Continuous beams are often used within RC structures, which are exposed to aggressive environmental impact. The use of the fiber-reinforced polymer (FRP) reinforcement in these objects and environments has a big significance, taking into account tendency of steel reinforcement to corrode. The main aim of these research studies is to estimate ability of continuous beams with glass FRP (GFRP) reinforcement to redistribute internal forces, as a certain way of ductility and desirable behaviour of RC structures. This paper gives the results of experimental research of seven continuous beams, over two spans of 1850 mm length, cross-section of 150 × 250 mm, that are imposed to concentrated forces in the middle of spans until failure. Six beams were reinforced with different longitudinal GFRP and same transverse GFRP reinforcements, and one steel-reinforced beam was adopted as a control beam. The main varied parameters represent the type of GFRP reinforcement and ratio of longitudinal reinforcement at the midspan and at the middle support, i.e., design moment redistribution. The results of the research have shown that moment redistribution in continuous beams of GFRP reinforcement is possible, without decreasing the load-carrying capacity, compared to elastic analysis. The test results have also been compared to current code provisions, and they have shown that the American Concrete Institute (ACI) 440.1R-15 well predicted the failure load for continuous beams with GFRP reinforcement. On the contrary, current design codes underestimate deflection of continuous beams with GFRP reinforcement, especially for higher load levels. Consequently, a modified model for calculation of deflection is proposed.
For RC structures, elements reinforced with steel reinforcement are still used nowadays. As preventing of steel reinforcement to corrode in RC structures could be expensive and very often without significant effects, FRP internal reinforcement is lately used as a replacement of steel reinforcement in RC structures, especially in aggressive environments. Nowadays, there is a significant number of structures such as garages, bridges, retaining walls, reservoirs, and marine objects, within which FRP reinforcement is successfully applied at RC structural elements. Continuous concrete beams are commonly used in some of these structures, especially in bridges, overpasses, marine structures, and parking garages. Additionally, continuous beams with FRP reinforcement can also find their application in facilities with magnetic scanning equipment, laboratories, airport towers, and MRI rooms in hospitals and other facilities with equipment requiring electrical and magnetic neutrality, where the presence of steel reinforcement can have an adverse effect on the usability of devices in these facilities.
Due to different mechanical and deformation characteristics of FRP reinforcement, as high tensile strength and low modulus of elasticity, the behavior of RC elements is considerably different compared to RC elements with steel reinforcement. Concerning the fact that FRP reinforcement demonstrates linear elastic behavior up to failure, meaning demonstrating lack of material nonlinearity, there is a question of ability of this material, in conjunction with concrete, to realize load redistribution in statically indeterminate structures [
So far, thorough theoretical and experimental research studies have been carried out on simple supported beams with FRP reinforcement in order to evaluate behavior regarding failure modes, load-carrying capacity, deflection, and cracks [
Certain experimental and theoretical research studies were also carried out on continuous beams with FRP reinforcement [
The approach that in continuous beams reinforced with FRP reinforcement, moment redistribution in critical sections is not allowed could be conservative [
After a literature review, it is concluded that, in very few number of experimental research studies on continuous beams with longitudinal FRP reinforcement, FRP reinforcement for stirrups was used [
The experimental program consisted of six continuous beams of total length of 3940 mm, at two equal spans of 1850 mm length, with a rectangular cross-section of 150 × 250 mm and with longitudinal and transverse GFRP reinforcement. In addition, a single beam with steel reinforcement was adopted as a control beam. All beams were examined up to failure, loaded by concentrated forces in the middle of both spans. The beams were divided into two series, with different GFRP longitudinal bars, and all were designed for a similar failure load. Dimensions and geometry of continuous beams and load disposition are given in Figure
Geometry and reinforcement details for tested beams (all dimensions are in mm).
Considering beams of Series 1, longitudinal reinforcement of beam G1-0 was designed for the elastic bending moments along the beam, while reinforcement of beams G1-15 and G1-25 was obtained for assumed moment redistribution at the middle support of 15% and 25%, respectively. For the beams G1-15 and G1-25, this meant smaller amount of reinforcement at the middle support and higher amount of reinforcement at the midspan compared to beam G1-0. In this way, for designed failure load, models with 0% (G1-0), 15% (G1-15), and 25% (G1-25) of designed moment redistribution from the middle support to the midspan were obtained. The reinforcement ratio of the beam G1-0 at the middle support has been chosen so as to be approximately 3 times higher than the balanced reinforcement ratio, which corresponds to recommendations of codes that beams with FRP reinforcement should be designed to experience concrete compression failure. In this way, it was provided, after the moment redistribution was made, that all cross-sections, in all beams of Series 1, were designed to have reinforcement ratio above the balanced reinforcement ratio. The control beam with steel reinforcement (S1-15) was designed to achieve the moment redistribution of 15% from the middle support to the midspan. The beams of Series 2 were designed in the same way as the beams of Series 1, only with different types of longitudinal GFRP reinforcement. Models with 0% (G2-0), 15% (G2-15), and 25% (G2-25) of designed moment redistribution from the middle support to the beam midspan were also adopted.
All beams were designed in accordance with ACI 440.1R-15 [
Reinforcement details and compressive strength of concrete for tested beams.
Beam | Day of testing | Middle support-top reinforcement | Midspan-bottom reinforcement | Concrete compressive strength | ||||||||
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Longitudinal reinforcement | EA (kN) | Reinforcement ratio (ACI) | Longitudinal reinforcement | EA (kN) | Reinforcement ratio (ACI) | |||||||
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S1-15 | 28 | 2Ø10 + 1Ø12 | 49526 | 0.82 | 1.26 | 0.65 | 2Ø12 + 1Ø10 | 54763 | 0.92 | 1.31 | 0.71 | 42.2 |
G1-0 | 29 | 3Ø14 | 20318 | 1.40 | 0.46 | 3.01 | 2Ø12 + 1Ø10 | 13558 | 1.00 | 0.57 | 1.75 | 42.2 |
G1-15 | 30 | 2Ø10 + 1Ø14 | 12604 | 0.86 | 0.53 | 1.63 | 2Ø12 + 1Ø14 | 17415 | 1.19 | 0.46 | 2.58 | 42.2 |
G1-25 | 30 | 2Ø10 + 1Ø12 | 11153 | 0.74 | 0.51 | 1.44 | 2Ø14 + 1Ø12 | 18867 | 1.25 | 0.45 | 2.80 | 42.2 |
G2-0 | 28 | 4Ø12 | 17654 | 1.11 | 0.33 | 3.35 | 2Ø10 + 2Ø9 | 10734 | 0.65 | 0.29 | 2.27 | 50.2 |
G2-15 | 29 | 3Ø10 + 1Ø12 | 12482 | 0.79 | 0.33 | 2.37 | 2Ø12 + 2Ø9 | 14182 | 0.86 | 0.32 | 2.69 | 50.2 |
G2-25 | 30 | 3Ø9 | 8033 | 0.48 | 0.29 | 1.70 | 3Ø12 + 1Ø10 | 15930 | 1.10 | 0.37 | 3.02 | 50.2 |
In these experimental research studies, two types of GFRP reinforcement were used: wrapped GFRP bars with 70% of longitudinal glass fibers (E-glass) in total volume, impregnated in the unsaturated polyester matrix for Series 1 (marked G1), and GFRP reinforcement with 75% of longitudinal glass fibers (E-glass), impregnated in an epoxy matrix for Series 2 (marked G2). GFRP reinforcement with polyester was wrapped in glass fibers, while GFRP reinforcement with epoxy resin was with rebars (Figure
Samples of GFRP longitudinal and transverse reinforcement.
Mechanical and deformation characteristics of GFRP reinforcement.
Diameter | Real area of bar A (mm2) | Tensile strength |
Yield strength |
Modulus of elasticity |
Ultimate strain |
---|---|---|---|---|---|
GFRP-1-Ø8 | 39.9 | 714.8 | — | 42640 | 16.8 |
GFRP-1-Ø10 | 70.6 | 703.1 | — | 41300 | 17.0 |
GFRP-1-Ø12 | 116.1 | 865.9 | — | 45832 | 18.9 |
GFRP-1-Ø14 | 152.8 | 813.5 | — | 44324 | 18.4 |
GFRP-2-Ø9 | 53.3 | 1170.4 | — | 50235 | 23.3 |
GFRP-2-Ø10 | 61.5 | 1059.3 | — | 43734 | 24.2 |
GFRP-2-Ø12 | 91.6 | 1060.4 | — | 48182 | 22.0 |
Steel-Ø10 | 78.5 | 639.5 | 509.6 | 188064 | 2.7a |
Steel-Ø12 | 113.1 | 622.5 | 452.7 | 176835 | 2.6a |
a
Two designed classes of concrete of 40 MPa and 45 MPa were used in experimental research studies, for the beams of Series 1 and Series 2, respectively, in order to provide a similar failure load for all beams. For each series of beams, concrete compressive strength after 28 days was obtained according to the investigation of 8 cubes of 150 mm edge, 8 cubes of 200 mm edge, and 17 cylinders of 150/300 mm dimension. Average values of test results of concrete compressive strength on cylinders 150/300 mm are given in Table
Continuous beams consisted of two equal spans placed on three supports over steel bearings. End supports were designed as horizontally movable, while the middle support was designed to prevent horizontal movement. Experimental models were examined in a closed frame which consisted of a unique system of horizontal beams and vertical ties. The load was placed over two hydraulic presses, capacity of 200 kN, in the middle of both spans.
Twelve electrical strain gauges were placed on longitudinal tension reinforcement in both bottom and upper zones of each beam. Also, three strain gauges were placed on compression reinforcement according to the scheme shown in Figure
Experimental setup and instrumentation for tested beams (all dimensions are in mm).
Equipped continuous beam before testing.
The load was applied to the beams as monotonically static growing load in increments from zero up to the failure. At the beginning of the tests, load was applied in increments of approximately 2-3 kN and after development of first cracks in increments of 5 kN. When approximately 80% of the estimated failure load was reached, again the load was applied in increments of 2-3 kN. Speed of load exposure of each increment was approximately 5 kN/min. All electronic data were collected in the computer using the data logger.
All beams of Series 1 and Series 2 were designed to experience concrete compression failure. Beam S1-15 demonstrated typical ductile flexural behavior, with high values of strains and deflections before failure. At the middle support, existing cracks experienced significant widths at failure. First, the tensile reinforcement yielded at the middle support, and after that, tensile reinforcement in the beam midspan. The failure of the beam G1-0 was initiated by concrete compression failure in the midspan in combination with shear, when one crack in the span diagonally propagated toward the load location. This leads to rupture of GFRP bars in the compressed zone by the dowel effect and one GFRP stirrup at the location of its bending. Within the beam G1-15, concrete compression failure took place at the middle support, when one crack near the support diagonally propagated toward the support. The failure of the beam G1-25 appeared at the same time, at the middle support, where concrete crushing was followed by the shear, and at the midspan, where concrete crushing was manifested by spalling of the cover in the extension of the diagonal crack that occurred in the exterior shear span.
The failure mode of the beams G2-0 and G2-25 was similar, initiated by concrete compression failure in the midspan in combination with shear, when one crack in the interior shear span diagonally propagated toward the load location. The failure at the midspan was followed by concrete crushing in the compression zone at the middle support within both beams. Also, in beam G2-25 at the same time, concrete crushing was manifested by spalling of the cover in the midspan in the extension of the diagonal crack that occurred in the exterior shear span. Within the beam G2-15, concrete compression failure took place at the middle support in combination with shear, with the characteristic bang. All longitudinal compressed and tensioned GFRP bars and stirrups ruptured at that section by the dowel effect. The failure modes of all beams are given in Figure
Failure modes of tested beams. (a) S1-15. (b) G1-0. (c) G1-15. (d) G1-25. (e) G2-0. (f) G2-15. (g) G2-25.
In Table
First crack loads and failure loads.
Beam | Load at first crack |
Failure load |
| |||
---|---|---|---|---|---|---|
Left midspan | Right midspan | Middle support | Midspan | Middle support | ||
S1-15 | 32 | 32 | 25 | 134.3 | 0.238 | 0.186 |
G1-0 | 15 | 13 | 13 | 115.6 | 0.112 | 0.112 |
G1-15 | 11 | 13 | 13 | 115.2 | 0.095 | 0.113 |
G1-25 | 13 | 13 | 13 | 119.6 | 0.109 | 0.109 |
G2-0 | 20 | 20 | 17 | 125.2 | 0.160 | 0.136 |
G2-15 | 17 | 17 | 17 | 124.9 | 0.136 | 0.136 |
G2-25 | 17 | 17 | 15 | 137.8 | 0.123 | 0.109 |
Appearance of new cracks and propagation of existing cracks in the beams of Series 1 with GFRP reinforcement stabilized at load that corresponded to approximately 60% of the failure load. The space between cracks, in average, was 120 to 180 mm, and it did not match the space between stirrups. It is evident that, in beams with GFRP reinforcement, a smaller number of cracks occurred than in the beam S1-15 with steel reinforcement, where the space between cracks was from 60 to 100 mm in the interior span, which basically matched the space between stirrups. This indicated poor bond strength between GFRP reinforcement and surrounding concrete, which caused great wideness on already developed cracks in critical sections. This phenomenon was also recorded by a few researchers who examined beams with FRP reinforcement [
Concerning beams of Series 2, with GFRP reinforcement with rebars, the number of cracks was greater with less widths, compared to the beams of Series 1. Cracks appeared in the sagging and in the hogging moment region until failure, regarding the fact that an utmost number of them formed until the load that corresponded to approximately 60% of failure load. The development of the cracks with increasing load corresponded fully to the adopted reinforcement in critical sections; i.e., the higher amount of reinforcement corresponded to a higher number of formed cracks in the section. The largest number of cracks at the middle support formed in the beam G2-0, with an extremely wide hogging zone where cracks appeared, due to the largest axial stiffness of the reinforcement compared to the beams G2-15 and G2-25. In the midspan, the most number of cracks, with the widest sagging zone, appeared in the beam G2-25, with the largest amount of reinforcement in the midspan. The development of the cracks in the beams of Series 2 was similar to that in the beam S1-15 and corresponded to the space between stirrups, indicating the good bond strength between GFRP reinforcement and surrounding concrete. The more pronounced diagonal cracks at higher load levels for the beams of Series 2, especially in the interior shear span, compared to the beam S1-15, indicated an increase of the shear stresses in the beams with GFRP reinforcement, which can be directly attributed to the use of GFRP stirrups, instead of steel stirrups. This was particularly pronounced in the beam G2-0, in which, due to the higher axial stiffness of reinforcement at the middle support and achieved “opposite” redistribution of internal forces (Section
In Figures
Load-maximum crack width relationship at the midspan for tested beams.
Load-maximum crack width relationship at the middle support for tested beams.
Within the midspan of the beams of Series 1 with GFRP reinforcement, the development of the maximum crack width was much equalized, until loads that corresponded to 40% of the failure load, regardless of the fact that reinforcement in the beam G1-0 had less stiffness (EA = 13558 kN) in relation to reinforcement of the beams G1-15 (EA = 17415 kN) and G1-25 (EA = 18867 kN). At the failure, the smallest maximum crack width in the midspan was in the beam G1-25 with the highest amount of reinforcement in the midspan. At the support, the influence of stiffness of reinforcement was evident, meaning that less axial stiffness of reinforcement at the middle support generated larger crack widths in the beams. Therefore, for the most of different load levels, the largest crack width was in the beam G1-25 (EA = 11153 kN) and the smallest was in the beam G1-0 (EA = 20318 kN).
Within the beams of Series 2, the axial stiffness of the GFRP reinforcement was clearly expressed on the maximum cracks width, both in the midspan and at the middle support. For initial load levels, the maximum crack width in the midspan was almost uniform for all beams of Series 2. For higher load levels, the beam G2-25 exhibited the smallest maximum crack width with the largest reinforcement axial stiffness in the midspan (EA = 15930 kN), and the beam G2-0 exhibited the largest maximum crack width, with the smallest reinforcement axial stiffness in the midspan (EA = 10734 kN). At the middle support, beam G2-25 exhibited the largest maximum crack width with the smallest reinforcement axial stiffness at the support (EA = 8033 kN), compared to the beams G2-0 (EA = 12482 kN) and G2-15 (EA = 17654 kN).
In Figure
Load-deflection relationship at the midspan for tested beams.
Within the beams of Series 1, for the same load level, as expected, deflections in beams with GFRP reinforcement were much higher than in the beam S1-15 with steel reinforcement. This is a result of larger crack widths in the beams with GFRP reinforcement, i.e., lower stiffness of the section. At higher load levels, the largest deflection exhibited the beam G1-0, with the smallest stiffness of the reinforcement in the midspan. The beams G1-15 and G1-25 practically had the same average values of deflection during the loading, which is expected due to a small difference in the stiffness of GFRP reinforcement in the beam midspan.
For the beams of Series 2, it could be seen that deflections were fairly uniform during loading, regardless of the different values of the axial stiffness of reinforcement in the critical sections. At higher load levels, the largest deflection was exhibited by beam G2-15 (EA = 14182 kN) and the smallest deflection was exhibited by beam G2-25 (EA = 15390 kN). Nonetheless, the beam G2-0 had significantly lower stiffness in the midspan (EA = 10734 kN), compared to beam G2-15, and also had a lower deflection. This is a consequence of a significant “opposite” moment redistribution from the midspan to the middle support (Section
Measuring of reactions served for determining internal forces, i.e., bending moments, along the continuous beam, based on which the process of moment redistribution was analyzed. Moment redistribution was obtained by comparing actual bending moments and those obtained by elastic analysis. In Table
Moments at failure, moments based on elastic analysis, and achieved percentage of moment redistribution.
Beam | Moments at failure (kNm) | Moments based on elastic analysis (kNm) | Achieved percentage of moment redistribution at failure (%) | |||
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Middle support | Left midspan | Right midspan | Middle support | Midspan | ||
S1-15 | 45.1 | 39.2 | 39.9 | 46.6 | 38.8 | 3.1 |
G1-0 | 40.3 | 32.7 | 33.9 | 40.1 | 33.4 | −0.5 |
G1-15 | 29.2 | 38.8 | 38.5 | 40.0 | 33.3 | 26.9 |
G1-25 | 33.8 | 38.2 | 38.7 | 41.5 | 34.6 | 18.5 |
G2-0 | 50.6 | 31.2 | 34.0 | 43.4 | 36.2 | −16.4 |
G2-15 | 35.3 | 41.9 | 38.3 | 43.3 | 36.1 | 18.5 |
G2-25 | 35.0 | 46.8 | 45.7 | 47.8 | 39.8 | 26.7 |
Development of bending moments and moment redistribution at the middle support for all beams depending on the applied load are given in Figures
Load-moment relationship at the middle support for tested beams.
Load-percentage of moment redistribution relationship at the middle support for tested beams.
Within the beam G1-0, that was designed based on elastic analysis, “opposite” moment redistribution is noticed with a highest value of 23% after appearance of first cracks. At failure, this value is significantly decreased, and it was 0.5%, which totally corresponded to designed values. “Opposite” redistribution caused the ratio between axial stiffness of tensile GFRP reinforcement between the middle support and midspan that was numbered 1.5. The beam G1-15 was designed to achieve redistribution of 15%, and it had the relation of axial stiffness of tensile reinforcement in the midspan and at the middle support of 1.38. At failure, moment redistribution at the middle support was significantly increased, and it was numbered 27%. Within the beam G1-25, designed to achieve moment redistribution of 25%, a failure moment redistribution of 18.5% was achieved, even though the relation between axial stiffness of reinforcement in critical sections was numbered 1.69. For the most levels during loading, the beam G1-25 had moment redistribution higher than 20% (Figure
Within the beam S1-15 with steel reinforcement, moment redistribution was expected after yielding of reinforcement at the support. Nevertheless, in the diagram of Figure
Within the beam G2-0, designed based on the internal forces obtained by the elastic analysis, at initial load levels, the increase of moment in the midspan was noticed, in relation to the moment at the middle support. After the appearance of the first cracks along the beam, a trend of moment redistribution was changed, i.e., a significant increase of the hogging moment in relation to the moment obtained by the elastic analysis. This trend of moment growth was kept to the failure of the beam. Therefore, “opposite” moment redistribution happened at failure of 16.4%, which was greatly influenced by the axial stiffness ratio of reinforcement at the middle support and in the midspan, which was numbered 1.65. Beams G2-15 and G2-25, designed to achieve moment redistribution, from the middle support into the midspan of 15% and 25%, achieved higher percentage of redistribution at failure of 18.5% and 26.7%, with ratio of axial stiffness of GFRP reinforcement between critical sections of 1.14 and 1.98, respectively. During the complete process of loading, beams had positive moment redistribution, which was the result of “set up” of the beams by means of adopted reinforcement, i.e., axial stiffness of reinforcement, which in the cracked section had a great contribution in the stiffness of critical sections, as previously discussed. Within the beams G2-15 and G2-25, an increase in the moment redistribution at failure was observed, probably as a result of the development of full nonlinearity of the compressed concrete.
In Figures
Load-strain relationship at the midspan for tested beams.
Load-strain relationship at the middle support for tested beams.
For the beams of Series 1, at load levels after appearance of first cracks, the strains in reinforcement in the midspan and at the support were higher in beams with GFRP reinforcement than in beam S1-15 with steel reinforcement. However, at higher load levels, after yielding of steel reinforcement, strain in the beam S1-15 significantly increased and overcame values of strains in beams with GFRP reinforcement. Comparing strains in reinforcement of the beams with strains in GFRP reinforcement, it is evident that, in the midspan, strains were highest in the beam G1-0 as a result of lowest stiffness of this reinforcement, while at the support, strains were highest in the beam G1-25, especially at higher load levels.
Within the beams of Series 2, quite uniform development of strains in the tensile reinforcement in the midspan, regardless of the significant differences in amount, i.e., the axial stiffness of adopted reinforcement of beams, could be noticed. At higher load levels, before failure, strains in the midspan of the beam G2-0 were about 10% higher than those in the beam G2-15 and in an average 5% higher compared to the beam G2-25. At the failure, strains in the beams G2-0 and G2-15 were practically identical, while the largest strains were in the beam G2-25, which reached the highest load-carrying capacity. As it has been already mentioned, this could be explained by achieved moment redistribution, which contributed that stiffness of reinforcement corresponded to higher bending moments. Because of that, lower strains did not correspond to higher amount of reinforcement, and vice versa. Comparing the strains in the tensile reinforcement at the middle support, the difference in strain values was significantly more pronounced than at the midspan. The higher strains were within the beam G2-25, and the smallest strains were within the beam G2-0, which fully corresponded to the adopted reinforcement at the middle support. The maximum tensile strains did not reach neither the ultimate values nor a single beam of Series 1 and Series 2, which responded to the fact that beams were designed to experience concrete compression failure. The largest measured strains at the middle support were in the beam G2-25 and amounted to 23‰, which was very close to the ultimate value of 23.3‰ shown in Table
From Figures
All beams were designed in accordance with the ACI 440.1R-15 [
Experimental and calculated failure loads for tested beams.
Beam | Failure load-load-carrying capacity (kN) |
| |||||
---|---|---|---|---|---|---|---|
Experiment | ACI | CSA | EC2 | Exp./ACI | Exp./CSA | Exp./EC2 | |
S1-15 | 134.3 | 90.1 | 88.8 | 90.1 | 1.49 | 1.51 | 1.49 |
G1-0 | 115.6 | 115.4 | 126.3 | 141.3 | 1.00 | 0.92 | 0.82 |
G1-15 | 115.2 | 111.8 | 123.0 | 137.0 | 1.03 | 0.94 | 0.84 |
G1-25 | 119.6 | 117.4 | 128.6 | 143.8 | 1.02 | 0.93 | 0.83 |
G2-0 | 125.2 | 113.6 | 128.0 | 145.2 | 1.10 | 0.98 | 0.86 |
G2-15 | 124.9 | 117.2 | 131.7 | 149.8 | 1.07 | 0.95 | 0.83 |
G2-25 | 137.8 | 110.8 | 125.2 | 141.7 | 1.24 | 1.10 | 0.97 |
From Table
Although the beams of Series 1 and Series 2 with GFRP reinforcement were designed to achieve similar failure loads, some higher values of failure loads were obtained for beams of Series 2. The reason for this phenomenon could be the sliding of GFRP reinforcement and surrounding concrete in the beams of Series 1. Reducing the amount of GFRP reinforcement at the middle support and increase in the midspan of continuous beams, as a result of designed moment redistribution, did not have influence on decrease of load capacity of continuous beams. Moreover, the beams G1-25 and G2-25, with the designed moment redistribution of 25%, achieved higher load capacity, compared to the beams with a designed moment redistribution of 0% and 15%, for 5% and 10%, respectively.
In the background of this paper, it is stated that, until now, as a result of a number of research studies on investigation of behavior of simple beams with FRP reinforcement, a number of expressions for determining deflection-load response is suggested. For calculation of deflection of continuous beams, loaded by concentrated forces at the middle of the span, the following equation obtained by elastic analysis is used:
ACI 440.1R-15 [
Calculation of deflection, in accordance with CSA S806-12 [
Habeeb and Ashour [
Kara and Ashour [
Ju et al. [
Load-deflection diagrams obtained by calculation according to expression (
Experimental and predicted deflection for tested beams. (a) G1-0. (b) G2-0. (c) G1-15. (d) G2-15. (e) G1-25. (f) G2-25.
In order to overcome stated shortcomings of the previous models, a modified model for calculation of deflection is proposed. The model is based on Branson’s equation used in ACI 440.1R-06 [
The very good match of the proposed model and experimental results was shown, both at lower and at higher load levels. The exception occurs in the beams G1-15 and G1-25 at loads immediately after cracking where a large increase in deflection happened, which indicates poor bond strength between GFRP bars and concrete in these beams. In particular, it is noted that the proposed model, using the coefficient
The subject of the experimental research shown in this paper is consideration of six continuous beams reinforced with GFRP reinforcement loaded by concentrated forces in the middle of the span, until failure, for different arrangements of reinforcement along the beam. Results of experimental research studies are compared to provisions of current regulations and codes by means of load capacity and load-deflection response. Based on these research results, following conclusions could be made: Continuous beams with GFRP reinforcement have the ability of moment redistribution in relation to moments obtained by linear elastic analysis, after appearance of first cracks in concrete. Values of moment redistribution dominantly depend on stiffness of critical sections at the support and in the midspan, which, primarily, due to wide and deep cracks, come to relation of axial stiffness of GFRP reinforcement in critical sections. Elastic redistribution of internal forces is based on this. Continuous beams with GFRP reinforcement show significant warnings before failure, in terms of large deflections and wide and deep cracks. It is specially defined additional curving of the deflection diagram at loads close to failure, as a result of the development of full nonlinearity of the compressed concrete. Reducing the amount of GFRP reinforcement at the middle support and increase of amount of GFRP reinforcement in the midspan of continuous beams, as a result of designed moment redistribution, in relation to the moments obtained by elastic analysis, do not have negative influence on load capacity of continuous beams and mainly influence the decrease of deflection. By increasing the designed moment redistribution to 25%, the load-carrying capacity increases by 5% to 10% in the beams with GFRP reinforcement. Wide and deep cracks that are formed in critical sections of continuous beams with wrapped GFRP bars with the unsaturated polyester matrix, in a smaller number compared to beams reinforced by steel reinforcement or GFRP bars with rebars and epoxy matrix, point out at poor bond strength between GFRP reinforcement and surrounding concrete. On the contrary, beams with GFRP reinforcement with rebars and epoxy matrix, based on development of the cracks of the beams, indicate very good bond strength between GFRP reinforcement and concrete. ACI 440.1R-15 [ Current codes ACI 440.1R-15 [
Applied load at the midpoint of each span (kN)
Load at failure (kN)
Load at the first crack (kN)
Experimental failure load (kN)
Calculated failure load (kN)
Beam span (mm)
Uncracked length at half of the beam (mm)
Moment of inertia of the gross section (mm4)
Cracked moment of inertia (mm4)
Effective moment of inertia (mm4)
Cracking moment (kNm)
Applied moment (kNm)
Cross-sectional area of tensile GFRP reinforcement (mm2)
Modulus of elasticity of GFRP reinforcement (MPa)
Modulus of elasticity of steel reinforcement (MPa)
Modulus of elasticity of concrete (MPa)
Tensile strength of GFRP reinforcement (MPa)
Yield strength of steel reinforcement (MPa)
Concrete compressive strength of cylinder (MPa)
Ultimate strain of GFRP reinforcement
Yield strain of steel reinforcement
Ultimate strain of concrete
GFRP reinforcement ratio
Balanced GFRP reinforcement ratio
Proposed reduction factor used in the calculation of the effective moment of inertia for the continuous beam with GFRP reinforcement
Proposed reduction factor used in the calculation of the effective moment of inertia for the continuous beam with FRP reinforcement
Reduction factor used in the calculation of the effective moment of inertia
Deflection at the midspan of the beam (mm).
The data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
The authors would like to express their gratitude to the Engineering Chamber of Montenegro for the financial support and China Road and Bridge Corporation Montenegro Branch for the donation of GFRP reinforcement for this research, and local construction companies from Montenegro for the donation of concrete admixtures. Also, the first author is grateful for the technical help of the Laboratory of the Faculty of Civil Engineering at the University of Montenegro.