Flexural Behavior of Self-Compacting PVA-SHCC Bridge Deck Link Slabs

Abstract

This paper studied the flexural behavior of bridge deck link slabs made with polyvinyl alcohol–strain-hardening cementitious composites (PVA-SHCC). The tensile and flexural properties of the self-compacting PVA-SHCC with four volume fractions, i.e., 0%, 1%, 1.5%, and 2%, were evaluated first. Next, using the similarity theory, composite models with a geometric similarity ratio of 1:5 were designed to represent the bridge deck with the link slabs. The models considered three materials for link slabs, including concrete, cement mortar, and self-compacting PVA-SHCC, and two different curing ages at 7 and 56 days. Bending tests were performed to investigate the flexural behavior of the models. Based on the fractal theory, the cracking characteristics of the models with different types of link slabs were compared, and the relationship between fractal dimensions and the flexural behavior of the models was studied. Numerical models were built to correlate with the results from the bending tests. It was illustrated that the flexural behavior of the self-compacting PVA-SHCC link slab is better than that of concrete and cement mortar link slabs, where the crack initiation and propagation can be postponed. The results can provide theoretical support and design guidance for the self-compacting PVA-SHCC bridge deck.

1. Introduction

When major girders are not continuous, a bridge deck link slab system can improve bridge continuity to provide more comfortable and safer conditions for drivers [1]. However, a bridge deck with a jointless link slab is usually the weak part of a bridge under long-term vehicle load, which results in cracking of the bridge deck and degradation of the bridge and highway system, especially in offshore bridges where there are more chloride ions to corrode the reinforcing steel [2,3]. While many structural maintenance and repair methods have been proposed and implemented, the inherent shortfall of concrete brittleness remains a hurdle. Therefore, new types of reinforced materials such as polypropylene fiber composites have been utilized to enhance the durability of reinforced concrete structures that can be used for both new and existing constructions [4,5,6].

Engineered cementitious composites (ECC), a type of high-performance fiber-reinforced cementitious composite (HPFRCC), was developed by Li and Leung [7]. It is mixed using short fibers and has obvious toughness properties. European scholars refer to cementitious composites with strain-hardening properties as strain-hardening cementitious composites (SHCC) [8,9]. Xu et al. [10,11] called it ultra-high-toughness cementitious composite (UHTCC) when the fiber volume fraction is less than 2.5% and the ultimate tensile strain is larger than 3%. Among different types of fibers, polyvinyl alcohol (PVA) fiber is promising because of its low modulus of elasticity, high tensile and ductile properties, and high resistance [12,13,14]. It provides a new direction for structural maintenance and repair. The chemical formula of polyvinyl alcohol is (C₂H₄O)_x [12]. Polyvinyl alcohol can be prepared by hydrolysis of polyvinyl acetate or other vinyl ester-derived polymers with formate or chloroacetate groups instead of acetate [12,13]. The PVA fiber used in this study was produced and sold by a Japanese chemical company (Kuraray Co., Ltd., Tokyo, Japan).

The mechanical properties of PVA-SHCC have been discussed and predicted by experimental study, theoretical calculation, finite element modelling, machine learning (ML), and artificial intelligence (AI). For material performance, the mechanical properties of PVA-SHCC were first tested under uniaxial and cyclic tension, bending, and impact loads [15,16,17]. These reports discussed the temperature response and time effect of PVA-SHCC and instituted a fuzzy-probabilistic durability concept for PVA-SHCC. Paul et al. [18] concluded that a high level of tensile strain was not associated with large crack width openings of PVA-SHCC. Their results indicated that slight and parallel cracks improved the integrity and ductility of reinforced concrete composites. Some research focused on the mechanical properties and durability of PVA-SHCC under aggressive environments [19]. The corrosion resistance of SHCC was tested and discussed through an experimental study [20]. For structural performance, PVA-SHCC was first used for bridge deck link slabs in a transportation infrastructure project in Grove Street Bridge, MI, USA, in 2005 [21]. By removing the existing expansion joints and replacing a portion of two adjacent decks with a section of ECC material on top of the joints, a continuous deck surface was constructed [22,23]. For theoretical calculation, the PVA-SHCC deck link slabs’ formulas for deformation and internal forces were deduced and instituted [24]. The calculation results indicated that PVA-SHCC bridge deck link slabs have higher cracking resistance than concrete bridge decks under fatigue load [25]. The cracking properties of new PVA-SHCC bridge deck link slabs were preliminarily studied by He et al. [26], but further research works should be performed. In recent years, machine learning (ML) and artificial intelligence (AI) algorithms have been utilized to accurately predict the mechanical behavior of materials, as well as to optimize the mechanical properties of structures. Some computational frameworks using surrogate models have been proposed to address stochastic, multi-scale issues in composite material design [27,28]. Artificial Neural Networks (ANN), Convolutional Neural Networks (CNN), and Forest Deep Neural Network (FDNN) can be utilized to analyze the mechanical properties of SHCC. Preliminary prediction models for compressive strength, tensile strength, and ductility of SHCC were built [29,30,31]. The characteristics of the dense microcracks and micropores of SHCC were identified and predicted by generative AI [32,33]. Because of harsh construction conditions, self-compacting concrete composites are suitable for bridge construction due to their high flowability and segregation resistance [23]. Therefore, bridge deck link slabs that have both self-compacting and strain-hardening properties are needed. However, self-compacting PVA-SHCC has not been deeply considered in existing research. The flexural performance and cracking mechanism of self-compacting PVA-SHCC bridge deck link slabs are still unclear and require further study.

Due to the limitations of former investigations, this study mainly focused on the flexural behavior and cracking characteristics of self-compacting PVA-SHCC bridge deck link slabs. A type of self-compacting PVA-SHCC was developed. Six composite models representing bridge decks were tested under bending loads. Three different types of link slabs (concrete, cement mortar, and self-compacting PVA-SHCC) and two curing ages (7 and 56 days) were considered. The cracking characteristics of the composite models were investigated via fractal theory. Numerical models were built.

2. Materials and Methods

2.1. Mechanical Properties of PVA-SHCC

2.1.1. Materials

KURALON™ K-II unoiled PVA fiber (Kuraray Co., Ltd.) was utilized in this study, with the mechanical properties shown in

Table 1. Properties of PVA fiber.

Density (g/cm3)	Diameter (μm)	Length (mm)	Strength (MPa)	Elasticity Modulus (GPa)	Elongation (%)
1.3	40	12	1560	41	6.5

. According to the relevant study [8,9,10,11,12], the percentage of PVA fiber should be lower than 2.5% due to the workability of SHCC. Therefore, four fiber volume fractions (0%, 1%, 1.5%, and 2%) were considered. Details of the mix proportions are shown in

Table 2. Mix proportions of self-compacting PVA-SHCC (kg/m³).

No.	Sand	Cement	Fly Ash	Water	Water Reducer	PVA	Cellulose
1	525.6	477.6	716.4	417.9	4.8	0 (0%)	0.333
2	520.3	472.8	709.3	404.2	11.8	13 (1%)	0.333
3	517.7	470.4	705.7	411.6	16.5	19.5 (1.5%)	0.333
4	515.1	468.1	702.1	396.4	16.4	26 (2%)	0.333

. By adjusting the proportions of fly ash and water reducer, the slump of the mixtures was larger than 550 mm, which indicates that the mixtures were self-compacting [34].

2.1.2. Tensile Properties

Twenty-four samples with dimensions of 400 × 80 × 10 mm³ were constructed [35]. The mixtures were mixed using a concrete mixer, removed from the forms after 24 h, and cured 28 days in standard conditions, where the relative humidity was larger than 90% and the temperature was 20 ± 3 °C. Tensile tests were conducted using the MTS CMT4000 machine (MTS Systems(China) Co., Ltd., Shanghai, China), as shown in Figure 1. The stress–strain curves of the self-compacting PVA-SHCC are shown in Figure 2.

From Figure 2, it can be observed that the mixtures without PVA resulted in brittle failure, where the tensile strength and ultimate tensile strain are approximately 0.6 MPa and 0.1%, respectively. The mixtures with PVA have obvious strain-hardening characteristics under tensile load, with ultimate tensile strains of approximately 0.9%, 2.6%, and 2.4% when fiber volume fractions are 1.0%, 1.5%, and 2.0%, respectively. The ductility ratio can be defined as:

μ = ε_u/ε_y

(1)

where ε_u and ε_y are the ultimate and yielding strains of PVA-SHCC, respectively. The average values of the ultimate tensile stress, the ultimate tensile strain, and μ for each mix proportion are listed in

Table 3. Ultimate tensile stress and strain of self-compacting PVA-SHCC.

Volume Addition Rates (%)	Ultimate Tensile Stress (MPa)	Yield Strain (%)	Ultimate Tensile Strain (%)	Ductility (μ = εu/εy)
0	0.623	0.051	0.095	1.86
1	1.108	0.083	0.913	11.02
1.5	1.201	0.112	2.596	23.18
2	1.359	0.102	2.431	23.76

.

The tensile cracks of PVA-SHCC are shown in Figure 3. The mixtures without PVA have only one crack under the ultimate tensile load, while the mixtures with PVA have more obvious parallel slight cracks as fiber volume fractions increase. Based on Table 3, the ultimate tensile strains of the mixtures with fiber volume fractions of 1.0%, 1.5%, and 2.0% are 9.6, 27.3, and 25.6 times the value of the mixture without PVA, respectively. Moreover, the μ of these mixtures are 5.9, 12.4, and 12.8 times the value of the mixture without PVA, respectively. The results indicate that the mixture has excellent tensile properties and ductility when the fiber volume fractions are 1.5% and 2.0%. It is noted that the μ of the mixture with a fiber volume fraction of 2.0% is only 2.5% greater than that of the mixture with a fiber volume fraction of 1.5%. In addition, the mixture with a fiber volume fraction of 2.0% has negative working performances, such as agglomeration and low flowability. Moreover, higher fiber volume fractions introduce higher cost. Therefore, 1.5% is best among all fiber volume fractions under tension load, which need to be verified by a bending test.

2.1.3. Flexural Behavior

Eighteen samples with dimensions of 400 × 100 × 10 mm³ were constructed with different fiber volume fractions (1%, 1.5%, and 2%). The bending test setup [35] and crack patterns of self-compacting PVA-SHCC are shown in Figure 4. To determine the strain of the tensile surface of samples at the mid-span, strain transducers were attached at point A (Figure 4). Figure 5 shows the stress–strain curves of self-compacting PVA-SHCC in the bending test at point A. The bending properties of self-compacting PVA-SHCC are listed in

Table 4. Bending properties of self-compacting PVA-SHCC.

	Items	Deflection (mm)	Quantity of Cracks	Max Width of Cracks (μm)	Ductility (μ = εu/εy)
Vol. (%)
1.0		4.6	14	98	10.15
1.5		5.0	17	105	14.32
2.0		5.3	19	102	15.60

.

From Table 4, it can be observed that the deflections and quantities of cracks increase as fiber volume fraction increases. The multipoint cracking mode is beneficial to enhance the durability and service life of cement-based composites. High toughness behavior is observed when the μ of self-compacting PVA-SHCC is larger than 10, where the sample with a fiber volume fraction of 2.0% has a deflection of 5.3 mm. The bending test results demonstrate that the mixture had excellent properties and ductility when the fiber volume fractions were 1.5% and 2.0%, which is similar to the conclusions drawn from the tensile tests. Therefore, the No. 3 mix proportion in Table 2 was used in the following composite model test.

2.2. Design of Composite Models

2.2.1. Similarity Design

Composite models were designed to represent the bridge deck and pavement shown in Figure 6 and Figure 7. Based on the Similarity Theory, geometric, physical, and boundary condition similarities can be built between the prototype structures and models [36]. Geometric similarities can be defined as follows:

S_A = S_l²; S_w = S_l³; S_I = S_l⁴; S_x = S_l·S_ε

(2)

where S_l, S_A, S_w, S_I, S_x, and S_ε are affinity constants of length, area, section modulus, moment of inertia, displacement, and normal strain, respectively. Physical similarity can be defined as the following:

S_μ = 1; S_ε = S_γ = 1; S_σ = S_E; S_τ = S_G

(3)

where S_μ, S_γ, S_σ, S_E, S_τ, and S_G are affinity constants of Poisson’s ratio, shear strain, normal stress, elastic modulus, shear stress, and shear modulus, respectively. Boundary condition similarity can be defined as follows:

S_P = S_σ·S_l²; S_M = S_σ·S_l³·σ

(4)

where S_P and S_M are affinity constants of concentrated load and concentrated moment, respectively. The affinity constant of stiffness has the following definition:

S_k = S_E·S_l

(5)

The ratio of geometric similarity was determined to be 1:5. The lateral inertia of vehicular loading was considered. The affinity constants of composite models based on the similarity design are shown in

Table 5. Affinity constants of composite models.

Types	Physical Quantity	Affinity Constants	Types	Physical Quantity	Affinity Constants
Physical	σ	1	Geometrical	l	1/5
	ε	1		x	1/5
	E	1		θ	1
	ν	1		A	1/25
Boundary condition	f	5		W	1/125
	P	1/25		I	1/625
	M	1/125		k	1/5
	J	1/125

.

2.2.2. Design of Composite Models

Based on the similarity design, the concrete beams were designed as two 1.2 m span rectangular sections with dimensions of 1200 × 100 × 120 mm³, where they were connected to a link slab [36,37]. The reinforcement ratio of the models for each span was 1.248%. Commercial concrete (C60) was used, with mix details shown in

Table 6. Mix proportions of concrete.

Cement (kg/m3)	Sand (kg/m3)	Stone (kg/m3)	Water (kg/m3)	Fly Ash (kg/m3)	Water Reducer (kg/m3)	W/C	W/B
450	678	1040	159	60.0	12.8	0.353	0.304

Note: W/C is the water–cement ratio; W/B is the water–binder ratio.

. The longitudinal reinforcements were HRB400 (φ16) hot-rolled reinforcing steel, with compressive and tensile strengths of 210 and 310 MPa, respectively. The stirrups were HPB335 (φ6) with a spacing of 100 mm (Figure 8). The concrete cover was 30 mm. The top surfaces of concrete beams were roughened by an angle grinder to enhance the bond strength of the interface between concrete and SHCC. The thickness of the SHCC link slab was 30 mm, which had two HRB400 (φ6) hot-rolled rebars in a longitudinal direction with a reinforcement ratio of 1.8%. No transverse reinforcement was provided. The joint width between the two spans was 15 mm, as shown in Figure 9. It should be noted that the inertial effect has significant impact on the deflection of thin rectangular plates like those in a bridge deck. The moving mass-beam-system models can be used to capture the dynamic behavior and responses of bridge decks under vehicular loading [38,39]. As an exploratory study, this paper focused on the bending characteristics of concrete beams and self-compacting PVA-SHCC link slabs. The dynamic vibration of self-compacting PVA-SHCC link slabs under the force of a moving mass should be considered in a further study.

2.3. Experimental Investigation

2.3.1. Test Setup

The specimens were divided into three series: C, M, and S, corresponding to bridge deck link slabs with concrete, cement mortar, and self-compacting PVA-SHCC, respectively. To determine the effect of early curing on the flexural behavior, two curing ages were considered (7 and 56 days). Details of the test conditions are shown in

Table 7. Test conditions.

Series	No.	Materials of Decks	Curing Age (Days)
C	C-1	Concrete	7
	C-2		56
M	M-1	Cement mortar	7
	M-2		56
S	S-1	Self-compacting PVA-SHCC	7
	S-2		56

.

2.3.2. Test Procedures

Bending tests were performed to investigate the flexural behavior of the models [39]. The experimental setup is shown in Figure 9. The setup ensured proper bonding between the concrete beams and the deck link slab, where zones A and B are no-stirrup bending zones (shown in Figure 8) and zone C is the jointless link slab. Six strain transducers were attached to the side surface of the composite models at each zone to record stain distributions under bending load [40].

3. Results and Discussions

3.1. Load–Deflection Relationships

After different curing times, composite models were taken out and bending tests were performed. The relationships between applied loads and average deflections of the two spans in composite models are shown in Figure 10. At the initial stage of the bending test, a linear relationship could be observed between the load and deflection. As the bending load increased, longitudinal reinforcements reached the yielding strength, and oblique cracks were generated and developed.

3.2. Ductility

Ductility, which needs to be carefully considered in the design of concrete structure, is a significant factor that affects the bending capacities of concrete beams [41,42]. High ductility is beneficial for the performance of bridge deck link slabs by providing an early warning before failure. Displacement ductility coefficient (μ) is used to evaluate the ductility of the models, which can be defined as follows:

μ = Δ_u/Δ_y

(6)

where Δ_u and Δ_y are the ultimate and yielding deflections of models, respectively. The results of μ are listed in

Table 8. Results of the bending test.

No.	Cracking Load (kN)	Ultimate Capacity (kN)	Ductility (μ = Δu/Δy)
C-1	20	71.4	1.26
C-2	15	81.7	1.51
M-1	17	64.3	1.74
M-2	10	91.1	1.60
S-1	17	73.8	1.69
S-2	15	92.2	1.83

. The ductility coefficients of series S and M are larger than that of series C. The ductility coefficient of S-2 is the largest in all specimens. It is indicated that the energy dissipation capacity of self-compacting PVA-SHCC bridge deck link slabs is high, which is more advantageous in terms of seismic performance.

3.3. Cracking and Ultimate Loads

The ultimate strengths (P_u) and cracking loads (P_cr) of composite models are listed in Table 8. The ultimate capacities of the composite models with a curing time of 56 days (average of C-2, M-2, and S-2) were 19.3% greater than those of the models with a curing time of 7 days (average of C-1, M-1, and S-1). The ultimate capacities of series S were the largest in all of the series, which demonstrated that the bending capacities of composite models increased when PVA-SHCC was used in bridge deck link slabs. However, the ultimate capacity of S-2 was only 4.8% greater than that of M-2; this was because the amount of reinforcing steel governed the capacity of this system. The aspect of the structural behavior that was more improved with SHCC was the serviceability, such as ductility (as shown in Table 8) and cracking characteristics.

3.4. Cracks and Failure Modes

Cracks in the bending zones (zones A and B) were recorded under various loading levels, as shown in Figure 11. The oblique cracks were observed when longitudinal reinforcements in the tensile zone yielded and concrete in the compressive zone crushed almost at the same time, which indicates that the typical failure mode of composite models was balanced, reinforced beam failure. In particular, as for series C and M, a major crack was generated and developed on the top surface of the jointless link slab (zone C), but many minor parallel cracks were observed in zone C of series S under the same loading level, which manifests that the link slab was damaged under the ultimate load. The link slabs were debonded from the composite model in M-1 but not in M-2, which indicates that the bond behavior between the link slabs and concrete was weak at the early curing age (7 days) under a bending load.

3.5. Cracking Characteristics of Jointless Link Slabs

3.5.1. Descriptions of Cracks

Cracks in the jointless link slabs (zone C) under the same bending load (70 kN) were recorded, as shown in Figure 12 and Figure 13. A major crack could be observed for each model. Many minor cracks, which were approximately parallel to the major crack, could be seen in series S, which is similar to the results from Figure 4. However, one or two minor cracks were observed in series C and M, and they were not parallel to the major crack.

3.5.2. Width and Depth of Cracks

The width, depth, and distance of the cracks in the jointless link slabs under the same load were measured and are recorded in Figure 14 and

Table 9. Properties of cracks in jointless link slabs (zone C).

No.	Materials	Quantities of Cracks	Width of Major Crack (mm)	Average Width of Cracks (mm)	Average Distance of Cracks (mm)	Ultimate Depth of Cracks (mm)	Df
C-1	Concrete	1	0.32	0.320	-	33.1	1.028
C-2	Concrete	1	0.38	0.380	-	30.5	1.060
M-1	Cement mortar	5	0.18	0.145	-	31.3	1.062
M-2	Cement mortar	2	0.22	0.160	-	23.6	1.036
S-1	PVA-SHCC	7	0.12	0.087	1.360	28.0	1.035
S-2	PVA-SHCC	9	0.14	0.105	1.473	25.5	1.087

, respectively. Series S had the most cracks, but with the smallest average width and ultimate depth among all series. The ultimate depth of cracks in series S (average of S-1 and S-2) was 15.9% and 2.8% smaller than those in series C and M, respectively.

The average width of cracks in series S (average of S-1 and S-2) was 72.6% and 37.0% smaller than those in series C and M, respectively. The cracks of the PVA-SHCC link slabs had the smallest width and depth, highest quantities, and parallelism under bending load, which indicates that the cracking property of the jointless link slabs can be improved using self-compacting PVA-SHCC.

3.5.3. Fractal Characteristics of Cracks

Fractal theory has been widely used to study fracture energy, cracking mode, and the porosity of engineering materials over the last two decades due to its high reliability [43]. In this study, the box-counting method [44] was used to obtain the distributions of cracks and fractal dimensions of link slabs. The initial side length (r) of the square mesh was 2 mm (L₀ = 2 mm), followed by square meshes with dimensions of 2L₀ × 2L₀, 4L₀ × 4L₀, 8L₀ × 8L₀, 16L₀ × 16L₀, etc. The quantities of the meshes that were occupied by the cracks were counted in every size of mesh, and the relationships between the quantities (N) and reciprocal of side length (1/r) were obtained. Figure 15 shows the relationships between logN and log(1/r). The fractal dimensions can be defined as follows:

D_f = logN/log(1/r)

(7)

As can be seen in Figure 15, the distributions of cracks in the link slabs had fractal characteristics because a liner relationship between logN and log(1/r) could be observed. The fractal dimensions are listed in Table 9.

3.5.4. Relationships between D_f and Bending Behavior

The relationships between D_f and P_u/μ are shown in Figure 16 and Figure 17, respectively. P_u and μ increased when D_f increased, following linear relationships. The relationship between P_u and D_f was clearer (R² was more than 0.9); however, the relationship between μ and D_f was not clear enough (R² was less than 0.7) because of the high discreteness of cracking propagation, which needed more experimental data to verify. These relationships could introduce a new approach to determine the bending behavior of bridge decks, which might be used for the health monitoring of a bridge. Further study is recommended.

3.6. Distribution of Strain

The strain distributions and tensile strains before crack generation in zone C of C-1, M-1, and S-1 at different loading levels are shown in Figure 18 and

Table 10. Tensile strains of link slabs before crack initiation (zone C).

No.	Materials of Decks	Cracking Load (kN)	Cracking Strain (10−3)
C-1	Concrete	14.70	0.715
M-1	Cement mortar	16.63	0.931
S-1	Self-compacting PVA-SHCC	38.34	8.963

, respectively. The cracking strain in zone C of S-1 was 8.963‰, which was 12.5 and 9.6 times those of C-1 and M-1, respectively. The cracking load of S-1 was 38.34 kN, which was 2.6 and 2.3 times those of C-1 and M-1, respectively. The results indicate that the self-compacting PVA-SHCC link slab had higher crack resistance, better integrity, and larger bending capacity than those of the concrete and cement mortar link slabs.

4. Finite Element Analysis

4.1. Finite Element Models

The geometry of the finite element (FE) model is shown in Figure 19. Materials used in modeling include concrete and PVA-SHCC. A concrete-damaged plasticity model was adopted to simulate cracking propagation until the failure of concrete beams. The yield function is as follows:

$$F(\overline{\sigma },{\tilde{\epsilon }}^{pl})={\displaystyle \frac{1}{1-\alpha }}(\overline{q}-3\alpha \overline{p}+\beta ({\tilde{\epsilon }}^{pl}){\widehat{\overline{\sigma }}}_{\max }-\gamma (-{\widehat{\overline{\sigma }}}_{\max }))-{\overline{\sigma }}_c^{pl}\le 0$$

(8)

where α, β, and γ are dimensionless material constants; ${\overline{\sigma }}_t^{ }$ and ${\overline{\sigma }}_c^{ }$ are effective tensile and compressive cohesion stresses, respectively; $\overline{q}$ is the Mises equivalent of effective stress; $\overline{p}$ is the effective hydrostatic pressures; and ${\tilde{\epsilon }}^{pl}$ is strain function, which is defined as the following:

$$\beta ({\tilde{\epsilon }}^{pl})={\displaystyle \frac{{\overline{\sigma }}_c({\tilde{\epsilon }}_c^{pl})}{{\overline{\sigma }}_t({\tilde{\epsilon }}_t^{pl})}}(1-\alpha )-(1+\alpha )$$

(9)

Based on the uniaxial tension and compression tests from this paper and others’ research [43,45], the typical stress–strain curves of PVA-SHCC are shown as dotted lines in Figure 20. To simplify the model, the following assumptions were made [46,47]: (1) stress–strain relationships of PVA-SHCC in tension and compression can be described as a bilinear curve in Figure 20 (left) and a trilinear curve in Figure 20 (right), respectively; (2) planes remain plane after deformation.

The tensile stress–strain relationship of PVA-SHCC can be expressed as follows:

$${\sigma }_t=\left\{\begin{array}{cc}\frac{{\sigma }_{tc}}{{\epsilon }_{tc}}\epsilon \hfill & 0\le \epsilon <{\epsilon }_{tc}\\ {\sigma }_{tc}+({\sigma }_{tu}-{\sigma }_{tc})(\frac{\epsilon -{\epsilon }_{tc}}{{\epsilon }_{tu}-{\epsilon }_{tc}})\hfill & {\epsilon }_{tc}\le \epsilon <{\epsilon }_{tu}\end{array}\right.$$

(10)

where σ_tc and σ_tu are cracking and ultimate tensile strengths, respectively, and ε_tc and ε_tu are cracking and ultimate tensile strains, respectively. The compressive stress–strain relationship of PVA-SHCC can be expressed as the following:

$${\sigma }_c=\left\{\begin{array}{cc} 2\frac{{\sigma }_{c0}}{{\epsilon }_{c0}}\epsilon \hfill & 0\le \epsilon <{\epsilon }_{c0}\\ \frac{2}{3}{\sigma }_{c0}+\frac{{\sigma }_{c0}}{2{\epsilon }_{c0}}(\epsilon -\frac{{\epsilon }_{c0}}{3})\hfill & \frac{1}{3}{\epsilon }_{c0}\le \epsilon <{\epsilon }_{c0}\hfill \\ {\sigma }_{c0}+({\sigma }_{cu}-{\sigma }_{c0})(\frac{\epsilon -{\epsilon }_{c0}}{{\epsilon }_{cu}-{\epsilon }_{c0}})& {\epsilon }_{c0}\le \epsilon <{\epsilon }_{cu}\end{array}\right.$$

(11)

where σ_c0 and σ_cu are compressive and ultimate compressive strengths, respectively, and ε_c0 and ε_cu are peak and ultimate compressive strains, respectively. This paper assumed σ_c0 = 2σ_cu and ε_c0 = 2/3ε_cu. The displacement was applied on the top of bridge deck at the same position with the bending test. The mechanical properties of concrete, reinforcing steel, and PVA-SHCC are listed in

Table 11. Mechanical properties of different materials.

Materials	Density (kg·m−3)	Elasticity Modulus (GPa)	Poisson’s Ratio	Yield Stress (MPa)	Plastic Strain (%)	Ultimate Tensile Stress (MPa)	Ultimate Tensile Strain (%)
Concrete	2400	32.5	0.2	N.A.	N.A.	1.150	0.010
Reinforcing steel	7800	210	0.3	300	0	N.A.	N.A.
PVA-SHCC	1900	21.0	0.25	N.A.	N.A.	1.201	2.596

Note: “N.A.” is “Not Applicable”.

, where the ultimate tensile stress and strain of PVA-SHCC are based on the test results from the Section 3.1.

4.2. Convergence Study

To ensure the FEM results were reliable and accurate, a convergence check was conducted in this paper. Coarse meshes usually produce inaccurate FEM results. The numerical solution tends toward a converged value when the number of elements increases. Five different numbers of elements were used in FE models. The influence of the elements’ density on ultimate load and ultimate deflection are listed in

Table 12. Results of mesh convergence check.

No. of Elements	Ultimate Load (kN)	Ultimate Deflection (mm)
2184	52.02	9.16
4294	72.08	13.73
6980	70.05	13.96
9832	69.76	14.07
12,766	69.47	14.11

. When the number of elements increased, ultimate load and ultimate deflection were similar. From the convergence check, 12766 C3D8R elements were chosen to produce reliable FEM results.

4.3. Comparisons between the Experimental and Modeling Results

To verify the accuracy of the FE model, load–deflection curves of the composite model with PVA-SHCC jointless link slabs from both testing (S-2) and the FE model are reported in Figure 21. Like the test results, the curve from FE model can also be divided into two stages. As the bending load increased, longitudinal reinforcements reached the yielding strength, and oblique cracks were generated and developed. The slight discrepancy between the test and FE results can be seen, especially in the plastic stage. The reason for this phenomenon is the self-compacting characteristics in different curing ages were not considered in the FE model. This FE model is appropriate for a specimen with the curing age at 56 days but is inappropriate to describe the change of characteristics under different curing ages. At the next research stage, more samples with different curing ages should be tested to build a more accurate prediction model for self-compacting SHCC. The constitutive relations of self-compacting SHCC used in the FE model were slightly different from that of the tested models. The deviation was less than 10%, which is acceptable.

The moment–curvature relationships between the testing and FE models are shown in Figure 22. The curves can be divided into two stages. At the first stage, the moment increased linearly with the curvature before the cracking moment of 14.1 kN·m was reached. Subsequently, the slope of the curve decreased at the second stage. When the tensile fiber strain of link slabs reached the ultimate tensile strain of PVA-SHCC, the maximum moment was obtained [47]. The ultimate moments of FE and test values are 126.4 kN·m and 113.9 kN·m, corresponding to curvatures of 10.39 × 10⁻⁵ mm⁻¹ and 9.80 × 10⁻⁵ mm⁻¹. The moment and curvature of the FE model were only 10.9% and 6.0% larger than those of the test values, which indicated that the FE model was valid and could be used in the following analysis.

4.4. Distributions of Stress and Strain in Beams

In this FE model, the plastic strain magnitude (PEMAG) was utilized to characterize the cracking trend. Typical distributions of the plastic strain magnitude of concrete beams with PVA-SHCC link slabs are shown in Figure 23. PEMAG concentrations could be observed in the tensile zone at the mid-span of the beams, which was consistent with the cracks of tested beams. Cracks were generated when the strain reached the ultimate tensile strain of the concrete. The quantities and lengths of the cracks increased with load increases, and then diagonal cracks appeared when the longitudinal steel bars yielded.

4.5. Distributions of Strain in Link Slabs

The PEMAG of concrete and self-compacting PVA-SHCC link slabs in zone C are shown in Figure 24 and Figure 25, respectively. The principal stress and plastic strain of the jointless link slabs under the ultimate bending load are listed in

Table 13. Ultimate principal stress and plastic strain of the bridge deck link slabs.

Materials	Ultimate Principal Stress (MPa)	Ultimate Plastic Strain (%)
PVA-SHCC	3.187	2.32
Concrete	3.336	0.098

.

As can be seen from Figure 24, PEMAG concentrations (2.73 × 10⁻⁴) were observed when the load reached 50% of the ultimate bending load, which indicated that cracks appeared in the concrete link slabs. Two main cracks were generated when the load reached the ultimate bending load, which was consistent with the cracks of Series C. From Figure 25, it can be seen that the distributions of strain in the self-compacting PVA-SHCC link slabs were uniform at different loading levels, which was consistent with the cracks of Series S. Finally, self-compacting PVA-SHCC link slabs failed when the load reached the ultimate bending load. The results were similar to the test results. They indicated that the crack resistance of self-compacting PVA-SHCC link slabs was higher than that of concrete link slabs under the bending load. The integrity and serviceability of composite models were improved. Therefore, the self-compacting PVA-SHCC in this study can be used as bridge deck link slabs to enhance the bending properties and crack resistance of a concrete bridge.

5. Conclusions

In this study, a new type of self-compacting PVA-SHCC was developed. The tensile and bending behaviors of self-compacting PVA-SHCC were studied. Six composite members were constructed and evaluated using bending tests. The bending behavior and cracking characteristics of the self-compacting PVA-SHCC link slabs were first evaluated by fractal theory. An FE model was developed. The testing and FE results were compared. Based on this study, the following conclusions can be drawn:

(1) The new self-compacting PVA-SHCC had a high ductility ratio (between 15 and 25) under both tensile and bending loads when the fiber volume fraction was between 1.5% and 2.0%. It is suitable to be used in both the model tests and structures.

(2) The ultimate bending capacity of Series S with self-compacting PVA-SHCC link slabs was 8.4% and 6.8% greater than those of Series C and M at the same curing age. The ductility coefficient of Series S was 27.1% and 8.3% greater than those of Series C and M at the same curing age. Cement mortar link slabs debonded from the concrete beam when they were cured for 7 days. This phenomenon did not happen when the composite members were cured for 56 days.

(3) The cracks in concrete (Series C) and cement mortar (Series M) link slabs were 3.65 and 1.59 times wider, respectively, and 1.19 and 1.03 times deeper, respectively, than those of self-compacting PVA-SHCC link slabs (Series S) under the ultimate bending load. The ultimate capacities and displacement ductility coefficients increased as D_f increased, following a linear relationship. The D_f of the composite members with self-compacting PVA-SHCC link slabs (Series S) was 1.6% and 1.1% larger than that of the composite members with concrete (Series C) and cement mortar (Series M) link slabs, respectively.

(4) Based on the strain distributions and cracking modes of FE models, the integrity of composite members was improved using self-compacting PVA-SHCC. The moment and curvature of FE models were 10.9% and 6.0% larger, respectively, than those from test results.

The flexural cracking mechanisms of self-compacting PVA-SHCC bridge deck link slabs can be revealed from the above conclusions, which provide theoretical support and design guidance for the PVA-SHCC with self-compacting characteristics. It is noted that the eccentric loading is an important issue in a tensile test of SHCC. A universal joint should be used at the ends of SHCC specimens to ensure axial tensile loading in further research. This study focused on the flexural behavior of PVA-SHCC structures. The torsional behavior of PVA-SHCC structures will be considered in further research.