Nonlinear Bond Model for the Dowel Action considering the Fatigue Damage Effect

To investigate the mechanical properties of dowel action under fatigue loads, 3 reinforced concrete specimens with different bar diameters (12mm, 20mm, and 25mm) were subjected to the fatigue loading and were designed to investigate the attenuation character of dowel action and the fatigue failure modes. +e load transfer mechanism of the bond was analyzed based on the 3D relative motions between reinforcing bars and subgrade concrete. Fatigue damage effects were considered in the model. A deterioration coefficient based on the deformation path was defined to represent the accumulation of fatigue damage. Verification of the model was conducted by comparing the analysis results with experimental data obtained in this study and from the literature, and satisfactory agreement was obtained.


Introduction
e dowel action is one of the main mechanisms of load transfer along the interfaces.ese interfaces and joints may turn out to be critical planes in the operation of the loadresisting mechanism and thus may govern the ultimate strength, ductility, and energy absorption capacity of an entire structure [1][2][3][4].As the important components of reinforced concrete structures, structural joints are directly subjected to shear loads of great magnitude and cycles.During their service life, the mechanical performances of dowel action gradually degrade due to the high cycle repetition of loads which might cause the sudden collapse of the entire structure.
erefore, the mechanical behavior of dowel action under fatigue loads should be taken into account.
In recent years, extensive experimental and analytical investigations have been carried out to study the mechanical properties of the dowel action [5][6][7].Vintzeleou and Tassios [8] experimentally investigated the behavior of dowels under cyclic deformation.Test results indicated that both the bearing capacity and the stiffness of dowel action show obvious degradation, especially in case of full reversed deformations.Mattock and Hawkins [9] carried out a series of high cycle fatigue tests of push-off specimens.e relation between dowel shear force and transverse shear slip for various cycle totals of specimen under different loading levels were obtained.
A number of finite element (FE) models have been developed for the analysis of the dowel action [10][11][12][13][14]. Soltani and Maekawa [1] developed a path-dependent RC interface model under a multidirectional displacement path.Moradi et al. [15] presented a simplified model for the dowel action by suggesting a simple formula for subgrade springs.Maekawa and Qureshi [16,17] proposed an enhanced computational model for the prediction of reinforcing bar behaviors under generic conditions of axial and transverse displacements.A unified model for RC interface stress transfer was presented based on a combination of a generic bar model and a physical model for aggregate interlock [18].An open-slip coupled model was developed to simulate the bond behavior of reinforcing bars in concrete [19] and was later extended to cover the cored reinforcement [20].e author developed a bond model considering the coupled damage effect.Both the effect of the crushing of the concrete supporting the bar and the strain effect are considered in the analysis [21].However, the mechanical behavior of dowel action under fatigue loads has yet to be considered.
Recently, research progress has been made on the fatigue life assessment of reinforced concrete structures with finite element methods (FEMs) [22].Nonlinear fatigue constitutive models of concrete intension, compression, and shear along crack planes were formulated and have been verified experimentally.To accelerate the computation for huge numbers of fatigue cycles, a direct path-integral scheme is also proposed for the fatigue simulation of long time range.
is framework has been further enhanced for the simulation of bridge slab performance under high cycle moving loads [23,24].erefore, these life assessment schemes could be used to solidify the quantitative background of the bond model considering the fatigue damage effect.
e objective of the present study is to develop a 3D bond model to predict the behavior of dowel action under fatigue loads.e effect of fatigue damage on the behavior of the dowel action is introduced, and a deterioration coefficient is chosen to characterize the damage level.ree block-type specimens were subjected to fatigue loads and were designed to investigate the attenuation character of dowel action and the fatigue failure modes.To demonstrate the versatility of the proposed model, verification is conducted by comparing analysis results with experiments.

Materials and Mixtures.
e concrete mixture shown in Table 1 was adopted in the present study.e components consisted of ordinary Portland cement (PO 42.5), coarse aggregate with a maximum gravel diameter of 20 mm, fine aggregate with a fineness modulus of 2.6, and polycarboxylatebased superplasticizer.e design yield strength of the bar is 400 MPa.

Specimen Design.
In this study, a direct shear test approach is chosen to control the load path and obtain the transverse displacement.ree block-type specimens with the dimension of 400 mm (length) × 300 mm (width) × 250 mm (height) were cast, with each specimen being reinforced with a deformed bar, as shown in Figure 1.
As summarized in Table 2, 3 bar diameters (12 mm, 20 mm, and 25 mm) were adopted in this study.e specimens were subjected to the fatigue loading to investigate the mechanism of fatigue deterioration and the fatigue failure modes.For the fatigue test, a repeated singlesided shear force was applied to the dowel bar.e maximum load level (V f2 ) was set equal to the 55% of the static capacity obtained from the monotonic tests [21], and the minimum load level was set as 1 kN (V f1 ) in order to keep the specimen stable.

Test Setup and Measurement Scheme.
e test setup is presented schematically in Figure 2. Two servohydraulic actuators with loading capacities of 250 and 150 kN were used in the loading frame, which allows both shear and axial forces to be applied to the dowel bar.e loading path is shown in Figure 3. Firstly, three static loading cycles were performed to eliminate the influence of the deformation of support device.en, a number of fatigue cycles (N � 100, 1000, 10000, 20000, 40000, 60000, 80000, and 100000) were carried out.A force-controlling mode was adopted for fatigue loading, and the loading frequency was set as 2 Hz.After 100000 fatigue cycles, the specimen was tested up to failure to measure the residual carrying capacity.
In all tests, the shear displacements and shear forces were continuously recorded during loading with a data acquisition frequency of 1 Hz.Furthermore, several LVDTs were used to monitor the concrete block at different positions and to measure possible rigid body displacements of the specimen with respect to the loading frame.

Experimental Results
. Figure 4 shows the dowel forcedowel displacement curves of specimen D12-F.e residual carrying capacity after 100000 fatigue cycles is 9.2 kN, less than the static capacity (11.6 kN for monotonic test [21]).It could be concluded that the accumulated fatigue damage due to fatigue loads will reduce the ultimate bearing capacity    Advances in Materials Science and Engineering were ruptured at the 68720 and 72485 fatigue cycles.e position of facture was located about 1 times the bar diameter away from the surface of the specimen.Figure 7 shows the typical failure mode of these specimens.It could be concluded that the fatigue failure mode of dowel action is the fatigue rupture of the dowel bar, and the fatigue life is determined by the fatigue properties of steel and concrete.
Figure 8 shows the incremental displacement against the number of fatigue cycles.ere exists a linear relation between incremental shear displacement and the logarithm of the number of fatigue cycles.
is is similar to the S-N diagram for the fatigue life of materials which is in agreement with previous published results [9].For specimens D20-F and D25-F, when the number of cycles is less than 40000, similar trends were presented to specimen D12-F, a linear relation between incremental shear displacement and the logarithm of the number of fatigue cycles was obtained.When the number of fatigue cycles is equal to 60000, there was a sudden increase in the shear displacement of the dowel bar and the stiffness degradation is obviously observed.
Based on the test data, the fatigue failure process of dowel action could be divided into two stages: the accumulation of fatigue damage and the fatigue rupture of dowel bar.In the former stage, the fatigue damage of materials gradually accumulated under fatigue loads.In the latter stage, the bearing section of dowel bar was fractured due to the growth and propagation of the microcracks in steel.

Mechanical Mechanism of ree-Dimensional Relative
Motions.To simulate the mechanical property of the dowel action, a three-dimensional bond model considering the coupled damage has been developed [21].In which, the three-dimensional relative and the damage effect caused by both the micro radial cracks and the localized crushing of the surrounding concrete were integrated.A discrete FE method is adopted in this research.Both the steel bar and concrete are modelled by three-dimensional solid elements, and interface elements with zero thickness are used to describe the effect of the bond transition zone [20,21].Fatigue loads carried by reinforcing bars are transferred through this transition zone to the surrounding concrete.
In this paper, the basic methodology of previous work [21] was used and the damage equation of fatigue loads was analyzed.As shown in Figure 9, the three-dimensional relative motions can be resolved into 3 directions, namely, the normal direction, hoop direction, and axial direction.In the normal direction, the opening and closing of the interface represent the compression and separation between the dowel bar and the surrounding concrete due to the bending deformation of the dowel bar under shear loading.e bond stress field can be computationally summarized as follows [21]: where τ a is the bond stress in the axial direction, σ n is the total normal stress, τ n is the hoop stress, δ is the relative slip in the hoop direction, θ is the inclined angle of the rib, L is 4 Advances in Materials Science and Engineering the spacing of the ribs in the axial direction, G n is the tangential stiffness in the hoop direction, σ a is the local normal contact stress considering the damage effect, σ n ′ is local normal compression stress, and τ a ′ is the friction stress along the rib.Details of the model are skipped here, but can be found in the previous work [21].
Experiment results proposed by Maekawa and Qureshi [16] showed that the dowel and pullout behaviors of embedded bar are mutually interactive and interdependent.
e pullout behavior is affected by the transverse shear, and the dowel action is in turn affected by the pullout behavior.
is mutual dependency is strongly influenced by the local plasticity resulting from the interaction between the interface and the embedded bar.As shown in Figure 10, both micro radial cracks due to axial slip and the localized crushing of the surrounding concrete due to shear displacement can damage the subgrade concrete.
erefore, in this model, both the local axis strain of rebar and the local compressive strain of concrete are adopted, and the local normal compression stress σ n ′ and local normal contact stress σ a can be expressed as where ε s and ε c denote the strain of the dowel bar and corresponding concrete, and λ c is the damage index due to the localized crushing of concrete and is determined as 3.5 according to previous analysis.Verification of the basic methodology has been conducted by comparing the analysis results with experimental data and previous published models [21].e distribution characteristics of the curvature and the local axial-bending stress of reinforcing bar are in agreement with previous published results [18,21].

Damage Evolution under Fatigue Loads.
When the reinforcing bar is subjected to fatigue loads, high-frequency interaction between concrete and steel bar will be caused.e fatigue damage accumulates during the interaction and causes the deterioration of the bond behavior.e process of fatigue damage accumulation actually represents the micro radial cracks and the localized crushing of the surrounding concrete, and it heavily depends on the path of the relative motions between concrete and reinforcing bar.erefore, a deterioration coefficient χ based on the deformation path could be defined to represent the accumulation of fatigue damage.
e effect of fatigue damage on the bond behavior could be considered by reducing the total normal stress σ n and the bond stress τ a using the deterioration coefficient χ: where τ af and σ nf are the bond stress in the axial direction and the total normal stress, considering the fatigue damage, and χ is the damage coefficient which can be computed as In (10), the path of the path of the relative motions between concrete and reinforcing bar is considered by introducing displacement parameters w and δ 2 .ζ is the acceleration factor used in numerical integration.
Logarithm of number of cycles (N)

Advances in Materials Science and Engineering
where K e f is the stiffness matrix of the bond model, considering the fatigue damage, and K e f is the stiffness matrix of the bond model regardless of the fatigue damage.
Altogether, the three-dimensional bond stress field considering the fatigue damage can be computationally summarized as follows:

Scheme of the Numerical Analysis for Fatigue Load
To simulate the dowel action under combined fatigue shear and pullout tension, the bond model is integrated into the path-dependent integral scheme [22,25].In this scheme, fatigue damage of concrete is considered as the timedependent plasticity and fracturing.e fatigue damage of concrete is formulated in terms of uniaxial compression, tension, and shear, respectively.
As shown in Figure 11, an elastoplastic and fracture model is used to simulate the uniaxial compression behavior of concrete: (7) where ε, ε e , and p are the total strain, elastic strain, and plastic strain, respectively; σ is the total stress; E 0 is the initial stiffness; and K c is the fracture parameter which can be computed as Figure 12 shows the tension model of concrete.A simple model is used to represent the tension nonlinearity: where σ T is the total tension stress, ε T is the total tension strain, and K T is the tensile fracture parameter that can be defined as where H is the instantaneous fracture related to tension softening, F is the time-dependent fracture due to creep effect, and G is the fatigue damage.
For the shear fatigue behavior of concrete, a shear transfer model, considering the crack roughness and contact friction, is adopted in this study, as shown in Figure 13.Similar to the deterioration coefficient, a fatigue modification factor is used: τ is the transferred shear stress under fatigue loads and δ 0 and ω 0 are the crack and slip, respectively.e fatigue modification X changes with regard to common logarithmic scale.

Verification
To verify the proposed bond model, specimens D12-F, D20-F, and D25-F are simulated, and the comparison of the FE analytical results and the experimental data is shown in Figure 14.Both the deterioration of stiffness and the ultimate load due to fatigue damage can be numerically simulated.
In addition, a series of interface-shear experiments are chosen to verify the applicability of the bond model [8].As shown in Figure 15, 2 specimens with different concrete cover are subjected to cyclic loads.A comparison of the analysis and the experimental results is shown in Figure 16.A good agreement between the analytical and experimental results is confirmed.

Conclusions
To predict the force-displacement characteristics (dowel stiffness) and strength of the dowel action under fatigue loads, a series of tests were carried out to determine the mechanism of dowel action fatigue deterioration and the fatigue failure modes.A nonlinear bond model considering the fatigue damage effects was developed.
e following conclusions can be drawn from the results: (1) e accumulated fatigue damage due to the fatigue loads will significantly reduce the ultimate bearing capacity of dowel action.e fatigue deterioration of dowel action should be considered in engineering design to ensure the safety of the structures.

( 4 )
A systematic experimental veri cation was performed to demonstrate the versatility of the model, and a reasonable accuracy was found.Datae data on gures and tables used to support the ndings of this study are included within the article.

Table 1 :
Mixture proportion of concrete.

Table 2 :
Details of specimens subjected to fatigue loads.