At present, several formulas have been proposed to predict the average crack spacing in the negative moment regions of continuous composite beams. However, the views on the influence factors of the existing formulas are not yet unified. The models employed by the design codes to calculate the average crack spacing were based on reinforced concrete structures, while the factors associated with composite beams were not fully taken into account. The common models in CEB-FIP (1978), China Code GBJ 10-89 (1989) and China Code GB 50010-2010 (2010) can be expressed, respectively, as follows: l cr = 2 ( c s + l s 10 ) + k 1 k 2 d eq ρ te , (1) l cr = 1 .1 ( 2.7 c s + 0.11 d eq ρ te ) ν ,   (2) l cr = 1.1 ( 1.9 c s + 0.08 d eq ρ te ) . (3)

An improved model based on China Code GBJ 10-89 (1989) was proposed by Wu et al. (1993), while the transverse reinforcement ratio and diameter were considered. The equation obtained was then as follows: l cr = 2.7 c s + 0.1 ρ te / d eq + 1.2 ρ Hte / d Heq . (4)

As the aforementioned models for average crack spacing showed no relation with the combination effect, the previous studies took the combined force ratio and spacing of stud connectors into consideration (Nie and Zhang, 1997; Yu and Guo, 2004), and two novel models were then proposed according to China Code GBJ 10-89 (1989) and China Code GB 50010-2010 (2010). The two equations are shown, respectively, as follows: l cr = 1 .1 ( 2.7 c s + 0.11 ρ te / d eq + 0.25 R p 2 / p ) ν   (5) l cr = 1.9 c s + 0.08 ρ te / d eq + 0.04 R p 2 / p . (6)

The development of cracking in slabs as part of composite beams is decisively influenced by the transverse reinforcement (Ramm and Elz, 2002). But this influence factor was not considered in the models mentioned earlier. A simple model which took the combined force ratio and transverse reinforcement spacing into account was established then by Zhang et al. (2011), while other important factors were not taken into account. The equation is given as follows: l cr = l a ( 1 − R p 3 ) . (7)

In the aforementioned equations, l _cr is the average crack spacing; k ₁ is the bonding performance coefficient of longitudinal tensile bars, and k ₁ = 0.4 is for a deformed bar; k ₂ is the influence coefficient of component stress distribution, and k ₂ = 0.25 is for a tension member; ν is the surface characteristics coefficient of longitudinal tensile bars, and ν = 0.7 is for deformed bar; c _s is the concrete cover thickness of reinforcement; l _s and l _a are the spacings of longitudinal reinforcement and transverse reinforcement, respectively; d _eq and d _Heq are the equivalent diameters of longitudinal tensile bars and transverse bars, respectively; ρ _eq and ρ _Heq are the tensile reinforcement ratio and transverse reinforcement ratio, respectively; R _p is the combined force ratio; and p is the spacing of stud connectors.

In summary, the average crack spacing in the negative moment regions of continuous composite beams is mainly related to the concrete cover thickness of reinforcement, equivalent diameters of longitudinal tensile bars, the tensile reinforcement ratio, the combined force ratio, the spacing of stud connectors, and transverse reinforcement. However, the influence factors considered in the existing formulas mentioned before, proposed by researchers and codes, were not comprehensive.

In this section, a total of 38 specimen tests from previous literature were summarized and analyzed to study the influence factors of average crack spacing, as can be seen in Table 1. According to Eq. 6, which was based on China Code GB 50010-2010 (2010), a modified formula of crack spacing was proposed by introducing the factor of transverse reinforcement spacing l _a and taking the consistency of dimension into consideration. The parameters were achieved by fitting the data listed in Table 1 with the procedure shown in Figure 1. In this way, the novel model obtained was then as follows: l cr = 1.9 c s + 0.08 0.25 ρ te / d eq + 0.05 R p 2 / p + 0.07 / l a . (8)

The physical meaning of Eq. 8 is consistent with Eqs. 1–7 which were empirical formulas derived from data fitting. To verify the validity of the modified model described before, existing experiments conducted by Song et al. (2021) and He et al. (2010) were selected and analyzed with other proposed models expressed in Eqs 3–7. As is shown in Table 2, the maximum error was controlled within 10% for Eq. 8. Generally, it can be seen that the results of the modified formula are in better agreement with the experimental results compared with existing models. Hence, it is effective to use the modified model to study the average crack spacing in the negative moment regions of continuous composite beams.

The fatigue stress state of the reinforcing bar in the concrete slab of composite beams under negative moment is basically the same as that of a reinforced concrete structure. With the increase of repeated cycles, the effective tensile area of the reinforcing bar will decrease, especially at the cracking location (Song, 2006). When the loss of cross-sectional area of the reinforcing bar in the process of fatigue loading conforms to Miner’s rule, the effective area of reinforcing bar at cracking position after n cycles can be determined by Eq. 21. Considering the degradation of the effective area of the reinforcing bar and the shear stiffness of studs under fatigue loading, an equation proposed by Fan and Nie (2005) can be modified and applied to the stress calculation of the reinforcing bar in the cracked section, which was defined as the following Eq. 22. In the equations given earlier, the parameters or coefficients can refer to the existing literature (Song et al., 2020). By solving Eq. 21 and Eq. 22, the cycle-dependent stress of the reinforcing bar can be obtained. A r f ( n ) = A r [ 1 − n / N f 3 ( 1 − σ s ( n ) / f sy ) ]   (21) σ s ( n ) = E s A s ( ε r 0 f I 0 f y 0 − ε s f I s y 0 r f ) [ ( A r f ( n ) + A s ) I s + A r f ( n ) A s y 0 2 ] y 0 r f . (22)

The mechanical approach to investigating the crack opening of a cracked beam is based on knowledge of the crack position (stabilized cracking) (Fabbrocino et al., 2007). Due to the randomness and complexity of the cracks’ development, the following two assumptions are made to simplify the model problem:

(1) The main crack occurs at the position of maximum bend.

Each sub-element is defined between two adjacent cracks, that is, on the average crack spacing l _c,r , as shown in Figure 3. The nodes for a sub-element discretization based on the finite difference method are also presented in Figure 3, as well as the details of the composite beams. According to a generic approach used in reinforcement concrete beams (Oliveira et al., 2008), the theoretical analysis of a sub-element can be referred to a sub-domain with a length of dx of the concrete slab, as shown in Figure 4, where the cases of the shear-slip behavior at the beam–slab interface and bond–slip behavior at the reinforcement–concrete interface are schematically reported together with some issues related to the boundary conditions. Then the axial force equilibrium of the bar and the cross-section under fatigue loading at a certain number of repeated cycles n can be expressed as follows: d σ u ( n ) A u f = ∓ π τ u f d u d x , (23) d σ b ( n ) A b f = ∓ π τ b f d b d x ,   (24) d σ c ( n ) A eff = ∓ n u A u f d σ u ( n ) ∓ n b A b d σ b   ( n ) − υ n d x , (25) where dσ _u(n) and dσ _b(n) are the stress increments under fatigue loading of reinforcement bars in the top layer and bottom layer, respectively; τ ^f _u and τ ^f _b are the bonding strengths under fatigue loading of reinforcement bars; d _u and d _b are the diameters of reinforcement bars; n _u and n _b are the numbers of reinforcement bars; dσ _c is the stress increments of the tension concrete under fatigue loading; A _eff is the effective tension area of the concrete slab; A ^f _u and A ^f _b are the areas of reinforcement bars under fatigue loading; dx is the length of the sub-domain of the concrete slab; and ∓ or ± depends on the position of the sub-domain.

According to Eqs 23, 24, the distribution coefficient of the tress increments of reinforcement bars can be defined as follows: ξ = d σ b ( n ) d σ u ( n ) = τ b f τ u f d u d b . (26)

Based on assumption 2, the following equation is obtained: s u f = h u h b s b f , (27) where s ^f _u and s ^f _b are the slips of reinforcing bars in the top layer, and bottom layer, respectively; h _u and h _b are the distances of the beam–slab interface to reinforcing bars.

Then the stress increments of the tension concrete can be expressed as follows: d σ c ( n ) = ∓ n u A u f d σ u ( n ) ∓ n b ξ A b d σ u   ( n )   − υ n d x A eff . (28)

The stress variable in a sub-element depends on the relative slip between reinforcing bars and surrounding concrete, that is, the difference in longitudinal displacements between them. Thus, this description can be given as follows: d s ( x ) d x = d [ u s ( x ) − u c ( x ) ] d x + ε sh = ε s ( x ) − ε c ( x ) + ε sh , (29) where u _s(x) and u _c(x) are the longitudinal displacements of reinforcement bars and surrounding concrete, respectively; ε _s(x) and ε _c(x) are, respectively, the strains of reinforcement bars and surrounding concrete; ε _sh is the shrinkage strain of concrete, and ε _sh = 310µ is for this work according to China Code GB 50917-2013 (2013).

According to the division for the sub-element with small length Δx (see Figure 4), the finite difference forms of <b> Eqs 23, 28, 29 are as follows: σ su , i + 1 ( j ) ( n ) = σ s u , i ( j ) ( n ) ∓ 4 d u τ su, i f ( j ) Δ x , (30) σ c , i + 1 ( j ) ( n ) = σ c , i ( j ) ( n ) + ± ( n u d u + n b ξ d b ) π τ su , i f ( j ) − υ i ( j ) A ef f Δ x ,   (31) s su , i + 1 f ( j ) = s su , i f ( j ) ∓ ( σ su , i + 1 ( j ) ( n ) + σ su , i ( j ) ( n ) 2 E s − σ c , i + 1 ( j ) ( n ) + σ c , i ( j ) ( n ) 2 E c + ε sh ) Δ x . (32)

The values of σ _su and σ _c in the i+1 section are determined from the values attained in the i section, by using the method of finite difference (Castel et al., 2012). Based on the bond–slip theory, the crack widths of steel–concrete composite beams under hogging moment can be expressed as follows: w i ( j ) ( n ) = { 2 s su , 0 f ( j )                                                                                       j = 1         ,         i = 0 s su , 0 f ( j − 1 ) + s su , m f ( j )                               j > 1           ,         i = 0 2 s su , m f ( j )                                                                                     j = k ,           i = m . (33)

The boundary conditions in the cracked section x=(j-1)l _cr of each sub-element numbered j applied to Eqs 31–33 can be expressed as follows: σ s, 0 ( j ) ( n ) ≠ 0 ;   σ c, 0 ( j ) ( n ) = 0. (34)

And the boundary condition at the abscissa x = jl _cr established as the condition of convergence can be given in the following form: σ c,m ( j ) ( n ) = 0. (35)

Figure 5 shows the flowchart of the procedure for numerical solution. Through giving a tentative value s _su,0, it is possible to predict the crack widths along the composite beam corresponding to a certain loading level. The specific steps are as follows:

1) Input basic information and data of composite beam. Calculate the average crack spacing l _cr according to Eq. 8. Divide a half-span structure into k sub-elements. Divide a sub-element into m sub-domains. Take the sub-element (i.e., numbered j) as the research object and input the repeated cycles n. Obtain the reinforcement stress σ _s,0(n) at the cracked section by combining Eq. 21 and Eq. 22. Calculate the concrete stress σ _c,0(n) = 0 at the cracked section by Eq. 34.

In order to obtain the experimental values of crack width in the negative moment regions of steel–concrete composite beams under fatigue loading, a total of three test beams were designed and manufactured. The specimens were placed upside down on two steel supports to simulate the action of a negative moment. Among them, the specimen numbered SCB1-1 was used for a static loading test, while the specimens numbered SCB1-2 and SCB1-3 were used for fatigue loading tests. Figure 6 shows the dimension details of the specimens. All test beams are equipped with stud connectors with a diameter, height, and spacing of 16 mm, 90 mm, and 100 mm, respectively. The ratio of longitudinal reinforcing bars is designed as 4.0% with a diameter of 16 mm. Material tests were conducted on concrete and steel before the formal loading of the specimens. The strength grade of the concrete was designed to be C50. The tensile reinforcement bars and steel plates used HRB400 and Q345, respectively (of the same factory batch). The axial tensile strength, elastic modulus, and the average result of compressive strength of concrete are 3.44 MPa, 3.47 × 10⁴ MPa, and 51.2 MPa, respectively. Meanwhile, the average results of the tensile yield strengths of the web (or top flange), bottom flange, and reinforcing bars are 443 MPa, 391 MPa, and 592 MPa, respectively. The setup of the fatigue test is illustrated in Figure 7. The fatigue load limits of SCB1-2 and SCB1-3 were designed as 25%F _u and 40%F _u, where F _u is the ultimate bearing capacity of SCB1-1 obtained from the static test. The loading ratio and frequency were set as 0.1 and 2 Hz, respectively. And the sine wave was used for the fatigue loading. The fatigue loading was suspended when the repeated cycles reached the specified times of 1 × 10⁴, 5 × 10⁴, 10 × 10⁴, 50 × 10⁴, 100 × 10⁴, 150 × 10⁴, 200 × 10⁴, and 250 × 10⁴. Then the static loading test was performed to measure the crack width by an electronic crack width measurement instrument. The typical experimental results are listed in Table 3, in which ΔF is the fatigue load amplitude, that is, F _max−F _min; F _cr is the cracking load in a static test; N _f is the fatigue life. The typical pattern of crack development is shown in Figure 8.

The relative slip between the reinforcing bar and concrete increases gradually under the action of fatigue load, resulting in the growth of crack width with repeated cycles. At present, there is no specific method to calculate the crack width in the negative moment regions of composite beams under fatigue loading. For reinforced concrete structures, the crack width under fatigue loading is generally expressed by the method under static loading through appropriate correction, that is, the experimental regression formula obtained by using the initial crack expansion coefficient (Song, 2006). According to the test results, a statistical empirical formula was given for the maximum crack width after n loading cycles: ω max = ω 0 ( 0.382 − 0.0227 lg n ) lg n , (36) where, ω ₀ is the initial maximum crack width, which can be calculated according to the conventional method under static loading. In this study, it is calculated according to the recommended formula in JTG D62-2004 (2004), which is expressed as follows: ω 0 = C 1 ′ C 2 ′ C 3 ′ σ s E s ( 30 + d eq 0.28 + 10 ρ ) , (37) ρ = A r + A p b h 0 + ( b f − b ) h f , (38) where A _p is the area of prestressed reinforcement in the tensile zone; b _f and h _f are the width and thickness of the tensile concrete slab; C ₁´ = 1.0 is for the deformed bar; C ₂´ = 1.0 is for load effect; C ₃´ = 1.2 is for the axial tension members.

In GB 50010-2010 (2010), the non-uniform coefficient φ of tensile rebar strain for reinforced concrete specimens under fatigue loading is equivalent to 1.0, that is, the bond between the rebar and the concrete fails completely, considering the fatigue effect. The calculation method is proposed based on the assumptions that the fatigue failure occurs in reinforced concrete structures or the loading cycles reaches 200 × 10⁴, and the maximum crack width can be expressed as follows: ω max = α cr σ s E s ( 1.9 c s + 0.08 d eq ρ eq ) , (39) where α _cr is the characteristic coefficient of tension members under the action of force; f _tk is the standard value of concrete tensile strength; for other coefficients, refer to the explanation given before.

Figure 9 shows the load–crack width response of the experimental test, the existing calculation methods, and the proposed model in this study. It can be found that the crack width measured in the test increased with the repeated cycles. It developed rapidly in the early loading process of about 10 × 10⁴ cycles, and then grew slowly. The crack width calculated according to Eq. 36 had a large deviation from the test results at different times of cyclic loading. The results of the maximum crack width by Eq. 39 were limited to a particular condition. In addition, the calculated values were not universal for the structure checking and were not in good agreement with the test results. As for the numerical model in this study, which fully takes the characteristics of the negative moment regions of composite beams into consideration, the computed results are in good agreement with the experimental values. And the accuracy was further verified by the tested values from other literature (Lin et al., 2013), as shown in Figure 10. As a result, the numerical model can well reflect the development trend of crack width in the process of fatigue loading and provide a reference for anti-fatigue design and checking calculation of composite beam in negative moment region.