Research Article  Open Access
Gongfa Chen, Zhuangcheng Fang, Shaodi Wang, Haibo Jiang, Hualian Liang, "Numerical Analysis on Shear Behavior of Joints under Low Confining and Eccentric Loads", Advances in Civil Engineering, vol. 2019, Article ID 4589824, 16 pages, 2019. https://doi.org/10.1155/2019/4589824
Numerical Analysis on Shear Behavior of Joints under Low Confining and Eccentric Loads
Abstract
The joints of precast concrete segmental beams (PCSBs), which are in complex stress status and susceptible to failure, are very important parts of the structure. In this paper, a finite element model was established to study the shear performance of singlekeyed joints. The plastic damage model was used to simulate the cracking of specimens. Three types of singlekeyed joints were investigated, including the dry joint with normal concrete (NC), dry joint with steel fiberreinforced concrete (SFRC), and epoxied joint with NC. The cracking pattern, ultimate shear strength, and loaddisplacement curve for these specimens were obtained. Based on these numerical simulation models, extended analyses in terms of low confining pressures and eccentric loads were performed. It has been found that the influence of fiberreinforced concrete should be considered. The ultimate shear strength of specimens reduced with the reduction of confining pressure. When an eccentric load was applied, a lower shear capacity would be obtained. Under the low confining stress, the AASHTO LRFD 2014 provision underestimated the shear strength of singlekeyed dry joints with both NC and SFRC, while the shear capacity of singlekeyed dry joints with both NC and SFRC has been overestimated under the eccentric loads.
1. Introduction
Precast concrete segmental beams (PCSBs), which are assembled from precast segments through posttensioned prestressing, are becoming more and more popular in modern bridge construction. With the advantages of rapid construction, good durability, and low life cycle cost, PCSBs have solved lots of problems in design, construction, and maintenance for bridge engineering. Previous studies have found that the shear behavior of joints in PCSBs was crucial for the safety of PCSBs. Hence, the design, construction, and quality of joints should be well controlled. Nowadays, two types of joints (dry joints and epoxy joints) have been widely used in engineering projects. For the former one, the joint is attached closely without any adhesive; for the latter one, epoxy resin is coated on the surface of the joints before assembling of PCSBs. Although dry joints and epoxy joints have many advantages, their shear performance is far inferior to that of monolithic joints [1]. Therefore, it is significant to properly design keyed joints with better shear performance and better structure. The shear performance of joints has been investigated under the disadvantageous condition of low confining stress and eccentric loading.
Lower horizontal confining pressure and larger eccentric loading, which may occur in bridge engineering, weaken the shear capacity of joints greatly. Usually, the capacity of joints was divided into two portions, i.e., the friction action of joints and shear resistance provided by the key. As shown in Figure 1, the bottom of joints is more likely to open in low confining pressure. This decreases the ultimate load of joints because of the weakening of both friction and shear resistance. When a larger eccentric load is applied, the additional bending moment formed by the eccentric load will be offset or even larger than that produced by horizontal restraint force. In this case, the joint will be opened, which results in the decrease of the shear capacity of joints. Besides, the prestress applied in fabrication and assembling, or the construction error at prefabrication, will also lead to the low confining stress on the shear keyed joints. This has greatly influenced construction of bridge engineering, yet few studies focused on it.
Recently, several investigations focused on the experimental and theoretical study of shear behavior of joints in PCSBs. The shear performance of epoxied joints was better than that of dry joints, even close to that of castinplace monolithic joints [1]. However, more brittle failure mode was observed for epoxied joints compared with dry joints [2]. Increasing the number of keys improved the shear strength of joints, but the strength of multikeyed dry joints was less than that of the total singlekeyed dry joints with the same quantity [3]. Changing the key geometry may also greatly influence the shear performance of joints [4]. Additionally, higher horizontal confining pressure or using a more suitable material (e.g., SFRC) also increased the strength of dry joints effectively. The limitation of current design codes such as AASHTO specifications and Spanish provision was investigated [5, 6]. Thus, some design formulas were proposed [6]. Nevertheless, a large amount of labor, resources, or equipment or a big place was demanded for experimental research. Even for theoretical analysis, necessary experimental data were required for verification.
Because of the limitation of experimental tests and theoretical analysis, numerical simulation became more and more popular to predict the stress, strain, energy, or damage for shear behavior of joints. Compared with experimental results, the numerical model was valid so that extended analysis can be conducted with the model [7]. Thus, some parameters that were hard to study by experimental methods can be investigated by numerical models [8–11]. Generally, the existing experimental, theoretical, and numerical investigations mainly focused on dry joints, but there are few studies which focused on epoxied joints or dry joints with SFRC which have better shear resistance comparatively. In addition, confining pressure greatly influences the shear capacity of joints.
In this paper, finite element simulations were performed, and the numerical model was validated by the experimental results from the study of Jiang et al. [12] and Zhou and Mickleborough [5]. In order to study the shear failure performance of joints under low horizontal confining pressure and eccentric loads, extended parametric analyses were performed. Finally, the parametric analysis results were compared with AASHTO provision. The results of this paper will provide important reference for experimental investigation, theoretical research, or engineering practice.
2. Experimental Model
2.1. Experimental Data
In order to improve the validity of the finite element model, previous experimental studies were referred, including the experimental results of monolithic joints with NC [12], singlekeyed dry joints with NC [12], singlekeyed dry joints with SFRC [12], and singlekeyed epoxied joints with NC [5]. The initial nomenclatures and parameters for each specimen are listed in Table 1. To get a unified format, all the specimens were renamed, as shown in Table 1. The redefined specimen designation indicated the type of joint, the type of concrete, and the confining pressure.
 
Note: is the cylinder compressive strength of concrete at the test day. 
2.2. Experimental Model Size
Details of experimental specimens and reinforcements are shown in Figure 2. It can be seen that the dimensions of specimens of monolithic joints were 540 mm × 340 mm × 100 mm, the same as those of singlekeyed dry joint specimens, while the dimensions of singlekeyed epoxied specimens were 620 mm × 500 mm × 250 mm, larger than those of singlekeyed dry joints.
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3. Numerical Model
3.1. Plastic Damage Constitutive Model of Concrete
In this study, the concrete damaged plasticity (CDP) model was used to simulate the elasticplastic behavior of concrete in ABAQUS 6.14. There were two types of failure modes, i.e., the tensile cracking failure and compressive crushing failure. For the finite element model in this paper, the dilation angle, the eccentricity of flow, and the viscosity coefficient were set as 31, 0.1, and 0.0001, respectively. Additionally, the ratio of uniaxial strength to biaxial strength (f_{b0}/f_{c0}) was 1.16.
As CDP cannot describe the initiation and development of fractures, to simulate the fracture of concrete, it was assumed that the fracture was initiated when the tensile strain in the tension stiffening stage was larger than ten times of standard concrete failure strain [10]. When reaching the ultimate tensile strain, the concrete has lost the resistance to cracking.
3.2. Constitutive Relationship of Concrete
According to the Chinese code for the design of concrete structures [13], the modulus of elasticity and Poisson’s ratio of concrete were determined by using the following formula:where is the average value of compressive strength of concrete. This code [13] also provided the stressstrain relationship of concrete to describe the behaviors of concrete under uniaxial tension and compression. The constitutive relationship of concrete under uniaxial compression is expressed aswhere is the damage evolution parameter of concrete under uniaxial compression, is the compressive strain of concrete, is the parameter for the descent segment in a constitutive relationship of concrete under uniaxial compression, is the uniaxial compressive strength of concrete and takes the average value of the compressive strength of concrete in this paper, and is the peak compressive strain corresponding to . For different compressive strengths shown in Table 1, corresponding values of and were acquired through linear interpolation based on the recommended information listed in Table 2.

The constitutive relationship of the concrete under uniaxial tension is expressed aswhere is the damage evolution parameter of the concrete under uniaxial tension, is the parameter for the descent segment in a constitutive relationship of concrete under uniaxial tension, is the uniaxial tension strength of concrete and takes the average value of the tension strength of concrete in this paper, and is the peak tensile strain corresponding to . Due to the lack of experimental results, the tension strength for different types of normal concrete was then taken as 10% of its compressive strength, which has also been adopted in the study by Shamass et al. [10]. Similarly, the values of and were calculated with linear interpolation based on the parameter values depicted in Table 3.

3.3. Constitutive Relationship of SFRC
Considering the influence of fiber factor F [14], the damage constitutive relationship of SFRC under uniaxial compression could be shown as follows:where is the modulus of elasticity of SFRC without damage; is the compressive strain of SFRC; is the bond coefficient of the fiber, which is related to the geometric properties of the fiber and taken as 1.2 in this study; and , , and are the volume fraction, length, and diameter of the fiber, respectively. Actually, this is a quite simplified way to derive the tensile behavior of SFRC. Some sort of inverse analysis based on bending tests should be used to derive the tensile properties of SFRC materials if possible.
3.4. Constitutive Relationship of Epoxy
For epoxied joint specimens in this study, the bonding strength of interface between the epoxy and concrete was 22 MPa, which was much higher than the tensile strength of concrete [5]. On the contrary, both the compressive and tensile strengths of the epoxy were much higher than those of concrete. As a result, all the epoxied specimens failed in shear or in tensile of concrete. To be simple, the epoxy in the finite element model was set as an elastic material with the elastic modulus of 4.826 GPa and Poisson’s ratio of 0.2, respectively [9].
3.5. Constitutive Relationship of Steel Rebar
An elasticplastic bilinear model was established to represent the constitutive relationship of the reinforcement. This stressstrain relationship could be expressed aswhere is the elastic modulus of the steel rebar, is the yield strength of the reinforcement, and is the strain at yield strength of the steel rebar.
3.6. Finite Element Modelling
A threedimensional solid model was used for simulation in this paper (Figure 3). In the model, both the concrete and epoxy layer were simulated as solid elements (C3D8R), while the reinforcement was simulated as a truss element (T3D2). Finer mesh was used for the most critical castellated keyed area. Especially, surfacetosurface contact interaction was adopted for the numerical model, using the finite sliding analysis procedure. Moreover, the contact surface associated with the lower portion (i.e., female part) was taken as the master surface, while the other one of the upper male section was defined as the slave surface. Based on the experimental results [5, 12], the friction coefficient between concrete surfaces was assumed as 0.6 for keyed joint specimens. Particularly, material nonlinear and contact nonlinear analyses were used in the numerical simulations. Additionally, all the translational degrees of freedom for the bottom surface were restricted.
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For the loading process, constant uniform pressure was applied on both sides of the model to simulate the horizontal confining stress. The confining stress values were 0.5, 1.0, and 2.0 MPa, respectively, covering the monolithic or singlekeyed dry joint area of 200 × 100 mm^{2}, as per Jiang et al. [12]. Similarly, for the case of singlekeyed epoxied joints, the confining pressure was applied covering a keyed area of 200 × 100 mm^{2}. The confining stress values were 1.0, 2.0, and 3.0 MPa, respectively, as per Zhou and Mickleborough [5]. The vertical displacementcontrolled loading was then applied on top of the model, which was simulated by creating a boundary condition moving vertically downward, with a prescribed displacement rate (0.3 mm/min) as adopted in the experiments done by Jiang et al. [12] and Zhou and Mickleborough [5].
4. Simulated Results and Discussion
The simulated results and experimental ones are compared on shear capacity (Table 4), loaddisplacement curves (Figure 4), and cracking development process (Figure 5). Especially, all the slippages of the joint were taken as the average deflections of that at the top and the bottom surface of the joint, which was consistent with the location of vertical LVDTs in the previous experimental study [5, 12].
 
Note: NC = normal concrete; SFRC = steel fiberreinforced concrete. 
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4.1. Comparison of Shear Capacity
Table 4 lists the experimental and simulated shear capacities for specimens with different types of joints. For monolithic joint specimens, the values of average and standard deviation of ratio of numerical results to experimental ones () were 0.94 and 0.03, respectively. This indicated that the numerical results for specimens with monolithic joints gave an acceptable conservative prediction. For singlekeyed dry joints (NC or SFRC) and singlekeyed epoxied joints (with the epoxy thickness of 1 mm, 2 mm, or 3 mm), the ultimate shear capacity obtained by finite element analysis was in good agreement with that by the test. The average value of for both the dry joint specimens and epoxied joint specimens was 1.00, and the standard deviation of for dry joint specimens and epoxied joint specimens was 0.02 and 0.04, respectively.
4.2. Comparison of LoadDisplacement Curves
Figure 4 shows the experimental and simulated loaddisplacement curves for specimens with different joints. The experimental results (EXP) were presented as the scattered dots, while the finite element model (FEM) analysis results were shown as the dashed or solid line. It can be found that the simulated curves and the test data points were approximately similar. For better understanding, the curve can be divided into three portions.
4.2.1. Portion I: Ascending Phase
In the beginning of this stage, all of the materials were in the linear elastic state and the relative displacement of the specimen increased with the increase of applied loading. After the first crack occurred, the stiffness of the specimen diminished, causing the slope of the loaddisplacement curve decrease gradually. At the end of this stage, the slope of the curve approached zero and the ultimate shear capacity was reached at the same time.
It can be found that the slope of test curves for monolithic joints was slightly larger than that of the simulation curves. This may be resulted from the difference between the adopted constitutive relation and the real stressstrain relation for concrete, which highlighted the importance of proper constitutive materials in numerical analysis. As for singlekeyed dry joint specimens (NC or SFRC), the test results were much close to the simulated curves, except for singlekeyed dry joint specimens with the confining pressure of 1.0 MPa. The stiffness of the experimental result for K1C501.0 was smaller than that of the numerical result, but the ultimate shear capacities are almost the same. This slight deviation of stiffness may be related to the neglect of brittleness and aggregate interlock effects of concrete.
4.2.2. Portion II: Descending Phase
With the shear failure of the joint, the load dropped suddenly and the fracture ran through the entire joint. Large plastic deformation and material damage were also evident. Especially, due to the limitation of the data acquisition instrument, only the data on ultimate load and the data after sudden dropping of the curve were recorded during the experiments. However, it should be noted that these data points were close to the numerical curve for all types of specimens. Thus, it is acceptable to use the simulated curves to represent the behavior of the descending portion.
4.2.3. Portion III: Residual Phase
After the material destroyed, the capacity of the joint has dropped to a constant value, which was called residual load. At this time, the shear capacity of the joint was mainly provided by the friction action of the shear plane, against the horizontal confining pressure. It can be seen that the numerical results can predict the residual load well.
In general, the experimental data points were distributed near the simulation curve, which indicated that those are in good agreement. The acceptable stiffness deviations in some tests probably resulted from the difference of material properties or the neglect of aggregate interlock effects of concrete under shear action. Furthermore, the unavoidable installation error, quality of models and fabrication technique, rotation of specimens during testing, or even the unfavorable cracking would also affect the validation of the numerical model.
4.3. Comparison of Cracking Process
Figure 5 shows the experimental and simulated cracking process for different joint types of specimens. In this figure, the cracking mode, development of cracking, and failure mode are presented. In order to express the cracking, the area is illustrated in gray where the tension strain has exceeded the ultimate tension strain of concrete.
In Figure 5(a), the cracking process for specimens with monolithic joints (M0C600.5) is analyzed. The gray area initially occurred at the joint bottom and then extended upward to the top. When approaching the ultimate load, the gray area has penetrated the entire shear area, which indicated that the monolithic joint has been completely fractured. This simulated cracking process was consistent with the experimental results, which was shown below the numerical results in the figure.
In Figure 5(b), specimen K1C500.5 was selected to analyze the cracking process of the singlekeyed dry joints. Both the numerical and experimental results showed a similar cracking process. The crack (gray area) occurred at the bottom corner of the key at about 45° with regard to the horizontal direction. As the load continued, a fracture initiated and new cracking appeared at the root of the key, extending upward to the top. Finally, the latter crack penetrated the key so that the joint has been completely sheared off and destroyed. The simulation was consistent with the development of corresponding test for the whole cracking process, including the cracking trend and the value of load at each phase. Particularly, though the corner cracks appeared firstly, they stopped to develop once the new cracks formed in the root area of the key.
In Figure 5(c), the crack initiation and development of singlekeyed epoxied joints (EA1C503.0) between experimental and simulated results are compared. The gray area mainly occurred on the bottom corner of the key at first, which indicated that the first crack appeared in this area. This crack extended to the top of the joint as shear load increased. Finally, similar to other types of joints, the crack penetrated the entire key when reaching the ultimate shear capacity. This cracking process also showed high consistency between test and finite element model results.
Generally, it is acceptable to use the numerical method to study the cracking process of all specimens. In experimental investigation, the crack developed rapidly, causing much difficulty to record the whole cracking process in detail. With the finite element model, this deficiency can be eradicated, and it can also comprehensively reveal the phenomenon of crack development.
5. Extended Parametric Analysis
As mentioned previously, both the confining pressure and eccentric loads were very important factors that cause the reduction of joint shear resistance. In this part, the influence of lower confining pressure and different loading positions on shear performance of joints was studied through finite element analysis.
5.1. Effect of Lower Confining Pressure
Due to the limitation of experimental investigation, the minimum horizontal confining pressure used in the previous study was generally 0.5 MPa [12]. In this part, the finite element model was used to analyze the shear performance of joints under the confining pressure varying from 0.1 MPa to 0.5 MPa.
5.1.1. Shear Capacity and Relative Displacement
The shear capacity and relative displacement for specimens under low horizontal confining pressure (from 0.1 MPa to 0.5 MPa) are shown in Table 5. To be clear, all the specimens were compared with the corresponding specimens with highest confining pressure (0.5 MPa) in the same series.
 
Note: is the numerical ultimate strength for the same joint type of specimen under 0.5 MPa confining stress; is the numerical slip at ultimate strength for the same joint type of specimen under 0.5 MPa confining stress; NC = normal concrete; SFRC = steel fiberreinforced concrete. 
For the monolithic joint specimen, when the confining pressure varied from 0.5 MPa to 0.1 MPa, the reduction of ultimate load and relative displacement was 6.9% and 7.2%, respectively. Though both the shear capacity and relative slip decreased with the decrease of the confining pressure at a low level, the effect of the confining stress was not significant.
When the confining pressure decreased from 0.5 MPa to 0.1 MPa, the ultimate load for singlekeyed dry joint specimens with NC also decreased. As the confining pressures decreased from 0.5 MPa to 0.2 MPa, the reduction of the ultimate load was not significant. However, when the confining pressure was reduced from 0.2 MPa to 0.1 MPa, the ultimate load decreased by 32.1%. In addition, when the confining pressure was reduced within the range of 0.5 MPa to 0.2 MPa, the relative displacement decreased accordingly, but the magnitude of reduction was not obvious. In contrast, when the horizontal confining pressure was reduced to 0.1 MPa, the relative displacement increased by 44.3%, compared with the confining pressure of 0.5 MPa.
As for singlekeyed dry joint specimens with SFRC at lower confining pressure (0.1 MPa to 0.5 MPa), some different conclusions can be drawn. A little reduction of the shear capacity and increase of the relative displacement can be noted. When the confining stress dropped to 0.1 MPa or 0.2 MPa, the ultimate load was greatly reduced and the relative slip was increased largely.
As shown in Table 5, the singlekeyed epoxied joints with 1 and 3 mm thickness epoxy are analyzed at different confining pressures from 0.1 MPa to 0.5 MPa. The result showed that as the confining pressure decreased, the ultimate load and relative displacement decreased correspondingly. Compared with the confining pressure of 0.5 MPa, the reduction of the ultimate load of the singlekeyed epoxied joints with 1 and 3 mm thickness epoxy was 13.6%, and the reduction of the relative displacement was 14.7% and 17.9%, respectively.
5.1.2. LoadDisplacement Relationship
The ultimate loaddisplacement curves for specimens at different confining pressures (0.1 MPa to 0.5 MPa) are shown in Figure 6.
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For the monolithic joint, no obvious difference can be seen for different confining pressure specimens. Therefore, it can be considered that the effect of reduction of the confining pressure from 0.5 MPa to 0.1 MPa on the shear performance of monolithic joints is not obvious.
It can be seen from Figure 6(b) that when the confining pressure decreased from 0.5 MPa to 0.2 MPa, the other characteristics and trends of the curves remained basically unchanged, except the maximum point and the plateau of the curves decreased correspondingly. However, when the confining pressure was reduced to 0.1 MPa, the corresponding curve trends changed largely, and the curves were more fluctuated. The curve had a short plateau before reaching the peak (ultimate load), and it also experienced a plateau before dropping to the stationary stage (residual load).
The ultimate loaddisplacement curves of singlekeyed dry joints with SFRC at different confining pressures (0.1 MPa to 0.5 MPa) are shown in Figure 6(c). Similar to the loadslip relationship of singlekeyed dry joints with NC, the corresponding curve changed significantly when the confining pressure reduced to 0.2 MPa. The slope of the curve decreased obviously at the plasticity stage, and the curve became relatively flat after reaching the ultimate load. Therefore, the shear capacity of singlekeyed dry joint specimens with SFRC decreased greatly when the confining pressure dropped to 0.2 MPa and 0.1 MPa.
For NC singlekeyed dry joint specimens with 0.1 MPa confined stress, the unusual phenomenon, as depicted in Figure 6(b), may also be related to the cracking process. The outward slip of the upper male portion resulted in the first constant level of loading. With the increase of vertical load, the sliding friction became large enough to stop the slip. Meanwhile, the bottom corner crack lengthened and widened, causing the slight rotation of the male part. At this time, the ultimate shear strength was reached and the root area of the key started to work. As a result, the key joint shear resistance and rotation existed together, leading to the second plateau. When the key root area cracks linked with each other (penetration), the dropping of the curve occurred. This can also be observed in the figure that both the former bottom corner cracks and latter root area cracks lengthened and widened. With the aggregate interlock action along the cracking concrete, the final shear resistance of the keyed joint was maintained at a constant level (residual stage) at last. The cause of abrupt change in SFRC singlekeyed dry joint specimens under low confining stress was similar to that with NC, as depicted in Figure 6(c). The better ductility of SFRC may be the reason for the sustained increase of shear capacity before reaching the peak load.
The ultimate loaddisplacement curves of singlekeyed epoxied joints with 1 and 3 mm thickness epoxy and regular concrete at different confining pressures (0.1 to 0.5 MPa) are shown in Figures 6(d) and 6(e). It can be seen that, after reaching the ultimate load, the curve in the descending stage showed fluctuation; that is, the curve first fell rapidly, then had a plateau or a slight rise, and finally descended again to a plateau slowly. It showed that the failure of singlekeyed epoxied joints exhibited more ductility under low confining pressures. In addition, it can be clearly seen that all the ultimate loaddisplacement curves of the model under low confining pressure had a very similar trend. The shear resistance of singlekeyed epoxied joints decreased gradually under the low confining pressures from 0.1 MPa to 0.5 MPa.
5.2. Effect of Eccentric Loads
As shown in Figure 1, an eccentric moment would exist on the joint, causing the rotation of both the specimen and experimental instrument. Considering the difficulty and safety of the experiment, almost all of the shear tests of joints loaded through the centric line of the top surface. In this part, the finite element model was used to analyze the shear performance of joints under loading with the eccentricity of 0 mm (centric line loading), 30 mm, 60 mm, and 90 mm.
5.2.1. Shear Capacity and Relative Displacement
The shear capacity and relative displacement for specimens under eccentric loads (with the eccentricity from 0 mm to 90 mm) are listed in Table 6. The simulated results were compared with those of the same specimen with centric loads.

Specimen M0C502.0 was chosen as a representative for parametric analysis on eccentric loads of monolithic joints. With the increase of eccentricity, both the ultimate strength and slip at the ultimate strength decreased.
For singlekeyed dry joints with NC or SFRC (based on specimens K1C502.0 and K1S502.0), the effect of eccentric load on the shear capacity and relative slip was similar. The farther the loading point away from the centric line, the smaller the ultimate load. On the contrary, the slip at the ultimate strength increased with the increase of eccentricity but leveled off or dropped down at larger eccentric loads.
As for the singlekeyed epoxied joint with NC (based on specimen EA1C502.0), it can be found in Table 6 that the ratio of the corresponding ultimate load of the model with the loading offset of 30 mm, 60 mm, and 90 mm to the ultimate load at the centric line position is 0.787, 0.589, and 0.394, respectively. This indicated that the shear capacity of epoxied joints decreased with the increase of loading eccentricity, while the slip at the ultimate strength increased with the increase of the eccentricity.
5.2.2. LoadDisplacement Relationship
The ultimate loaddisplacement curves for specimens under eccentric loads (with the eccentricity from 0 mm to 90 mm) are shown in Figure 7.
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For monolithic joints (M0C502.0), the trend of the loaddisplacement curve was similar for different loading positions. But the peak load and residual load of joints were lower than those of joints under larger loading eccentricity.
Based on specimen K1C502.0, the loaddisplacement relationship of singlekeyed dry joints with NC has been studied. The curve changed obviously when the eccentricity was 60 mm or 90 mm. In this case, the curve has only the ascending and residual phases and no descending phase. Similarly, the singlekeyed dry joint specimens with SFRC have twostage loadslip curves when the eccentric load distance was larger than 30 mm.
This unusual phenomenon may be related to the final failure modes of singlekeyed dry joint specimens under different eccentric loads, which are also shown in Figure 7(b). For specimens loaded at the centric line or at the eccentricity of 30 mm, the joints failed in shear of the key. Especially, slight opening of the joint for specimens under less than 30 mm eccentric loading, resulting from the rotation of the male part, can be observed. This demonstrated that the eccentric loads generated an additional moment, which will counteract the restraint action of horizontal confining pressure. Because of the resistance from horizontal confining pressure, specimens under 30 mm eccentric loading had a similar loaddisplacement relationship to those with centric loading. However, when the eccentricity increased to 60 mm or 90 mm, the moment generated by the eccentric load was large enough to counteract the confine formed by the horizontal restraint force. As a result, the male part slides outwards as a rigid body and the shear capacity of the joint was mainly provided by the friction force. It can also be found in Figure 7(b) that the specimens under 60 mm or 90 mm eccentric loading failed in crushing of concrete below the key, and no crack appeared on it.
As for singlekeyed epoxied joint specimens, different loaddisplacement curves can be found for specimens under 60 mm or 90 mm eccentric loading. After reaching the peak load, the shear capacity decreased slightly. With the increase of displacement, the load rose again or even exceeded the initial peak value. Finally, the shear capacity of joints decreased slowly.
For this different curve characteristic of singlekeyed epoxied joint specimens under larger eccentric compression, it may be related to the cracking process. The cracking process of singlekeyed epoxied joint specimens at the loading eccentricity of 90 mm is also shown in Figure 7(d). When the eccentric load is applied, additional moment will generate on the top area of the joint. Due to the cohesion of epoxy, tensile cracks were formed in this area but without the joint opening. Then, the crack developed downwards and began to generate in the key. When the key was penetrated by the crack, the curve reached the first peak load. At this time, the key was sheared off and the tensile stress was redistributed, causing the slight decrease of the loaddisplacement curve. As the crack developed toward the bottom of the joint, the curve rose slowly. When reaching the ultimate shear capacity, the crack penetrated the entire joint, but no any damage can be seen in the epoxy layer. Thus, it can be found that, under large eccentric loads, the breaking of the singlekeyed epoxied joint was from the top to the bottom. At the final residual stage, the shear capacity of the joint may be related to the friction between cracked concrete surfaces under confining pressure.
6. Comparison of the Simulated Results with AASHTO Provision
Currently, the calculation of shear capacity of the dry joint in design was usually based on the provision in the AASHTO 2003 specification. This equation was computed as a function of the root area of the key (m^{2}), the cylinder compressive strength of the male part (MPa), the horizontal confining pressure (MPa), and the area of the friction contact on the joint surface (m^{2}):
The comparison of the AASHTO design formula with the parametric analysis results is listed in Table 7. For this table, the average values of for NC specimens or SFRC specimens were 0.89 and 0.77, respectively. This result indicated that the formula of AASHTO 2003 specification underestimated the ultimate loads of singlekeyed dry joints under low confining pressure for both NC and SFRC. On the contrary, the values of for NC specimens were always larger than those for fiberreinforced concrete specimens. This highlighted the necessity of considering the concrete type in joint design.
 
Note: NC = normal concrete; SFRC = steel fiberreinforced concrete. 
Table 7 also shows the comparison between the AASHTO results and simulated results of singlekeyed dry joint specimens under eccentric loading. It can be seen that the AASHTO formula overestimated the ultimate shear capacity of singlekeyed dry joints for either NC or SFRC. Since the AASHTO formula neglected the influence of loading position on the joints, the calculated ultimate load was the same for the identical specimen under different eccentric loading. However, the ultimate load of the model was reduced due to the local bending moment caused by eccentric loading in the simulation, resulting in a smaller ultimate load than that estimated by the AASHTO formula. In order to draw a reliable and reasonable conclusion, more experimental results should also be considered. In general, the AASHTO formula should be reduced when calculating the shear capacity of singlekeyed dry joints under an eccentric load.
7. Conclusion
In this paper, the numerical simulations were performed to study the effect on shear behavior of different singlekeyed joints in PCSBs by extended parameters of low confining pressure and eccentric loads. Based on the simulation results, the main conclusions were drawn as follows:(1)Simulated ultimate shear capacity, loaddisplacement curve, and cracking process for monolithic joint, singlekeyed dry joint, and singlekeyed epoxied joint specimens were in good agreement with the experimental results. This indicated that the proposed finite element model may be able to predict accurately the shear behavior of these joints.(2)For lower horizontal confining pressures, the decrease of confining pressure (from 0.5 MPa to 0.1 MPa) has a little influence on the shear capacity of monolithic joint specimens (7.2%) and singlekeyed epoxied joint specimens (13.6%), but significant decrement can be noted for singlekeyed dry joints with NC (32.1%) and with SFRC (20.0%).(3)Similar loaddisplacement relationship can be seen for monolithic joint specimens or singlekeyed epoxied joint specimens when the confining pressure reduced from 0.5 MPa to 0.1 MPa, while a large change in the loaddisplacement relationship can be observed for singlekeyed dry joint specimens with NC or SFRC for 0.2 MPa confining pressure.(4)The reduction of the shear capacity was obvious for different types of specimens with increasing loading eccentricity. For the eccentricity larger than 60 mm, an obvious change of the loadslip curve can be seen in singlekeyed specimens.(5)The AASHTO formula underestimated the shear capacity of specimens under lower confining pressure (average of 0.83) but overestimated the shear capacity of joints at eccentric loads (average of 1.60). On the contrary, the AASHTO formula neglects the influence of concrete type, while it has been proved to be a vital variable. Thus, it is suggested that the AASHTO design code should consider the effect of eccentric load and concrete type.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest for this article.
Acknowledgments
This research was funded by the National Natural Science Foundation of China (51778150), Natural Science Foundation of Guangdong Province, China (2016A030313699), and Science and Technology Planning Project of Guangzhou City (201804010422). The authors gratefully acknowledge these generous supports.
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Copyright © 2019 Gongfa Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.