Flexural design of concrete beams reinforced with FRP rebars

Abstract The corrosion of steel rebars is the main cause of reinforced concrete degradation, which results in increasing costs with structural rehabilitation and repairs. As a solution, corrosion resistant rebars, such as those of FRP – Fiber-reinforced polymer –, have been used to replace conventional steel. This paper describes the development of a design program that calculates the flexural FRP reinforcement of T-shape beams. The possibilities as regards the neutral axis position, failure mode and concrete linear or non-linear behavior define the design scenarios for which their respective equations were deduced. The flexural strengths computed using the deduced equations showed agreement with experimental results for 125 beams, validating the proposed methodology. Since FRP rebars are vulnerable to creep rupture, the sustained stresses must be lower than the maximum allowed by ACI 440.1R-15, which may require increases in areas, modifying the flexural strength. Therefore, the equations to compute the new neutral axis depth and flexural strength based on the adjusted area were deduced and implemented in the computational program. Subsequently, this paper presents design examples considering all scenarios for which the equations were deduced. The design of one T-section considering different FRP rebars combined to normal and high-performance concretes is also reported. The results showed that beams reinforced with aramid and glass FRP required large areas to avoid creep rupture, whereas the areas of those reinforced by carbon FRP rebars were considerably small; however, they exhibited small curvatures and fragile failure when under-reinforced.


RESEARCH SIGNIFICANCE
As previously mentioned, the costs with structural repairs have considerably increased, which suggests the gradual replacement of steel by non-metallic reinforcements such as FRP. However, there is no code approaching the design of FRP flexural members, which means the responsibility for structural safety and functionality is entirely attributed to the designer [6]. Furthermore, in spite of the excellent design examples presented by ACI 440.1R-15, this guideline does not address the design of T-shape sections. All examples refer to rectangular sections, with priority to compressioncontrolled members.
T-shape beams often occur in practice, given the need to consider the slab contribution to the flexural strength. Their design with FRP rebars is not as simple as that of steel, especially for under reinforced cross-sections where the parameters α and λ of the concrete simplified stress block are unknown. There are several design approaches that depend on the neutral axis position associated to FRP and concrete simultaneous failure, as well as that related to the initiation of the concrete non-linear behavior at the most compressed fiber [8].
Therefore, the development of computational programs incorporating the FRP constitutive models and safety factors to a particular reinforced concrete code has the potential to familiarize students and engineers with the design of Tshape beams reinforced with FRP, given the need to replace steel by durable and sustainable materials. Furthermore, they are able to calculate the same cross-section for different FRP types and concrete strength grades, electing the design that combines structural safety and functionality to economic solutions.

SCOPE
This paper incorporates the FRP parameters provided by ACI 440.1R-15 [3] to the Brazilian code NBR 6118:2014 [9], deducing the formulations to calculate the FRP longitudinal reinforcement of tension and compression-controlled T-shape sections. The design assumptions accounted for failure due to concrete crushing and FRP rupture, neutral axes on the flange or web and concrete exhibiting linear or non-linear behavior. Additionally, considering that the reinforcement areas may be adjusted to avoid creep rupture, the assumptions and formulations to compute the final flexural strength are also described. Those formulations are validated by comparing the flexural capacities obtained experimentally to those predicted by the proposed procedures.
The primordial objective was to computationally implement those formulations, developing a design program that calculates the longitudinal FRP reinforcement of T-sections under different scenarios of failure mode, neutral axis position and concrete linear or non-linear behavior. The objective was to identify effective combinations of FRP and concrete strengths resulting in balanced sections, ductility and proper use of materials' mechanical properties. Furthermore, this research aimed to evaluate changes in failure modes, flexural strengths and curvatures caused by adjustments in the FRP areas to avoid creep rupture.

DESIGN FOR THE ULTIMATE LIMIT STATE FOR FLEXURE
In order to develop the formulations for the design, the concrete constitutive model of NBR 6118:2014 [9] as well as its parameters were utilized. Figure 1 illustrates the parabola-rectangle model defined by Equations 1 and 2. � n � if 0 ≤ c < c2 (1) cd = 0.85fcd if c2 ≤ c< cu (2) The FRP RC cross-sections fail due to concrete crushing or reinforcement rupture. ACI 440.1R-15 [3] introduces the concept of balanced reinforcement ratio, for which both failure modes occur simultaneously. The balanced area A b is associated to the neutral axis position x b and to the balanced moment M b . Unlike ACI 440.1R-15, Barbosa [8] compares the design moment M d to M b to define the failure mode. If M d < M b , the cross-section is tension-controlled, whereas M d ≥ M b indicates compression-control. Since concrete and FRP reaches their ultimate strains ε cu and ε fud simultaneously, x b is obtained through compatibility as:  [9] design procedures, the adjusted strength for environmental conditions still needs another factor. As indicated by the Brazilian Recommended Practice for FRP RC structures CT 303 [10], the value of 1.30 was adopted. Thus, the design tensile strength f fud accounts for the environment and eventual uncertainties covered by this factor. Figures 2 and 3 illustrate balanced sections for which the concrete simplified stress block reaches the flange and web, respectively, resulting in two different approaches to calculate M b . If λ u x b < h f , the calculation of the balanced moment accounts only for the flange compressed area. Otherwise, if λ u x b ≥ h f , both flange and web compressed areas are considered. Equations 4 and 5 define the balanced moments for both scenarios, following the order they were mentioned:  If the balanced block depth λ u x b does not reach the web and the design bending moment is lower than the balanced one, the cross-section is tension-controlled and the actual stress block λ u x associated to M d is smaller than the flange thickness [7). Moreover, if the cross-section is too under-reinforced, the FRP rebars may fail before the concrete exhibits non-linear behavior. As a result, the values of α u and λ u no longer applies, which suggests a linear constitutive model for concrete [3]. Therefore, this scenario results in two design approaches distinguished by the concrete behavior. In contrast, if M d ≥ M b , the actual stress block may reach the web or not, leading to other two approaches [8].
Conversely, if the balanced stress block λ u x b reaches the web and M d < M b , there are two possibilities as regards the stress block associated to the design moment: λ u x < h f or λ u x ≥ h f . Furthermore, for each of these two possibilities, concrete may behave linearly or not, resulting in other four design approaches. However, if M d ≥ M b , the actual stress block reaches the web [8]. Therefore, there are nine different design approaches considering only the ultimate limit state for flexure. All formulations are deduced as follows:

Scenario 1 -Balanced block on the flange and tension-controlled section
This scenario is characterized by λ u x b < h f and M d < M b , which implies that λ u x < h f . However, given the assumption of concrete behaving linearly, it is necessary to define the neutral axis position and the bending moment from which linearity no longer applies. NBR 6118:2014 [9] allows considering the linear stress-strain relationship for stresses under 50% of the concrete compressive strength. Accordingly, this research adopted the maximum stress of 0.5(0.85f cd ), which provides the strain ε clin from Equation 1 as ε c2 (1-0.5 1/n ). Therefore, the neutral axis x lin related to the linearity limit and obtained through strain compatibility is: Since the balanced block lies on the flange and the section is tension-controlled, x lin is smaller than the flange thickness [8]. Figure 4 illustrates the equilibrium and compatibility conditions to calculate the moment M lin associated to x lin considering the linear stress-strain relationship for concrete. Thus, if the design moment M d is lower than the reference moment M lin provided by Equation 7, the linear constitutive model applies [8].
The secant elasticity modulus E lin adopted for the linear approach corresponds to the slope of the line connecting the origin to the point (ε clin , 0.425f cd ). In order to find the unknown neutral axis depth x, the values of M lin and x lin in Equation 7 are replaced by M d and x, respectively. The stress 0.425f cd , in turn, is replaced by the product of the secant modulus E lin and the most compressed fiber strain ε t , written as a function of x. As a result, there is a cubic equation whose solution within the interval 0 < x < h f corresponds to the neutral axis depth: Equation 8 is solved through the Newton-Raphson Method, initially assuming that x = 1.5x b . The iterative process ends once the error |x i+1 -x i | reaches 10 -5 , which means the neutral axis depth has been found. Negative roots or values exceeding x lin are computationally disregarded. By imposing equilibrium, the required area A f to resist M d is found through: Conversely, if the design moment M d is higher than M lin , the linear approach no longer applies seeing that the stress in the most compressed fiber exceeds 50% of 0.85f cd . Although concrete does not fail, ACI 440.1R-15 recommends using, as a conservative approach, the simplified stress block associated to the crushing of the concrete. In this scenario, the parameters α cu and λ u are calculated as follows, in consonance with NBR 6118:2014 [9]: By imposing the equilibrium conditions illustrated in Figure 5, the neutral axis and the required FRP area are computed as follows: Regardless on the concrete behavior, the strain at the most concrete compressed fiber ε t is computed through compatibility as:

Scenario 2 -Balanced block on the web and tension-controlled section
If the balanced stress block reaches the web and the cross-section is tension-controlled, there are four possibilities: first, the actual stress block is on the flange and the concrete linear approach applies; second, the stress block remains on the flange but concrete exhibits non-linear behavior; third, the stress block reaches the web while concrete behaves linearly and fourth, the linear approach no longer applies for the stress block on the web [8].
Furthermore, the reference neutral axis x lin may be either on the flange or web, which results in two different methods to compute the reference moment M lin . If x lin < h f , M lin is determined from Equation 7; otherwise, the compressive stresses on the flange and web must be considered as shown in Figure 6. First, it is necessary to determine the resulting compressive force F lin from Equation 15 and its center y � lin from Equation 16. Therefore, the reference moment M lin is computed as the product of F lin and the moment arm d -y � lin .
If x lin < h f and M d < M lin , the neutral axis x and the reinforcement area A f are computed through Equations 8 and 9, the same as for scenario 1. Conversely, if x lin ≥ h f and M d < M lin , the neutral axis x associated to M d may be on the flange or web. Initially, it is assumed that x < h f , for which Equation 8 applies. If the value found for x is smaller than the flange thickness, the assumption is correct, and the required area is computed through Equation 9. Nonetheless, if x ≥ h f , the assumption is incorrect, and the value of x is not valid. Consequently, the calculation of the neutral axis depth and the required FRP area must account for the compressive stresses on the flange and web as shown in Figure 6. Thus, the variables x lin and ε clin in Equations 15 and 16 are replaced by the neutral axis depth x and the strain at the most compressed fiber ε t , respectively. Since the strain ε t can be written as a function of x, the neutral axis depth becomes the only unknown variable. As a result, the expression F c (d -y � c ) = M d results in a third-degree equation described as follows: Equation 20 is solved exactly as Equation 8, through the Newton-Raphson Method, initially arbitrating x as 1.5x b and ceasing the iterative process once the error becomes smaller than 10 -3 . Entering the value of x in F c leads to the resulting compressive and tensile forces, which allows determining the required FRP area as: However, if x lin < h f and M d ≥ M lin , concrete behaves non-linearly and the simplified stress block related to M d may reach either only the flange or the web. First, it is assumed that λ u x < h f so that x is computed through Equation 12. If the found neutral axis depth is smaller than h f /λ u , the assumption is confirmed, and the reinforcement area computed through Equation 13. In contrast, if x ≥ h f /λ u , the assumption is incorrect, and the calculation of x needs to account for both flange and web compressed areas. Figure 7 illustrates the cross-section analysis for two bending moments: M dw and M df , accounting for the areas b w x and (b f -b w )h f , respectively. They are directly determined as: Since the neutral axis position relies on the compressed area b w x and the moment M dw , its depth x is computed as follows: The areas A fw and A ff illustrated in Figure 7 are calculated to resist the moments M dw and M df , respectively. Thus, the total reinforcement area A f corresponds to: This approach is also valid if x lin ≥ h f and M d ≥ M lin considering that concrete exhibits non-linear behavior and the stress block λ u x associated to M d reaches the web. For all these possibilities, the concrete most compressed fiber ε t is determined in the same manner as for the scenario 1, through Equation 14.

Scenario 3 -Balanced block on the flange and compression-controlled section
If the balanced stress block lies on the flange and the design moment is higher than the balanced one, the crosssection is over-reinforced and the stress block λ u x can reach the web or not. First, it is assumed that λ u x < h f , with x obtained from Equation 12. If the value found for x confirms this assumption, the next step consists of computing the reinforcement stress f f . Since FRP exhibits linear elastic behavior, the compatibility conditions illustrated in Figure 8 allows determining f f directly from its strain as: The required area A f found through equilibrium is calculated as follows: However, if the neutral axis depth obtained from Equation 12 is equal to or higher than h f /λ u , the assumption is incorrect, and the stress block reaches the web. Figure 9 depicts the cross-section analysis for the calculation of the neutral axis depth and reinforcement area.  The moments M df , M dw as well as the neutral axis depth x are obtained from Equations 22, 23 and 24. This approach is the same as that of tension-controlled sections with stress block reaching the web and concrete behaving non-linearly. Nonetheless, the reinforcement stress is not equal to f fud since the FRP rebars do not fail. Therefore, the stress in the FRP layer is obtained from Equation 26, and the total reinforcement area A f from:

Scenario 4 -Balanced block on the web and compression-controlled section
Unlike the scenario 3, λ u x b ≥ h f and M d ≥ M b , which ensures that the stress block depth λ u x is equal to or larger than the flange thickness. Therefore, the compatibility and equilibrium conditions for this scenario are also illustrated in Figure 9. The neutral axis depth x and the reinforcement area A f , in turn, are computed through Equations 24 and 28, respectively.

CHECKING FOR FRP CREEP RUPTURE
The required area A f to meet the ultimate limit state for flexure may not be enough to avoid creep rupture due to sustained stresses [8]. ACI 440.1R-15 establishes that such stresses must not exceed 20, 30 and 55% of the tensile strength f fu for GFRP, AFRP and CFRP, respectively. Accordingly, the tensile strength f fu for this verification accounts only for the environmental conditions, not incorporating other safety factors.
To determine the sustained stresses f fs , the load combination defined as almost permanent by NBR 6118:2014 [9] was implemented. Additionally, Equations 29 to 31 developed by Ghali and Favre [11] were used to calculate the neutral axis depths x cr under service conditions as shown in Figure 10. The parameter η f , in turn, refers to the modular ratio E f /E cs .
x cr = -a 2 +�a 2 2 -4a 1 a 3 2a 1 The cracking moment of inertia I cr depends on the neutral axis position. Equations 33 and 34 apply for x cr < h f and x cr ≥ h f , respectively [12]. To find the sustained stress f fs , ACI 440.1R-15 and NBR 6118:2014 [9] adopt the linear approach defined in Equation 35. The moment M apc refers to the almost permanent load combination. 2 (33) If the sustained stress exceeds the maximum allowed by ACI 440.1R-15, areas of 0.001 cm 2 are progressively incremented to A f , for which Equations 29 to 35 are computationally solved for each adjustment. The sustained stress decreases continuously, and the final adjusted area A adj is that making the sustained stress equal to or slightly lower than the maximum allowed. Thus, the area A adj meets both limit states for flexure and creep rupture.

DETERMINATION OF THE FLEXURAL STRENGTH
Because of increments in FRP areas to meet both limit states, the flexural strengths and the neutral axis depths increase. Consequently, the simplified stress block previously located on the flange may reach the web, the failure mode may switch from tension to compression-controlled and for the cross-sections that remain tension-controlled, the concrete linear behavior may no longer apply [8].
In order to determine the failure mode, the adjusted area is compared to the balanced one A b and, in case of tensioncontrol, A adj is compared to A lin , the area from which the concrete linear approach no longer applies. There are four scenarios as regards the determination of the flexural strength, explained as follows:

Tension-Control and balanced block on the flange
If A adj < A b , the cross-section is under-reinforced with failure characterized by the FRP rupture. Additionally, if the balanced block depth is smaller than the flange thickness, the stress block associated to the adjusted area λ u x adj does not reach the web. Furthermore, if A adj < A lin , the concrete stress-strain relationship can be considered as linear [8].
By imposing equilibrium in Figure 4, the area A lin for which the stress in the most compressed fiber corresponds to 50% of 0.85f cd is: If A adj < A lin , the adjusted neutral axis depth x adj associated to A adj is computed by imposing the equilibrium and compatibility conditions illustrated in Figure 4, which leads to Equations 37 and 38. Once x adj is found, the flexural strength M r is, thus, obtained from Equation 39.
Conversely, if A adj ≥ A lin , the simplified stress block represents the concrete constitutive model seeing that the linear approach no longer applies. By imposing the equilibrium conditions shown in Figure 5, the adjusted neutral axis and the flexural strength are determined as follows:

Tension-control and balanced block on the web
If A adj < A b and the balanced stress block reaches the web, the block associated to the adjusted area as well as the reference neutral axis x lin can be located either on the flange or web. If x lin < h f and A adj < A lin , the neutral axis associated to A adj is on the flange [8]. Therefore, the adjusted neutral axis and the flexural strength are obtained from Equations 37, 38 and 39.
However, if x lin < h f and A adj ≥ A lin , concrete exhibits non-linear behavior and the adjusted stress block may be located either on the flange or web. First, it is assumed that λ u x adj < h f , which allows computing the adjusted neutral axis and flexural strength through Equations 40 and 41, respectively. However, if the value found for x adj is equal to or higher than h f /λ u , the assumption is incorrect; the stress block reaches the web [8]. Therefore, both flange and web compressed areas must be considered. Imposing equilibrium and compatibility for the cross-section illustrated in Figure 7 leads to the correct values of x adj and M r , computed as follows: In contrast, if x lin ≥ h f , Equation 36 no longer applies to compute A lin since the linear distribution of the compressive stresses extends to the web, as shown in Figure 6. Therefore, A lin corresponds to the ratio between F lin , defined in Equation 15, and the FRP design tensile strength f fud . If A adj < A lin , the adjusted neutral axis may be on the flange or web. The cross-section analysis illustrated in Figure 4 as well as Equations 37 and 38 determine x adj for the assumption x adj < h f . However, if the solution of such equations provides x adj ≥ h f , the assumption is invalid and the correct value of x adj is obtained considering the cross-section analysis shown in Figure 6.
Subsequently, by imposing that the compressive force F adj equals to the ultimate reinforcement load A adj f fud , the adjusted neutral axis depth is computed as follows: x adj = a 1 �� The compressive force center y � adj associated to x adj is obtained from Equation 16, replacing x lin by x adj . Therefore, the adjusted flexural strength corresponds to: Finally, if x lin ≥ h f and A adj ≥ A lin , the stress block associated to A adj reaches the web and the concrete exhibits nonlinear behavior. Thus, the adjusted neutral axis depth and the flexural strength are obtained from Equations 42 and 43, according to the cross-section analysis illustrated in Figure 7.

Compression-control and balanced block on the flange
If the balanced block is located on the flange and the section is compressed-controlled, the stress block associated to A adj may reach the web or not. First, it is assumed that λ u x adj < h f , corresponding to the analysis illustrated in Figure  8. Since the reinforcement does not fail, its strain is unknown, written as a function of x adj [8]. Therefore, since the resulting compression and tension forces are equal, the adjusted neutral axis is obtained as follows: If the value found for x adj confirms the assumption that λ u x adj < h f , the flexural strength is: Conversely, if x adj ≥ h f /λ u , the assumption is incorrect and Equations 49 and 50 do not apply. It is necessary to consider the compressive stresses on the flange and web as shown in Figure 9, establishing equilibrium of forces and strain compatibility. Thus, x adj is computed as: The flexural strength is obtained from Equation 43, the same as for tension-controlled sections whose neutral axis is on the web and concrete behaves non-linearly [8].

Compression-control and balanced block on the web
If the balanced block reaches the web and the cross-section is compression-controlled, the adjusted neutral axis depth is larger than the balanced one, which means that λ u x adj ≥ h f as well. Therefore, x adj is computed through Equations 51, 52 and 53 while the flexural strength through Equation 43. The only difference from the previous scenario is that the designer knows for sure that the stress block reaches the web.

EXPERIMENTAL VALIDATION
To validate the proposed methodology, the design equations were used to inversely compute the flexural capacities of 125 beams to posteriorly compare with experimental results. The details of all specimens are shown in Table 1, where the reference in brackets indicates the experimental program related to a group of specimens. The T-section dimensions are given as bw/bf and h/h f in the fields corresponding to b and h, respectively. The abbreviations TC and CC refer to the tension and compression-controlled failure modes, respectively.
The concrete compressive strengths were obtained experimentally, mostly from testing cylinders in uniaxial compression after 28 days. The majority of experimental programs obtained the FRP mechanical properties from direct tensile tests; others provided only the manufacturer data, as indicated with a * in Table 1. The majority of the beams were tested under four-point bending, with the load applied at a steady rate of 0.8 to 1.2 mm/min or at steps of 2 to 5 kN. Those from [25] and [20], in turn, were tested under three-point loading. All specimens exhibited flexural failure either due to FRP rupture or crushing of the concrete.
Since the experimental flexural capacities are influenced by the actual mechanical properties of materials, the reduction factors for the concrete compressive and FRP tensile strengths were not included in the analytical analysis. The term 0.85 in αcu was also suppressed to account for the short-term loading inherent to the experimental programs. Moreover, since some beams had multiple reinforcement layers, the compatibility and equilibrium equations were adapted to account for different reinforcement distributions. The theoretical and experimental ultimate moments were plotted along with the identity line, and the accuracy of the analytical model assessed through the coefficient of determination R 2 .

RESULTS AND DISCUSSION
This section addresses two aspects of the results: the experimental validation of the proposed methodology and the application of the design program, considering different examples for each scenario described in Section 4. Table 1 summarizes this comparison for each group of beams, considering different concrete grades and FRP types, as well as beams of rectangular and T-sections. The average ratio between theoretical and experimental moments M th /M exp corresponds to 1.0006, with mean deviation of 0.10 and coefficient of determination R 2 = 0.962, which suggests agreement of the analytical methods with respect to experimental results. Figure 11 illustrates the scattering of the data in relation to the identity line.

Program Interface
Using the developed design program, Figure 12 illustrates the calculation of a T-shape section reinforced with AFRP rebars. The user chooses the FRP type and inserts its mechanical properties. Yet, they are not able to define tensile strengths and elasticity moduli out of the intervals defined by Table 4.2.1 of ACI 440.1R-15 for each FRP type. For this example, in particular, the design bending moment corresponds to 98% of the balanced one, which indicates proper use of AFRP and concrete mechanical properties. Nonetheless, since the sustained stress due to the almost permanent load combination is higher than the maximum allowed by ACI 440.1R-15, the required area was increased by 1% to avoid creep rupture. This increment in the FRP area was not enough to switch the failure mode from tension to compression-controlled, resulting in a flexural strength 0.8% higher. This slight increase was not enough to characterize waste of the AFRP rebars' mechanical properties.
If a high-performance concrete with fck = 90 MPa is used, the cross-section becomes too under-reinforced, with flexural strength corresponding to 38% of the balanced moment. As a result, there is waste of the concrete's mechanical properties since the strain at the top corresponds to only 29% of the ultimate strain ε cu . In conclusion, the grade that best fits the reinforcement type for this particular load condition is the grade C20, which allows taking advantage of both concrete and FRP mechanical properties.

Examples considering different design scenarios
Since there are several approaches to compute the required FRP area, Table 2 describes the design of the same Tshape section illustrated in Figure 12, considering all the possibilities presented in Section 4. The approach 1A and 1B refer to the first scenario, considering the concrete linear and non-linear behavior, respectively. In turn, 2A indicates concrete linearity and x < h f , whereas 2B, non-linearity. Additionally, both 2C and 2D refer to x ≥ h f with concrete behaving linear and non-linear, respectively. Regarding the third scenario, the approaches 3A and 3B refer to λ u x < h f and λ u x ≥ h f , respectively; whereas the only possibility for the fourth scenario, 4A, corresponds to λ u x b ≥ h f and M d ≥ M b . Regarding the cases 1A and 1B, the results showed that the balanced block is located on the flange when highperformance concretes are used in conjunction with FRP rebars exhibiting large ultimate strains. If the cross-section is tension-controlled, the concrete linear stress-strain relationship is more likely to apply if the applied bending moment is considerably lower than the balanced one, especially if high-performance concretes are used.
In contrast, the second scenario is characterized by the use of FRP with low deformability compared to first scenario. As a result, the balanced block reached the web. However, since the design bending moments regarding 2A and 2B are significantly lower than the balanced ones, their respective stress blocks fell on the flange. Even though the design moments corresponding to 2C and 2D are lower than the balanced ones, their respective neutral axes are positioned on the web. However, the linear approach applied only for 2B.
Alike the possibilities 1A and 1B, the association of high-performance concretes with large deformability FRP caused the balanced block to reach only the flange for cases 3A and 3B. In contrast, the applied moments are higher than the balanced ones, making stresses blocks fall on the flange and web, respectively. Regarding 4A, the association of concrete and FRP is similar to that of scenario 2, with the balanced block reaching the web.
The design for the ultimate limit state and verification for creep rupture are described in Table 3. The adjustments in AFRP and GFRP areas for 1A, 2A and 2B resulted in deeper neutral axes xadj and increased top concrete strains ε c , not reducing the reinforcement strains ε f . As a result, the cross-section curvatures φ slightly increased, improving ductility. The CFRP RC compression-controlled sections 3A and 3B, in turn, exhibited the largest curvatures while the tension-controlled 2C, the smallest.
Since the cross-sections 2A, 2B and 2C are tensile-controlled with small curvatures, the designer may either increase the amount of reinforcement up to the balanced area A b or decrease the compressive strength f ck , which would lead to larger curvatures. However, if the section becomes compression-controlled, a further increase in A f or decrease in f ck would deepen the neutral axis and reduce the FRP strain, leading to smaller curvatures.  Table 3. Design for the ultimate limit state and checking for creep rupture for all approaches Case x (cm) Af (cm 2 ) Mapc (kNm) ffs (MPa) Aadj (cm 2 ) Ab (cm 2 ) Failure xadj (cm) εc (‰) εf (‰) Mr (kNm) 10 3 φ (rad/m)