Strength of Concrete Slabs Reinforced with Carbon Fiber Reinforced Polymer under Torsion

4.2.1. Concrete

The model of Concrete Damage Plasticity (CDP) available in ABAQUS was used to simulate the concrete's response under loading. This model considers the degradation of concrete and accounts for two modes of failure: crushing under compression and cracking under tension. Several parameters are required to define the concrete's plasticity model, including the dilation angle (ψ), the compressive stresses ratio in biaxial and uniaxial states (fbo / fco), the plastic potential eccentricity, the yield surface shape factor (K), and the viscosity parameter. The CDP parameters applied in the current study are presented in Table 2 [14-17].

The concrete stress-strain relationship in compression was modeled utilizing the following equations (1a-d) [18]:

(1a)

(1b)

(1c)

(1d)

The parameters R_ε=4 and R_σ=4 can be applied, σ_c represents the effective stress, ε_c denotes the effective strain, ε₀ is the strain of concrete at peak stress, set at 0.003, and E_c stands for the initial modulus of elasticity. Fig. (2) illustrates a typical stress-strain curve in compression (for f’c = 30 MPa) as relevant to this study.

The concrete stress-strain relationship in tension has been described using the following equations (2 a,b) [19-22]:

(2a)

(2b)

ε_t refers to the tensile strain of concrete, f_t represents the tensile strength of the concrete, and ε_cr is the strain of concrete at peak stress, set at 0.00015 in the current study. Fig. (3) displays a typical stress-strain relationship in tension relevant to the current study.

The damage parameter (d_t) accounts for the concrete's degradation under tensile stress and is determined as the ratio of the post-cracking to the ultimate stresses [14, 17]. Moreover, a value of 0.8 was used for the compressive stiffness recovery factor, reflecting the significant recovery of stiffness that occurs when cracks close as the loading transitions from tension to compression.

4.2.2. FRP Reinforcement

FRP bars were modeled utilizing a linear elastic-brittle material property, characterized by a tensile strength of 1700 MPa and a modulus of elasticity of 150 GPa [23]. The reinforcement was incorporated into the concrete model as an embedded region.

4.2.3. Steel Reinforcement

The steel bars were modeled utilizing a linear elastic-perfect plastic material property characterized by a modulus of elasticity of 200 GPa [14].

4.2.4. Plates

The plates utilized beneath the applied loads and at supports were assigned linear elastic material properties, having a modulus of elasticity of 200 GPa. The interaction between the concrete and plates was established using “Tie constraints” [14].

5.2.1. Influence of FRP Bar Size on the Torsional Behavior of Slabs

Data on torsional properties are provided in Table 3. These data include the torsional cracking moment (T_cr), the torsional ultimate moment (T_ult), and the change in the values of these moments with the increase in FRP bar size. The torsional cracking moment exhibited a slight increase with larger FRP bar sizes across the various groups. Specifically, an increase of approximately 3%–4% was observed when using 12 mm FRP bars instead of 8 mm bars, aligning with the findings of Ibraheem and Mukhlif [8], who also reported a limited effect of reinforcement on the cracking load. However, the effect of FRP bar size on the torsional ultimate moment was more pronounced compared to its effect on the torsional cracking moment. The final column of Table 3 in each group highlights the percentage of the torsional ultimate moment for the slab to that of the reference case with 8 mm FRP bars. An increase in the FRP bar size led to a corresponding rise in the torsional ultimate moment, with enhancements ranging from 8% to 21% when 12 mm bars replaced 8 mm bars in the specimens. The highest observed increase was 21%, occurring in the group of slabs with a compressive strength of concrete of 50 MPa, where the torsional ultimate moment rose from 26.4 kN·m/m for S-50-8 to 32 kN·m/m for S-50-12.

Fig. (5) shows the T-θ charts for the slabs. These charts are similar for the different FRP bar sizes of the slabs.

5.2.2. Influence of Compressive Strength of Concrete on the Torsional Behavior of Slabs

Table 4 presents the torsional cracking moment and torsional ultimate moment for all the slabs, arranged to show the influence of the compressive strength of concrete on these two moments. The slabs were arranged in three groups, each with a different FRP bar size. Every group included seven slabs with different compressive strengths of concrete. Torsional cracking moment significantly increased with an increase in compressive strength of concrete of different groups. This moment increased by 53% for different groups due to the influence of using compressive strength of concrete of 50 MPa instead of 21 MPa. For each group, the final column in Table 4 provides the percentage of the torsional ultimate moment of the slabs to that of the reference case with a compressive strength of concrete of 21 MPa. The torsional ultimate moment showed a substantial increase with the rise in the compressive strength of concrete. This moment increased between 27% to 42% by using the compressive strength of concrete of 50 MPa instead of 21 MPa in different groups. In this regard, the maximum increase in torsional ultimate moment was 42% (from 22.5 kN.m/m for S-21-12 to 32 kN.m/m for S-50-12), which was for the slabs in the group reinforced with 12 mm bar size.

Fig. (6) illustrates the T-θ charts for the slabs, organized to highlight the impact of the compressive strength of concrete on this relationship. The charts reveal a similar T-θ relationship across different compressive strengths of concrete of the slabs, with an increase in both the torsional cracking and ultimate moments as the compressive strength of concrete increases. The T-θ curves in both Figs. (5 and 6) exhibit a nearly bilinear behavior. The initial segment of the curve is approximately linear with a steeper slope, indicating the stiffness of the uncracked slab. Beyond a certain load level, the curve transitions to another linear segment with a gentler slope, representing the reduced stiffness of the cracked slab. These findings are consistent with previous experimental studies by Rashid [25], Zhang [26], and Adam [27], which reported similar trends in load-deflection curves for slabs reinforced with FRP under various loading conditions.

5.2.3. Torsional Behaviour of Slabs with FRP Reinforcement Compared to Slabs with Steel Reinforcement

A comparison of the torsional behavior of slabs with FRP bars reinforcement and those with steel bars reinforcement was conducted. Table 5 presents the torsional cracking moment and torsional ultimate moment for two groups of slabs, both having a compressive strength of concrete of 30 MPa. Group (G3) consisted of slabs reinforced with FRP bars in three different sizes, while group (G8) consisted of slabs reinforced with steel bars of similar sizes to those in (G3). The torsional cracking and ultimate moments for group (G3) were comparable to those of group (G8). This indicates that, in addition to the beneficial properties of FRP materials compared to steel, the torsional behavior and strength of slabs with FRP bar reinforcement are similar to those with steel bar reinforcement. These findings not only bridge a critical gap in understanding the torsional behavior of FRP-reinforced slabs but also pave the way for their broader adoption in applications requiring enhanced durability and performance under torsional loads. It is also worth noting that, as shown in Table 5, the torsional ultimate moment of slabs reinforced with FRP is 91–92% of that of slabs reinforced with steel bars. These findings are in close agreement with the findings of Zhang [26], which reported a 91% capacity for CFRP-reinforced slabs compared to steel-reinforced slabs under centrally concentrated loading.

5.2.4. Verification of the FEA Model

To validate the FEA model and assess its accuracy, the results for slab S-30-8 were compared with the experimental findings of two slabs derived from the P-d relationship curves presented by Lopes et al [5]. These experimental slabs share the same dimensions, support conditions, loading, and reinforcement bar size as slab S-30-8 analyzed in the current study. The comparison results are provided in Table 6. Notably, these experimental slabs are reinforced with steel bars, as no experimental data currently exists for FRP-reinforced slabs under torsion. The comparison focused on the maximum applied load (P_max) and the corresponding vertical displacement (d_pmax). As shown in Table 6, the results of the current study closely match the experimental data of Lopes et al The P_max obtained in this study was 52.8 kN, which represented 96% of the average P_max (55 kN) from the experimental slabs. The slightly lower P_max observed in this study aligns with previous findings indicating that CFRP reinforcement slightly reduces the load capacity of slabs compared to steel reinforcement [26]. Similarly, the d_pmax recorded in this study was 273 mm, representing 99.3% of the average d_pmax (275 mm) from Lopes et al's experiments. It is worth noting that the d_pmax value in this study, although not explicitly detailed in earlier sections, corresponds to a slab rotation (θ) of 0.0446 rad/m at the maximum torsional moment for slab S-30-8, as shown in Figs. (5 and 6). As described in Section 5.2, the relationship between rotation and displacement was calculated using Equation (4).