Finite Element Analysis of Continuously Reinforced Bonded Concrete Overlay Pavements Using the Concrete Damaged Plasticity Model

Abstract

In this study, cracking patterns and widths were analytically investigated in a continuously reinforced bonded concrete overlay (CRBCO), as they developed due to temperature change and drying shrinkage, as the environmental load for the sustainable management of deteriorated concrete pavements. The parameters of the concrete damaged plasticity (CDP) model used for the nonlinear finite element analysis (FEA) of the continuously reinforced concrete pavement were determined through comparison of the FEA results with the field crack survey results so as to be used in the nonlinear FEA of the CRBCO pavement. The total temperature change, which combines the actual temperature change with the temperature change converted from the drying shrinkage, considering stress relaxation, was adopted in the FEA as the environmental load applied to the CRBCO pavement. The locations and movements of the reflection and transverse cracks in CRBCO were investigated via FEA. The reflection cracks occurred in the overlay at all of the joints of the existing pavement. Only one secondary crack, with a width that was 5–6 times narrower than that of the reflection cracks, occurred between adjacent reflection cracks under various conditions. Thus, the crack width of the CRBCO was predominantly affected by joint movement in the existing pavement. In addition, the crack widths predicted by the CDP model were narrower than those predicted using the elastic model by approximately 10%. Therefore, crack movement in a CRBCO pavement can be reasonably predicted by the CDP model.

1. Introduction

In Republic of Korea, asphalt overlays have primarily been used for the large-scale repair of deteriorated concrete pavement. However, this method has a critical disadvantage in that the movement of joints or cracks in the existing concrete slab, due to changes in temperature and humidity, causes reflection cracks at the bottom of the asphalt overlay. Therefore, asphalt overlays can only be used in a limited manner unless this issue is resolved, despite the effectiveness of the method in terms of the cost and the duration of the construction [1].

Alternatively, concrete overlays can be used instead of asphalt overlays. Concrete overlays are categorized as bonded concrete overlays (BCOs) or unbonded concrete overlays, based on the bonding conditions between the overlay layer and the existing pavement’s surface. BCOs are categorized into two types: ordinary BCOs and continuously reinforced bonded concrete overlays (CRBCOs).

Generally, continuously reinforced concrete pavements (CRCPs) exhibit good performance when the bearing capacity of the underlying layers is sufficient [2,3,4,5,6]. As shown in Figure 1, lean concrete, mainly used as a subbase for concrete pavements in Republic of Korea, has a relatively high bearing capacity, providing appropriate conditions for CRCPs [7]. In addition, a sound concrete slab can be used as the underlying layer of a CRBCO after the milling of the deteriorated part [8]. Therefore, CRBCOs offer an appropriate alternative to large-scale repair methods for deteriorated concrete pavements in Republic of Korea.

Although the development of reflection cracks at existing pavement joints or crack locations cannot be avoided in a CRBCO, a crack naturally caused by an environmental load can be tolerated because it is a type of CRCP, in which random cracks are permitted unless their widths are too large. Therefore, CRBCOs should only be designed to prevent cracks with large widths that exceed the criterion. The propagation of reflection and transverse cracks that develop in a CRBCO can vary according to the conditions of the underlying layers of the pavement and the local climate. However, no study has been conducted on CRBCOs in Republic of Korea because this method has not yet been implemented. Therefore, in the present study, the development and propagation of cracks were analyzed in a CRBCO using nonlinear finite element analysis (FEA), considering the conditions of the pavement structure using lean concrete as a subbase and the local climate in Republic of Korea.

2. Determination of the Parameters of the Concrete Damaged Plasticity Model

In contrast to jointed plain concrete pavements (JPCPs), transverse cracks occur at random locations on the surface of a CRCP that does not have a joint. Therefore, conducting FEA to predict the occurrence and propagation of transverse cracks in CRCPs is crucial for engineers involved in design, construction, and management. The Mohr–Coulomb, Drucker–Prager, cohesive zone, concrete damaged plasticity (CDP), and extended finite element method (XFEM) models have been primarily used in commercial FEA programs to analyze cracks in concrete [9,10,11,12,13,14,15].

Among the various models, the CDP model has been frequently used to analyze the damage and fracturing of concrete [13]. The CDP model, based on the studies of Kupfer et al. [16] and Lubliner et al. [17], is a constituent model for concrete materials that improves on the Drucker–Prager model. In addition to the analysis of the macroscopic fracture behavior of concrete structures, the model has also been used in the analysis of mesoscale cracks by defining concrete as heterogeneous [18,19,20,21].

The decrease in the strength of the concrete due to stress exceeding the yield strength should be considered in the analysis of cracks in a CRCP. Moreover, tension stiffening (Figure 2a) and compression hardening (Figure 2b) should be defined in the CDP model to predict the tensile and compressive failure of a CRCP, considering the decrease in the strength of the concrete caused by the cracks. Tension-stiffening data are required to depict the strain-softening behavior of cracked concrete in FEA software, such as ABAQUS, as graphically displayed in Figure 2b. The compression hardening behavior beyond the elastic state is described in Figure 2a [13]. The parameters given in Figure 2 can be formulated using Equation (1) [13,22]. The general parameter values of the strength decrease, as used in the CDP model, were provided by Dassault Systemes [22].

$$\begin{array}{c}{\epsilon }_{oc}^{el}=\frac{{\sigma }_c}{E_o}{\epsilon }_c^{~pl}={\epsilon }_c^{~in}-\frac{d_c}{\left(1-d_c\right)}\frac{{\sigma }_c}{E_o}\\ {\epsilon }_{ot}^{el}=\frac{{\sigma }_t}{E_o}{\epsilon }_t^{~pl}={\epsilon }_t^{~in}-\frac{d_t}{\left(1-d_t\right)}\frac{{\sigma }_t}{E_o}\end{array}$$

(1)

The material properties used in the CDP model, such as the dilation angle, eccentricity, biaxial-to-uniaxial compressive strength ratio (fbo/fco), hardening parameter (K), and viscosity parameter, were defined to obtain realistic results for cracks that had developed in reinforced concrete [23]. In this study, a dilation angle of 35° was determined by comparing the range between 31° and 42°, as suggested by Genikomsou and Polak [23], using the experimental and numerical results obtained by Najafgholipour et al. [24]. The fbo/fco ratio (1.16) was determined based on studies by Kupfer et al. [16] and Lubliner et al. [17]. A hardening parameter of 2/3 was determined based on research by Lubliner et al. [17], Ren et al. [25], and Najafgholipour et al. [24]. In addition, an eccentricity coefficient of 0.1 was determined, based on the work of Ren et al. [25] and Othman and Marzouk [26]. Various other values have also been suggested based on the literature [13,23,27,28,29,30].

3. Quantification of Environmental Load

Generally, cracks develop in a CRCP owing to changes in the temperature and humidity of the concrete. Therefore, the temperature change and drying shrinkage of a concrete slab should be predicted for use in the analysis of cracks that develop in a CRCP. The temperature of a CRBCO, as measured in the field, is required for the prediction of behavior caused by temperature changes. However, these data are unavailable in Republic of Korea because the respective method has not yet been implemented. Therefore, in this study, the temperature measured at a pavement slab with a thickness of 500 mm, constructed in Incheon, Republic of Korea, was used to determine temperature changes according to the depth of the CRBCO pavement.

Five thermometers, with 100 mm spacing at depths of 50–450 mm, were used to measure the concrete temperature according to the slab depth, as shown in Figure 3a. The temperature, measured in 2018, when the yearly temperature range was the largest in the last 20 years in the region, was further analyzed. The yearly temperature range according to the slab depth is presented in Figure 3b. In the case of the 50 mm slab depth, the lowest temperature over the last 20 years was −14.4 °C, and the highest temperature was 45.5 °C, with a range of 61 °C. The temperature range increased in areas closer to the surface, from 40 °C at the bottom to 61 °C at the top, as shown in Figure 3c.

A CRBCO pavement is a new, continuously reinforced concrete layer placed on top of an existing concrete pavement. The expansion and shrinkage caused by temperature affect the existing slab and CRBCO in the same way because a fully bonded state is assumed. However, drying shrinkage should be considered because the existing slab occurs only in the CRBCO layer, in a state where drying shrinkage is completed, as shown in Figure 4. In this study, the drying shrinkage occurring in the CRBCO layer was calculated as a converted equivalent temperature change.

The drying shrinkage of the CRBCO was predicted using Yang’s model (Equation (1)) based on the thickness and age of the overlay layer, as shown in Figure 5a [31]. The drying shrinkage increased with age, reaching a maximum of approximately 500 $\mathsf{\mu }\mathsf{\epsilon }$ although it varied according to thickness.

$${\epsilon }_{sh}=\frac{a_{1}t}{a_2+t}\times \left[1+a_3{e}^{-a_{4}V/S}\right],$$

(2)

where

• ${\epsilon }_{sh}$ is the drying shrinkage strain;
• $t$ is the age (in days);
• $V/S$ is the volume–surface ratio (in mm); and
• $a_1, a_2, a_3, a_4$ are the model parameters.

Figure 5. Predicted drying shrinkage of CRBCO according to its thickness and age. (a) Before considering stress relaxation; (b) After considering stress relaxation.

Figure 5. Predicted drying shrinkage of CRBCO according to its thickness and age. (a) Before considering stress relaxation; (b) After considering stress relaxation.

The drying shrinkage increased gradually with the age of the CRBCO, which differed from the temperature change, which occurred over daily and yearly cycles. Therefore, stress relaxation caused by the creep of concrete should be considered in the long-term drying shrinkage of CRBCO. Previous studies have suggested that the range of stress relaxation is between 20% and 73% of the stress developed by the constraint of the behavior of concrete structures owing to drying shrinkage [32,33,34,35,36,37].

The drying shrinkage of the CRBCO for a design lifespan of 20 years was predicted. As displayed in Figure 5a, approximately 90% of the total drying shrinkage that occurred during the entire lifespan was predicted to develop within 1 year, whereas 98% occurred within 4 years. The stress relaxation coefficient of 57%, suggested by Beushausen and Alexander [38] for BCO, based on an experiment, was considered, as shown in Figure 5b, for the tensile stress caused by the long-term drying shrinkage of the CRBCO layer.

The drying shrinkage, considering the stress relaxation that developed during a certain period, was converted into an equivalent temperature change by dividing it by the coefficient of thermal expansion, as shown in Equation (3):

$$△T_{sh}=\frac{{\epsilon }_{sh}}{{\alpha }_c},$$

(3)

where

• $△T_{sh}$ is the temperature change equivalent to drying shrinkage (°C), and
• ${\alpha }_c$ is the coefficient of thermal expansion of concrete (/°C).

Figure 6 depicts the total temperature change by combining the yearly temperature change and drying shrinkage equivalent temperature change, considering the stress relaxation according to the depth of the CRBCO pavement, with various CRBCO thicknesses. The total temperature change decreased slightly with an increase in the CRBCO thickness, as the volume-to-surface ratio also increased, as shown in Equation (1). The total temperature change of −85.7 °C was calculated at the top surface of the CRBCO with a thickness of 80 mm, as shown in Figure 6a. This result is similar to the total temperature change of −89.1 °C, suggested by Ha et al. [39], achieved by combining the temperature change and the drying shrinkage of concrete.

4. Analysis of Cracking Pattern

4.1. Determination of the Viscosity Parameter of CRCP by FEA

A reflection crack in the CRBCO can develop at the joint location of the existing pavement because the CRBCO is constructed on top of deteriorated JPCP. A transverse crack can develop between adjacent reflection cracks when the environmental load is sufficiently large. However, the cracking pattern has not been observed under the pavement structure and regional climate conditions in Republic of Korea because CRBCO has not yet been deployed in Republic of Korea. Therefore, the feasibility of simulating the cracking pattern in CRBCO using FEA with the CDP model was investigated for the CRCP.

Michał and Andrzej [29] emphasized that the viscosity parameter is the most critical parameter of the CDP model for producing realistic results in the analysis of crack development in reinforced concrete. Therefore, the appropriate viscosity parameter value for the FEA of CRBCO in this study was determined to be within the range of 0.00001–0.01. The viscosity parameter value was determined by comparing the crack survey results with the FEA results for the CRCP deployed on the test road of the Korea Expressway Corporation (KEC), located in Yeoju, Republic of Korea. Figure 7a shows a cross-sectional view of CRCP. The average crack spacing of the CRCP measured in 2011, 9 years after construction in 2002, was 1.42 m, as shown in Figure 7b [40].

The CRCP with a steel ratio of 0.6% was modeled to have a length of 30 m, as shown in Figure 8, referring to the design of the CRCP in the KEC test road. The four sides of the model could move freely in all directions as boundary conditions. The friction behavior between the concrete surface layer and the asphalt separation layer was simulated using a friction coefficient of 0.2. An elastic foundation with a spring constant of 0.03 was applied under the asphalt separation layer [41]. The elastic modulus of the new concrete pavement slab, as suggested by Tayabji and Okamoto [42], was applied to the concrete surface layer of the FE model, and Poisson’s ratio, the unit weight, and the coefficient of thermal expansion of the concrete surface layer were determined based on a study by Park et al. [41,43]. The material properties for the FEA of the CRCP are listed in

Table 1. Material properties of CRCP for the FE model.

Property	Surface Layer	Separation Layer	Rebar
Elastic modulus (MPa)	35,000	3000	200,000
Poisson’s ratio	0.18	0.35	0.3
Unit weight (kgf/m3)	2400	2300	7850
Coefficient of thermal expansion (/°C)	10.7 × 10−6	20.0 × 10−6	10.0 × 10−6

.

Transverse cracks develop at random locations in a CRCP or CRBCO, owing to environmental loads, such as temperature change and drying shrinkage, because joints are not installed in pavements. Therefore, applying the actual environmental load to the model is necessary to realistically analyze cracking in CRCPs and CRBCOs.

The average ambient temperature in October 2002, when the CRCP was constructed on a KEC test road in Yeoju, Republic of Korea, was 11 °C. The lowest temperature recorded between the construction and the final crack survey in May 2011 was −22.8 °C. Thus, the temperature change at the top of the concrete surface layer was determined to be −33.8 °C.

The drying shrinkage of −394 µɛ was calculated using Equation (1) for the concrete surface layer that had aged for 9 years when the final crack survey was conducted [31]. The drying shrinkage, considering a stress relaxation value of −169.4 µɛ, was realized by applying a stress relaxation coefficient of 57%, as suggested by Beushausen and Alexander [38] for a BCO.

Using Equation (2), from the drying shrinkage with stress relaxation, a temperature change of −15.83 °C was determined. At the top of the concrete surface layer, a total temperature change of −49.83 °C was calculated by combining the actual temperature change of −34 °C and the temperature change equivalent to the drying shrinkage of −15.83 °C, considering stress relaxation.

The temperature of the CRCP was not measured at the time of the final crack survey although it was measured for a few years after the construction of the KEC test road. Therefore, the total temperature change in the CRCP, according to depth, was predicted, as shown in Figure 9, considering the total temperature change of −49.83 °C at the top surface of the CRCP, as calculated above; the trend in the temperature change distribution in the concrete slab is shown in Figure 3c.

The viscosity parameter should be determined empirically because the crack pattern changes depend on it in an analysis using the CDP model [13]. Therefore, a preliminary analysis needed to be performed to determine the viscosity parameters to be applied in this study.

FEA was performed for the CRCP according to the viscosity parameter by applying the total temperature change, presented in Figure 9, as the environmental load. Cracks did not appear on the pavement surface when the viscosity parameter was 0.01 because the damage was extensively dispersed across the entire pavement surface, as shown in Figure 10a. Cracks began to appear when the viscosity parameter decreased to 0.005, as shown in Figure 10b. Furthermore, cracks developed when the viscosity parameter was reduced to 0.001, as shown in Figure 10c, which is consistent with the findings of Michał and Andrzej [29].

A total of 22 cracks appeared in the CRCP when the viscosity parameter was set at 0.001, as shown in the full-scale model (Figure 11). The distance between the first and the last crack, in the model with a length of 30 m, was 29.4 m; hence, the average crack spacing was 1.4 m. Therefore, the viscosity parameter of 0.001 was determined for the FEA of the CRBCO because the predicted average crack spacing was similar to that of the actual crack spacing surveyed for the CRCP in the KEC test road.

4.2. Cracking Patterns in CRBCO from FEA using CDP Model

As shown in Figure 12, an FE model of the CRBCO with a length of 12 m was developed to analytically investigate the cracking pattern in the CRBCO pavement. Two joints with a spacing of 6 m were modeled symmetrically in the existing pavement. The existing concrete slab surface and CRBCO bottom were modeled as completely adhering. The existing concrete slab bottom and lean concrete subbase top were modeled to demonstrate the friction behavior. In addition, an elastic foundation was modeled at the bottom of the lean concrete subbase.

The elastic moduli of the overlay layer and existing slab were determined as 35,000 and 28,000 MPa, respectively, after implementing the design method of the Portland Cement Association (PCA) for newly constructed and existing pavement slabs. The material properties of the subbase and rebar were determined according to a study by Park et al. [41,43]. The material properties of the CRBCO pavement used in the FEA are listed in

Table 2. Material properties for FE model of CRBCO.

Property	CRBCO	Existing Slab	Subbase	Rebar
Elastic modulus (MPa)	35,000	28,000	3500	200,000
Poisson’s ratio	0.15	0.15	0.15	0.3
Unit weight (kgf/m3)	2400	2400	2400	7850
Coefficient of thermal expansion	10.7 × 10−6	10.7 × 10−6	10.7 × 10−6	10.0 × 10−6

.

The cracking pattern of a CRBCO can vary according to design variables, such as the thickness of the CRBCO, the thickness of the existing slab after milling, and the steel ratio.

Table 3. Range of CRBCO design variables for cracking pattern analysis.

Variable	Range
	Minimum	Middle	Maximum
Overlay thickness (mm)	80	110	150
Existing slab thickness (mm)	130	200	300
Steel ratio (%)	0.25	0.50	0.73

presents the ranges of the design variables determined in this study. The thickness of the CRBCO was determined as being 80–150 mm, considering a BCO thickness of 80–150 mm, as suggested by the design method of the American Association of State Highway and Transportation Officials (AASHTO), a minimum BCO thickness of 70 mm, as suggested by the PCA, and a minimum thickness of the reinforced BCO of 76 mm, as suggested by the Texas Department of Transportation (TxDOT) [42,44,45]. The thickness of the existing slab was determined to be between 130 and 300 mm, considering the concrete pavement slab thickness and milling depth used for Korean expressways. The ratio of steel installed in the CRBCO was determined as being between 0.25% and 0.73%, according to the CRBCO thickness and existing slab thickness, considering the CRCP manual provided by the Federal Highway Administration (FHWA) [46]. The rebar spacing was calculated using the steel ratio and the rebar diameter.

The cracks that developed in the CRBCO constructed on the deteriorated JPCP were analyzed using the quantified environmental load, shown in Figure 6c. In the FE model, the thickness of the existing slab after milling was 200 mm, and the thickness of the overlay was 100 mm. In addition, the steel ratio was 0.5%. In most cases of analysis, transverse cracks developed near the center between the adjacent reflection cracks, as shown in Figure 13b, after the occurrence of reflection cracks in the overlay layer immediately above the joint of the existing pavement, as shown in Figure 13a. No additional transverse cracks were observed at any other random locations.

The stress and strain at the cracking locations were investigated to indirectly compare the widths of the reflection and transverse cracks at the top surface of the CRBCO. When the elastic model was used, the stress and strain increased continually, even after the stress of an element in the FE model, which increased with the external load, exceeded the strength. However, when the CDP model is used, a crack is considered as occurring at the element when the stress at the location exceeds the strength. Consequently, when the CDP model was used in FEA, the strain increased at the location of the crack, whereas the stress decreased.

As shown in Figure 14, reflection cracks developed in the CRBCO at the joints of the existing pavement, which were 3000 mm from the center of the FE model in both directions. Consequently, a transverse crack with a 20% strain of reflection cracks developed near the center between the adjacent reflection cracks. The larger strain at the locations of the reflection cracks was caused by the widening of the joint, owing to the decrease in the total temperature of the concrete, as shown in Figure 9.

5. Analysis of Crack Width of CRBCO

The appropriateness of the results predicted by FEA using the CDP model was verified by comparing the widths of the reflection cracks predicted by the CDP and elastic models. Figure 15 presents the FE model of the CRBCO for the elastic analysis, which did not use the CDP model. Elastic analysis was performed by modeling the reflection and transverse cracks in the overlay layer in advance because elastic analysis cannot predict the occurrence of any crack due to the application of an external load. The material properties are listed in Table 3, and the previously described boundary conditions were applied to the FE models using the elastic and CDP models.

The same environmental load was used for both the linear and nonlinear FEA, as shown in Figure 6c. When the elastic model was used, the width of the reflection crack at the top of the overlay layer became wider than those at other depths because of the upward curling of the existing slab caused by the environmental load, as shown in Figure 6c, for which the temperature at the top decreased more than that at the bottom of the pavement, as can be seen in Figure 16a. In contrast, when the CDP model was used, the deformation increased at the locations of the reflection cracks, particularly at the top of the overlay layer, as depicted in Figure 16b.

The widths of the reflection cracks predicted by the elastic and CDP models were compared. FEA was performed considering steel ratios of 0.56%, 0.45%, and 0.37%, as calculated for the 16 mm diameter rebar with 120, 150, and 180 mm spacing, respectively. The crack width increased linearly as the steel ratio decreased in both the elastic and CDP models, as shown in Figure 17. The crack width predicted by the CDP model was slightly narrower than that predicted by the elastic model, by 7–10% (0.04–0.06 mm). Thus, the appropriateness of the FEA results obtained using the CDP model was verified.

6. Conclusions

• In this study, crack patterns and widths were predicted via FEA using the CDP model. First, the parameter values considered in the CDP model were determined, based on previous studies. In contrast with other parameters, a range instead of a specific value was suggested for the viscosity parameter. Therefore, a viscosity parameter of 0.001, which predicts an average transverse crack spacing of 1.42 m for the CRCP constructed on the KEC test road in Republic of Korea, was determined by performing FEA using the CDP model.
• FEA was further employed to investigate the cracking pattern in the CRBCO using the parameters of the CDP model. In contrast to the CRCP, in which transverse cracks occurred randomly, reflection cracks developed immediately above the joints of the existing pavement. In addition, a traverse crack (secondary crack) occasionally appeared near the center of the adjacent reflection cracks. The occurrence of the reflection crack appeared to be directly affected by the shape of the joint of the existing slab, curled upward owing to the environmental load. Meanwhile, the occurrence of the secondary crack was primarily affected by the tensile stress, which developed in the CRBCO owing to the environmental load. The reflection crack width was approximately 5–6 times wider than the traverse crack, and as a result, the width of the crack that developed in the CRBCO was predominantly affected by the behavior of the joints of the existing slabs.
• By simulating the reflection and transverse cracks in the model, FEA using the elastic model was performed to verify the appropriateness of the FEA results obtained using the CDP model. The crack width predicted by the CDP model was slightly narrower than that predicted by the elastic model, by 7–10%, according to the steel ratio. Unless the crack was modeled in advance, the elastic model could not simulate the propagation of cracks caused by an external load because the occurrence of cracks could not be analyzed. Therefore, crack movement can be predicted by predicting the location of the crack and modeling the crack at the locations in advance in the FEA using the elastic model. However, both the occurrence and propagation of cracks in CRBCO can be observed via FEA using the CDP model. In addition, it was verified that the predicted crack width, according to the steel ratio, was reasonable, as compared with that predicted by the elastic model.
• In this study, the crack patterns of CRBCO were predicted using FEA. Moreover, it was possible to identify the key points in the maintenance of the CRBCO pavement. The reflection crack that developed immediately above the joint of the existing slab was the dominant type of crack in the CRBCO. Therefore, crack patterns should be considered for the appropriate design, construction, and management of CRBCO pavements.