Behavior of non-prismatic RC beams with conventional steel and green GFRP rebars for sustainable infrastructure

This study presents an experimental and finite element analysis of reinforced concrete beams with solid, hollow, prismatic, or non-prismatic sections. In the first part, a total of six beams were tested under four-point monotonic bending. The test matrix was designed to provide a comparison of structural behavior between prismatic solid and hollow section beams, prismatic solid and non-prismatic solid section beams, and prismatic hollow and non-prismatic hollow section beams. The intensity of shear was maximum in the case of prismatic section beams. The inclusion of a tapered section lowered the demand for shear. In the second part, Nonlinear Finite Element Modeling was performed by using ATENA. The adopted modeling strategy resulted in close agreement with experimental crack patterns at ultimate failure. However, the ultimate failure loads predicted by nonlinear modeling were generally higher than their corresponding experimental results. Whereas in the last part, the developed models were further extended to investigate the effect of the strength of concrete and ratio of longitudinal steel bars on the ultimate load-carrying capacity and cracking behavior of the reinforced concrete beams with solid, hollow, prismatic, or non-prismatic sections. The ultimate loads for each beam predicted by the model were found to be in close agreement with experimental results. Nonlinear modeling was further extended to assess the effects of concrete strength and longitudinal reinforcement ratio on failure patterns and ultimate loads. The parametric study involved beams reinforced with glass fiber-reinforced polymer (GFRP) bars against shear and flexural failure. In terms of ultimate load capacities, diagonal cracking, and flexural cracking, beams strengthened with GFRP bars demonstrated comparable performance to the beams strengthened with steel bars.

Bridges are a vital part of the infrastructure of any country.Bridges with long spans are often constructed in modern construction works.One of the key concerns in long-span bridges is related to self-weight.The use of non-prismatic sections along the span can significantly lower the self-weight of the structure.In addition, novel aesthetic designs are possible from an architectural standpoint [1][2][3] .The use of non-prismatic beams can help decrease the clear ceiling heights: a desirable feature from the cost perspective.The use of hollow-section members has also been a practice to reduce the self-weight, with their applications in almost every kind of structure [4][5][6][7] .Reinforced concrete (RC) beams with hollow sections provide a viable solution for the passage of sewage or transmission lines [8][9][10] .The behavior of RC beams with non-prismatic or hollow sections has been investigated by numerous researchers separately.Abbas et al. 9 studied the behavior of high-strength concrete hollow beams strengthened with steel fibers.Vijayakumar and Madhavi 11 tested RC beams with an 80 mm circular opening along the neutral axis.The cracking, as well as the ultimate load of hollow beams without hybrid fiber

Experimental program
Test matrix and analysis program.The present study investigated the structural response of beams with non-prismatic, hollow, or a combination of both sections.In the first part, a total of six beams were tested.The details of the tested beams are presented in Table 1.Three beams, i.e., P-S-01, NP-S-02, and NP-S-03, were constructed with solid sections.The remaining three beams, i.e., P-H-04, NP-H-05, and NP-H-06, were constructed with hollow sections.P-S-01 and P-H-04 were constructed with prismatic sections.Beams NP-S-02, NP-S-03, NP-H-05, and NP-H-06 incorporated non-prismatic sections along their spans.However, the length of the nonprismatic span in specimens NP-S-02 and NP-H-05 was smaller than that of the specimens NP-S-03 and NP-H-06.In the second part, the experimentally tested beams (Table 1) were analyzed in computer program ATENA Table 1.Details of the beam specimen of experimental program.www.nature.com/scientificreports/("Finite element modeling").Whereas in the last part, parametric finite element modeling was performed to study the effect of the strength of concrete and longitudinal reinforcement ratio on the ultimate load-carrying capacity and cracking behavior of prismatic and non-prismatic RC beams with solid and hollow sections.The developed finite element models were further extended, and a total number of 12 RC beams were modeled in ATENA.The details of parametric RC beams are shown in Table 2.In beams P-S-01-HSC, NP-S-02-HSC, NP-S-03-HSC, P-H-04-HSC, NP-H-05-HSC, and NP-H-06-HSC strength of concrete was increased to 30 MPa to study the effect of strength of concrete on the ultimate load carrying capacity and cracking behavior of prismatic and non-prismatic RC beams with solid and hollow section.Whereas in beams P-S-01-HRR, NP-S-02-HRR, NP-S-03-HRR, P-H-04-HRR, NP-H-05-HRR, and NP-H-06-HRR, two deformed bars of diameter 25 mm were used to assess the effect of high reinforcement ratio.In the following section, the finite element analysis results of these beams were compared with the beams P-S-01, NP-S-02, NP-S-03, P-H-04, NP-H-05, and NP-H-06.Further, finite element modeling was extended to study the behavior of non-prismatic solid and hollow beams reinforced with glass fiber-reinforced polymer (GFRP) bars.It is important to note that the type of bars for the shear and flexural reinforcement were GFRP bars.Two sizes of GFRP bars were considered for the bottom longitudinal reinforcement, i.e., diameters of 16 mm and 25 mm, whereas the concrete strength was kept at 15 MPa.Typical GFRP bars are shown in Fig. 1.

Details of beams.
Each beam had a total length of 1700 mm, a height of 250 mm, and a width of 150 mm.
The structural details of all beams are shown in Fig. 2. All beams were longitudinally reinforced with two 16 mmdeformed beams on the tension side and two 12 mm-deformed bars on the compression side.Four stirrups of 6 mm-round bars were provided within the midspan at a spacing of 100 mm.The longitudinal reinforcement was provided with 90 • hooks near beam ends.Specimen P-S-01 had a prismatic solid section.Specimen NP-S-02 had a prismatic solid section of 500 mm at its ends, followed by a tapered solid section of 200 mm, and a prismatic solid section again within the midspan for 300 mm.Specimen NP-S-03 had an increased tapered length of 400 mm.As a result, the depth of the section at midspan was 150 mm in contrast to the depth of 200 mm in   Material properties.The mechanical properties of steel reinforcement were estimated by following the recommendations of ASTM E8/E8M-21 42 .Standard tensile tests by using three samples were carried out to determine the stress-strain curves of all steel bars.Table 3 summarizes the estimated properties of steel reinforcement.Concrete cylinders were fabricated by pouring concrete in three equal layers with each layer subjected to   Test setup and instrumentation.Each beam was subjected to a monotonic four-point bending, as shown in Fig. 4. The load was applied by using a hydraulic jack, and the intensity of the applied load at each time instant was monitored by using a load cell.The vertical deflection of the beam was measured by using three displacement transducers.Two additional displacement transducers were attached to the ends to measure any unwanted support uplift.One strain gauge was mounted at the top longitudinal bar at midspan, whereas another strain gauge was mounted to the bottom longitudinal bar.

Experimental results
Failure modes.The failure of all beams is shown in Fig. 5.The failure of Specimen P-S-01 was accompanied by flexural cracks near its soffit.At a load of 30 kN, flexural cracks started to appear and propagated toward the neutral axis.The increase in load resulted in the appearance of additional flexural cracks.At a load of around 62 kN, an inclined crack appeared staring from the left loading plate towards the left support.Consequently, the beam lost its capacity in a brittle manner.Specimen P-S-01 was unable to demonstrate any significant vertical  www.nature.com/scientificreports/left loading plate was unable to propagate to the left support.As a result, a ductile response was exhibited by Specimen NP-S-03.
The failure of Specimen P-H-04 was identical to that of Specimen P-S-01.A wide shear crack was observed near the load of 42 kN.Following that, the load capacity dropped suddenly.Specimen NP-H-05 exhibited an identical response to that of Specimen NP-S-02.Several flexural cracks appeared within the tension zone before the formation of the final shear crack at a load of 65 kN.Finally, the failure of Specimen NP-H-06 was identical to that of Specimen NP-S-03.As shown in Fig. 5, Specimen NP-H-06 exhibited several inclined cracks, but the formation of these cracks was slow and did not result in a sudden drop in its load capacity.
Ultimate load and deflection.A summary of the ultimate load and deflection is presented in Table 4.
Specimen P-S-01 was able to withstand a maximum load of 62.82 kN, whereas the corresponding deflection was 2.51 mm.Specimen NP-S-02 demonstrated a higher peak load of 70.47 kN than that Specimen P-S-01.In addition, the deflection at peak load was significantly higher as well.The length of the tapered section was further increased in Specimen NP-S-03.This resulted in a further increase in deflection at ultimate load, whereas the peak sustained load least among specimens with solid section.It can be observed that by increasing the tapered length along the beam, the flexibility of the beam increased.A similar trend was also observed in specimens with hollow sections.A comparison of the peak loads of specimens P-S-01 and NP-S-02 suggests that a hollow section beam demonstrated a 32.81% lower capacity than that of the same beam but with a solid section (see Fig. 6).On the contrary, this reduction in peak capacity was insignificant in the case of the non-prismatic section.For instance, the difference in the peak sustained loads of specimens NP-S-02 and NP-H-05 was 9.85% (see Fig. 6).The same difference was reduced to 2.17% in the case of specimens NP-S-03 and NP-H-06 (see Fig. 6).Hence, it can be concluded that beams with hollow non-prismatic sections can have the same capacity as those of the same beams but with solid sections.However, the beams with prismatic sections can have significantly different capacities depending upon the presence of longitudinal openings.
Load-deflection curves.The load-deflection curves of all beams are shown in Fig. 7.It is to be noted that the initial stiffness of load-deflection curves was estimated by joining a line from origin to the point that corresponded to the 40% peak capacity 44 , as shown in Fig. 8a, whereas the comparison of computed initial stiffness is shown in Fig. 8b,c.Several important conclusions can be drawn: (1) the initial stiffness was maximum in the case of prismatic sections, i.e., for specimens P-S-01 and P-H-04, (2) the inclusion of non-prismatic section reduced the initial stiffness.This reduction was proportional to the length of the non-prismatic section along the beam.For instance, NP-S-02 had lower initial stiffness than that P-S-01, whereas NP-S-03 had the least initial stiffness among specimens P-S-01, NP-S-02, and NP-S-03, (3) ductility of beams increased as the length of the non-prismatic section along the beam increased.This suggests that a beam with a non-prismatic section is less   Strain of steel bars.The strain gauge measurements along the top and bottom longitudinal bars are shown in Fig. 9.The maximum positive strain in specimens P-S-01, NP-S-02, and NP-S-03 were 1034 microns, 2016 microns, and 1430 microns, respectively.This shows that the inclusion of a tapered section allowed the beam to achieve a better structural response.The maximum positive strain in specimens P-H-04, NP-H-05, and NP-H-06 were 752 microns, 960 microns, and 1287 microns, respectively.The inclusion of openings also reduced the maximum positive strain along longitudinal bars.For instance, the maximum positive strain along longitudinal bars in specimens P-S-01 and P-H-04 were 1034 microns and 752 microns, respectively.The maximum negative strain in specimens P-S-01, NP-S-02, and NP-S-03 were -432 microns, -2020 microns, and -2016 microns, respectively.Similarly, the maximum negative strain in specimens P-H-04, NP-H-05, and NP-H-06 were − 374, − 326, and − 3620 microns, respectively.Clearly, a tapered beam was able to withstand higher steel strains both on the negative and positive sides.However, solid section beams were able to achieve higher negative strains than those of hollow section beams.

Accumulated dissipated energy.
The total energy dissipated by each specimen was computed by integrating the areas under load-deflection curves.The last column of Table 4 presents the computed accumulated dissipated energy of all specimens.For beams with solid sections, the least dissipated energy belonged to Specimen P-S-01, with a value of 319.45 kN-mm.The inclusion of a tapered section in Specimen NP-S-02 increased this value to 647.46 kN-mm.The length of the tapered section was further increased in Specimen NP-S-03.Correspondingly, the accumulated dissipated energy increased to 1041.87 kN-mm.For hollow section beams, the least dissipated energy also belonged to Specimen P-H-04, i.e., with a prismatic section.The inclusion of tapered sections in Specimens NP-H-05 and NP-H-06 increased the dissipated energy to 1397.78 kN-mm and 975.29 kN-mm, respectively.Thus, in general, beams with non-prismatic sections were able to demonstrate higher energy dissipation capacities than beams with prismatic sections.Further, the energy dissipation capacities of beams with non-prismatic hollow sections were higher than beams with non-prismatic solid sections.

Finite element modeling
The nonlinear finite element modeling was performed by using ATENA 45 .In the first step, FEM was performed and validated against experimental results of the present study, similar to existing studies 46,47 .Then, FEM was extended to study the effects of concrete strength glass fiber-reinforced bars on the structural performance of beams.The concept of smeared cracking is utilized in ATENA for nonlinear modeling.The 8-node solid element CC3DNonLinCementitious2 was used to model concrete.The global mesh size was kept at 0.015 m.The uniaxial stress-strain relation shown in Fig. 10 was used for concrete.Linear elastic behavior was assumed for concrete before cracking in tension, whereas a fictitious model based on crack-opening law and fracture energy was utilized for post-cracking in tension.The ascending compressive branch was modeled by using the recommendations of CEB-FIP 48 .The steel reinforcing bars were modeled by using a multilinear curve to allow for modeling various stages along the constitutive stress-strain relation of steel bars.The multilinear curve was assigned to built-in truss materials in ATENA.The definition of a typical multilinear curve for steel bars in ATENA is shown in Fig. 11.The slip between steel bars and concrete was neglected by assuming a perfect bond.In this study, parametric analysis was conducted to select appropriate mesh size to accommodate the accuracy of results, analysis time and storage capacity of the computer.Based on the parametric analysis, global mesh size was selected as 0.05 m was selected in the ATENA (Fig. 12).The meshing and reinforcement details are shown in Fig. 13.In the past 49,50 , linear stress versus strain responses were observed for GFRP rebars, therefore in this study, the GFRP rebars were modelled as linear curve using built-in truss element CCReinforcement.
The comparison of crack patterns at the ultimate failure of beams is shown in Fig. 14.In general, the adopted modeling strategy resulted in close agreement with experimental crack patterns for all beams.Inclined shear cracks between loading points and support plates were observed in solid and hollow prismatic beams.The cracks observed in non-prismatic beams were lesser than those in prismatic beams.This was confirmed by their corresponding experimental results.Table 5 summarizes the comparison of experimental and analytical ultimate loads of all beams.The ultimate loads predicted by the nonlinear modeling were generally higher than their corresponding experimental values.
Parametric finite element modeling.Although many studies have reported the behavior of non-prismatic RC beams, however, the effect of the strength of concrete was not studied for different types of non-prismatic RC beams with solid and hollow sections.Further, the effect of the longitudinal reinforcement ratio was also not considered in the previous studies.Therefore, in this section, parametric finite element modeling was performed to study the effect of the strength of concrete and longitudinal reinforcement ratio on the ultimate load-carrying capacity and cracking behavior of prismatic and non-prismatic RC beams with solid and hollow sections.The developed finite element models were further extended, and a total number of 12 RC beams were modeled in ATENA.The details of parametric RC beams are shown in Table 2.The results of parametric finite element analysis are discussed in the following sections.Figure 16 presents the comparison of FEM results with actual concrete strength (left column) and FEM results with high concrete strength (right column).By increasing the concrete strength, the moment capacity of concrete was expected to increase.Consequently, more flexural cracks within the tension zone were expected.This was confirmed by FEM results (see right column of Fig. 16) with increased intensity of flexural cracks.However, the main crack orientation remained the same, indicating that failure patterns would not alter by increasing the concrete strength.In general, wider cracks were observed for high-strength concrete.
Effect of reinforcement ratio.The ultimate loads increased as the longitudinal reinforcement ratio increased (see Fig. 17).The increase in longitudinal reinforcement ratio had a similar effect on crack patterns and their width as that of the concrete strength.Wider and more intense cracks were observed by increasing the longitudinal reinforcement ratio.The results of the second parametric study are shown in Fig. 18.The beams were again expected to sustain higher moment capacities as a result of an increased cross-sectional area of steel bars.This was reflected in FEM results in terms of more flexural cracks (see right column of Fig. 18).More importantly, the failure crack patterns were not changed.Thus, it can be established that the failure patterns observed in the present study will not change if the longitudinal reinforcement ratio is increased.Further, the failure patterns were found to be independent of the size of longitudinal bars.
Effect of GFRP rebars.A comparison of the ultimate loads predicted by FEM for beams with steel bars and different sizes of GFRP bars is shown in Fig. 19.By comparing the response beams with steel and GFRP bars of the same diameter, a close agreement between the ultimate capacities was observed, with a difference of only 23%.This difference was further reduced when the diameter of GFRP bars was increased to 25 mm.The comparison of cracking patterns predicted by FEM for steel and GFRP bars of 16 mm diameter is shown in Fig. 20.It is recalled that GFRP bars were used for both the longitudinal and shear reinforcement.The cracking patterns for beams with steel and GFRP bars were comparable, with GFRP bars showing a slightly larger number of diagonal cracks.Thus, it can be inferred that GFRP bars tend to resist shear forces in a similar way as that of steel bars.The number and extent of flexural cracks in GFRP-strengthened beams were similar to that of steel-reinforced beams.A similar behavior was observed in the case of beams longitudinally reinforced with GFRP bars of 25 mm diameter (see Fig. 21).The failure of beam is initiated by the yielding of steel bars in steel reinforced beams, whereas the fracture of GFRP bars control the failure in GFRP strengthened beams.Further, the higher number of cracks in GFRP strengthened beams can be associated with the lower stiffness of GFRP bars as compared to steel bars, similar observations were made elsewhere 51,52 .3. The peak load was higher in solid section beams as compared to that hollow section beams.This difference in peak loads was prominent in prismatic beams.4. Beams with non-prismatic sections were able to demonstrate higher energy dissipation capacities than beams with prismatic sections.Further, the energy dissipation capacities of beams with non-prismatic hollow sections were higher than beams with non-prismatic solid sections.www.nature.com/scientificreports/ 5.The adopted finite element modeling strategy resulted in close agreement with experimental crack patterns at ultimate failure.However, the ultimate failure loads predicted by nonlinear modeling were generally higher than their corresponding experimental results.6.The results of the parametric study showed that the ultimate loads of non-prismatic solid and hollow sections RC beams were increased as the strength of concrete and reinforcement ratio of longitudinal steel bars were increased.7. Beams strengthened with GFRP bars were able to demonstrate capacities that were comparable to that of steel-reinforced beams.

Figure 3 .
Figure 3. Construction of RC beams (a) typical Styrofoam in prismatic beam, (b) typical styrofoam in steel bars, (c) construction of RC beams.

Figure 8 .
Figure 8.(a) Definition of initial stiffness, (b) comparison of initial stiffness of solid section beams, and (c) comparison of initial stiffness of hollow section beams.

Figure 9 .
Figure 9.Comparison of longitudinal bar strains on tension and compression sides.

Figure 11 .
Figure 11.Definition of multilinear curve to model stress-strain behavior of steel bars.

Figure 12 .
Figure 12.Typical display of mesh size in ATENA.

Figure 14 .
Figure 14.Comparison of experimental and analytical crack patterns at the ultimate stage.

Figure 15 .
Figure 15.Comparison of ultimate loads predicted by FEM for actual and high concrete strength.

Figure 16 .
Figure 16.Comparison of FEM cracking results (left column) actual concrete strength and (right column) concrete strength doubled.

Figure 17 .
Figure 17.Comparison of ultimate loads predicted by FEM for actual and high longitudinal reinforcement ratio.

Figure 18 .
Figure 18.Comparison of FEM cracking results (left column) actual longitudinal bars and (right column) 25 mm longitudinal bars.

Figure 19 .
Figure 19.Comparison of ultimate loads predicted by FEM for steel bars and GFRP bars.

Figure 20 .
Figure 20.Comparison of FEM cracking results (left column) actual beams and (right column) beams with 16 mm GFRP bars.

Figure 21 .
Figure 21.Comparison of FEM cracking results (left column) actual beams and (right column) beams with 25 mm GFRP bars.

Table 3 .
Mechanical properties of steel reinforcement.

Table 4 .
Summary of ultimate load, deflection, and dissipated energy.

Table 5 .
Comparison of experimental and FEM predicted ultimate loads.