Basalt fibre continuous reinforcement composite pavement reinforcement design based on finite element model

The shrinkage of concrete materials due to temperature drops will cause relative slippage between the CRC and base layers. According to the shear test and regression statistics results, the distribution of frictional shear stress caused by slip is usually divided into linear, bilinear and hyperbolic models [4]. They are different due to different types of grassroots. We use a bilinear slip model closer to the actual situation, as shown in Figure 1. Comprehensively consider the critical displacement value u_o of various types of bases. Suppose the amount of slippage u_c ≤ u_o when the crack occurs, that is, the shear stress τ_c = k_cu_c, where k_c is the shear slip coefficient between the CRC layer and the base layer.

Fig. 1

Relative slippage occurs between the concrete and the reinforcement due to the uncoordinated deformation. We use the shear stiffness k_s to characterise the bond-slip relationship. It is related to the material properties and surface structure of the bar. According to the results of the indoor pull-out test of this subject, the typical bond-slip curve of BFRP bars is similar to the bond failure of steel bars [5]. The result is shown in Figure 2. When the relative slip is 0–0.5 mm, the relationship between bonding stress and relative slip is approximately linear. Therefore, we can adopt a simplified linear relation expression τ_s = k_s(u_c − u_s), where τ_s is the bonding stress and u_s is the displacement of the contact interface.

Fig. 2

The CRC board temperature drop and shrinkage model is shown in Figure 3. We analyse the micro-element section with the length of the concrete slat dx along the direction of travel. The origin of the coordinates is set at the midpoint of the slat. It is assumed that the concrete’s stress and the reinforcement are uniforms in the cross-section, and the deformation of the two perpendiculars to the axial direction is ignored [6]. For the strain caused by the drying and shrinkage of concrete, we can calculate it according to the humidity of the environment and using empirical formulas. The boundary conditions for solving the equation are assumed as follows. (1) At the crack surface, the tendon displacement is zero and the concrete stress is zero. (2) In the middle of the slab, the displacement of the bars is zero. After finishing the force balance equation of the slat model and the physical equation of concrete and reinforcement, the following formula can be obtained.

Fig. 3

In the formula a₁ = πd_sk_s/(A_cE_c),a₂ = bk_c/(A_cE_c),a₃ = πd_sk_s/(A_sE_s), b is the rib spacing, d_s is the diameter, E_c,E_s is the elastic modulus of concrete and reinforcement, respectively, A_s is the cross-sectional area of a single rib and A_c is the cross-sectional area of the concrete slab.

Suppose ΔT is the temperature drop. ε_sh is the concrete drying shrinkage strain. a_c is the longitudinal linear expansion coefficient of the bar. a_c is the longitudinal linear expansion coefficient of concrete. σ_c is the concrete stress. σ_s is the stress of the bar. We substitute the boundary conditions and finally simplify the resulting displacement stress expression as follows: (3) us=F1b1sh(r1x)+F2b2sh(r3x) {u_s} = {F_1}{b_1}sh({r_1}x) + {F_2}{b_2}sh({r_3}x) (4) uc=F1sh(r1x)+F2sh(r3x) {u_c} = {F_1}sh({r_1}x) + {F_2}sh({r_3}x) (5) σc=Ec(F1r1ch(r1x)+F2r3ch(r3x)+acΔT+εsh) {\sigma _c} = {E_c}({F_1}{r_1}ch({r_1}x) + {F_2}{r_3}ch({r_3}x) + {a_c}\Delta T + {\varepsilon _{sh}}) (6) σs=Es(F1b1r1ch(r1x)+F2b2r3ch(r3x)+asΔT) {\sigma _s} = {E_s}({F_1}{b_1}{r_1}ch({r_1}x) + {F_2}{b_2}{r_3}ch({r_3}x) + {a_s}\Delta T) where (7) r3=12[a1+a2+a3−(a1+a2+a3)2−4a2a3] {r_3} = \sqrt {{1 \over 2}[{a_1} + {a_2} + {a_3} - \sqrt {{{({a_1} + {a_2} + {a_3})}^2} - 4{a_2}{a_3}} ]} (8) r1=12[a1+a2+a3+(a1+a2+a3)2−4a2a3] {r_1} = \sqrt {{1 \over 2}[{a_1} + {a_2} + {a_3} + \sqrt {{{({a_1} + {a_2} + {a_3})}^2} - 4{a_2}{a_3}} ]} (9) F1=(acΔT+εsh)b2sh(r3L)b1r3sh(r1L)ch(r3L)−b2r1ch(r1L)sh(r3L) {F_1} = {{({a_c}\Delta T + {\varepsilon _{sh}}){b_2}sh({r_3}L)} \over {{b_1}{r_3}sh({r_1}L)ch({r_3}L) - {b_2}{r_1}ch({r_1}L)sh({r_3}L)}} (10) F2=−(acΔT+εsh)b1sh(r1L)b1r3sh(r1L)ch(r3L)−b2r1ch(r1L)sh(r3L) {F_2} = {{- ({a_c}\Delta T + {\varepsilon _{sh}}){b_1}sh({r_1}L)} \over {{b_1}{r_3}sh({r_1}L)ch({r_3}L) - {b_2}{r_1}ch({r_1}L)sh({r_3}L)}} (11) b1=(a1+a2−r12)/a1,b2=(a1+a2−r32)/a1 {b_1} = ({a_1} + {a_2} - r_1^2)/{a_1},{b_2} = ({a_1} + {a_2} - r_3^2)/{a_1}

From the force balance analysis, it is easy to know that the stress of the reinforcement is the smallest at the midpoint of the slab and the stress of the concrete is the largest here. The stress of the reinforcement increases gradually away from the midpoint [7]. It reaches the maximum value at the crack end x = ±L, and the concrete stress is reduced to zero. The calculation method of the crack spacing L of the secondary cracking under the action of temperature drop and dry shrinkage is as follows: It can be back-calculated using the maximum concrete stress in the middle of the slab as the critical condition. At the same time, we can further obtain the analytical formula for the control stress of the bar: (12) σc−max=σc|x=0=ft {\sigma _{c - \max}} = {\sigma _c}{|_{x = 0}} = {f_t} (13) σs−max=σs|x=0.5L {\sigma _{s - \max}} = {\sigma _s}{|_{x = 0.5L}}

The shrinkage of the concrete slab on both sides of the crack is the same. The crack width ω is equal to the shrinkage of the concrete at the cracked place, that is, two times the displacement of the concrete at that place: (14) ω=2uc|x=0.5L=2F1(1−b1)sh(r1x)+2F2(1−b2)sh(r3x) \omega = 2{u_c}{|_{x = 0.5L}} = 2{F_1}(1 - {b_1})sh({r_1}x) + 2{F_2}(1 - {b_2})sh({r_3}x)

The analytical expressions of crack width, crack spacing and stress of the reinforcement in the continuous reinforcement design control index of the CRC layer can be obtained [8]. For cement stabilised bases, since the frictional shear stress can easily reach the limit value, we should judge the correctness of the base slip relationship assumption based on the calculation result of u_c.

To verify the validity of the analytical method, we established a finite element model to simulate the temperature field cooling. The base layer and the CRC layer adopt a linear shear-slip relationship. Based on the known crack spacing, we obtain the concrete stress in the slab, the crack width and the BFRP tendons’ control stress.

According to the symmetry, we use the plate length L for analysis. We arrange two BFRP ribs symmetrically in the transverse direction to consider the interaction between the ribs. We use an 8-node reduced integral solid element (C3D8R) for the BFRP rib, CRC and base layers. At the same time, we encrypt the BFRP rib mesh, as shown in Figure 4. Assuming that each part of the material is uniform linear elasticity, the concrete and BFRP tendons have orthotropic expansion characteristics, and only the longitudinal expansion coefficient is given. We use cohesive contact to simulate the bond-slip characteristics between BFRP bars and concrete and ignore the lateral deformation caused by the embedding of the bars [9]. We use general static analysis to achieve uniform temperature drop simulation by setting different temperature field values. At the bottom of the model, the intermediate phase displacement constraints are imposed on both lateral sides and the longitudinal ends of the base layer.

Fig. 4

The length of the slats used by the finite element and analytical methods is L = 1.8 m. The longitudinal BFRP ribs in the board are arranged at 1/2 the height of the CRC board. The horizontal spacing and board margin are 8 cm. The longitudinal bending stiffness between BFRP bar and concrete is 20 GPa/m. The bottom friction coefficient of the CRC layer is 40 MPa/m. The temperature field decreases evenly by 30°C. Compared with the finite element results, we use the analytical method to calculate separately [10]. The shrinkage strain ε_sh takes two working conditions of 0 and 10⁻⁵. The material parameters used in our analysis are shown in Table 1. In Table 1, H is the thickness, E is the elastic modulus, α is the linear expansion coefficient and nu; is the Poisson’s ratio.

Due to the limited range of influence of the base layer and BFRP bars, the stress and displacement of each element at the same section of the CRC layer are not uniform. For example, the longitudinal displacement value at the bottom of the CRC layer is more significant than that near the BFRP ribs. The CRC layer is 1/4 height from the bottom and 1/2 of the horizontal element node as the output position of the representative value of σ_c−max the stress in the middle of the concrete slab. The displacement of the concrete node at the crack end B, which is 1/2 of the crack width, is w. The average stress of the BFRP tendon element is used as the control stress σ_s−max. The comparison of the three maximum results is shown in Table 2.

From the results in Table 2, it can be seen that the crack control index results of the analytical method and the finite element method are consistent without accounting for shrinkage strain. The error is <7%. The displacement value of the concrete at the crack, which is 1/2 of the width value, does not exceed the critical slip value of most base courses. This is in line with the assumption of the slip relationship at the grassroots level [11]. When we use the analytical method to calculate the shrinkage strain, each index value has increased by about 35%. The concrete stress in the slab has exceeded the ultimate tensile strength, indicating that the effect of concrete drying shrinkage on the transverse cracks of the CRC layer cannot be ignored.

In summary, the analytical crack calculation proposed by us can more accurately predict the actual road conditions of the BFRP bar continuously reinforced pavement under the conditions of uniform temperature drop and dry shrinkage. This can provide a reference for the continuous reinforcement design of BFRP bars.

We use the analytical method to analyse the influence of the elastic modulus of BFRP bars on the crack mode of the CRC layer. The BFRP bars used are two types of thread bars with the elastic modulus of 40 and 60 GPa, and the yield strength is 510 MPa. In addition, the comparative calculation of the steel bar shows that the elastic modulus of the steel bar is 200 GPa and the yield strength is 335 MPa. The linear expansion coefficient of the bar is 9×10⁻⁶°C, the diameter is 12 mm and the bending stiffness is 20 GPa/m. The horizontal spacing of longitudinal bars adopts eight types between 8 cm and 22 cm, and the reinforcement ratio is 0.18–0.51%. The shrinkage factor of concrete is 0.0002. The rest of the parameters are the same as the values taken while verifying the above-mentioned finite element model. The calculation results of crack width, spacing and BFRP tendon control stress are shown in Figure 5.

Fig. 5

It can be seen from Figure 5 that the three indexes of BFRP tendons are the same as the steel bars along with the change of the longitudinal bar spacing, and the crack width and spacing are more significant than the calculated values of the bars [12]. For the same type of reinforcement, the crack spacing and width of the BFRP bars are more significant than the calculated values of the steel bars. Taking the longitudinal rib spacing of 16 cm as an example, when the elastic modulus of BFRP ribs changes from 60 MPa to 40 MPa, the crack width increases by about 15.3%, the maximum crack spacing increases 14.8% and the control stress decreases by 3.4%. When the spacing of longitudinal bars is <16 cm, the restraining effect of BFRP bars on the concrete gradually weakens with the spacing increase, and the effect of elastic modulus on the crack spacing and width becomes more significant. When the longitudinal rib spacing is >16 cm, the calculation result of the control stress is only a theoretical value. This can be used to reflect the overall trend of change.

The characteristics of low elastic modulus of BFRP tendons result in large pavement crack spacing and width, and its joints have inadequate load transfer capacity and are prone to breakage. We take the crack width as the primary control target, so we need to choose a material with a higher elastic modulus.

Let’s take a diameter of 12 mm and a modulus of elasticity of 40 GPa as an example. The bonding stiffness of BFRP tendons was selected as 15, 20, 25 and 30 GPa/m to analyse the influence on the control index, as shown in Figure 6. The parameter settings of the transverse spacing of longitudinal ribs and other materials are the same as the elastic modulus analysis in Section 4.1.

Fig. 6

The results in Figure 6 show that under the condition that the reinforcement is not changed, the bonding stiffness decreases and the crack spacing and width increase, and the increase in the two is equivalent. When the reinforcement ratio is low, the bonding stiffness has a more significant impact on the stress of the BFRP tendons. For example, when the spacing is 8 cm (reinforcement ratio is 0.50%), the crack spacing and width increase of the remaining three bonding stiffness are 9%, 20% and 36%. On the whole, the stress increase of BFRP tendons is less affected by the bonding stiffness.

According to the results of laboratory tests, it can be known that the bonding stiffness of BFRP bars is related to the strength of concrete, the diameter of the bars and the thickness of the protective layer. The bonding stiffness is usually about 25 GPa/m. Therefore, to control the width and spacing of cracks, BFRP bar material and surface structure choice should ensure that the bending stiffness is not <20 GPa/m in actual engineering.

We analyse the influence of the reinforcement ratio on the three major indicators. The article takes three kinds of BFRP bars with diameters of 12, 14 and 16 mm as examples, and the lower value of 40 GPa is selected for the modulus of elasticity. The transverse spacing of the longitudinal ribs and the material parameters are the same as the elastic modulus analysis. The reinforcement ratio ρ is 0.33–0.90%. The calculation result is shown in Figure 7.

Fig. 7

As shown in Figure 5, the crack spacing, width and control stress of the two types of BFRP tendons increase approximately linearly with the spacing of longitudinal tendons. For example, when the spacing is 8–22 cm, the crack spacing, width and control stress of BFRP tendons are increased by 1.4 times. The three major indicators of BRFP reinforcement in Figure 6 change with the fluctuation of the reinforcement ratio, and the overall trend becomes smaller as the reinforcement ratio increases. The diameter of the reinforcement limits the size of the reinforcement ratio. This shows that simply controlling the reinforcement ratio in the crack control design cannot achieve the desired effect. This requires careful consideration of the design of the reinforcement scheme. Three sets of data are selected from the analytical calculation results, and the reinforcement ratio of each set is close. It was shown in Table 3. Through analysis, it is found that the crack width, spacing and control stress are smaller under the reinforcement scheme with small diameter and small spacing. And the smaller the reinforcement ratio, the more significant the difference. Therefore, the diameter and spacing of the control ribs should not be too large to obtain a better crack control effect.

The current ‘Code’ mainly provides three control standards for reinforcement design for continuous reinforced cement concrete pavement. (1) The average spacing of transverse cracks is not >1.8 m. (2) The average width of the cracks at the embedding depth of the longitudinal steel bars is not >0.5 mm. (3) The tensile stress of the steel bar does not exceed the yield strength of the steel bar.

According to the conclusion that the limit slip of BFRP bar and concrete linear bonding is about 0.5 mm according to the experiment in this paper, we use the analytical method to calculate the crack width limit to be about 1.0 mm. Considering the absence of rain corrosion in BFRP and the reduction of road cost, we can appropriately relax the cracks and spacing limits given in the existing ‘Code’ in the design of BFRP reinforcement. Therefore, we propose to increase the limit of crack width to 1.0 mm, the corresponding limit of average crack spacing is 2.0 m and the reinforcement ratio of the CRC layer is not <0.5%. The actual project must consider various factors such as specific traffic volume and traffic axle load, pavement functional requirements, pavement structure composition, and short-term and long-term economic rationality to select a reasonable reinforcement ratio.