Ultimate flexural capacity prediction of assembled steel-UHTCC composite bridge decks

Abstract

Steel-concrete composite bridge decks (CBDs) have been widely used in practical bridge structures. In recent years, a novel assembled CBD comprising of sectional steels and perfobond rib (PBL) shear connectors was recently developed for eliminating the on-site welding requirements and reducing the steel consumption. By using ultra-high toughness cementitious composite (UHTCC) as the surface layer, the durability of the assembled CBDs could be further improved. In this paper, analytical and numerical investigations on the flexural behavior and ultimate capacity prediction of the assembled steel-UHTCC CBDs are performed. Firstly, a theoretical model considering various degrees of shear connection and the shear-slip effect of PBL shear connectors was developed to predict the ultimate flexural capacities of assembled steel-UHTCC CBDs with PBL shear connectors under both sagging and hogging moments. Subsequently, finite element (FE) models of assembled steel-UHTCC CBDs with various numbers of PBL shear connectors and profile steel parts in shapes of T, Z, B and H were established using ABAQUS, and validated against existing experimental results. Finally, the proposed theoretical model was validated using FE simulation results, with deviations not exceeding 10% and most of them are within 5%, demonstrating the accuracy of the proposed theoretical model. This paper provides a practical method to calculate flexural capacity and optimize PBL shear connector layouts, significantly improving the cost efficiency when designing CBD structure.

Keywords

ultra-high toughness cementitious composite composite bridge deck perfobond rib (PBL) shear connectors theoretical model finite element analysis

Introduction

In recent decades, the increasing demands of transportation have necessitated the rapid development of long-span bridges, thereby imposing significant challenges on bridge engineering. For their superior sectional stiffness compared to orthotropic steel bridge decks (OSBD), steel-concrete composite bridge deck (CBD) structures have garnered attention, which mitigate fatigue in steel components (Qin et al., 2022; Zhu et al., 2020). But the traditional CBDs involve large amount of steel plates and shear connectors welding, which is costly, complex, and difficult to ensure quality. The assembled CBDs offer a solution to the problem.

However, the implementation of assembled steel-concrete CBDs in long-span bridge decks encounter several challenges. The high weight and low tensile strength of conventional concrete often result in lower crack resistance and necessitate a thicker cover layer of over 120 mm (Su et al., 2018). Additionally, the ultimate capacity of CBDs is significantly influenced by the strength of shear connection between steel and concrete (Walter et al., 2007), often resulting in the extensive use of shear studs. The proliferation of studs increases steel consumption and introduces additional fatigue details at the welding area. To address cracking issues, the use of new materials in pavement layers has been explored. Shao et al. introduced ultra-high-performance concrete (UHPC) in the construction of steel-UHPC CBD structures. (Feng et al., 2021; Luo et al., 2019; Shao et al., 2023). The exceptional strength and ductility of UHPC can significantly reduce the thickness of pavement layers to about 50 mm. However, there are still risks of cracking in the hogging moment regions within steel-UHPC CBDs (Nasrin and Ibrahim, 2021; Shao et al., 2018). To reduce steel consumption and welding areas, researches have focused on innovative shear connectors, particularly the Perfobond rib (PBL) (Karam et al., 2020; Sun et al., 2023). PBL connectors are composed of perforating rebars and concrete tenons cast through rib perforations, noted for their high ductility, load-bearing capacity, and fatigue resistance when used in composite structures (He et al., 2013; Kim et al., 2011; Li et al., 2018; Liu et al., 2013). However, the application of PBL in traditional steel-concrete CBDs presents some challenges. The PBL ribs will split the concrete initially, increasing the risk of cracking during service. This indicates a compatibility issue between PBL connectors and concrete.

The aforementioned issues might be addressed through the use of ultra-high toughness cementitious composite (UHTCC). UHTCC was initially proposed as engineered cementitious composite (ECC) by Li and Leung, (1992). In China, Xu et al. (2012) optimized ECC material by replacing large volumes of cement with fly ash and successfully developed UHTCC with characteristic of tensile strain hardening. Table 1 summarizes a comparison of typical material properties between UHTCC and normal concrete. While the elastic modulus of UHTCC is similar to the common concrete, it exhibits much higher tensile strength and crack resistance (Yu et al., 2024). Xu et al. (2019) demonstrated that UHTCC can achieve a peak tensile strain of 3–6%, which is 200–700 times higher than that of ordinary concrete. Under tensile stress, UHTCC develops multiple micro-cracks with a narrow crack width below 0.1 mm. Tong et al. investigated the flexural behavior of steel-UHTCC CBDs with stud shear connectors and found that the use of UHTCC can not only effectively reduce fatigue stress amplitude in steel parts but also achieve ultra-high toughness in CBD structures (Tong et al., 2023; Tong et al., 2023; Tong et al., 2025; Wang et al., 2023). Moreover, the research suggested that the UHTCC cover thickness for CBD structures can range from 45 to 50 mm, enabling weight reduction and cost saving in engineering projects (Huang et al., 2017; Xu et al., 2021; Yin et al., 2023).

Table 1.

Comparison of material properties between concrete, UHPC and UHTCC.

Material	Modulus of elasticity (GPa)	Ultimate tensile strength (MPa)	Peak tensile strain	Peak compressive strain
Concrete	28–34.5	1–3	0.01%	0.2%
UHPC	40–50	6–8	0.15%	0.2%–0.6%
UHTCC	15–20	4.5–6	3% – 6%	0.5%–0.6%

To facilitate easier assembly and enhance mechanical properties (Chen et al., 2024) proposed novel assembled CBDs incorporating UHTCC and specially designed profiled steel parts. The cold-formed duplicate profiled steel parts in various shapes (T, Z, B, and H) are pre-drilled with PBL holes and then welded together, reducing the amount of steel plates welding. By threading the reinforced bars through the reserved holes, the PBL connectors can be formed above the steel plate, as shown in Figure 1. Four-point bending tests were conducted on four types of proposed assembled steel-UHTCC CBDs to study the flexural performance of assembled steel-UHTCC CBDs under sagging moment, as illustrated in Figure 2.

Figure 1.

Proposed assembled steel-UHTCC CBD.

Figure 2.

Schematic diagram of experimental instrument.

The research indicated that UHTCC, due to its excellent crack resistance, has better compatibility with PBL connectors compared to other types of concrete, and can effectively reduce cracking caused by PBL separation. The combination of UHTCC and PBL in assembled CBDs can achieve better engineering convenience, higher load capacity and durability (Chen et al., 2024). Profit by their excellent performance, the assembled steel-UHTCC CBDs have great application value in engineering. The research on flexural performance of the assembled CBDs is of great significance for the design and application of this new type of CBD.

Existing research on the assembled steel-UHTCC CBDs primarily employs over-dense PBL shear connectors to achieve complete shear connections. A reduction in PBL shear connectors could enhance construction efficiency and mitigate fatigue risks caused by excessive welding, but limited attention has been given to the flexural behavior of steel-UHTCC CBDs under partial shear connections. Consequently, systematic exploration on how various degrees and shear-slip effect of shear connection influence flexural performance is critical for improving the cost-effective and fatigue-resistant CBD systems.

In this study, a theoretical model is developed to predict the ultimate flexural capacities of assembled steel-UHTCC CBDs with PBL shear connectors under both sagging and hogging moments, considering various degrees of shear connection and the shear-slip effect of PBL shear connectors. Additionally, FE models of novel assembled steel-UHTCC CBDs with various numbers of PBL shear connectors and profile steel parts in shapes of T, Z, B and H are established to validate the accuracy of the theoretical analysis.

Theoretical analysis

Calculation method of shear force

Based on the existing studies, the flexural capacity of assembled steel-UHTCC CBDs is influenced by the interfacial shear connection degree between the UHTCC slab and steel parts. The frictional force between UHTCC and steel parts was neglected in comparison to the shear force provided by PBL connections. Consequently, CBDs were considered as laminated decks when there is no shear connection at the interface.

The schematic diagram of the PBL connector is shown in Figure 3. The shear force was simplified as concentrated forces provided by each PBL shear connector. The ultimate shear capacity of an individual PBL shear connector V_u,1 can be expressed as

V_{u, 1} = τ_{br} A_{b} + κ (A_{c} f_{c} + 1.55 A_{s} f_{y})

(1)

where A_c and A_s are the cross-sectional areas filled by UHTCC and steel bar in a single PBL hole; f_c is the compressive strength of UHTCC; f_y is the yield strength of the perforating reinforced bar; t_br is the residual interfacial bond strength for UHTCC, expressed by equation (2); A_b is the effective bonded area, expressed in equation (3); the coefficient κ represents the shape characteristics of PBL ribs, as expressed in equation (4).

τ_{br} = (α_{1} ρ_{st} + α_{2} \frac{V_{f} L_{f}}{ϕ_{f}} + α_{3}) f_{ts}

(2)

where α₁, α₂, α₃ are the coefficients determined by the push-out tests, taken as 0.03, 0.018, and 0.031 respectively, based on the experimental data of (Tan et al., 2022); ρ_st is the sectional reinforcement ratio of the transverse reinforced bar; f_ts is the splitting tensile strength of UHTCC; V_f, L_f, ϕ_f represent the volume content, average length, and normalized diameter of PVA fibers, respectively.

A_{b} = 2 (L_{p} - n D) B_{p}

(3)

where L_p and B_p respectively denote the embedded length and height (or width) of the PBL rib.

κ = 2.1 \sqrt{\frac{t_{p}}{D}}

(4)

where t_p is the thickness of the PBL rib.

Figure 3.

Stress state of UHTCC elements at mid-span section under incomplete shear connection.

The shear-slip behavior of PBL was simplified to an ideal elastoplastic relationship (Kang et al., 2014), characterized as: (1) when the shear force is less than V_u,1, the shear force of the PBL connector is linearly related to its slip deformation; (2) once the shear force reaches V_u,1, the PBL shear connector transitions into a phase of plastic deformation, at which point the shear force ceases to increase with further slip.

Due to the shear-slip effects, the plane section assumption cannot be directly applied to the cross-section of steel-UHTCC composite structures under various degrees of shear connection strength (Pan et al., 2016). Consequently, a theoretical model based on partial interaction theory (Johnson and Lee, 1977; Wang et al., 2019) was proposed, as illustrated in Figure 4. The following assumptions were involved:

(1) The plane section assumption is applied to the steel part and UHTCC slab separately.

(2) Both the steel and UHTCC are considered to be ideal elastic-plastic materials.

(3) Since the UHTCC and steel part are vertically laminated, the curvature and rotation of the them at the same section are equal. Meanwhile, there is slippage along the longitudinal direction of CBD at the interface between UHTCC and steel part.

(4) After yielding, cracking in the UHTCC at mid-span leads to a transition in interfacial behavior: plastic deformation of the UHTCC results in a reduction in the growth rate of shear-slip deformation at the steel-UHTCC interface. Thus, Post-yield interfacial shear forces are approximated as the elastic-stage limit value.

Figure 4.

Analysis of partial interaction on steel-UHTCC CBDs under four-point bending.

An isolated micro beam element in elevation was shown in Figure 4, with a length of dx. In Figure 4, L_s represents the shear span length, and L_b represents the length of the pure bending region. The slip equation can be expressed as

v_{L} l_{s} = k S (x)

(5)

where v_L is the longitudinal shear force per unit length; l_s is the longitudinal spacing of PBL shear connectors; k is the shear stiffness of a single PBL shear connector, and its value can refer to the research of (Yang and Chen, 2018); S(x) represents interfacial slippage between steel part and UHTCC.

According to equation (5), K can be defined as the longitudinal shear stiffness of the interface, expressed as

K = v_{L} |_{S = 1} = \frac{k}{l_{s}}

(6)

v_{L} = K S (x)

(7)

As show in Figure 4, the relationship between N(x) and v_L can be expressed as

v_{L} = \frac{d N}{d x}

(8)

Under the action of external bending moment M, the steel beam is subjected to the combined action of compressive force N_s and bending moment M_s. Concurrently, UHTCC slab is subjected to the combined action of tensile force N_c and bending moment M_c. The axial force N_s and N_c are numerically equal and denoted as N. Based on the equilibrium of bending moments, the relationship between M and N can be expressed as

M = M_{c} + M_{s} + N \cdot d_{c}

(9)

where d_c represents the distance between the neutral axis of the UHTCC slab and steel part.

In the elastic stage, the effect of vertical uplift is negligible, and consequently, the steel part and the UHTCC slab must exhibit the same curvature, denoted as ϕ. The moment-curvature relations can be given by the simple beam theory as

M_{s} = E_{s} I_{s} ϕ

(10)

M_{c} = E_{c} I_{c} ϕ

(11)

where E_c and E_s are elastic modulus of UHTCC slab and steel part, respectively; I_c and I_s are inertia moment of UHTCC slab and steel part, respectively.

The definition of slip strain is obtained by solving the first derivative of the relative slip S(x) with respect to x. Derived from equations (7) and (8), the slip strain is given by

ε_{slip} = \frac{d S (x)}{d x} = \frac{1}{K} \frac{d^{2} N}{d x^{2}}

(12)

The slip strain ε _slip is the sum of the effects of axial force N and moment (M_c + M_s), expressed as

ε_{slip} = ε_{slip 1} + ε_{slip 2}

(13)

The slip strain caused by axial force N is expressed as

ε_{slip 1} = N (\frac{1}{E_{c} A_{c}} + \frac{1}{E_{s} A_{s}})

(14)

The slip strain caused by bending moment (M_c + M_s) is expressed as

ε_{slip 2} = - \frac{d_{c}}{Σ E I} (M - N d_{c})

(15)

Σ E I = E_{c} I_{c} + E_{s} I_{s}

(16)

By considering equations (12)–(15), the differential equation can be obtained in a standard form as

\frac{d^{2} N}{d x^{2}} - λ^{2} N = K Δ

(17)

δ = \frac{1}{E_{c} A_{c}} + \frac{1}{E_{s} A_{s}} + \frac{{d_{c}}^{2}}{Σ E I}

(18)

λ^{2} = K δ

(19)

Δ = - \frac{M d_{c}}{Σ E I}

(20)

where λ represents the combined eigenvalue of the CBDs, as expressed in equation (19); δ is the comprehensive flexibility coefficient between the steel part and UHTCC slab, as expressed in equation (18).

For four-point bending component, the external bending moment of shear span and pure bending section can be expressed as

M (x) = {\begin{cases} \frac{P L_{s}}{2} (0 \leq x \leq \frac{L_{b}}{2}) \\ \frac{P}{2} (\frac{L}{2} - x) (\frac{L_{b}}{2} < x \leq \frac{L}{2}) \end{cases}

(21)

L_{s} + \frac{L_{b}}{2} = \frac{L}{2}

(22)

where P represents the combined force located on two loading points.

By substituting equation (21) into equation (17), N at shear span and pure bending region can be solved, and the curvature ϕ can be calculated. The detailed solution process has been provided in Appendix A. By substituting x = 0 into the solution, the shear force N₀ and curvature ϕ₀ at the mid span position can be obtained and expressed as

N_{0} = \frac{(e^{η (\frac{1}{2 γ} - 1)} - e^{η (\frac{1}{2 γ} + 1)}) γ L d_{c} + (1 - 2 γ) L d_{c} η (1 + e^{\frac{η}{γ}})}{2 η δ (1 + e^{\frac{η}{γ}}) \sum E I} P

(23)

ϕ_{0} = \frac{P L_{b} / 2 - N_{0} d_{c}}{\sum E I}

(24)

γ = \frac{L_{s}}{L}

(25)

η = λ \cdot L_{s}

(26)

where γ is the shear span ratio of the steel-UHTCC CBDs, expressed as equation (25); η is the dimensionless composite degree of steel-UHTCC CBDs, expressed as equation (26). The coefficient K_n can be used to represent the linear relationship between the mid span shear force N₀ and the load P, as shown in equation (28). This coefficient is related to the section size, material properties, and loading position of the component.

N_{0} = K_{n} P

(27)

K_{n} = \frac{(e^{η (\frac{1}{2 γ} - 1)} - e^{η (\frac{1}{2 γ} + 1)}) γ L d_{c} + (1 - 2 γ) L d_{c} η (1 + e^{\frac{η}{γ}})}{2 η δ (1 + e^{\frac{η}{γ}}) \sum E I}

(28)

As stated in Assumption (4), the post-yield interface shear force can be estimated as equivalent to the interface shear force observed at the end of the elastic stage. According to the study conducted by (Maghsoudi and Bengar, 2006), the yield deformation is defined as the moment when the outermost fiber of the steel ribs yields. Therefore, the yield load P_y of the four specimens can be interpreted as the minimum load which makes the outermost fiber of the steel ribs yield. Under sagging moment, the yield position is located at the bottom edge of the steel part; while under hogging moment, it is located at the top edge of the steel part. The stress at this position on the mid-span section can be expressed as

σ_{s} = \frac{N_{y}}{A_{s}} + \frac{M_{s} y_{s}}{I_{s}}

(29)

where σ_s is the yield strength of the steel; N_y is the shear force at the mid-span interface when yielding occurs; and y_s is the distance from the centroid of the steel part to the calculated edge.

By combining equations (9)–(11), and (27)–(29), the solution for the yield load P_y and the shear force N_y can be obtained as

P_{y} = \frac{2 A_{s} Σ E I σ_{s}}{2 Σ E I K_{n} + A_{s} E_{s} (- 2 d_{c} K_{n} + L_{b}) y_{s}}

(30)

N_{y} = K_{n} P_{y}

(31)

When the load reaches the ultimate capacity of the CBDs, the interface shear force N_u can be considered equivalent to the interface shear force at the end of the elastic stage of structure, denoted as N_y. However, there is an upper limit to the shear bearing capacity of a single PBL connector. In cases where the number of PBL connectors is very small, the shear force acting on the PBL connector can easily reach its ultimate bearing capacity. At this point, the PBL connector undergoes plastic deformation, rendering the previously mentioned shear-slip stiffness theory inapplicable. In this case, the sum of the ultimate bearing capacity of all the PBL connectors on the whole span should be calculated as the interfacial shear force. In addition, theoretically, the shear force transmitted by PBL connectors cannot exceed the ultimate compressive (or tensile) capacity of UHTCC slab. Especially under hogging moment, the UHTCC slab will soon reach its ultimate compressive (or tensile) capacity and occurs plastic failure. In summary, the interface shear force under sagging and hogging moments in the ultimate state can be calculated using equation (32).

N_{u} = {\begin{cases} \min {N_{y}, n V_{u, 1}, σ_{cp} b t_{c}} for sagging moment \\ \min {N_{y}, n V_{u, 1}, σ_{tu} b t_{c}} for hogging moment \end{cases}

(32)

where n is the total number of PBL shear connectors; V_u,1 is the ultimate sheer capacity of a single PBL shear connector; σ_cp and σ_tu respectively refer to the peak compressive and tensile stress of UHTCC; the width of the UHTCC slab is represented by b; the thickness of the UHTCC slab is represented by t_c.

Calculation of ultimate flexural capacity

The Z-shaped profiled steel section has been used as an example for calculation. The detailed dimensions of the components can be found in Figure 5. Steel sections of the other shapes can be calculated by adjusting some of the cross-sectional dimensions shown in Figure 5.

Figure 5.

Schematic diagram of the dimensions of the cross-section used to calculate the capacity.

Showing the case of plastic neutral axis below, within, and above the steel plate respectively, Figure 6 illustrates the stress distributions across sections of the steel parts. In Figure 6, N_ts and N_cs are the tensile and compressive forces produced by tensile and compressive regions of steel part, respectively; A_s1 and A_s2 represent areas of tensile and compression zones of the steel part; h_0s represents the distance from the plastic neutral axis of the steel part to its lower edge; y_s1 and y_s2 respectively are the distances from the centroid of the cross-section in the tensile and compressive regions to the plastic neutral axis of the steel part; M_us is the bending moment contributed by the steel part under the ultimate moment.

Figure 6.

Schematic diagram for calculating theoretical capacity of steel part with different plastic neutral axis position. (a) Under the steel plate. (b) Within the steel plate. (c) Above the steel plate.

According to the equilibrium condition of axial force, the compressive and tensile stress of steel part should satisfy equation (34) and (35).

N_{u} = N_{ts} - N_{cs}

(33)

M_{su} = N_{ts} y_{s 1} + N_{cs} y_{s 2}

(34)

where h_0s, y_s1 and y_s2 can be obtained from equation (34). By substituting the results into equation (35), M_us can be obtained. In this study, as steel is isotropic material, the calculation formula for the ultimate sagging and hogging moments is the same without considering the buckling of steel ribs. For the three types of sections in which the plastic neutral axis is located in different positions, the calculation processes and expressions for M_us and h_0s are listed in Appendix B.

The sectional stress distributions of the UHTCC slab are exhibited in Figure 7. N_tc and N_cc are the tensile and compressive force produced by tensile and compressive regions of the UHTCC slab, respectively; A_c1 and A_c2 represent areas of tensile and compression zones of the of the UHTCC slab; h_0c represents the distance from the plastic neutral axis of the UHTCC slab to its bottom edge; y_c1 and y_c2 respectively represent the distances from the centroid of the cross-section in the tensile and compressive regions to the plastic neutral axis of the UHTCC slab; M_u is the bending moment contributed by the UHTCC slab under the ultimate moment.

Figure 7.

Schematic diagram for calculating theoretical capacity of UHTCC part.

The compressive and tensile stress of the UHTCC slab should satisfy equation (35) and (36).

N_{u} = N_{cc} - N_{tc}

(35)

M_{cu} = N_{tc} y_{c 1} + N_{cc} y_{c 2}

(36)

Similar to the calculation method of steel parts, M_cu can be calculated by combining equations (35) and (36). Since UHTCC is not isotropic material, it has different expressions for M_cu under sagging and hogging moments, detailed in Appendix B. In the end, by substituting equations (32), (34), and (36) into equation (9) , the ultimate moment M_u of assembled CBD can be obtained.

The relationship between the ultimate flexural capacity and the degree of shear connection, as derived from theoretical models, is illustrated in Figure 8. For assembled CBDs of various sizes and shapes, it is challenging to reflect the relationship between the degree of shear connection and ultimate flexural capacity solely through the number of PBL shear connectors. To conveniently evaluate the effectiveness of PBL shear connector numbers in improving the shear connection degree, many existing studies (Chen et al., 2024; Tong et al., 2023) defined the normalized PBL shear connector numbers α as

α = {\begin{cases} \frac{V}{V_{u}} = \frac{n}{n_{t}^{c}} for sagging moment \\ - \frac{V}{V_{u}} = - \frac{n}{n_{t}^{c}} for hogging moment \end{cases}

(37)

where V denotes the shear force of the shear span transferred by PBL connectors under sagging or hogging moment; V_u denotes the theoretical maximum shear capacity under sagging moment, expressed as equation (38); n_t^c refers to the threshold number of PBL shear connectors to reach the theoretical maximum shear capacity.

V_{u} = n_{t}^{c} V_{u, 1} = σ_{cp} b t_{c}

(38)

where the ultimate shear capacity of a single PBL shear connector is denoted by V_u,1, which is expressed in equation (1); σ_cp refers to the peak compressive stress of UHTCC; the width of the UHTCC slab is represented by b, and the thickness of the UHTCC slab is represented by t_c.

Figure 8.

Theoretical curve to predict flexural capacity under sagging and hogging moment.

Finite element analysis and verification

Element selection and interaction settings

To validate the accuracy of the theoretical model, approximately 100 FE models were established. As shown in Figure 9(a), the shell element with four nodes calculated with reduced integration (S4R) was used to simulate the profiled steel parts and steel diaphragms (Zhang et al., 2025). The UHTCC slabs were simulated using the three-dimensional solid element with eight nodes calculated with reduced integration (C3D8R) (Tong et al., 2024). Considering that rebars primarily bear shear forces, the two-node linear beam element in space (B31) was used for simulation (Chen et al., 2025).

Figure 9.

Schematic diagram of FE model. (a) Mesh condition. (b) Interaction settings.

The interaction settings between components are shown in Figure 9(b). For the contact surfaces between the UHTCC slabs and the profiled steel parts, a “penalty” friction formula (with a friction coefficient of 0.2) was used for “contact” in the tangential direction, and “hard” contact was applied in the normal direction (Yu et al., 2025). For the PBL shear connectors, a small portion of steel plates near the PBL holes were embedded in the UHTCC slabs to simulate the interactions. The reinforced bars were also embedded in the UHTCC slabs.

Constitutive model

The constitutive material models for reinforced bars and steel plates were simplified (Wu et al., 2025; Yu and Tong et al., 2025), as shown in Figure 10(a). As depicted in Figure 10(b), the compressive constitutive model of UHTCC includes both ascending and descending stages, following the study of (Zhou et al., 2015). The relationship between stress σ and strain ε during the ascending stage under compression is defined by equation (39).

σ = {\begin{cases} E_{0} ε 0 < ε < ε_{0.4} \\ (1 - γ) E_{0} ε ε_{0.4} < ε < ε_{0} \end{cases}

(39)

where E₀ is the elastic modulus of UHTCC, expressed by equation (40); ε_0.4 denotes the strain at 40% of ultimate strength; ε₀ denotes the strain at ultimate strength; and coefficient γ can be expressed by equation (41).

E_{0} = 1500 {(f_{c}^{'})}^{0.638}

(40)

γ = α \frac{ε E_{0}}{f_{c}^{'}} - β

(41)

where f´_c represents cylindrical strength of UHTCC; parameters α and β are found to be 0.308 and 0.124 by test results (Chen et al., 2024).

Figure 10.

Simplified constitutive material models. (a) Steel plates and reinforced bars. (b) UHTCC.

Since the ultimate tensile strain of UHTCC can generally reach 3%–6% (Xu et al., 2019), which exceeds the deformation capacity of CBD structures at failure, the tension softening is usually ignored in the tensile constitutive model of UHTCC (Wang et al., 2023). The post-peak tensile behavior of the UHTCC was characterized using a bilinear curve, based on analysis of the test results conducted by (Zhou et al., 2015), expressed as

σ = {\begin{cases} E_{tc} ε 0 < ε < ε_{tc} \\ σ_{tc} + E_{tu} (ε - ε_{tc}) ε_{tc} < ε < ε_{tu} \end{cases}

(42)

where σ_tc, ε_tc, and E_tc respectively represent the stress, strain, and elastic modulus of UHTCC at the onset of first cracking; E_tu is the elastic modulus of UHTCC during the tensile strain hardening phase.

Mesh sensitivity test

Nine FE examples of W-Z specimen with different mesh sizes (among 10-150 mm) were established for the mesh sensitivity test, as shown in Figure 11. It can be seen that the FE models can well simulate the ultimate capacity of assembled steel-UHTCC CBDs when the mesh size was smaller than 75 mm. Based on the results of the mesh sensitivity test, the element sizes for the UHTCC slabs and profiled steel parts were set as 25 mm, while the element size for reinforced bars was set as 2 mm.

Figure 11.

Ultimate load of steel-UHTCC CBDs with different mesh sizes.

Verification of FE models

To verify the accuracy, FE models of four different types of proposed assembled steel-UHTCC CBDs were established and compared with experimental results (Chen et al., 2024), as shown in Figure 12(a)–(d). The thickness of the UHTCC slabs is 45 mm and the diameter of PBL shear connectors is 12 mm. Each specimen is welded from three duplicate profiled steel sections. Additionally, FE models of steel-UHTCC CBDs with stud shear connectors from experimental study of (Tong et al., 2023) were developed and validated to further confirm the accuracy of FE modeling method, as shown in Figure 12(f)–(g).

Figure 12.

Comparison of load-deflection curves between FE models and tests. (a) W-T. (b) W-Z. (c) W-B. (d) H-B. (e) A1 (Tong et al., 2023). (f) B1 (Tong et al., 2023). (g) U1 (Tong et al., 2023).

The load-deflection curves of the FE models and the experimental results match well. It is evident that all errors are within 10%, with the discrepancy in ultimate load generally being less than 5%. This minor deviation between the FE simulation results and experimental results suggests that the model can be further utilized to validate the accuracy of theoretical analysis.

Accuracy of flexural resistance formulas

To simulate the impact of different degrees of shear connection on the ultimate flexural capacity, numerous FE models with four different types of profiled steel parts were established. Figure 12 presents a comparison between the results of FE simulations and theoretical calculations. In this comparison, the number of PBL shear connectors, n, is transformed into a normalized parameter α, as expressed in equation (39), to contrast the differences in flexural capacity under different degrees of shear connection (Figure 13).

Figure 13.

Comparison of relationship between different shear connection degrees and ultimate flexural capacity between FE models and theoretical model. (a) W-T. (b) W-Z. (c) W-B. (d) H-B.

The curves indicate that under sagging moments, the ultimate flexural capacity M_u increased more rapidly when the number of PBL shear connectors is very low (α = 0 - 0.5). In this range, the shear connectors theoretically reach their ultimate shear capacity and undergo plastic deformation. When α exceeding 0.5, the growth rate of M_u slows down as the shear connectors work within their elastic range. With the increase in shear-slip stiffness at the interface between UHTCC and steel plates, the efficiency of internal stress distribution within the composite structure is mildly enhanced, which leads to a slight increase in the overall ultimate flexural capacity. Maintaining the normalized PBL shear connector number α between 0.5 and 1 is optimal to achieve a fine ultimate flexural capacity of assembled steel-UHTCC CBDs while ensuring economic efficiency in engineering applications. For instance, reducing α from 1.5 to 0.5 in a W-T CBD could cut down PBL shear connector counts by 67%, significantly decrease the steel usage and welding time while retaining 96% of the flexural capacity, thereby decrease on-site construction requirements, which is expected to save costs. In addition, reducing unnecessary cutting and welding can also enhance the fatigue resistance of steel structures.

The positions of yield load and ultimate load under sagging moments, as determined by theoretical analysis, are illustrated on the load-displacement curves obtained from FE simulations in Figure 14.

Figure 14.

Comparison of theoretical ultimate and yield load and load-deflection of FE model (W-Z, α = 1.5).

When under hogging moments, the UHTCC slabs will be subjected to tension, crack or even fracture. Given that the tensile strength of UHTCC is significantly lower than its compressive strength, the contribution of UHTCC plates to flexural resistance is extremely limited. Therefore, under hogging moments, only a minimal number of PBL shear connectors are required to achieve full shear connection, and further increasing the number of PBL shear connectors does not significantly affect the ultimate flexural capacity of assembled steel-UHTCC CBDs.

Detailed theoretical and simulation results (due to space limitations, only the example of α being a multiple of 0.5 is shown), along with their comparisons, are presented in Table 2 and Figure 15. It can be observed that the discrepancy between the theoretical analyses and FE simulation results is relatively minor, not exceeding 10%, with most deviations within 5%, demonstrates the accuracy of the theoretical model.

Table 2.

Comparison of results between FE module and theoretical analysis (unit: kN).

Section shape	Moment direction	α	M _u,FE	M _u,theory	Error
W-T	Sagging moment	0	186.565	194.470	4.24%
		0.5	210.463	212.902	1.16%
		1	215.931	218.051	0.98%
		1.5	218.284	220.100	0.83%
	Hogging moment	0	−184.541	−188.564	2.18%
	Hogging moment	−1	−183.726	−193.935	5.38%
W-Z	Sagging moment	0	86.670	89.478	3.24%
		0.5	106.497	104.719	−1.67%
		1	107.551	106.481	−0.99%
		1.5	108.279	107.126	−1.06%
	Hogging moment	0	−86.551	−88.920	2.74%
	Hogging moment	−1	−90.365	−93.711	3.70%
W-B	Sagging moment	0	319.294	296.925	−7.01%
		0.5	357.912	346.207	−3.27%
		1	360.181	354.132	−1.68%
		1.5	361.433	357.283	−1.15%
	Hogging moment	0	−306.324	−295.762	−3.54%
	Hogging moment	−1	−310.276	−302.167	−2.55%
B-H	Sagging moment	0	276.190	289.205	4.71%
		0.5	300.891	307.296	2.13%
		1	308.677	313.326	1.51%
		1.5	309.499	314.918	1.75%
	Hogging moment	0	−276.222	−287.887	3.92%
	Hogging moment	−1	−277.756	−293.140	5.45%

Figure 15.

Accuracy of theoretical formula compared to simulated values. (a) W-T. (b) W-Z. (c) W-B. (d) H-B.

Conclusion

The flexural behavior of assembled steel-UHTCC CBDs using PBL shear connectors under different degrees of shear connection has been studied through theoretical analysis and FE simulation. The following conclusions can be summarized.

(1) A theoretical formula to calculate the ultimate flexural capacity of assembled steel-UHTCC CBDs based on partial interaction theory has been developed. It accounts for the shear-slip effects of PBL shear connectors and is capable of predicting the ultimate flexural capacity of assembled steel-UHTCC CBDs under both sagging and hogging moments.

(2) FE models of a novel assembled steel-UHTCC CBD were established using ABAQUS to investigate their flexural behavior. This model was compared with experimental results to validate the accuracy of the FE simulations.

(3) A comparison and validation between the theoretical analysis and FE simulation results have been conducted, confirming the accuracy of the theoretical model. The discrepancy between the theoretical analysis and FE simulation results is relatively minor, not exceeding 10%, with most deviations within 5%.

(4) Based on the results from FE simulations, it has been identified that maintaining the normalized PBL shear connector number α between 0.5 and 1 in engineering applications is optimal to achieve a fine ultimate flexural capacity of assembled steel-UHTCC CBDs while giving consideration to economic efficiency and fatigue resistance.

Footnotes

ORCID iD

Jing-Zhong Tong

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study was funded by the Natural Science Foundation of Zhejiang Province (Grant no. LR24E080002), the National Natural Science Foundation of China (Grant no. 51978607, 52108180).

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

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