Flexural performance of concrete beams reinforced with steel–FRP composite bars

Abstract

Flexural performance of concrete beams reinforced with steel–FRP composite bar (SFCB) was investigated in this paper. Eight concrete beams reinforced with different bar types, namely one specimen reinforced with steel bars, one with fiber-reinforced polymer (FRP) bars and four with SFCBs, while the last two with hybrid FRP/steel bars, were tested to failure. Test results showed that SFCB/hybrid reinforced specimens exhibited improved stiffness, reduced crack width and larger bending capacity compared with FRP-reinforced specimen. According to compatibility of strains, materials’ constitutive relationships and equilibrium of forces, two balanced situations, three different failure modes and balanced reinforcement ratios as well as analytical technique for predicting the whole loading process are developed. Simplified formulas for effective moment of inertia and crack width are also proposed. The predicted results are closely correlated with the test results, confirming the validity of the proposed formulas for practical use.

1 Introduction

The use of FRP bars as an alternative reinforcement in concrete structures has shown to be a valid way to overcome the durability issues of steel-reinforced concrete structures, resulting from the corrosion of steel reinforcement. Over the past two decades, the performance of FRP-reinforced concrete structures has received significant attentions, and this new reinforced system is being widely used in marine structures, hydraulic structures, high-speed railway and subway structures. Masmoudi [1] experimentally and theoretically investigated the effect of FRP reinforcement ratio on cracking, deflection, bending capacity and modes of failure of concrete beams. Grace [2] carried out the experimental study on the mechanical behavior of reinforced concrete beams strengthened by FRP laminates. Pecce [3] discussed the structural behavior, such as curvature, deflection, and crack spacing and width, of concrete beams reinforced with glass FRP bars and analyzed verifications at ultimate and serviceability conditions. Aiello [4] investigated the deformability of concrete flexural members reinforced with FRP rebars. Gravina [5] conducted comparative tests on simply supported and continuous concrete beams reinforced with FRP bars to predict the bending moment distribution. Xue [6] proposed a calculation method for deflections of concrete beams reinforced with FRP rebars. Qi [7] conducted an experimental study on cracking patterns, load–deflection response, load–moment relationship, internal force redistribution and ultimate load of continuous beams reinforced with FRP rebars. Tu [8] conducted an experimental study on simply supported beams reinforced with bonded AFRP tendons, and an effective calculation method for the ultimate load-carrying capacities was derived. Skuturna [9] studied the design methods for calculating the load-carrying capacity of reinforced concrete elements in flexure strengthened with external FRP reinforcement. Lapko [10] reported that much lesser cross-sectional stiffness of basalt BFRP bars produces higher deflections and crack widths compared to the beams reinforced with steel bars of the same cross section. Zhou [11] investigated the in-plane seismic behavior of eight unreinforced masonry walls before and after being retrofitted by BFRP.

Mahroug [12] showed that continuously supported BFRP-reinforced concrete slabs exhibited larger deflections, wider cracks and brittle failure compared with the counterpart reinforced with steel. Such behavior is attributed to the fact that FRP reinforcing bars exhibit a linear elastic stress–strain relationship up to failure without any yielding, causing brittle failure without enough warning to RC structures user. Therefore, few suggestions were recently proposed to improve the ductility of FRP-reinforced concrete members.

A hybrid system consisting of both FRP and steel reinforcement was introduced to improve reinforced concrete element ductility and durability [13,14,15,16,17,18,19,20,21]. In such reinforcement system, FRP reinforcement is located in the outer layer, whereas steel bars are embedded more deeply, achieving larger cover concrete, combining the advantages of FRP and steel reinforcement simultaneously by improving durability and ductility as well as reducing deflection and crack width. Lau [13] proposed that steel longitudinal reinforcement should be added to form a hybrid FRPRC beam to improve its ductility. Huang [14] conducted an experimental and theoretical study on the mechanical behaviors of steel–GFRP-reinforced concrete beams. Ge [15, 16] investigated the flexural behavior of hybrid FRP/steel-reinforced concrete beams and ECC–concrete composite beams. The test results showed that the ductility of hybrid reinforced composite beams is higher than that of traditional RC beams and formulas for cracking, yield and ultimate moments as well as deflections of hybrid reinforced beams are developed. Kara [17] presented a numerical method for estimating the curvature, deflection and moment capacity of hybrid FRP/steel-reinforced concrete beams. Refai [18] reported the structural performance of concrete beams reinforced with steel and GFRP hybrid reinforcement. Yoo [19] investigated the flexural behavior of UHPFRC beams reinforced with GFRP rebars and steel/GFRP hybrid reinforcements. Sun [20] conducted an experimental study on the flexural behavior of concrete beams reinforced with bundled hybrid steel/FRP bars. Maranan [21] investigated the flexural behavior of geopolymer concrete beams reinforced with a hybrid GFRP and steel bars.

Wu et al. [22] recently introduced a newly developed steel–FRP composite bar (SFCB), composed of inner steel bar wrapped by FRP, combining the advantages of the two materials. SFCBs exhibit high strength, good ductility, high elastic modulus, stable secondary stiffness and excellent corrosion resistance. Uniaxial static and cyclic tensile tests of SFCB showed a bilinear stress–strain relationship before FRP rupture. After the inner steel yielded, SFCB displayed a stable post-yielding stiffness. Few investigations showed the effectiveness of SFCB as structural reinforcement for concrete structures [23, 24]. However, further experimental and computational investigations of structural elements reinforced with SFCBs are essential to better understand their structural behavior and to encourage their use in real structures.

In this paper, static flexural experiments of SFCB, steel, FRP and hybrid RC beams were conducted and compared. The effect of reinforcement form and ratio on the bending capacity, crack width and deflection of RC specimens was studied. Theoretical analysis based on strain compatibility, realistic constitutive relationships and forces equilibrium is also conducted to predict the failure modes, bending capacity, crack width and deflections. Taking the properties of SFCB materials into consideration, modified formulas for the crack width and effective moment of inertia of SFCB RC flexural components are also proposed.

2 Experimental design

2.1 Mechanical performance of material

2.1.1 Concrete

The main concrete ingredients were Portland cement (CEM 42.5), medium sand of grain diameter 0.35–0.5 mm, gravel of maximum size 15 mm and tap water. The mass ratio of water, cement, sand and gravel was 0.39:1.0:1.29:2.88, i.e., 168, 432, 558 and 1242 kg, respectively, for one cubic meter of concrete. The concrete compressive test was conducted at the same time as the flexural experiment of RC beams. The mean compressive strength f_{cu, mv} of concrete obtained from testing three 150 × 150 × 150 mm [25] cubes was 43.85 MPa. The modulus of elasticity E_c (= 10²/(2.2 + 34.7/f_cu)), compressive strength f_c (= 0.88α_c1α_c2f_cu) and tensile strength f_t (= 0.348α_c2f^0.55_cu) are 33.43 GPa, 28.65 MPa and 2.80 MPa, respectively [26], where α_c1 is the strength ratio of concrete prism to concrete cube and α_c2 is the brittleness reduction coefficient of high-strength concrete.

2.1.2 Longitudinal reinforcements

Five different types of longitudinal reinforcements were used, namely steel, basalt FRP (BFRP) and three SFCBs with different arrangements. Table 1 and Fig. 1a present details of the reinforcements used, where d and d_s are the full and steel diameters, OFT represents the out-wrapping fiber type, t_f is the thickness of outer FRP material, d_r and s_r are the depth and spacing of ribbed ribs, respectively. All reinforcing bars were of 12 mm diameter. Reinforcing bars A and E were fully made of steel and BFRP, respectively, whereas the other three bars had an inner steel core of diameter d_s, externally wrapped with FRP layer of thickness t_f achieving the full diameter d.

Table 1 Details of longitudinal reinforcement

Fig. 1

Tested longitudinal reinforcements

Tensile properties of FRP and SFCB reinforcements were obtained by testing three specimens for each type [27]. The total length, anchorage and free length of the specimen were 1200, 400 and 400 mm, respectively, as shown in Fig. 1b. Figure 2 and Table 2 present the tensile stress–strain curves and mechanical properties of tested bars, respectively; the values of modulus of elasticity as well as the strengths are the mean values from three test specimens. E_I and E_II are the moduli of bars before and after yielding, respectively; f_u and f_y are the ultimate and yield stress of bars, respectively. As reinforcing bars E are purely made from FRP material, only E_I is measured and presented.

Fig. 2

Tensile stress–strain curves of tested bars

Table 2 Mechanical performance of tested bars

As can be seem from Fig. 2 and Table 2, the three SFCBs, all, exhibit stable secondary stiffness. At the beginning, the load was carried by the outside fiber and inner steel simultaneously; the higher the proportion of inner steel, the higher the elastic modulus E_I and yield strength f_y. The SFCB appeared to yield when its strain is up to about 0.002 (the yield strain of inner steel). This is illustrated by the fact that SFCB had fewer, but stably increasing, stress increments with the same strain increment, which means that SFCB exhibited stable stiffness after inner steel yielded (post-yield stable secondary stiffness). Because the inner steel had already yielded, the load was mainly resisted by the outside fiber. As the load increased, the loading capacity reached its peak when the fiber fractured at the middle part of the specimen [22].

Figure 3 presents the typical failure mode of each type of refinements. In all specimens, failure occurred at the middle region of the specimens, indicating that the end anchorage was effective. The steel–FRP composite bar underwent a threadlike blowout fracture, illustrating that the resin component of SFCB had better coupling performance with fiber [22].

Fig. 3

Failure modes of tested bars

2.1.3 Bond-slip behavior

Standard pullout experiments were conducted to test the bond-slip behavior between the five reinforcing bars and concrete [28]. Figure 4a–c represents the schematic diagram of pullout specimen, pullout setup and bond stress–strain curves, respectively.

Fig. 4

Pullout test

Plastic pipes were embedded to reduce the local stress at the load end during testing. The loading was applied by a hydraulic jack and measured by a load sensor; the slip of the free end was measured by a displacement sensor attached to the bar.

Table 3 presents the bond strengths of each reinforcing bar with concrete, while Fig. 5 shows the tested bond-slip specimens, where the bond strength of concrete and reinforced bars, τ, is the stress corresponding to a bond slip of 1.0 mm [29].

Table 3 Bond properties of reinforcement

Fig. 5

Failure mode of tested bond-slip specimens

All specimens’ anchorage lengths were 5d to promote the same failure mode of pullout. As can be seen from Table 3, the bond strength ratio τ_i/τ_A of SFCBs/BFRP bar to steel bar is all approximately equal to 1.0, illustrating that the ribbed SFCBs/BFRP bar has very similar bond strength as ribbed steel bar.

Table 4 presents the anchorage length with respect to the bar diameter, n_a = L_a/d, where L_a is the anchorage length, of various reinforced bars. The anchorage length is obtained by considering the balance failure case, where bar pullout and rupture simultaneously occur; f_uπd²/4 = τπdL_a, d is the bar diameter, τ is the bond strength of concrete and reinforced bars, f_u is the tensile strength of the reinforced bars and L_a is the anchorage length of the reinforced bars. Rearranging the above formula, the anchorage length with respect to the bar diameter can be obtained, n_a = L_a/d = f_u/4τ, as presented in Table 4.

Table 4 Details of designed specimen

2.2 Specimens design and testing program

In total, eight specimens were tested to investigate the flexural performance, one steel RC specimen, one FRP RC specimen, two hybrid steel/FRP RC specimens and the other four SFCB RC specimens. Details of specimens are shown in Table 4, and reinforcement details are presented in Fig. 6. The cross section width is b =120 mm and height is h = 180 mm. The height of the centroid of reinforced bars to the cross section extreme tensile fiber is h_r = 25 mm, and the effective height of cross section is h₀ = 175 mm. ρ represents the practical reinforcement ratio, ρ = A/(bh₀), A is the cross section area of reinforced bars; ρ_nE represents the nominal elastic modulus reinforcement ratio, for SFCB RC specimen, ρ_nE = ρE_I/E_s, and for FRP/steel hybrid RC specimens, ρ_nE = ρ_s + ρ_fE_f/E_s; ρ_ns represents the nominal strength reinforcement ratio, for SFCB RC specimens, ρ_ns = ρf_sfu/f_y, and for FRP/steel hybrid RC specimens, ρ_ns = ρ_s + ρ_ff_fu/f_y. F_u represents the tensile capacity of longitudinal reinforcement, for SFCB RC specimens, F_u = f_sfuA_sf, for FRP RC specimens, F_u = f_fuA_f, for steel RC specimens, F_u = f_uA_s and for hybrid RC specimens, F_u = f_fuA_f + f_uA_s. A_f and A_s are the cross section areas of FRP and steel reinforcement, respectively. ρ_f and ρ_s are the reinforcement ratio of FRP and steel reinforcement, respectively. f_fu and f_sfu are the ultimate stress of FRP and SFCB reinforcement, respectively. f_u and f_y are the ultimate and yield stress of steel bars, respectively. E_f, E_s and E_I are the elastic modulus of FRP, steel and SFCB before the yielding of inner steel, respectively.

Fig. 6

Schematic diagram of test specimens

2.3 Loading and testing program

Figure 7 presents the loading and testing system. The moment was controlled by a distribution beam on the top of the specimen and applied by a hydraulic jack, while its value was tested by a load sensor (measurement resolution 0.10 kN, accuracy ± 0.10 kN) laid on the top of the jack. Electrical resistance strain acquisition instrument TDS-530 was used to capture loading at various stages. Dial indicators (type I, measurement resolution 0.01 mm, accuracy ± 0.02 mm) were located at mid-span, supporting and loading points. Cracks distributed between the pure flexural span were marked, and their widths (at the height the same as the centroid of reinforcement) were measured and recorded at various loading stages by crack width measuring instrument KON-FK(B) (measurement resolution 0.02 mm, accuracy ± 0.02 mm). Dial indicators equidistant (type II, measurement resolution 0.001 mm, accuracy ± 0.003 mm) with 200 mm measurement length along the height direction (10, 45, 90, 135 and 170 mm, respectively) on the side of mid-span cross section were pasted to measure the average concrete strains under various applied loads.

Fig. 7

Loading and testing system

3 Experimental results and analysis

3.1 Distribution of concrete average strain

Figure 8 presents the distribution of concrete average strain along the cross section height at various loading stages, where M_u is the experimental ultimate bending moment.

Fig. 8

Distribution of concrete average strain

As observed from Fig. 8, the depths of neutral axis gradually move up with the increase in loading. The average concrete strain along the cross section height is almost a linear distribution, illustrating the validity of the plane section assumption that will be used later in the analytical development below. The depth of neutral axis of steel-reinforced specimen BA is the smallest, that of FRP-reinforced specimen BB is the largest, whereas that of SFCB/hybrid reinforced specimens is in between, reflecting reinforcing bars’ modulus as measured above.

3.2 Moment–deflection curves

Figure 9 presents the moment–deflection curves at mid-span, where d_lim is the deflection limit under serviceability state (that is 3.75 mm) [30] for all test specimens.

Fig. 9

Moment–deflection curves

As observed from Fig. 9, the beams tested display different features, depending on the characteristics of the reinforcement used. For beams reinforced with FRP bars, two distinct stiffnesses are clearly identified, namely before cracking of concrete and after cracking until concrete crushing. On the other hand, the loading process of SFCB/hybrid RC specimens obviously presents three stages: Stage 1: from initial loading until concrete cracking; Stage 2: after concrete cracking to yielding of SFCB reinforcement (inner steel); Stage 3: stable secondary stiffness after yielding until concrete crushing. The beam reinforced with steel bars exhibited similar behavior to that of SFCB/hybrid RC specimens, but after yielding, the beam stiffness was almost flat. The deflections of SFCB/hybrid RC specimens increase gradually with an increase in the applied load after the yielding of SFCB/steel. For steel reinforcement almost fully plastic after yielded, the deflections of steel RC specimens increase dramatically even the load does not increase.

Table 5 presents the comparison of crack, yield and ultimate moments of all specimens, where the yield moment of specimen BA and ultimate moment of specimen BB are taken as control moments: cracking moment M_cr corresponding to the crack of tensile concrete, yield moment M_y corresponding to the yielding of tensile steel or SFCB, and ultimate moment M_u corresponding to the crushing of compressive concrete.

Table 5 Comparison of capacity for bending

As observed from Table 5, all specimens have similar cracking moment; the yield moments increase with an increase in nominal elastic modulus reinforcement ratio. Ultimate moment increases (M_{u, BF} = 15.80 kN m < M_{u, BE} = 18.90 kN m) with an increase in the nominal elastic modulus reinforcement ratio (ρ_{nE, BF} = 0.41% < ρ_{nE, BE} = 0.69%) while specimens having similar nominal strength reinforcement ratio (ρ_{nf, BF} = 1.94% ≈ ρ_{nf, BE} = 1.98%).

Table 6 presents the comparison of deflections under quasi-permanent moments combination, where FRP RC specimen BB is taken as a control specimen. d_q,BB is the deflection corresponding to the quasi-permanent moments combination of control specimen BB (that is M_{q, BB} = 10.37 kN m). M_d,lim is the moment corresponding to the deflection limit d_lim. η_d,u = M_d,lim/M_u is the bending capacity utilization coefficient controlled by the deflection limit under serviceability state.

Table 6 Comparison of deflections and crack widths

As observed from Table 6, the deflections of specimens with high nominal elastic modulus reinforcement ratio (specimens BA and BG) are less than the deflection limit under serviceability state. The deflection of steel and FRP RC specimens corresponding to the quasi-permanent moments combination of control specimen BB d_q,BB is the smallest and largest, respectively; that of SFCB and hybrid reinforced specimens is between them; meanwhile, the deflection decreases with the increase in nominal elastic modulus reinforcement ratio. The bending capacity utilization coefficient controlled by the deflection limit under serviceability state η_d,u of steel and FRP RC specimens is the highest and lowest, respectively, and that of SFCB and hybrid RC specimens is between them; meanwhile, the coefficient increases with the increase in nominal elastic modulus reinforcement ratio.

3.3 Cracks and failure modes

Figure 10 presents the moment–crack width curves, where ω_lim is the crack width limit under serviceability state (that is 0.32 mm) [26, 31] for all specimens. Table 7 presents the comparison of crack widths under the quasi-permanent moments combinations, where FRP RC specimen BB is taken as a control specimen and ω_q,BB is the crack width corresponding to the quasi-permanent moments combination of control specimen BB. M_ω,lim is the moment corresponding to the crack width limit. η_ω,u = M_ω,lim/M_u is the bending capacity utilization coefficient controlled by the crack width limit under serviceability state.

Fig. 10

Moment–crack width curves

Table 7 Comparison of tested and predicted mechanical property of SFCB

As observed from Table 7, the crack widths of specimens with high nominal elastic modulus reinforcement ratio (specimens BA,BD, BE, BG and BH) are less than the crack width limit under serviceability state. The crack width of steel and FRP RC specimens corresponding to the quasi-permanent moments combination of control specimen BB ω_q,BB is the smallest and largest, respectively; that of SFCB and hybrid RC specimens is between them; meanwhile, the crack width decreases with the increase in nominal elastic modulus reinforcement ratio. The bending capacity utilization coefficients controlled by the crack width limit under serviceability state η_ω,u of steel and FRP RC specimens are the highest and lowest, respectively, and those of SFCB and hybrid RC specimens are between them; meanwhile, the coefficients increase with the increase in nominal elastic modulus reinforcement ratio.

Figure 11 presents specimens’ failure modes.

Fig. 11

Specimens’ failure modes

Typical appropriate reinforced flexural failure of concrete crushing occurred for all specimens. For steel RC specimen BA, as the steel reinforcement yielded, crack width and deflection significantly increased, until, finally, the top extreme concrete compressive fiber reached its ultimate strain and crushed. For SFCB RC specimens (BC, BD, BE and BF) and hybrid RC specimens (BG and BH), the development rates of crack width and deflection after yielding of steel also increase but at a slower rate compared with specimen BA as the external FRP wrapping was able to achieve a more stable behavior. However, as the load increased, the top extreme concrete compressive fiber reached its ultimate strain and crushed. For FRP RC specimen BB, the crack width and deflection were clearly larger than other specimens and eventually failed due to concrete crushing at the top extreme concrete compressive fiber. For all specimens, no signs of bond-slip occurred between the reinforcing bars and concrete.

4 Analysis of SFCB RC beams

4.1 Basic assumptions

4.1.1 Material constitutive model

• (1) Concrete

The constitutive relationship of concrete [26] is shown in Fig. 12.

Fig. 12

Constitutive relationship of concrete

The compressive constitutive relationship of concrete can be represented by Eq. (1):

$$\sigma_{\text{c}} = \left\{ {\begin{array}{*{20}l} {f_{\text{c}} \left( {1 - \left( {1 - \frac{{\varepsilon_{\text{c}} }}{{\varepsilon_{\text{co}} }}} \right)^{2} } \right),} \hfill & {0 \le \varepsilon_{\text{c}} \le \varepsilon_{\text{co}} } \hfill \\ {f_{\text{c}} ,} \hfill & {\varepsilon_{\text{co}} < \varepsilon_{\text{c}} \le \varepsilon_{\text{cu}} } \hfill \\ \end{array} } \right.$$

(1)

where ε_c and σ_c are the compressive strain and corresponding stress in concrete, respectively; f_c is the concrete compressive strength; ε_co and ε_cu are the strain, while the stress up to compressive strength and the ultimate compressive strain, respectively.

The tensile constitutive relationship of concrete can be represented by Eq. (2):

$$\sigma_{\text{ct}} = \frac{{f_{\text{ctu}} }}{{\varepsilon_{\text{ctu}} }}\varepsilon_{\text{ct}} ,\quad 0 \le \varepsilon_{\text{ct}} \le \varepsilon_{\text{ctu}}$$

(2)

where ε_ct and σ_ct are the tensile strain and corresponding stress in concrete, respectively, f_ctu and ε_ctu are the ultimate tensile strength and corresponding strain, respectively.

• (2) SFCBs

Figure 13 represents the simplified tensile constitutive relationship [22] of SFCB. It can be expressed by Eq. (3), where ε_sf and σ_sf are the tensile strain and corresponding stress in SFCB material, E_I and E_II are the modulus of elasticity before and after yielding of inner steel, respectively, f_sfy and f_sfu are yield and ultimate strength, respectively, of SFCB, and ε_sy and σ_fu are yield strain of inner steel and ultimate tensile strain of out-wrapped FRP material, respectively.

$$\sigma_{\text{sf}} = \left\{ {\begin{array}{*{20}l} {E_{\text{I}} \varepsilon_{\text{sf}} ,} \hfill & {0 \le \varepsilon_{\text{sf}} \le \varepsilon_{\text{sy}} } \hfill \\ {f_{\text{sfy}} + E_{\text{II}} \left( {\varepsilon_{\text{sf}} - \varepsilon_{\text{sy}} } \right),} \hfill & {\varepsilon_{\text{sy}} < \varepsilon_{\text{sf}} \le \varepsilon_{\text{fu}} } \hfill \\ \end{array} } \right.$$

(3)

Fig. 13

Tensile constitutive relationship of SFCB

According to the principle of composite materials, E_I = a_sE_s + a_fE_f, E_II = a_fE_f, where a_f and a_s are the cross section area ratio of out-wrapped FRP, A_f, and inner steel A_s cross section areas, respectively, to the total bar area A_sf, a_f = A_f/A_sf, a_s = A_s/A_sf.

A comparison of predicted and tested mechanical properties of SFCB is shown in Fig. 14 and Table 7, where the subscripts e and p represent the experimental and predicted values, respectively, r_mv and r_cov represent the mean value and coefficient of variation of the ratios of predicted value to experimental value, respectively.

Fig. 14

Comparison of tested and predicted stress–strain curves

As observed from Fig. 14 and Table 7, the predicted mechanical properties of SFCB show good agreement with that obtained from experiments.

4.1.2 Basic assumptions

The following assumptions have been taken into account in the analysis presented below:

• No slip occurs between SFCB/steel/FRP bars and surrounding concrete, i.e., perfect bond.

• The assumption of plane section at various loading stages is valid.

• The loading process of SFCB-reinforced flexural specimen exhibits three distinct stages as observed in experiments and described in Sect. 2.2, whereas failure occurs when either the extreme concrete compressive fiber or tensile SFCB reaches their respective ultimate strain.

4.2 Analysis of SFCB RC cross sections

4.2.1 Failure modes

Based on the materials’ constitutive relationship, three failure modes and their corresponding strain distribution can be identified for SFCB-reinforced concrete specimens as shown in Fig. 15, where x_c and h_t are the height of compressive (neutral axis depth) and tensile concrete zone, respectively, and the subscripts 1 and 2 represent the balanced failure 1 and 2, respectively. h₀, h_r and h₀ are defined in Sect. 1.2, while ε_c, ε_cu, ε_sfy, ε_sf and ε_sfu are defined in Sect. 3.1. ① Compressive failure before SFCB reinforcement yielding (over-reinforced case): ε_c = ε_cu and ε_sf < ε_sfy. ② Compressive failure 2 after yielding of SFCB: ε_c = ε_cu and ε_sfy ≤ ε_sf < ε_sfu. ③ Tensile failure: ε_c < ε_cu and ε_sf = ε_sfu.

Fig. 15

Strain distribution under balanced failure modes

As presented in Fig. 15, if ε_c = ε_cu and ε_sf = ε_sfy simultaneously take place, balanced failure 1 occurs; if ε_c = ε_cu and ε_c = ε_cu simultaneously take place, balanced failure 2 occurs.

According to triangle similarity, the relative compression height ξ, which is defined as the depth of concrete compression zone x_c to the cross section effective height h₀, ξ = x_c/h₀, can be expressed as follows:

Balanced failure 1:

$$\xi_{\text{cb1}} = \frac{{x_{\text{c1}} }}{{h_{ 0} }} = \frac{{\varepsilon_{\text{cu}} }}{{\varepsilon_{\text{cu}} + \varepsilon_{\text{sfy}} }}$$

(4)

Balanced failure 2:

$$\xi_{\text{cb2}} = \frac{{x_{\text{c2}} }}{{h_{ 0} }} = \frac{{\varepsilon_{\text{cu}} }}{{\varepsilon_{\text{cu}} + \varepsilon_{\text{sfu}} }}$$

(5)

If ξ > ξ_cb1, failure mode ① occurs; if ξ_cb2 ≤  ξ  ≤  ξ_cb1, failure mode ② occurs; and if ξ  < ξ_cb2, failure mode ③ occurs.

Considering equilibrium of compression force resisted by concrete and tensile force by SFCB, the following expression can be obtained:

$$\int_{{h_{\text{t}} }}^{h} {\sigma_{\text{c}} (x)b} {\text{d}}x = \sigma_{\text{sf}} A_{\text{sf}}$$

(6)

where σ_c(x) is the concrete compressive stress corresponding to the fiber at height x, A_sf and σ_sf are the SFCB cross-sectional area and tensile stress, respectively.

For balanced failure 1, h_t = ε_cuh₀/(ε_cu + ε_sfy).

$$f_{\text{c}} bh_{ 0} \frac{{\varepsilon_{\text{cu}} - \varepsilon_{\text{co}} /3}}{{\varepsilon_{\text{cu}} + \varepsilon_{\text{sfy}} }} = f_{\text{sfy}} A_{\text{sf}}$$

(7)

Equation (7) can be transformed into the expression of reinforcement ratio, as follows:

$$\rho_{\text{sf,b1}} = \frac{{f_{\text{c}} }}{{f_{\text{sfy}} }} \cdot \frac{{\varepsilon_{\text{cu}} - \varepsilon_{\text{co}} /3}}{{\varepsilon_{\text{cu}} + \varepsilon_{\text{sfy}} }}$$

(8)

For balanced failure 2, x_c = ε_cuh₀/(ε_cu + ε_sfu).

$$f_{\text{c}} bh_{ 0} \frac{{\varepsilon_{\text{cu}} - \varepsilon_{\text{co}} /3}}{{\varepsilon_{\text{cu}} + \varepsilon_{\text{sfy}} }} = f_{\text{sfu}} A_{\text{sf}}$$

(9)

Equation (9) can be transformed into the expression of reinforcement ratio, as follows:

$$\rho_{\text{sf,b2}} = \frac{{f_{\text{c}} }}{{f_{\text{sfu}} }} \cdot \frac{{\varepsilon_{\text{cu}} - \varepsilon_{\text{co}} /3}}{{\varepsilon_{\text{cu}} + \varepsilon_{\text{sfu}} }}$$

(10)

ρ_sf,b1 and ρ_sf,b2 are defined as the maximum and minimum balanced reinforcement ratio, respectively. So, if ρ_sf >  ρ_sf,b1, failure mode ① occurs; if ρ_sf,b2 ≤  ρ_sf ≤  ρ_sf,b1, failure mode ② occurs; if ρ_sf< ρ_sf,b2, failure mode ③ occurs.

4.2.2 Experimental verifications

The maximum and minimum balanced reinforcement ratios for SFCB RC specimens can be obtained from formulas (8) to (10). Comparisons of tested and predicted failure modes are presented in Table 8. CFM and EFM indicate the calculated and experimental failure modes, respectively. SY, SNY and CC indicate steel yielding, steel not yielded and concrete crushing, respectively.

Table 8 Comparisons of experimental and predicted failure modes

As can be seen from Table 8, the predicted failure modes show good agreement with that observed in the experiments.

4.3 Section analysis of the whole loading process

Cross section analysis of the whole loading process of SFCB RC flexural specimens is developed. It is primarily based on materials’ constitutive models, plane section assumption, triangle similarity and equilibrium of force. Figure 16a–e represents the diagrammatic sketches of cross section, strain distribution and stress distribution of stages (I), (II) and (III), respectively.

Fig. 16

Cross section strain distribution at different failure modes

4.3.1 Stage (I): elastic stage

In this stage, the strain in tensile concrete, SFCB and compressive concrete meets the following conditions: 0 < ε_ct ≤ ε_ctu, 0 < ε_sf < ε_sfy and 0 < ε_c < ε_co, respectively.

The strain in the fiber at any height of cross section can be expressed as follows:

$$\varepsilon (x) = \left\{ {\begin{array}{*{20}l} {\varepsilon_{\text{ct}} (h_{\text{t}} - x)/h_{\text{t}} ,} \hfill & {0 \le x \le h_{\text{t}} } \hfill \\ {\varepsilon_{\text{ct}} (x - h_{\text{t}} )/h_{\text{t}} ,} \hfill & {h_{\text{t}} < x \le h} \hfill \\ \end{array} } \right.$$

(11)

The strain in the top outmost concrete fiber is ε(h)= ε_ct(h − h_t)/h_t, while the strain in SFCB reinforcement is ε_sf =  ε_ct (h_t − h_s)/h_t and the corresponding stress is σ_sf =  E_Iε_ct(h_t − h_s)/h_t.

As the concrete compressive force is equal to the resultant tension of tensile concrete and SFCB, the following formula can be listed:

$$\int_{0}^{{h_{\text{t}} }} {\sigma_{\text{ct}} (x)b} {\text{d}}x + \sigma_{\text{sf}} A_{\text{sf}} - \int_{{h_{\text{t}} }}^{h} {\sigma_{\text{c}} (x)b} {\text{d}}x = 0$$

(12)

Substituting Eq. (11) into Eq. (12), Eq. (12) can be converted to a function of h_t. By computer iterative calculation, h_t varying at loading stage (I) can be calculated. Taking the neutral axis as inertia axis, the cross section moment can be listed as follows:

$$M = \int_{0}^{{h_{\text{t}} }} {\sigma_{\text{ct}} (x)b(h_{\text{t}} - x)} {\text{d}}x + \sigma_{\text{sf}} A_{\text{sf}} (h_{\text{t}} - h_{s} ) + \int_{{h_{t} }}^{h} {\sigma_{\text{c}} (x)b(x - h_{\text{t}} ){\text{d}}x}$$

(13)

When ε_ct = ε_ctu, cracking moment M_cr can be obtained by Eqs. (11) to Eq. (13).

4.3.2 Stage (II): from cracking of concrete to the yielding of SFCB

In this stage, the strain in tensile concrete, SFCB and compressive concrete meets the following conditions: ε_ct > ε_ctu, 0 < ε_sf ≤ ε_sfy and 0 < ε_c ≤ ε_cu, respectively.

The strain of the fiber at any height of cross section can be conveyed as follows:

$$\varepsilon (x) = \left\{ {\begin{array}{*{20}l} {\varepsilon_{\text{sf}} (h_{\text{t}} - x)/(h_{\text{t}} - h_{\text{s}} ),} \hfill & {0 \le x \le h_{\text{t}} } \hfill \\ {\varepsilon_{\text{sf}} (x - h_{\text{t}} )/(h_{\text{t}} - h_{\text{s}} ),} \hfill & {h_{\text{t}} < x \le h} \hfill \\ \end{array} } \right.$$

(14)

The strain in the top outmost concrete fiber is ε(h)= ε_sf (h − h_t)/(h_t–h_s). The strain and stress in SFCB are ε_sf and σ_sf= E_Iε_sf.

As the pressure bore by compressive concrete is equal to the tension bore by tensile SFCB, the following formula can be listed:

$$\sigma_{\text{sf}} A_{\text{sf}} - \int_{{h_{\text{t}} }}^{h} {\sigma_{\text{c}} (x)b} {\text{d}}x = 0$$

(15)

Substituting Eq. (14) into Eq. (15), Eq. (15) can be converted to the function about h_t. By computer iterative calculation, h_t for varying loading at stage (II) can be calculated. Taking the neutral axis as inertia axis, the cross section moment is listed as follows:

$$M = \sigma_{\text{sf}} A_{\text{sf}} (h_{\text{t}} - h_{s} ) + \int_{{h_{t} }}^{h} {\sigma_{\text{c}} (x)b(x - h_{\text{t}} ){\text{d}}x}$$

(16)

When ε_sf = ε_sfy, yield moment M_y can be obtained by Eq. (14) to Eq. (16).

4.3.3 Stage (III): from the yielding of SFCB to failure

In this stage, the strain in tensile concrete, SFCB and compressive concrete meets the following conditions: ε_ct > ε_ctu, ε_sfy < ε_sf ≤ ε_sfu and 0 < ε_c ≤ ε_cu, respectively.

The strain of the fiber at any height of cross section can be conveyed as follows:

$$\varepsilon (x) = \left\{ {\begin{array}{*{20}l} {\varepsilon_{\text{c}} (h_{\text{t}} - x)/(h - h_{\text{t}} ),} \hfill & {0 \le x \le h_{\text{t}} } \hfill \\ {\varepsilon_{\text{c}} (x - h_{\text{t}} )/(h - h_{\text{t}} ),} \hfill & {h_{\text{t}} < x \le h} \hfill \\ \end{array} } \right.$$

(17)

The strain in SFCB is ε_sf= ε_c (h_t − h_s)/(h − h_s), and its corresponding stress is σ_s= f_sfy + E_II(ε_sf − ε_sfy).

Substituting Eq. (17) into Eq. (15), Eq. (15) can be converted to the function about h_t. By computer iterative calculation, h_t for varying loading at stage (III) can be calculated. The cross section moment can be obtained by Eq. (16).

When ε_c = ε_cu, ultimate moment M_u can be obtained by Eq. (15) to Eq. (17).

4.4 Comparisons of predicted and tested results

Comparisons of predicted cracking, yield and ultimate moment and tested results are presented in Table 9, where the subscripts e and p represent the experimental and predicted value, respectively, and r_mv and r_cov represent the mean value and coefficient of variation of ratios of predicted value to experimental value, respectively.

Table 9 Comparison of tested and predicted moment capacity

As observed from Table 9, the predicted bending capacity shows good agreement with tested results, illustrating the validity of the developed formulas.

5 Stiffness and deflection

For SFCB RC flexural specimens, after cracking, the test specimens exhibited two distinct flexural stiffnesses, before and after yielding of SFCB. So, the effective moment of inertia should be divided into two cases: I) before yielding of SFCB and II) after yielding of SFCB. I_g is the gross moment of inertia, I_{e, I} and I_{e, II} are the effective moment of inertia [30] before and after the yielding of SFCB, respectively. M, M_cr and M_y are the applied, cracking and yield moments, respectively. I_{cr, I} and I_{cr, II} are the cracked moment of inertia before and after, respectively, the yielding of SFCB. n_I and n_II are the modular ratio of SFCB before and after yielding, respectively, to concrete. k_I and k_II are the ratio of the height of concrete compressive zone before and after the yielding of SFCB, respectively, to effective height of cross section.

$$I_{\text{e,I}} = \left( {\frac{{M_{\text{cr}} }}{M}} \right)^{3} I_{\text{g}} + \left[ {1 - \left( {\frac{{M_{\text{cr}} }}{M}} \right)^{3} } \right]I_{\text{cr,I}} \le I_{\text{g}}$$

(18)

$$I_{\text{e,II}} = \left( {\frac{{M_{\text{y}} }}{M}} \right)^{3} I_{\text{y}} + \left[ {1 - \left( {\frac{{M_{\text{y}} }}{M}} \right)^{3} } \right]I_{\text{cr,II}} \le I_{\text{y}}$$

(19)

$$I_{\text{y}} = \left( {\frac{{M_{\text{cr}} }}{{M_{\text{y}} }}} \right)^{3} I_{\text{g}} + \left[ {1 - \left( {\frac{{M_{\text{cr}} }}{{M_{\text{y}} }}} \right)^{3} } \right]I_{\text{cr,I}}$$

(20)

$$I_{\text{cr,I}} = \frac{{bh_{0}^{3} }}{3}k_{\text{I}}^{3} + n_{\text{fI}} A_{\text{sf}} h_{0}^{2} (1 - k_{\text{I}} )^{2}$$

(21)

$$I_{\text{cr,II}} = \frac{{bh_{0}^{3} }}{3}k_{\text{II}}^{3} + n_{\text{fII}} A_{\text{sf}} h_{0}^{2} (1 - k_{\text{II}} )^{2}$$

(22)

$$k_{\text{I}} = \sqrt {2\rho n_{\text{I}} + (\rho n_{\text{I}} )^{2} } - \rho n_{\text{I}}$$

(23)

$$k_{\text{II}} = \sqrt {2\rho n_{\text{II}} + (\rho n_{\text{II}} )^{2} } - \rho n_{\text{II}}$$

(24)

If M ≤ M_cr, I_e= I_g. If M_cr < M ≤ M_y, the overall flexural stiffness E_cI_{e, I} is between E_cI_g and I_{cr, I}. If M_y < M ≤ M_u, the overall flexural stiffness E_cI_{e, II} is between E_cI_y and I_{cr, II}.

The comparisons of predicted and tested moment–deflection curves of SFCB RC specimens are presented in Fig. 17.

Fig. 17

Comparison of predicted and tested moment–deflection curves

As observed from Fig. 17, the moment–deflection curves predicted by the modified formulas considering the mechanical properties of SFCB RC flexural specimens, especially under the service loading stage (about 40–70% bending capacity), fit well with the experimental results, confirming the validity of the developed formulas for practical use.

6 Crack width

The following formula is recommended by ACI 440.1R-06 [30] to predict the crack width of FRP RC flexural components:

$$w = 2\frac{{f_{\text{f}} }}{{E_{\text{f}} }}\beta k_{\text{d}} \sqrt {d_{\text{c}}^{ 2} { + }s^{ 2} / 4}$$

(25)

where w is the extreme crack width (the point at the tensile edge of cross section); f_f and E_f are the tensile stress and elastic modulus of FRP reinforcement, respectively; β is the ratio of the height of tensile zone to the distance between the neutral axis and centroid of reinforcement; k_b is a coefficient related to the bond property of reinforced bars and surrounding concrete; d_c is the concrete thickness of protective layer of FRP reinforcement, that is, the distance between the tensile edge of section and the centroid of FRP reinforcement; s is the spacing of reinforced bars.

For SFCB-reinforced concrete flexural specimens, the above formula for crack width is modified to Eq. (26) to account for mechanical characteristics of SFCB RC specimens as well as the location at which the crack width is calculated. The strain ε_sf in SFCB is calculated from Eq. (27):

$$w = 2\varepsilon_{\text{sf}} k_{\text{d}} \sqrt {d_{\text{c}}^{ 2} { + }s^{ 2} / 4}$$

(26)

$$\varepsilon_{\text{sf}} = \frac{M}{{E_{\text{c}} I_{\text{e}} (1 - k)h_{0} }}$$

(27)

The comparisons of predicted and tested moment–crack width curves of SFCB RC specimens are presented in Fig. 18.

Fig. 18

Comparison of predicted and tested moment–crack width curves

Figure 18 indicates that the moment–crack width curves predicted by the modified formulas show good agreement with tested results, especially under the service loading stage (about 40–70% bending capacity), illustrating the validity of the proposed formulas for engineering application.

7 Conclusions

Experimental and theoretical analyses of structural behavior of SFCB RC beams are carried out. The following conclusions may be drawn:

• 1. The predicted mechanical properties of SFCBs obtained by the principle of composite materials are in good agreement with the test results, and the ribbed SFCBs/BFRP bars showed comparable bond behavior to that of ribbed steel bars.

• 2. As SFCB has the characteristic of stable secondary stiffness, the loading processes of SFCB/hybrid RC specimens obviously present three stages. With an increase in nominal elastic modulus reinforcement ratio, the bending capacity increases gradually.

• 3. Deflections and crack widths of specimens with high nominal elastic modulus reinforcement ratio are less than the corresponding limits under serviceability state. Under the serviceability state, crack widths and deflections of steel and FRP RC specimens are the smallest and largest, respectively, while those of SFCB and hybrid RC specimens are in between; meanwhile, their values decrease with an increase in nominal elastic modulus reinforcement ratio.

• 4. For the bending capacity utilization coefficient controlled by the deflection/crack width limit under serviceability state, the values of steel and FRP RC specimens are the highest and lowest, respectively, while those of SFCB and hybrid RC specimens are between them and increase with the increase in nominal elastic modulus reinforcement ratio.

• 5. Based on strains compatibility, material’s constitutive models and forces equilibrium, failure modes, balanced failure states and balanced reinforcement ratios as well as analytical technique for predicting the whole loading process are also developed, displaying good agreement with test results.

• 6. On the base of ACI design guidelines and taking the mechanical characteristics of SFCB RC beams into consideration, formulas for effective moment of inertia and crack width are proposed, showing good agreement with experimental results, illustrating their validity.