. The first mix binder type was made by utilising 100% cement volume. The remaining two types were blended cement system containing 10% volume of silica fume or metakaolin with 90% cement volume. For each of three binder mixes, 7 beams of the size 100 × 200 × 1500 mm were cast. The first beam (labelled as control beam) contained no fibres, while the remaining 6 beams were cast using FRC containing a volume fraction of 1%, 2%, and 3% PVA and basalt fibres. All 21 beams were tested to failure under three-point flexural loading. A complete detail of the mix design, investigation of the flexure test results of the beams, has been investigated by the authors and can be found in [
Table
Material and mixture composition used in the experimental program.
Materials | Series “P” | Series “S” | Series “M” | ||||||
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Ordinary Portland cement (OPC) | 450 kg/m3 | 405 kg/m3 | 405 kg/m3 | ||||||
Silica fume | — | 45 kg/m3 | — | ||||||
Metakaolin | — | — | 45 kg/m3 | ||||||
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Fine aggregate | 670 kg/m3 | ||||||||
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Coarse aggregates | 600 kg/m3 (aggregate size: <10 mm), 500 kg/m3 (aggregate size: 10 to 20 mm) | ||||||||
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Fibre content | 0 | 13 kg/m3 | 26 kg/m3 | 0 | 13 kg/m3 | 26 kg/m3 | 0 | 13 kg/m3 | 26 kg/m3 |
26 kg/m3 | 52 kg/m3 | 26 kg/m3 | 52 kg/m3 | 26 kg/m3 | 52 kg/m3 | ||||
39 kg/m3 | 78 kg/m3 | 39 kg/m3 | 78 kg/m3 | 39 kg/m3 | 78 kg/m3 | ||||
(PVA) | (Basalt) | (PVA) | (Basalt) | (PVA) | (Basalt) |
Ordinary Portland cement (OPC) was used in all mixes. Kuraray RF 4000 polyvinyl alcohol (PVA) fibres and chopped basalt fibres were used as micro reinforcements for the concrete. The physical properties of both fibres are given in [
The aim of uniaxial compression testing of FRC cylindrical specimens was to define a stress-strain law to simulate the complete compression behaviour of the FRC analysed. Therefore, 100 × 200 mm size cylindrical specimens were tested under deformation control condition. Testing detail and complete procedure of the compressive stress-strain modelling are given in [
Compression test results of plain concrete with and without fibres at 28 days.
Series | Cylinder IDs | Experimental results | Calculated results | Avg. | ||||||||
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“P” | P-0 | Cylinder 1 | 74.53 | 0.002372 | 0.012199 | 43,135 | 41,754 | 31,426 | 0.159 | 73.43 | — | |
Cylinder 2 | 72.34 | 0.002204 | 0.011287 | 42,709 | 42,301 | 32,829 | 0.149 | — | — | |||
PP-1 |
Cylinder 1 | 75.61 | 0.002687 | 0.022323 | 43,329 | 40,022 | 28,139 | 0.186 | 73.97 | 0.455 | ||
Cylinder 2 | 72.34 | 0.002646 | 0.021088 | 42,696 | 39,220 | 27,339 | 0.178 | — | 0.455 | |||
PP-2 | Cylinder 1 | 75.23 | 0.002936 | 0.024516 | 43,243 | 38,562 | 25,618 | 0.167 | 77.06 | 0.909 | ||
Cylinder 2 | 78.88 | 0.003078 | 0.019840 | 43,933 | 38,955 | 25,629 | 0.184 | — | 0.909 | |||
PP-3 | Cylinder 1 | 74.56 | 0.002998 | 0.022076 | 43,102 | 38,065 | 24,871 | 0.236 | 75.74 | 1.364 | ||
Cylinder 2 | 76.92 | 0.002826 | 0.022500 | 43,553 | 39,631 | 27,217 | 0.225 | — | 1.364 | |||
PB-1 | Cylinder 1 | 74.20 | 0.002335 | 0.019264 | 43,059 | 41,909 | 31,778 | 0.186 | 74.48 | 13.89 | ||
Cylinder 2 | 74.76 | 0.002406 | 0.019029 | 43,167 | 41,574 | 31,073 | 0.178 | — | 13.89 | |||
PB-2 | Cylinder 1 | 78.02 | 0.002469 | 0.019525 | 43,772 | 42,208 | 31,599 | 0.167 | 78.39 | 27.78 | ||
Cylinder 2 | 78.77 | 0.002544 | 0.016495 | 43,913 | 41,932 | 30,965 | 0.184 | — | 27.78 | |||
PB-3 | Cylinder 1 | 77.34 | 0.002560 | 0.014745 | 43,631 | 41,351 | 30,210 | 0.236 | 76.34 | 41.67 | ||
Cylinder 2 | 75.35 | 0.002412 | 0.016063 | 43,443 | 41,525 | 30,709 | 0.225 | — | 41.67 | |||
“S” | S-0 | Cylinder 1 | 82.33 | 0.002319 | 0.014198 | 44,590 | 44,850 | 35,500 | 0.174 | 81.17 | — | |
Cylinder 2 | 80.01 | 0.002376 | 0.022226 | 44,168 | 43,592 | 33,675 | 0.229 | — | — | |||
SP-1 | Cylinder 1 | 81.58 | 0.002600 | 0.019254 | 44,442 | 43,580 | 33,379 | 0.170 | 82.17 | 0.455 | ||
Cylinder 2 | 82.75 | 0.002811 | 0.019839 | 44,653 | 41,491 | 29,438 | 0.199 | — | 0.455 | |||
SP-2¥ | Cylinder 1 | 84.51 | 0.002660 | 0.017231 | 44,954 | 42,967 | 31,772 | 0.299 | 83.39 | 0.909 | ||
Cylinder 2 | 82.27 | 0.002351 | 0.019068 | 44,552 | 44,545 | 34,993 | 0.237 | — | 0.909 | |||
SP-3 | Cylinder 1 | 82.24 | 0.002693 | 0.020304 | 44,534 | 42,040 | 30,538 | 0.283 | 82.01 | 1.364 | ||
Cylinder 2 | 81.78 | 0.002521 | 0.017923 | 44,452 | 43,060 | 32,441 | 0.297 | — | 1.364 | |||
SB-1 | Cylinder 1 | 83.64 | 0.002464 | 0.022310 | 44,812 | 44,103 | 33,943 | 0.170 | 82.64 | 13.89 | ||
Cylinder 2 | 81.65 | 0.002430 | 0.01131 | 44,454 | 43,711 | 33,601 | 0.199 | — | 13.89 | |||
SB-2 | Cylinder 1 | 82.39 | 0.002571 | 0.022026 | 44,575 | 42,908 | 32,046 | 0.299 | 83.51 | 27.78 | ||
Cylinder 2 | 84.63 | 0.002633 | 0.019979 | 44,975 | 43,186 | 32,143 | 0.237 | — | 27.78 | |||
SB-3 | Cylinder 1 | 81.83 | 0.002672 | 0.015939 | 44,459 | 42,046 | 30,624 | 0.283 | 81.98 | 41.67 | ||
Cylinder 2 | 82.14 | 0.002727 | 0.018415 | 44,516 | 41,797 | 30,122 | 0.297 | — | 41.67 | |||
“M” | M-0 | Cylinder 1 | 85.23 | 0.002381 | 0.011832 | 45,109 | 45,308 | 35,798 | 0.149 | 84.85 | — | |
Cylinder 2 | 84.47 | 0.002316 | 0.017960 | 44,973 | 45,613 | 36,478 | 0.190 | — | — | |||
MP-1 | Cylinder 1 | 84.91 | 0.002619 | 0.012112 | 45,039 | 43,378 | 32,422 | 0.166 | 84.19 | 0.455 | ||
Cylinder 2 | 83.46 | 0.002400 | 0.012304 | 44,781 | 44,552 | 34,777 | 0.191 | — | 0.455 | |||
MP-2 | Cylinder 1 | 84.42 | 0.002654 | 0.018596 | 44,938 | 42,977 | 31,807 | 0.216 | 85.48 | 0.909 | ||
Cylinder 2 | 86.53 | 0.002678 | 0.017777 | 45,309 | 43,468 | 32,312 | 0.217 | — | 0.909 | |||
MP-3 | Cylinder 1 | 81.74 | 0.002711 | 0.015828 | 44,444 | 41,774 | 30,152 | 0.253 | 82.91 | 1.364 | ||
Cylinder 2 | 84.07 | 0.002778 | 0.017877 | 44,862 | 42,071 | 30,264 | 0.259 | — | 1.364 | |||
MB-1 | Cylinder 1 | 85.71 | 0.002552 | 0.017760 | 45,179 | 44,108 | 33,586 | 0.166 | 84.80 | 13.89 | ||
Cylinder 2 | 83.89 | 0.002437 | 0.014156 | 44,858 | 44,398 | 34,425 | 0.191 | — | 13.89 | |||
MB-2 | Cylinder 1 | 86.20 | 0.002530 | 0.017112 | 45,251 | 44,420 | 34,070 | 0.216 | 87.06 | 27.78 | ||
Cylinder 2 | 87.91 | 0.002607 | 0.018787 | 45,549 | 44,392 | 33,723 | 0.217 | — | 27.78 | |||
MB-3 | Cylinder 1 | 87.30 | 0.002535 | 0.015883 | 45,429 | 44,725 | 34,437 | 0.253 | 87.56 | 41.67 | ||
Cylinder 2 | 87.85 | 0.002760 | 0.012901 | 45,525 | 43,319 | 31,831 | 0.259 | — | 41.67 |
(a) Strain corresponding to peak stress,
For plain concrete labelled as “control” and FRC, two equations are proposed for the current experimental results of the control [
Description of the proposed compressive stress-strain model.
As there is no significant influence of fibre addition on the ascending branch of the compressive stress-strain curve, therefore, the following expression was found suitable for the current experimental data presented in Table
For
In (
For
For FRC containing PVA fibres, the suggested expression for descending branch
For FRC containing basalt fibres, the expression for descending branch
Note that in (
For FRC, the designed compression stress distribution is simplified using conversion factors “
Equivalent rectangular compressive stress block (adopted from ACI 363R-92 [
The coefficient
Estimated coefficients
Series | Cylinder IDs | Coefficients |
Coefficients |
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Cylinder 1 | Cylinder 2 | Cylinder 1 | Cylinder 2 | ||||||||||||||
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“P” | P-0 | 0.424 | 0.640 | 0.68 | 0.85 | 0.469 | 0.559 | 0.54 | 0.94 | 0.491 | 0.597 | 0.55 | 0.98 | 0.504 | 0.551 | 0.59 | 0.84 |
PP-1 | 0.373 | 0.699 | 0.84 | 0.75 | 0.377 | 0.671 | 0.80 | 0.75 | 0.436 | 0.646 | 0.67 | 0.87 | 0.440 | 0.639 | 0.65 | 0.88 | |
PP-2 | 0.359 | 0.713 | 0.89 | 0.72 | 0.348 | 0.698 | 0.90 | 0.70 | 0.399 | 0.718 | 0.81 | 0.80 | 0.393 | 0.696 | 0.80 | 0.79 | |
PP-3 | 0.348 | 0.668 | 0.86 | 0.70 | 0.368 | 0.689 | 0.84 | 0.74 | 0.427 | 0.627 | 0.66 | 0.85 | 0.400 | 0.716 | 0.81 | 0.80 | |
PB-1 | 0.433 | 0.63 | 0.65 | 0.87 | 0.416 | 0.656 | 0.67 | 0.83 | 0.447 | 0.648 | 0.65 | 0.89 | 0.470 | 0.641 | 0.58 | 0.94 | |
PB-2 | 0.391 | 0.72 | 0.83 | 0.78 | 0.392 | 0.702 | 0.76 | 0.78 | 0.470 | 0.626 | 0.63 | 0.94 | 0.440 | 0.688 | 0.66 | 0.88 | |
PB-3 | 0.390 | 0.68 | 0.78 | 0.78 | 0.417 | 0.628 | 0.64 | 0.83 | 0.423 | 0.747 | 0.75 | 0.85 | 0.651 | 0.673 | 0.44 | 1.30 | |
“S” | S-0 | 0.454 | 0.580 | 0.57 | 0.91 | 0.431 | 0.601 | 0.63 | 0.86 | 0.432 | 0.625 | 0.65 | 0.86 | 0.410 | 0.655 | 0.72 | 0.82 |
SP-1 | 0.385 | 0.679 | 0.79 | 0.77 | 0.365 | 0.682 | 0.84 | 0.73 | 0.419 | 0.687 | 0.74 | 0.84 | 0.444 | 0.648 | 0.66 | 0.89 | |
SP-2 | 0.374 | 0.719 | 0.86 | 0.75 | 0.382 | 0.697 | 0.82 | 0.76 | 0.369 | 0.762 | 0.93 | 0.74 | 0.406 | 0.686 | 0.76 | 0.81 | |
SP-3 | 0.375 | 0.683 | 0.82 | 0.75 | 0.388 | 0.690 | 0.80 | 0.78 | 0.449 | 0.703 | 0.70 | 0.90 | 0.410 | 0.699 | 0.77 | 0.82 | |
SB-1 | 0.412 | 0.64 | 0.70 | 0.82 | 0.435 | 0.575 | 0.56 | 0.87 | 0.399 | 0.679 | 0.68 | 0.80 | 0.398 | 0.655 | 0.70 | 0.80 | |
SB-2 | 0.382 | 0.72 | 0.84 | 0.76 | 0.390 | 0.653 | 0.71 | 0.78 | 0.329 | 0.776 | 0.78 | 0.66 | 0.350 | 0.595 | 0.72 | 0.70 | |
SB-3 | 0.376 | 0.69 | 0.82 | 0.75 | 0.370 | 0.727 | 0.83 | 0.74 | 0.367 | 0.774 | 0.77 | 0.73 | 0.375 | 0.669 | 0.76 | 0.75 | |
“M” | M-0 | 0.399 | 0.805 | 0.91 | 0.80 | 0.403 | 0.801 | 0.89 | 0.81 | 0.472 | 0.767 | 0.73 | 0.94 | 0.454 | 0.664 | 0.66 | 0.91 |
MP-1 | 0.378 | 0.716 | 0.85 | 0.76 | 0.404 | 0.650 | 0.72 | 0.81 | 0.518 | 0.771 | 0.67 | 1.04 | 0.439 | 0.691 | 0.71 | 0.88 | |
MP-2 | 0.373 | 0.690 | 0.83 | 0.75 | 0.373 | 0.705 | 0.85 | 0.75 | 0.373 | 0.729 | 0.88 | 0.75 | 0.412 | 0.729 | 0.80 | 0.82 | |
MP-3 | 0.371 | 0.718 | 0.87 | 0.74 | 0.363 | 0.712 | 0.88 | 0.73 | 0.386 | 0.746 | 0.87 | 0.77 | 0.409 | 0.717 | 0.79 | 0.82 | |
MB-1 | 0.384 | 0.72 | 0.84 | 0.77 | 0.417 | 0.635 | 0.65 | 0.83 | 0.323 | 0.704 | 0.70 | 0.65 | 0.372 | 0.558 | 0.64 | 0.74 | |
MB-2 | 0.402 | 0.65 | 0.73 | 0.80 | 0.385 | 0.656 | 0.72 | 0.77 | 0.448 | 0.678 | 0.68 | 0.90 | 0.330 | 0.558 | 0.72 | 0.66 | |
MB-3 | 0.387 | 0.71 | 0.82 | 0.77 | 0.37 | 0.693 | 0.80 | 0.74 | 0.410 | 0.712 | 0.71 | 0.82 | 0.335 | 0.641 | 0.81 | 0.67 |
The aim of the uniaxial tension testing of the FRC specimens was to define a stress-strain law to simulate the complete tensile behaviour. Therefore, 70 × 70 × 600 mm size notched specimens were tested under deformation control condition. The testing detail and the complete procedure of the tensile stress-strain modelling are given in [
Tension test results at 28 days.
Series | Specimen ID | Direct tension test result | |||||||
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At cracking | Postcracking | At cracking | Postcracking | ||||||
Load |
Elongation (mm) | Load |
Elongation (mm) | Stress |
Strain |
Stress |
Strain |
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“P” | P-0 | 26.30 | 0.02428 | — | — | 5.367 | 0.0001214 | — | — |
PP-1 | 26.44 | 0.02321 | 10.617 | 0.0363 | 5.396 | 0.0001160 | 2.167 | 0.0001817 | |
PP-2 | 30.91 | 0.02483 | 17.442 | 0.1015 | 6.308 | 0.0001242 | 3.559 | 0.0005077 | |
PP-3 | 26.63 | 0.02539 | 21.677 | 0.1009 | 5.436 | 0.001269 | 4.424 | 0.0005047 | |
PB-1 | 27.27 | 0.02428 | — | — | 5.564 | 0.0001214 | — | — | |
PB-2 | 31.23 | 0.02578 | — | — | 6.374 | 0.0001289 | — | — | |
PB-3 | 27.95 | 0.03046 | — | — | 5.703 | 0.0001523 | — | — | |
“S” | S-0 | 27.02 | 0.02482 | — | — | 5.515 | 0.0001241 | — | — |
SP-1 | 29.15 | 0.02201 | 10.72 | 0.04157 | 5.949 | 0.0001100 | 2.188 | 0.0002078 | |
SP-2 | 30.73 | 0.03151 | 15.672 | 0.08832 | 6.271 | 0.0001576 | 3.198 | 0.0004416 | |
SP-3 | 31.24 | 0.03703 | 18.623 | 0.08505 | 6.375 | 0.0001851 | 3.801 | 0.0004253 | |
SB-1 | 29.76 | 0.030212 | — | — | 6.073 | 0.0001511 | — | — | |
SB-2 | 31.37 | 0.036938 | — | — | 6.402 | 0.0001847 | — | — | |
SB-3 | 28.80 | 0.024795 | — | — | 5.879 | 0.0001240 | — | — | |
“M” | M-0 | 30.24 | 0.02505 | — | — | 6.172 | 0.0001252 | — | — |
MP-1 | 31.35 | 0.02745 | 11.928 | 0.04299 | 6.399 | 0.0001373 | 2.434 | 0.0002150 | |
MP-2 | 32.08 | 0.03542 | 17.861 | 0.04268 | 6.547 | 0.0001771 | 3.645 | 0.0002134 | |
MP-3 | 29.84 | 0.03377 | 19.28 | 0.06781 | 6.089 | 0.0001688 | 3.935 | 0.0003390 | |
MB-1 | 29.16 | 0.02525 | — | — | 5.952 | 0.0001262 | — | — | |
MB-2 | 33.53 | 0.03408 | — | — | 6.843 | 0.0001704 | — | — | |
MB-3 | 34.68 | 0.03097 | — | — | 7.078 | 0.0001549 | — | — |
For the prediction of the tensile stress-strain
General response of concrete under direct tension [
Considering the idealized stress-strain behaviour shown in Figure
In (
In (
For the postcracking behaviour, the mathematical model can be found in [
In general, a beam element subjected to flexural loads may experience three basic stages of stress states, as shown in Figure Stage I: linear elastic stress-strain relationship in both compression and tension, that is, stress state before the material cracks in tension Stage II: linear elastic stress-strain relationship in compression while tension is in inelastic distribution, that is, the stress state after the material has reached cracking strain before compression strain (corresponding to inelastic stress) is reached Stage III: inelastic stress distribution for both compression and tension stress-strain fields, that is, the stress state after the material has cracked in tension and crushed in compression
Three stages of stress-strain response in flexure. (a) Stage I: elastic behaviour in compression and tension. (b) Stage II: elastic behaviour in compression and plastic behaviour in tension (cracking stage). (c) Stage III: nonlinear behaviour in compression and plastic behaviour in tension (ultimate stage).
The elastic stage and cracking stage models are bit similar. The difference between these two stages is that concrete tensile strain
At first, the results were estimated for the cracking stage or Stage II condition (refer to Figure
In (
Using (
In (
Using the approach similar to Eurocode 2 [
In (
After determining the neutral axis depth, the next step was to determine the tension force
Using tension force
In (
The results of the ultimate loads were found to be close to the numerical results. Thus, proposed rectangular stress block mode concept can be used for the practical designing of the structural member made of FRC.
The flexure testing of FRC reinforced with PVA and basalt fibres is shown in Figure
Typical loading arrangement of beams [
Comparison of the predicted and experimental results in flexure (at cracking).
Specimen IDs | Experimental results | Beam cross section | Beam span ( |
Predicted results at cracking | Predicted/experimental flexural load | Proposed stress distribution at cracking | |||||||
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Cracking load (kN) | Deflection (mm) | Strain at cracking | Width |
Height |
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Area of tension zone mm2) |
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Lever arm (mm) | Flexural load at cracking (kN) | ||||
P-0 | 10.45 | 0.227 | 0.00001216 | 100 | 200 | 1400 | 5.103 | 100×100 | 25.515 | 133.33 | 9.72 | 0.93 |
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PP-1 | 11.68 | 0.262 | 0.00000877 | 100 | 200 | 1400 | 5.687 | 100×100 | 28.435 | 133.33 | 10.83 | 0.93 | |
PP-2 | 12.10 | 0.289 | 0.00000075 | 100 | 200 | 1400 | 5.689 | 100×100 | 28.445 | 133.33 | 10.84 | 0.90 | |
PP-3 | 10.88 | 0.282 | 0.00001581 | 100 | 200 | 1400 | 5.301 | 100×100 | 26.505 | 133.33 | 10.10 | 0.93 | |
PB-1 | 10.70 | 0.221 | 0.00001900 | 100 | 200 | 1400 | 5.359 | 100×100 | 26.795 | 133.33 | 10.21 | 0.95 | |
PB-2 | 12.44 | 0.272 | 0.00000152 | 100 | 200 | 1400 | 5.390 | 100×100 | 26.95 | 133.33 | 10.27 | 0.83 | |
PB-3 | 10.99 | 0.230 | 0.00000023 | 100 | 200 | 1400 | 5.201 | 100×100 | 26.005 | 133.33 | 9.91 | 0.90 | |
S-0 | 10.99 | 0.232 | 0.00001114 | 100 | 200 | 1400 | 5.488 | 100×100 | 27.44 | 133.33 | 10.45 | 0.95 | |
SP-1 | 11.82 | 0.255 | 0.00000102 | 100 | 200 | 1400 | 6.062 | 100×100 | 30.31 | 133.33 | 11.55 | 0.98 | |
SP-2 | 12.459 | 0.263 | 0.000001416 | 100 | 200 | 1400 | 6.064 | 100×100 | 30.32 | 133.33 | 11.55 | 0.93 | |
SP-3 | 11.462 | 0.254 | 0.000014 | 100 | 200 | 1400 | 5.658 | 100×100 | 28.29 | 133.33 | 10.78 | 0.94 | |
SB-1 | 11.93 | 0.243 | 0.00000710 | 100 | 200 | 1400 | 5.746 | 100×100 | 28.73 | 133.33 | 10.94 | 0.92 | |
SB-2 | 12.53 | 0.283 | 0.00000237 | 100 | 200 | 1400 | 5.756 | 100×100 | 28.78 | 133.33 | 10.96 | 0.87 | |
SB-3 | 11.43 | 0.271 | 0.000002083 | 100 | 200 | 1400 | 5.559 | 100×100 | 27.795 | 133.33 | 10.59 | 0.93 | |
M-0 | 11.833 | 0.241 | 0.000000416 | 100 | 200 | 1400 | 5.963 | 100×100 | 29.815 | 133.33 | 11.36 | 0.96 | |
MP-1 | 12.08 | 0.230 | 0.00001316 | 100 | 200 | 1400 | 6.265 | 100×100 | 31.325 | 133.33 | 11.93 | 0.99 | |
MP-2 | 12.804 | 0.250 | 0.0000129 | 100 | 200 | 1400 | 6.267 | 100×100 | 31.335 | 133.33 | 11.94 | 0.93 | |
MP-3 | 11.61 | 0.243 | 0.00001007 | 100 | 200 | 1400 | 5.851 | 100×100 | 29.255 | 133.33 | 11.14 | 0.96 | |
MB-1 | 12.31 | 0.260 | 0.00000403 | 100 | 200 | 1400 | 5.947 | 100×100 | 29.735 | 133.33 | 11.33 | 0.92 | |
MB-2 | 14.134 | 0.292 | 0.00000533 | 100 | 200 | 1400 | 5.961 | 100×100 | 29.805 | 133.33 | 11.35 | 0.80 | |
MB-3 | 14.557 | 0.300 | 0.000003728 | 100 | 200 | 1400 | 5.747 | 100×100 | 28.735 | 133.33 | 10.95 | 0.75 |
The flexure strength of FRC beams reinforced with PVA and basalt fibres (without reinforcing bars) was estimated for cracking stage (Stage II) using flexural model shown in Figure
The load-deflection response of all simply supported FRC beams is discussed in detailed by the authors in [
Three material types were defined as follows: Concrete material (for plain and FRC) Steel (to model loading rod and supports) Fibre reinforcement (for FRC only)
For plain and FRC beams, material properties were essentially required. In order to define plain and FRC material, the results of the material properties obtained from the compressive and tensile stress-strain curve models were used as input parameters for the FEA. The summary of input parameters is given in Table
Summary of the initial material input parameters, FEA parameters, and results.
Series | Beam IDs | Cylinder(a) compressive strength |
Trial parameters | FE results for the monitoring points | ||||||
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Elastic(b) modulus |
Tensile(c) strength ( |
Fracture(d) energy |
Prescribed(e) displacement (mm) | 1 | 2 | 3 | 4 | |||
Load (kN) | Deflection (mm) | Maximum(f) strain at bottom (mm/mm) | Maximum strain at top (mm/mm) | |||||||
“P” | P-0 | 73.43 | 43.00 | 3.62 | 0.11 | 0.01 | 10.416 | 0.2199 | 0.0001238 | 0.00009936 |
PP-1 | 73.97 | 38.00 | 4.68 | 0.0068 | 0.01 | 11.968 | 0.2599 | 0.0001151 | 0.0001189 | |
PP-2 | 77.06 | 36.00 | 4.908 | 0.0163 | 0.01 | 12.048 | 0.2899 | 0.0001239 | 0.0001284 | |
PP-3 | 75.74 | 32.50 | 3.8 | 0.09 | 0.01 | 10.264 | 0.275 | 0.0001693 | 0.0001266 | |
PB-1 | 74.48 | 41.00 | 4.45 | 0.061 | 0.01 | 10.632 | 0.2199 | 0.00009742 | 0.0001001 | |
PB-2 | 78.39 | 40.00 | 4.58 | 0.125 | 0.01 | 12.352 | 0.2699 | 0.0001582 | 0.0001227 | |
PB-3 | 76.34 | 41.70 | 4.26 | 0.0853 | 0.01 | 10.982 | 0.2299 | 0.0001326 | 0.0001043 | |
“S” | S-0 | 81.17 | 42.50 | 4.80 | 0.06 | 0.01 | 11.512 | 0.2299 | 0.0001019 | 0.0001045 |
SP-1 | 82.17 | 39.2 | 4.75 | 0.0193 | 0.01 | 11.594 | 0.2499 | 0.0001107 | 0.0001141 | |
SP-2 | 83.39 | 38.3 | 4.82 | 0.0693 | 0.01 | 11.788 | 0.2599 | 0.0001151 | 0.0001188 | |
SP-3 | 82.01 | 42.00 | 3.5 | 0.15 | 0.01 | 11.214 | 0.2499 | 0.0001119 | 0.0001666 | |
SB-1 | 82.64 | 41.00 | 4.71 | 0.009 | 0.01 | 11.85 | 0.2599 | 0.0001573 | 0.0001181 | |
SB-2 | 83.51 | 38.00 | 4.60 | 0.115 | 0.01 | 12.168 | 0.2799 | 0.0001660 | 0.0001276 | |
SB-3 | 81.98 | 37.00 | 4.55 | 0.09 | 0.01 | 11.420 | 0.2699 | 0.0001615 | 0.0001232 | |
“M” | M-0 | 84.85 | 41.80 | 4.95 | 0.04 | 0.01 | 11.820 | 0.2399 | 0.0001064 | 0.0001091 |
MP-1 | 84.19 | 44.00 | 4.90 | 0.064 | 0.01 | 11.906 | 0.2299 | 0.0001020 | 0.0001043 | |
MP-2 | 85.48 | 43.00 | 5.25 | 0.06 | 0.01 | 12.684 | 0.2499 | 0.0001108 | 0.0001135 | |
MP-3 | 82.91 | 41.25 | 3.60 | 0.1 | 0.01 | 10.474 | 0.2399 | 0.0001352 | 0.00001032 | |
MB-1 | 84.80 | 40.00 | 5.15 | 0.04 | 0.01 | 12.268 | 0.2599 | 0.0001152 | 0.0001185 | |
MB-2 | 87.06 | 43.00 | 5.72 | 0.112 | 0.01 | 14.11 | 0.2899 | 0.0001816 | 0.0001312 | |
MB-3 | 87.56 | 40.50 | 5.98 | 0.05 | 0.01 | 14.326 | 0.2999 | 0.0001329 | 0.0001367 |
The beams were modelled as a three dimensional solid object characterized by a material model “SBETA” by defining the user-defined material properties and material laws proposed by the researchers earlier [
It is to be noted that the aim of this investigation was to assess the accuracy of the predictive models as illustrated earlier for the compressive and tensile stress-strain behaviour of FRC containing PVA and basalt fibres in predicting the flexural response of plain and FRC beams using finite element analysis approach, which will be assessed by comparing with the experimental results of the flexural behaviour of 21 FRC beams.
All beams were modelled in full-scale mode. The dimensions of the modelled beam were 100 × 200 × 1500 mm. It was assumed that FRC beams were simply supported at the bottom on semicircular steel plates, and the displacement in the direction of the beam length was set as zero (refer to Figure
Finite element meshes and monitoring point definition.
Typical crack pattern at beam failure.
The support restrained was modelled as typical roller and hinged; that is, the movement of modelled beam was vertically and horizontally restrained. The material properties given in Table The first monitoring point was located at the middle of the beam at the top of the loading rod to monitor the maximum load attained by the beam before failure. The second monitoring point was located at the middle of the beam near its bottom surface, where largest vertical displacements can be expected. The deflections will be monitored at this location. The third and fourth monitoring points were located at 30 mm on the right and left side of the middle of the beam near its bottom surface to monitor the lateral displacement (similar to the tested beam given in Nasir et al. 2016). The reason for selecting 30 mm was the length of strain gauges used to measure the strain at the middle of the bottom face. The actual length of the strain gauges was 60 mm. Therefore, the results of strain in the FE analysis were calculated by placing 2 monitoring points for displacements 60 mm apart, by subtracting their values and dividing by 0.06 m.
For the fracture mechanism, smeared crack approach was adopted. A typical critical stress location failure pattern in the analysed beam, representative of all FRC beams, is shown in Figure Maximum load (monitoring point 1) versus deflection (monitoring point 2) Failure mode (or crack pattern at failure) and the post-peak response of the concrete in the presence and absence of the fibres Lateral displacement between monitoring points 3 and 4, which were 60 mm apart. By dividing lateral displacement as given by ATENA 3D with 60 mm, strain was estimated corresponding to the maximum load versus deflection
The comparison of the 3D nonlinear finite element analysis results with the experimental results is presented in the forthcoming section; however, the FE analysis results and final parameters to obtain these results are summarised in Table
The comparison of the finite element analysis (FEA) results with the experimental and mathematical results is presented in Table
Comparison of finite element (FE) analysis and experimental results.
Series | Beam IDs | Finite element analysis (FEA) results | Experimental results | Mathematical model results | Experimental/FEA results | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Load (kN) | Deflection (mm) | Tensile strain (mm/mm) | Load (kN) | Deflection (mm) | Tensile strain (mm/mm) | Load (kN) | Load (kN) | Deflection (mm) | Tensile strain (mm/mm) | ||
“P” | P-0 | 10.416 | 0.2199 | 0.0001238 | 10.45 | 0.227 | 0.0001310 | 9.72 | 0.9967 | 0.9687 | 0.9450 |
PP-1 | 11.67 | 0.2599 | 0.0001151 | 11.68 | 0.262 | 0.0001243 | 10.83 | 0.9991 | 0.9920 | 0.9260 | |
PP-2 | 11.954 | 0.2799 | 0.0001239 | 12.10 | 0.289 | 0.0001286 | 10.84 | 0.9879 | 0.9685 | 0.9635 | |
PP-3 | 10.264 | 0.275 | 0.0001693 | 10.88 | 0.282 | 0.0001708 | 10.10 | 0.9434 | 0.9752 | 0.9912 | |
PB-1 | 10.632 | 0.2199 | 0.0000974 | 10.70 | 0.221 | 0.0001034 | 10.21 | 0.9936 | 0.9950 | 0.9422 | |
PB-2 | 12.352 | 0.2699 | 0.0001582 | 12.44 | 0.272 | 0.0001590 | 10.27 | 0.9929 | 0.9923 | 0.9950 | |
PB-3 | 10.982 | 0.2299 | 0.0001326 | 10.99 | 0.230 | 0.0001326 | 9.91 | 0.9993 | 0.9996 | 1.0000 | |
“S” | S-0 | 10.85 | 0.2299 | 0.0001019 | 10.99 | 0.232 | 0.0001020 | 10.45 | 0.9873 | 0.9909 | 0.9990 |
SP-1 | 11.594 | 0.2499 | 0.0001107 | 11.82 | 0.255 | 0.0001297 | 11.55 | 0.9809 | 0.9800 | 0.8535 | |
SP-2 | 11.788 | 0.2599 | 0.0001151 | 12.459 | 0.263 | 0.0001468 | 11.55 | 0.9461 | 0.9882 | 0.7841 | |
SP-3 | 11.214 | 0.2499 | 0.0001119 | 11.462 | 0.254 | 0.0001736 | 10.78 | 0.9784 | 0.9839 | 0.6446 | |
SB-1 | 11.60 | 0.2399 | 0.0001063 | 11.93 | 0.243 | 0.0001610 | 10.94 | 0.9723 | 0.9872 | 0.6602 | |
SB-2 | 12.168 | 0.2799 | 0.0001660 | 12.19 | 0.283 | 0.0001699 | 10.96 | 0.9982 | 0.9890 | 0.9770 | |
SB-3 | 11.420 | 0.2699 | 0.0001615 | 11.43 | 0.271 | 0.0001666 | 10.59 | 0.9991 | 0.9959 | 0.9694 | |
“M” | M-0 | 11.820 | 0.2399 | 0.0001064 | 11.833 | 0.241 | 0.0001258 | 11.36 | 0.9989 | 0.9954 | 0.8458 |
MP-1 | 11.906 | 0.2299 | 0.0001020 | 12.08 | 0.230 | 0.0001431 | 11.93 | 0.9856 | 0.9996 | 0.7128 | |
MP-2 | 12.684 | 0.2499 | 0.0001108 | 12.804 | 0.250 | 0.0001838 | 11.94 | 0.9906 | 0.9996 | 0.6028 | |
MP-3 | 10.474 | 0.2399 | 0.0001108 | 11.06 | 0.243 | 0.0001737 | 11.14 | 0.8882 | 0.9461 | 0.9505 | |
MB-1 | 12.268 | 0.2599 | 0.0001152 | 12.31 | 0.260 | 0.0001549 | 11.33 | 0.9966 | 0.9996 | 0.7437 | |
MB-2 | 14.11 | 0.2899 | 0.0001816 | 14.134 | 0.292 | 0.0001831 | 11.35 | 0.9983 | 0.9928 | 0.9918 | |
MB-3 | 14.326 | 0.2999 | 0.0001329 | 14.557 | 0.300 | 0.0001528 | 10.95 | 0.9841 | 0.9997 | 0.8698 |
The comparison of load-deflection curves and strain profile, obtained from the ATENA 3D program as a result of finite element analysis, with the experimental results is shown and discussed in the forthcoming section.
Figure
Experimental versus simulation load-deflection curve for series “P.” (a) Beam “P-0” (without fibres). (b) Beam “PP-1” (1% PVA fibres). (c) Beam “PB-1” (1% basalt fibres). (d) Beam “PP-2” (2% PVA fibres). (e) Beam “PB-2” (2% basalt fibres). (f) Beam “PP-3” (3% PVA fibres). (g) Beam “PB-3” (3% basalt fibres).
Experimental versus simulation load-deflection curve for series “S.” (a) Beam “S-0” (without fibres). (b) Beam “SP-1” (1% PVA fibres). (c) Beam “SB-1” (1% basalt fibres). (d) Beam “SP-2” (2% PVA fibres). (e) Beam “SB-2” (2% basalt fibres). (f) Beam “SP-3” (3% PVA fibres). (g) Beam “SB-3” (3% basalt fibres).
Experimental versus simulation load-deflection curve for series “M.” (a) Beam “S-0” (without fibres). (b) Beam “SP-1” (1% PVA fibres). (c) Beam “SB-1” (1% basalt fibres). (d) Beam “SP-2” (2% PVA fibres). (e) Beam “SB-2” (2% basalt fibres). (f) Beam “SP-3” (3% PVA fibres). (g) Beam “SB-3” (3% basalt fibres).
The goodness of fit of the load-deflection curves shown in Figures
In the present paper, a flexural capacity model based on traditional equivalent rectangular stress block concept for FRC beams reinforced with PVA and basalt fibres is assessed through the experimental results of 21 beams. The model parameters for both compressive and tensile stresses were defined by allowing the conversion of the stress-strain curve models, which were suggested by the authors in earlier studies, into equivalent rectangular stress blocks, similar to Eurocode 2 and ACI 318-08.
Furthermore, the successful applicability and suitability of the analytical compressive and tensile stress-strain curve models of PVA and basalt fibre-reinforced concrete have been investigated through FEA approach in 3D nonlinear finite element analysis program ATENA 3D. The results of FEA showed approximately similar stiffness as exhibited by the tested FRC beams reinforced with 0 to 3% of PVA and basalt fibres. The results of the parameters were also close to the one obtained as a result of experimental testing.
The comparison of the results of load-deflection curves, strain profiles, and failure modes of FRC beams obtained as a result of finite element results using “ATENA 3D program” showed close agreement with the experimental results.
Initial modulus of elasticity calculated from compressive stress-strain curve
Secant modulus of elasticity calculated from compressive stress-strain curve
Modulus of elasticity of FRC
Modulus of elasticity of the composite
Modulus of elasticity of the concrete matrix without fibre
Modulus of elasticity of FRC
Secant modulus of elasticity
Modulus of elasticity of fibre
Strain
Strains corresponding to
Compressive strain corresponding to the maximum stress
Strain corresponding to the effective peak compressive strength of concrete,
Compressive strain corresponding to the limiting stress
Concrete cracking strain
Ultimate compressive strain
Ultimate tensile strain corresponding to
Ultimate compressive strain assumed as 0.0035
Flexural load at ultimate
Flexural load at cracking
Maximum compressive stress of plain concrete
Maximum compressive stress of FRC
Maximum tensile stress of plain concrete
Maximum tensile strength/stress of FRC
Tensile stress after cracking
Flexural capacity or moment capacity of FRC at stress
Material parameter in compressive stress-strain curve modelling
Load corresponding to cracking
Root mean square errors
Absolute fraction of variance
Volume fraction of the fibres
Material parameter used for predicting tensile stress-strain curves.
The authors declare that there are no conflicts of interest regarding the publication of this paper.