Experimental-Numerical Analysis of the Effect of Bar Diameter on Bond in Pull-Out Test Experimental-Numerical Analysis of the Effect of Bar Diameter on Bond in Pull-Out Test

: Bar diameter is one of the basic factors affecting bond behavior, which is still of interest due to opposing opinions regarding its effect on bond behavior in the pull-out test. This paper presents an experimental and numerical bond analysis of ribbed reinforcing bar in concrete. The aim was to experimentally evaluate the effect of bar diameter on the bond behavior in the pull-out test and to perform numerical simulations of the conducted experiments in ABAQUS to verify their convergence to the obtained experimental results. The experiments used concrete of C35/45 grade and B500SP reinforcing steel bars of three diameters: 10, 12, and 16 mm. FEA simulations employed the Concrete Damaged Plasticity (CDP) material model and the Contact Cohesive Behavior (CCB) method to model the concrete–bar interface. The study shows that bar diameter signiﬁcantly affects the bond, both on the bond stress–slip relationship and the type of bond failure, as well as on the bond strength. FEA simulations correctly reﬂected the bond behavior observed in the specimens. The analytical models presented estimates that were too conservative regarding the maximum bond stress relative to the experimental results. Abstract: Bar diameter is one of the basic factors affecting bond behavior, which is still of interest due to opposing opinions regarding its effect on bond behavior in the pull-out test. This paper presents an experimental and numerical bond analysis of ribbed reinforcing bar in concrete. The aim was to experimentally evaluate the effect of bar diameter on the bond behavior in the pull-out test and to perform numerical simulations of the conducted experiments in ABAQUS to verify their convergence to the obtained experimental results. The experiments used concrete of C35/45 grade and B500SP reinforcing steel bars of three diameters: 10, 12, and 16 mm. FEA simulations employed the Concrete Damaged Plasticity (CDP) material model and the Contact Cohesive Behavior (CCB) method to model the concrete–bar interface. The study shows that bar diameter significantly affects the bond, both on the bond stress–slip relationship and the type of bond failure, as well as on the bond strength. FEA simulations correctly reflected the bond behavior observed in the specimens. The analytical models presented estimates that were too conservative regarding the maximum bond stress relative to the experimental results.


Introduction
Concrete is one of the most commonly used materials in construction. It is characterized by low tensile strength to compressive strength. Therefore, reinforcing bars are used in structural elements with tension zones. The interaction of concrete and bars allows forces to be transferred from the reinforcement to the surrounding concrete. This cooperation is known as bond. Nowadays, ribbed steel reinforcing bars are widely employed in engineering practice, which owe their high bond to concrete mainly to the ribs. According to [1], about 80% of the bond is due to the mechanical interlocking of the bar in the concrete, while the remaining 20% comes from the action of adhesion and friction. The force acting on the bar induces two forces around the rib: a pressure force (F P ) perpendicular to the rib surface and a friction force (F F ) parallel to the rib surface. The consequent resultant force (F R ) can then be decomposed into two components: a horizontal component (F b ) that transfers forces from the bar to the concrete and causes compression of the concrete between the ribs, and a vertical component (F s ) responsible for splitting the concrete (Figure 1).

Introduction
Concrete is one of the most commonly used materials in construction. It is characterized by low tensile strength to compressive strength. Therefore, reinforcing bars are used in structural elements with tension zones. The interaction of concrete and bars allows forces to be transferred from the reinforcement to the surrounding concrete. This cooperation is known as bond. Nowadays, ribbed steel reinforcing bars are widely employed in engineering practice, which owe their high bond to concrete mainly to the ribs. According to [1], about 80% of the bond is due to the mechanical interlocking of the bar in the concrete, while the remaining 20% comes from the action of adhesion and friction. The force acting on the bar induces two forces around the rib: a pressure force (F P ) perpendicular to the rib surface and a friction force (F F ) parallel to the rib surface. The consequent resultant force (F R ) can then be decomposed into two components: a horizontal component (F b ) that transfers forces from the bar to the concrete and causes compression of the concrete between the ribs, and a vertical component (F s ) responsible for splitting the concrete (Figure 1). The fundamental quantities of bond analysis are the bond stress (τ b ) and slip (s), which is the relative displacement between the concrete and the bar. Most often, one of the basic The fundamental quantities of bond analysis are the bond stress (τ b ) and slip (s) which is the relative displacement between the concrete and the bar. Most often, one o the basic results of bond analysis is the bond stress-slip curve τ b s . Rehm proposed tha this curve is a fundamental law for bond, like the stress-strain curves for concrete or stee [2]. The shape of the curve indicates the type of bond failure. There are two types of bond failure: pull-out failure (POF) and splitting failure (SF) [3,4]. An example of τ b s curves depending on the type of failure is shown in Figure 2. The red curve in Figure 2 shows an example of the τ b s relationship for pull-ou failure. This situation occurs when the bar in the concrete is well confined. Then, the fail ure is caused by the horizontal component of the bond force. The initial, linear part of the graph (black section) corresponds to the range of adhesion action. As the slip increases microcracks are formed, which are manifested by the nonlinearity of the τ s curve unti the maximum value of the bond stress, called bond strength (τ b,max ), is reached. Once the peak is exceeded, shearing of the concrete between the bar ribs takes place. The final stage is the friction at the concrete-concrete interface, i.e., the meeting of the concrete between the ribs and the concrete surrounding the bar, which results in stabilization of the curve The other two curves in Figure 2 represent the bond stress-slip relationship for the split ting failure. In this situation, the bar confinement is insufficient to achieve the stress τ b,max Beyond the linear range, the increasing slip induces radial and tangential stresses in the concrete due to the vertical component of the bond force, causing damage. This leads to the complete failure of the bond (green curve) when the bar is poorly confined in the con crete, or to a significant decrease in bond stresses, followed by a residual ability of the concrete and bar to cooperate when the confinement is moderate (blue curve).
One of the most commonly performed bond experiments is the pull-out test (POT) The purpose of the POT is to evaluate the effect of selected factors on the bond. The main test results are the τ b s relationship and the maximum bond stress. This test has been performed since the 1940s. [5,6]. Currently, it is mostly used to analyze the influence o innovative materials on bond behavior (new generation concretes, non-metallic reinforce ment, etc.) [7][8][9][10], as well as to evaluate the influence of accidental factors affecting the cooperation of the concrete and the bar [11][12][13][14]. The main disadvantages of the POT in clude lack of representation of the tensile zone of the RC members (the bar is in tension while the concrete is in compression), slip-dependent bond function (bond stress is the same in each section), unprecedented in reinforced concrete confinement of the bar, lead ing to pull-out failure. The disadvantages above prevent the direct use of POT results in actual RC members [15]. The red curve in Figure 2 shows an example of the τ b (s) relationship for pull-out failure. This situation occurs when the bar in the concrete is well confined. Then, the failure is caused by the horizontal component of the bond force. The initial, linear part of the graph (black section) corresponds to the range of adhesion action. As the slip increases, microcracks are formed, which are manifested by the nonlinearity of the τ b (s) curve until the maximum value of the bond stress, called bond strength (τ b,max ), is reached. Once the peak is exceeded, shearing of the concrete between the bar ribs takes place. The final stage is the friction at the concrete-concrete interface, i.e., the meeting of the concrete between the ribs and the concrete surrounding the bar, which results in stabilization of the curve. The other two curves in Figure 2 represent the bond stress-slip relationship for the splitting failure. In this situation, the bar confinement is insufficient to achieve the stress τ b,max . Beyond the linear range, the increasing slip induces radial and tangential stresses in the concrete due to the vertical component of the bond force, causing damage. This leads to the complete failure of the bond (green curve) when the bar is poorly confined in the concrete, or to a significant decrease in bond stresses, followed by a residual ability of the concrete and bar to cooperate when the confinement is moderate (blue curve).
One of the most commonly performed bond experiments is the pull-out test (POT). The purpose of the POT is to evaluate the effect of selected factors on the bond. The main test results are the τ b (s) relationship and the maximum bond stress. This test has been performed since the 1940s. [5,6]. Currently, it is mostly used to analyze the influence of innovative materials on bond behavior (new generation concretes, non-metallic reinforcement, etc.) [7][8][9][10], as well as to evaluate the influence of accidental factors affecting the cooperation of the concrete and the bar [11][12][13][14]. The main disadvantages of the POT include lack of representation of the tensile zone of the RC members (the bar is in tension, while the concrete is in compression), slip-dependent bond function (bond stress is the same in each section), unprecedented in reinforced concrete confinement of the bar, leading to pull-out failure. The disadvantages above prevent the direct use of POT results in actual RC members [15].
Bond is affected by many factors. Among the most important of these are the concrete strength (compressive/tensile/shear strength), as well as those related to the bar Downloaded from mostwiedzy.pl confinement in the concrete, which consists of the thickness of the bar cover, the presence of transverse reinforcement or the presence of external compressive/tensile stresses [16], and the embedded length. The geometry of the rebar also has a significant effect on the bond. Besides the relative rib area (f R ), the bar diameter bar is also important, as it is used in many issues related to the interaction of concrete and reinforcement, including in determining the bond strength needed to determine development length and splice length in Eurocode 2 [17] and fib Model Code 2010 [4]. In the standard [18], the bar diameter determines the parameters related to the geometry of its ribbing. The same standard [18] includes a procedure for conducting a pull-out test (based on the recommendations of RILEM [19]), which introduces representative diameters (d b ) for each of four bar-size groups. These groups are presented in Table 1. Based on Table 1, it can be concluded that, for example, for the group of medium diameters, it is sufficient to conduct the test only for a diameter of 16 mm and assume that the results obtained will correspond to the bond behavior for the other diameters in this group. In pull-out tests, the bond length usually depends on the bar diameter. Most often, this length does not exceed five times the bar diameter (l b = 5d b ). These samples are known as short specimens, which make it possible to assume that the bond stresses along the bond length have the same value [20][21][22][23].
In the literature, one can find papers in which the influence of the bar diameter on the bond behavior was investigated. However, there is no shortage of controversy, because in the work [24], it was found that increases of the diameter of the bar increase its bond, while in the publication [25], it was noted that, with an increase of the diameter of the ribbed bar, its bond decreases due to the lack of linearity between the increase of the relative rib area to the diameter. In addition, there is also the issue of the difference in the cooperation with concrete of thin [26] and large-diameter bars [27]. The difficulties in comparing test results due to differences in the conduct of the POT regarding test pieces (bond length, dimensions of concrete block) or additional factors affecting bond (concrete strength, material of rebar) are also worth keeping in mind. Taking all of this into account, it is difficult to say unequivocally how bar diameter affects bond behavior in the pull-out test.
Performing a numerical simulation of a pull-out test allows for a reliable representation corresponding to the actual course of the study [28,29], and allows for obtaining and observing results that cannot be obtained through experimentation (e.g., stress distribution in concrete). Computer programs based on the Finite Element Method (FEM) make analyses available using advanced material models and efficient ways to model the concrete and bar interaction, thereby allowing convergence with experimental results. There are two methods for POT simulation. The first method uses the actual shape of the reinforcement, that is, it models a ribbed bar. Then the bond properties related only to friction and adhesion are determined, since the mechanical interlocking phenomenon is due to the geometry of the bar [30,31]. The second method involves the modeling of a specimen with a plain bar, connected to the concrete by elements with specific properties to simulate the bond [32,33]. Then, it is necessary to use a bond model describing the relation τ b (s). In this paper, the second method is employed for numerical simulation of the pull-out test.
This paper covers the scope of the experimental and numerical analysis of bond. The experimental analysis consisted of pull-out tests to evaluate the effect of diameter on the Downloaded from mostwiedzy.pl bond behavior of steel ribbed bars in concrete under monotonic loading in this test. The test pieces were made without additional confinement and from typical materials that are most commonly found in construction practice today, so the only variable parameters of the test were the bar diameters, which were 10, 12, and 16 mm. On the other hand, the numerical analysis was intended to reflect the performance of the experimental specimens subjected to the pull-out test in terms of the adopted material models of concrete and steel, as well as to evaluate the assumed modeling method of the concrete-bar interface and bond behavior from the obtained empirical results. The investigation presented in the paper is a part of a broad research plan devoted to a comprehensive experimental-numerical bond analysis of specimens at various levels of observation of the concrete-rebar interaction [34]. The influence of basic factors affecting bond, such as the bar confinement in the concrete and the stress state of the specimen, will be analyzed, employing a pull-out test at different bond lengths and a beam test. The research plan aims, among other things, to develop a method for using the results of the pull-out test on short specimens in practical engineering applications affected by bond.

Concrete
Plain concrete grade C35/45 according to the standard [35] was adopted for the study. The following were used in the concrete mix: 0-2 mm fine aggregate, 2-8 and 8-16 mm coarse aggregate, Portland cement of strength class 42.5 MPa (CEM I 42.5R), meeting the requirements of the standard [36], and water. In addition, a superplasticizer was used to obtain the consistency of the concrete mix for convenient concreting of the specimens. After a trial series of concrete batches, the concrete mix composition shown in Table 2 was determined experimentally. For each concrete batch intended for the POT test pieces, additional specimens were made to assess selected concrete parameters: cubes of 150 × 150 × 150 mm and cylinders of 150 mm diameter and 300 mm height. The concrete compressive strength (f c ) was determined on three cubes and three cylinders according to standard [37]. The concrete tensile strength by splitting (f ct ) was specified on three cubes according to standard [38]. The secant modulus of elasticity of concrete (E c,s ) was tested according to standard [39] by means of Method B. Six cylinders were used for this purpose: three for testing the modulus and three as companion specimens to define the upper-stress limit during the test. The resulting modulus of elasticity values were treated as a mean value of the modulus of elasticity (E cm ). In addition, the consistency of fresh concrete was also determined in accordance with the standard [40]. The tests of the aforementioned strength parameters were carried out for each concrete batch 28 days after the specimens were made. Based on these, the concrete parameters shown in Table 3

Reinforcing Steel
Ribbed steel reinforcing bars with diameters of 10, 12, and 16 mm were accepted for testing. Bars were made of steel grade B500SP, which is often used in engineering practice in Poland. Reinforcing steel B500SP is steel with increased ductility (class C according to the standard [17]), which means that the ratio of characteristic strength values (f uk ) to the yield strength of reinforcing steel (f yk ) is in the range from 1.15 to 1.35, and characteristic strain corresponding to the steel strength (ε uk ) is no less than 7.5%. The value of f yk for B500SP steel is in the range from 500 to 625 MPa. The requirements for the ribs and details of their geometry are given in Figure 3 and Table 4, according to the standard [41].

Reinforcing Steel
Ribbed steel reinforcing bars with diameters of 10, 12, and 16 mm were accepted for testing. Bars were made of steel grade B500SP, which is often used in engineering practice in Poland. Reinforcing steel B500SP is steel with increased ductility (class C according to the standard [17]), which means that the ratio of characteristic strength values (f uk ) to the yield strength of reinforcing steel (f yk ) is in the range from 1.15 to 1.35, and characteristic strain corresponding to the steel strength (ε uk ) is no less than 7.5%. The value of f yk for B500SP steel is in the range from 500 to 625 MPa. The requirements for the ribs and details of their geometry are given in Figure 3 and Table 4, according to the standard [41].  The yield strength (f y ), ultimate strength (f u ), ductility (k), and modulus of elasticity (E s ) were assessed for the described bars. The specimens were 400 mm in length. The elongation of the bar was measured using an extensometer at a measuring length of 50 mm. Three bars for each diameter were tested according to the standard [42]. The results of the rebar testing are shown in Table 5. Based on the results, it was found that the strength parameters meet the requirements for B500SP steel.  The yield strength (f y ), ultimate strength (f u ), ductility (k), and modulus of elasticity (E s ) were assessed for the described bars. The specimens were 400 mm in length. The elongation of the bar was measured using an extensometer at a measuring length of 50 mm. Three bars for each diameter were tested according to the standard [42]. The results of the rebar testing are shown in Table 5. Based on the results, it was found that the strength parameters meet the requirements for B500SP steel.

Experimental Analysis
The pull-out test was employed for an experimental analysis of bond. This test involves pulling a rebar anchored in a concrete block, and measuring the pull-out force (F) and the corresponding slip (s) at the same time. The bar diameter was the variable parameter of the experiment. Three diameters were considered: 10, 12, and 16 mm. Three series of POTs, one for each diameter, were performed. The series consisted of six specimens. Concrete and reinforcing steel were used for the test pieces, the parameters of which are described and listed above in Tables 2-5.
The specimens consisted of a concrete cubic block with a side length of 160 mm, which is ten times the diameter of the largest bar, and an anchored bar over a bond length equal to five times the bar diameter. The adoption of such a block size was aimed at observing the effect of bar confinement on the type of bond failure. A plastic tube with an outer diameter of 30 mm was placed on the unbonded part of the bar. To ensure the constant position of the bar in the block during concreting and compaction of the fresh concrete, the free space in the tube was filled with foam sealant. A rubber plug placed at the end of the tube filled with foam sealant to prevent the undesirable flow of concrete into its interior, therefore, the length l b was not increased. The described method of securing the unbonded part of the bar was developed based on our own idea, which was verified during the POT pre-series. The concreting direction was perpendicular to the bar axis. An aluminum channel was glued to the side surface of the concrete block, upon which, the arm of an extensometer measuring slip rested, thus allowing for the measurement of the displacement of the concrete block in relation to the bar. Figure 4 shows a POT specimen. Details of the POT series are shown in Table 6.

Experimental Analysis
The pull-out test was employed for an experimental analysis of bond. This test involves pulling a rebar anchored in a concrete block, and measuring the pull-out force (F) and the corresponding slip (s) at the same time. The bar diameter was the variable parameter of the experiment. Three diameters were considered: 10, 12, and 16 mm. Three series of POTs, one for each diameter, were performed. The series consisted of six specimens Concrete and reinforcing steel were used for the test pieces, the parameters of which are described and listed above in Tables 2-5.
The specimens consisted of a concrete cubic block with a side length of 160 mm which is ten times the diameter of the largest bar, and an anchored bar over a bond length equal to five times the bar diameter. The adoption of such a block size was aimed at observing the effect of bar confinement on the type of bond failure. A plastic tube with an outer diameter of 30 mm was placed on the unbonded part of the bar. To ensure the constant position of the bar in the block during concreting and compaction of the fresh concrete, the free space in the tube was filled with foam sealant. A rubber plug placed at the end of the tube filled with foam sealant to prevent the undesirable flow of concrete into its interior, therefore, the length l b was not increased. The described method of securing the unbonded part of the bar was developed based on our own idea, which was verified during the POT pre-series. The concreting direction was perpendicular to the bar axis. An aluminum channel was glued to the side surface of the concrete block, upon which, the arm of an extensometer measuring slip rested, thus allowing for the measurement of the displacement of the concrete block in relation to the bar. Figure 4 shows a POT specimen Details of the POT series are shown in Table 6.  Downloaded from mostwiedzy.pl The test pieces were placed in a specially designed steel cage, mounted in a testing machine (Zwick Roell Z400). A steel bar welded to the top plate of the cage was clamped in the upper, movable part of the machine. The rebar of the specimen was clamped in the lower, stationary part of the machine. The plates were connected by a 16 mm diameter bar ending in threads and nuts with washers. Under the specimen, a pad was placed to minimize friction between the concrete block and the steel plate, which would cause additional confinement of the concrete and the bar. The pad consisted of two layers: a bottom layer of 3 mm thick rubber and an upper layer of 6 mm thick polyethylene sheet. A schematic and actual view of the test stand is shown in Figure 5.
2, x FOR PEER REVIEW 7 of 23 The test pieces were placed in a specially designed steel cage, mounted in a testing machine (Zwick Roell Z400). A steel bar welded to the top plate of the cage was clamped in the upper, movable part of the machine. The rebar of the specimen was clamped in the lower, stationary part of the machine. The plates were connected by a 16 mm diameter bar ending in threads and nuts with washers. Under the specimen, a pad was placed to minimize friction between the concrete block and the steel plate, which would cause additional confinement of the concrete and the bar. The pad consisted of two layers: a bottom layer of 3 mm thick rubber and an upper layer of 6 mm thick polyethylene sheet. A schematic and actual view of the test stand is shown in Figure 5.  The load was applied to the specimens by the lower cage plate (the cage plates were thick enough to be assumed non-susceptible) by a monotonic displacement increment at a rate of 1 mm/min, to ensure static loading of the test piece. Controlling the load by displacement made it possible to observe bond behavior after the maximum value of bond stress was reached, which would not have been possible if the load was controlled by a constant increment of force. The test was carried out up to a maximum slip value of 30 mm, or until the force dropped below 1% of the maximum force achieved in the test. All experiments were performed 28 days after the specimens were made. The result of the test was a curve of the local bond stress-slip relationship τ b s . The bond stress τ b was determined according to the formula: The measured slip value s was reduced by the elongation of the portion of the bar that was not anchored in concrete (l i ), determining the actual slip value at the i-th step The load was applied to the specimens by the lower cage plate (the cage plates were thick enough to be assumed non-susceptible) by a monotonic displacement increment at a rate of 1 mm/min, to ensure static loading of the test piece. Controlling the load by displacement made it possible to observe bond behavior after the maximum value of bond stress was reached, which would not have been possible if the load was controlled by a constant increment of force. The test was carried out up to a maximum slip value of 30 mm, or until the force dropped below 1% of the maximum force achieved in the test. All experiments were performed 28 days after the specimens were made. The result of the test was a curve of the local bond stress-slip relationship τ b (s). The bond stress τ b was determined according to the formula: The measured slip value s was reduced by the elongation of the portion of the bar that was not anchored in concrete (l i ), determining the actual slip value at the i-th step according to the formula: where: F i -pull-out force of the bar in the i-th step; F 0 -initial pull-out force; l-initial length of the bar not anchored in concrete; ∆l i−1 -increment of the length of the unanchored part of the bar in step i − 1; E s -elastic modulus of the reinforcing steel; A s -cross-sectional area of the bar. Formula (2) is true under the assumption that the reinforcing steel operates in the linear elastic range.

Numerical Analysis
Numerical analysis of the pull-out test was performed in the ABAQUS program. To carry out the FEM simulation, it was necessary to define material models of concrete and reinforcing steel reflecting their actual behavior in the experiments. In addition, it was also necessary to choose a method. The bond properties associated with the chosen method had to be determined, as well.
The elastic range and plastic range were defined in the concrete material model using the Concrete Damaged Plasticity (CDP) model. A dilatation angle of 35 • and a viscosity parameter of 0.008 were used to describe the complex stress state, while default values were used for the other parameters of the CDP model (Table 7). On the other hand, the stress-strain relationship for compression and stress-crack opening for tension were employed to describe the behavior of concrete in the uniaxial state ( Figure 6), as well as parameters corresponding to concrete grade C35 from MC2010 [4] ( Table 8). The exception was the fracture energy (G F ), the value of which was assumed according to the ABAQUS documentation [43]. In addition, the stress-strain relationship for compression was set to be linear, up to a stress value equal to 40% of the concrete compressive strength, which determined the obtaining of a new value for the secant modulus of concrete (E cm ).
where: F i -pull-out force of the bar in the i-th step; F 0 -initial pull-out force; l-initial length of the bar not anchored in concrete; Δl i−1 -increment of the length of the unanchored part of the bar in step i − 1; E s -elastic modulus of the reinforcing steel; A s -crosssectional area of the bar. Formula (2) is true under the assumption that the reinforcing steel operates in the linear elastic range.

Numerical Analysis
Numerical analysis of the pull-out test was performed in the ABAQUS program. To carry out the FEM simulation, it was necessary to define material models of concrete and reinforcing steel reflecting their actual behavior in the experiments. In addition, it was also necessary to choose a method. The bond properties associated with the chosen method had to be determined, as well.
The elastic range and plastic range were defined in the concrete material model using the Concrete Damaged Plasticity (CDP) model. A dilatation angle of 35° and a viscosity parameter of 0.008 were used to describe the complex stress state, while default values were used for the other parameters of the CDP model (Table 7). On the other hand, the stress-strain relationship for compression and stress-crack opening for tension were employed to describe the behavior of concrete in the uniaxial state ( Figure 6), as well as parameters corresponding to concrete grade C35 from MC2010 [4] ( Table 8). The exception was the fracture energy (G F ), the value of which was assumed according to the ABAQUS documentation [43]. In addition, the stress-strain relationship for compression was set to be linear, up to a stress value equal to 40% of the concrete compressive strength, which determined the obtaining of a new value for the secant modulus of concrete (E cm ).   Downloaded from mostwiedzy.pl Parameters of the elastic range were defined in the material model of reinforcing steel. The stress-strain relationship was assumed to be linear. The value of the modulus of elasticity was taken depending on the diameter of the bar, according to Table 5.
The bond behavior in the pull-out test is described by the local bond stress-slip relationship. The numerical POT simulations employ the bond model represented in MC2010 [4] (Figure 7). For monotonic loading, bond stress values are calculated according to Equations (3)-(6). Table 9 defines the characteristic points of the model. Justification of the rules and assumptions of the presented model is given in ref. [16]. Parameters of the elastic range were defined in the material model of reinforcing steel. The stress-strain relationship was assumed to be linear. The value of the modulus of elasticity was taken depending on the diameter of the bar, according to Table 5.
The bond behavior in the pull-out test is described by the local bond stress-slip relationship. The numerical POT simulations employ the bond model represented in MC2010 [4] (Figure 7). For monotonic loading, bond stress values are calculated according to Equations (3)-(6). Table 9 defines the characteristic points of the model. Justification of the rules and assumptions of the presented model is given in ref. [16]. For 0 ≤ s ≤ s 1 : For s 1 < s ≤ s 2 : For s 1 < s ≤ s 2 : For s 3 < s: For 0 ≤ s ≤ s 1 : For s 1 < s ≤ s 2 : For s 1 < s ≤ s 2 : For s 3 < s: The Contact Cohesive Behavior (CCB) method available in ABAQUS was employed to model the concrete-bar interface. This method allows for an efficient representation of the bond behavior considering the bond stress-slip law. With CCB, it is possible to simulate Downloaded from mostwiedzy.pl the linear-elastic range of the τ b (s) relationship, as well as the initiation and evolution of damage using the traction-separation law. The linear-elastic traction-separation law for the uncoupled stiffness option is expressed by the equation: The meaning of the subscripts (n, s, t) from Equation (7) for the directional designation in the CCB was shown in Figure 8.  The Contact Cohesive Behavior (CCB) method available in ABAQUS was em to model the concrete-bar interface. This method allows for an efficient represent the bond behavior considering the bond stress-slip law. With CCB, it is possible t late the linear-elastic range of the τ b s relationship, as well as the initiation and tion of damage using the traction-separation law. The linear-elastic traction-sep law for the uncoupled stiffness option is expressed by the equation: The meaning of the subscripts (n, s, t) from Equation (7) for the directional d tion in the CCB was shown in Figure 8. For the pull-out test simulations, "Traction" is interpreted as bond stress, an aration" as slip, allowing the local bond model to be used directly in describing b havior. The stiffness K is equal to the stiffnesses K ss and K tt . It is determined as f where: τ b,lin -the limiting bond stress terminating the linear relationship τ b s , s slip corresponding to the stress τ b,lin . According to Keuser et al. [44], the stiffnes determined by the formula: In addition to the definition of bond stiffness itself, it is also necessary to de the parameters associated with its damage. The first parameter is the initiation of d In the simulations performed, the Maximum Stress criterion (MAXS) was used. W criterion is met, the damage is initiated. The next stage is the development of dam the ABAQUS software for CCB, this is done through the Damage Evolution Law. describes the rate at which the cohesive stiffness is degraded. Degradation is con by a scalar variable (D) representing the overall damage at the contact point. The For the pull-out test simulations, "Traction" is interpreted as bond stress, and "Separation" as slip, allowing the local bond model to be used directly in describing bond behavior. The stiffness K is equal to the stiffnesses K ss and K tt . It is determined as follows: where: τ b,lin -the limiting bond stress terminating the linear relationship τ b (s), s lin -the slip corresponding to the stress τ b,lin . According to Keuser et al. [44], the stiffness K nn is determined by the formula: K nn = 100·K.
In addition to the definition of bond stiffness itself, it is also necessary to determine the parameters associated with its damage. The first parameter is the initiation of damage. In the simulations performed, the Maximum Stress criterion (MAXS) was used. When the criterion is met, the damage is initiated. The next stage is the development of damage. In the ABAQUS software for CCB, this is done through the Damage Evolution Law. The law describes the rate at which the cohesive stiffness is degraded. Degradation is considered by a scalar variable (D) representing the overall damage at the contact point. The variable D takes a value from 0 (no damage) to 1 (full damage). Thus, the D value increases as Downloaded from mostwiedzy.pl the damage progress. In this situation, the form of Traction-Separation Law expresses the formula: Damage Evolution controlled by displacement was adopted for simulations. The tabular method of the softening was employed. It was therefore necessary to determine the values of D and the corresponding inelastic slips (s in ). The value of the degradation variable, which is dependent on the slip value, can be calculated according to the formula: where: τ b (s) is calculated according to Equations (3)-(6); K is determined from Equation (8).
The slip s in is obtained by subtracting the slip corresponding to the stress τ b.lin from the total slip s: An axisymmetric model was used for numerical simulations in ABAQUS. For the calculations, the ABAQUS Standard solver was employed. Both the concrete block and the rebar were discretized using 4-node bilinear axisymmetric quadrilateral (CAX4) finite elements. Three finite element sizes were adopted in the model: 1.0, 2.0 and 4.0 mm. Figure 9 shows the finite element mesh of the specimen for POT-12 series. For the other cases, the model differed only in bar diameter. D takes a value from 0 (no damage) to 1 (full damage). Thus, the D value increases as the damage progress. In this situation, the form of Traction-Separation Law expresses the formula: Damage Evolution controlled by displacement was adopted for simulations. The tabular method of the softening was employed. It was therefore necessary to determine the values of D and the corresponding inelastic slips (s in ). The value of the degradation variable, which is dependent on the slip value, can be calculated according to the formula: where: τ b s is calculated according to Equations (3)-(6); K is determined from Equation (8). The slip s in is obtained by subtracting the slip corresponding to the stress τ b.lin from the total slip s: An axisymmetric model was used for numerical simulations in ABAQUS. For the calculations, the ABAQUS Standard solver was employed. Both the concrete block and the rebar were discretized using 4-node bilinear axisymmetric quadrilateral (CAX4) finite elements. Three finite element sizes were adopted in the model: 1.0, 2.0 and 4.0 mm. Figure  9 shows the finite element mesh of the specimen for POT-12 series. For the other cases, the model differed only in bar diameter.

Local Bond Stress-Slip Relationship
Based on the series of pull-out tests, the curves of the local stress-slip relationship were created. Figures 10-12 show the course of representative τ b s curves being the average of the results obtained from six specimens from the series together with the envelope.

Local Bond Stress-Slip Relationship
Based on the series of pull-out tests, the curves of the local stress-slip relationship were created. Figures 10-12 show the course of representative τ b (s) curves being the average of the results obtained from six specimens from the series together with the envelope. Buildings 2022, 12, x FOR PEER REVIEW 12 of 23   The shape of the τ b s curve for the POT-10 and POT-12 series (Figures 10 and 11) indicates that the specimens were damaged by pull-out, while the POT-16 series specimens ( Figure 12) were damaged by splitting (cf. Figure 2). The use of different bar sizes and bond lengths dependent on bar diameter resulted in variable conditions of bar confinement in concrete, which led to different types of bond failure. According to MC2010 [4], pull-out failure occurs when the ratio of the concrete bar cover to the bar diameter is no less than five. As seen in Table 6, this condition was met by specimens of the POT-10 and POT-12 series, which proves the correctness of the aforementioned notation. However, it should be noted that this notation probably refers to the situation when the bond length is five times the bar diameter. A similar assumption was made in MC2010 [4] for the semi-empirical formula for the bond stress at splitting failure (τ bu,split ). Examples of specimens after testing are shown in Figure 13.   The shape of the τ b s curve for the POT-10 and POT-12 series (Figures 10 and 11) indicates that the specimens were damaged by pull-out, while the POT-16 series specimens ( Figure 12) were damaged by splitting (cf. Figure 2). The use of different bar sizes and bond lengths dependent on bar diameter resulted in variable conditions of bar confinement in concrete, which led to different types of bond failure. According to MC2010 [4], pull-out failure occurs when the ratio of the concrete bar cover to the bar diameter is no less than five. As seen in Table 6, this condition was met by specimens of the POT-10 and POT-12 series, which proves the correctness of the aforementioned notation. However, it should be noted that this notation probably refers to the situation when the bond length is five times the bar diameter. A similar assumption was made in MC2010 [4] for the semi-empirical formula for the bond stress at splitting failure (τ bu,split ). Examples of specimens after testing are shown in Figure 13.   The shape of the τ b s curve for the POT-10 and POT-12 series (Figures 10 and 11) indicates that the specimens were damaged by pull-out, while the POT-16 series specimens ( Figure 12) were damaged by splitting (cf. Figure 2). The use of different bar sizes and bond lengths dependent on bar diameter resulted in variable conditions of bar confinement in concrete, which led to different types of bond failure. According to MC2010 [4], pull-out failure occurs when the ratio of the concrete bar cover to the bar diameter is no less than five. As seen in Table 6, this condition was met by specimens of the POT-10 and POT-12 series, which proves the correctness of the aforementioned notation. However, it should be noted that this notation probably refers to the situation when the bond length is five times the bar diameter. A similar assumption was made in MC2010 [4] for the semi-empirical formula for the bond stress at splitting failure (τ bu,split ). Examples of specimens after testing are shown in Figure 13. The shape of the τ b (s) curve for the POT-10 and POT-12 series (Figures 10 and 11) indicates that the specimens were damaged by pull-out, while the POT-16 series specimens ( Figure 12) were damaged by splitting (cf. Figure 2). The use of different bar sizes and bond lengths dependent on bar diameter resulted in variable conditions of bar confinement in concrete, which led to different types of bond failure. According to MC2010 [4], pull-out failure occurs when the ratio of the concrete bar cover to the bar diameter is no less than five. As seen in Table 6, this condition was met by specimens of the POT-10 and POT-12 series, which proves the correctness of the aforementioned notation. However, it should be noted that this notation probably refers to the situation when the bond length is five times the bar diameter. A similar assumption was made in MC2010 [4] for the semi-empirical formula for the bond stress at splitting failure (τ bu,split ). Examples of specimens after testing are shown in Figure 13.  Figure 14 shows representative τ b s curves for the conducted POT series. The initial, linear-elastic range is almost identical for all shown curves, indicating similar bond stiffness. In the non-linear range before the peak, the first discrepancies in their course appeared, especially for the POT-16 series, whose specimens reached the maximum bond stress in this range. For the POT-12 series, a faster degradation of post-peak bond stress is evident compared to the POT-10 series. This represents a faster degradation of bond stiffness due to the shearing-off of the concrete between the ribs. This is the result of greater bar confinement, owing to the thicker concrete cover. The smaller diameter affected the slower degradation of stiffness after the peak. This may also be influenced by the slightly different ribbing of the bar, which depends on the diameter. The τ b s curves of the POT-10 and POT-12 series start to stabilize at a slip value of 15 mm, reaching residual bond stress (τ bf ), which is about 20% of the maximum bond stress. Table 10 shows the extreme and mean maximum bond stresses obtained from the experiments, which were determined according to Equation (1). In addition, standard deviations (σ) and coefficients of variation (CV) were given.  In the non-linear range before the peak, the first discrepancies in their course appeared, especially for the POT-16 series, whose specimens reached the maximum bond stress in this range.  Figure 14 shows representative τ b s curves for the conducted POT series. The initial, linear-elastic range is almost identical for all shown curves, indicating similar bond stiffness. In the non-linear range before the peak, the first discrepancies in their course appeared, especially for the POT-16 series, whose specimens reached the maximum bond stress in this range. For the POT-12 series, a faster degradation of post-peak bond stress is evident compared to the POT-10 series. This represents a faster degradation of bond stiffness due to the shearing-off of the concrete between the ribs. This is the result of greater bar confinement, owing to the thicker concrete cover. The smaller diameter affected the slower degradation of stiffness after the peak. This may also be influenced by the slightly different ribbing of the bar, which depends on the diameter. The τ b s curves of the POT-10 and POT-12 series start to stabilize at a slip value of 15 mm, reaching residual bond stress (τ bf ), which is about 20% of the maximum bond stress. Table 10 shows the extreme and mean maximum bond stresses obtained from the experiments, which were determined according to Equation (1). In addition, standard deviations (σ) and coefficients of variation (CV) were given. For the POT-12 series, a faster degradation of post-peak bond stress is evident compared to the POT-10 series. This represents a faster degradation of bond stiffness due to the shearing-off of the concrete between the ribs. This is the result of greater bar confinement, owing to the thicker concrete cover. The smaller diameter affected the slower degradation of stiffness after the peak. This may also be influenced by the slightly different ribbing of the bar, which depends on the diameter. The τ b (s) curves of the POT-10 and POT-12 series start to stabilize at a slip value of 15 mm, reaching residual bond stress (τ bf ), which is about 20% of the maximum bond stress. Table 10 shows the extreme and mean maximum bond stresses obtained from the experiments, which were determined according to Equation (1). In addition, standard deviations (σ) and coefficients of variation (CV) were given. Downloaded from mostwiedzy.pl The differences between the extreme values for specimens of the POT-10 and POT-12 series are small, as evidenced by the low coefficient of variation-4.0 and 2.9%, respectively. The low variability is also evidenced by the envelopes of the results for the series shown in Figures 10 and 11. Such a small scatter in the results confirms the proper preparation of the test pieces, especially in terms of ensuring the assumed bond length. Larger scatter in the extreme maximum bond stresses occurred in the POT-16 series, as the CV for these specimens was 10.9%. This is an acceptable value, given the probabilistic nature of bond. The steeper slope of the green curve representing the POT-16 series in Figure 14 is relative to the other two. This is the effect of averaging results characterized by greater scatter. For the POT-16 series, the mean maximum bond stress is lower, and is about 70% of the mean stresses of the POT-10 and POT-12 series, which is due to a different type of bond failure.

Maximum Bond Stress
The effect of bar diameter on the experimental results obtained is insignificant, as the difference between the mean strength values of the POT-10 and POT-12 series specimens was 0.16 MPa. The bond length, depending on the bar diameter, allowed the specimen to obtain almost identical maximum adhesion stresses for the pull-out failure. This is an important observation in terms of the division into groups of bars of different diameters made in the pull-out test procedure in the standard [18] (see Table 1). Thus, it can be assumed that, in the case of providing greater confinement of a 16 mm diameter bar in concrete (for example, through a larger size of the concrete block), the specimen would be damaged by pull-out, and the maximum bond stresses would be like those obtained for the POT-10 and POT-12 series. Similar values of the maximum bond stress in the POT for various diameters and bond lengths, depending on d b , were observed in the papers [45,46]. Table 11 shows the extreme and mean slips along with σ and CV and the pull-out forces corresponding to the maximum bond stresses. Larger mean slip s max values were achieved in the POT-10 series specimens than in the POT-12 series. This was most likely thanks to better confinement of the bar by the concrete cover. For the POT-16 series, the mean slip value was about half as much, which was due to the splitting failure. As the bar diameter increased, the coefficient of variation increased, which did not exceed 30% for any series.

Slip and Maximum Pull-Out Force
While the mean bond strengths for the POT-10 and POT-12 series were almost the same, the values of the pull-out forces (F max ) differed significantly. From the obtained results, it can be seen that the larger the bar diameter, the higher the force required to pull the bar out of the concrete. This is an obvious conclusion, since the bond lengths were determined by the bar diameter (the larger the diameter, the greater the bond length). The determination of the bond stresses according to Equation (1) is based on the assumption that the pull-out force is uniformly distributed on the lateral bar surface. This allows qualitative comparison of bond for different bar diameters and bond lengths.

Bond Parameters in Numerical Simulations
One method to define the bond behavior in numerical simulations of a pull-out test in ABAQUS using the CCB is to use a bond model that defines a local bond stress-slip relationship. For this purpose, the model described in MC2010 [4] was employed. Based on the relationship presented in Figures 10-12, it was considered that the curves for the POT-10 and POT-12 series are described by the curve corresponding to the pull-out failure, and for the POT-16 series by the curve corresponding to the splitting failure without stirrups (see Figure 7). "Good bond conditions" were assumed for all series. In Table 12, the values of the characteristic points of the bond stress-slip curves obtained from the experiments and from the MC2010 bond model were compared. The parameters from the experiments were adjusted to best match the bond model curve with the representative curves from the experiments. By means of Equations (3)-(6), τ b (s) curves were created using the characteristic point values given in Table 12, which are shown in Figure 15. FEM simulations were performed in the slip range from 0 to 15 mm.
While the mean bond strengths for the POT-10 and POT-12 series were almost the same, the values of the pull-out forces (F max ) differed significantly. From the obtained results, it can be seen that the larger the bar diameter, the higher the force required to pull the bar out of the concrete. This is an obvious conclusion, since the bond lengths were determined by the bar diameter (the larger the diameter, the greater the bond length). The determination of the bond stresses according to Equation (1) is based on the assumption that the pull-out force is uniformly distributed on the lateral bar surface. This allows qualitative comparison of bond for different bar diameters and bond lengths.

Bond Parameters in Numerical Simulations
One method to define the bond behavior in numerical simulations of a pull-out test in ABAQUS using the CCB is to use a bond model that defines a local bond stress-slip relationship. For this purpose, the model described in MC2010 [4] was employed. Based on the relationship presented in Figures 10-12, it was considered that the curves for the POT-10 and POT-12 series are described by the curve corresponding to the pull-out failure, and for the POT-16 series by the curve corresponding to the splitting failure without stirrups (see Figure 7). "Good bond conditions" were assumed for all series. In Table 12, the values of the characteristic points of the bond stress-slip curves obtained from the experiments and from the MC2010 bond model were compared. The parameters from the experiments were adjusted to best match the bond model curve with the representative curves from the experiments. By means of Equations (3)-(6), τ b s curves were created using the characteristic point values given in Table 12, which are shown in Figure 15. FEM simulations were performed in the slip range from 0 to 15 mm.

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Many proposals of formulas for determining the maximum bond stresses have been described in the literature. Most of them consist of the product of the square root of the mean concrete compressive strength and a coefficient, which takes different forms depending on the approach used. In the formula proposed in MC2010 [4] for pull-out failure, the coefficient is a constant value. The same applies to the formula described by Harajli et al. [47]. In the Australian standard AS 3600 [48], the coefficient depends on the ratio of the bar cover thickness to the bar diameter, while in the formula presented by Orangun et al. [49], the coefficient additionally depends on the ratio of the bar diameter to the bond length. This formula was further developed by Darwin et al. [50], who also considered the effect of the ratio of maximum to minimum bar cover thickness. There are also proposals to determine the maximum bond stress for POF based on the product of the mean concrete compressive strength and a coefficient, which was proposed by Huang et al. [51]. In addition, MC2010 [4] also provides a separate formula for calculating the maximum stress τ b for a splitting failure. Table 13 summarizes the aforementioned formulas for determining maximum bond stress along with the values, as well as the values obtained from experiments. In brackets, the percentages of maximum bond stress to experimental results were given. The best agreement with the experimental results of the maximum bond stresses for the pull-out failure was obtained by the formula given by Huang et al. [51] (72%), and for splitting failure, by the formula proposed by Darwin et al. [50] (72%). In contrast, the least convergent results were obtained by the formula according to AS 3600 [48] for POF (52% and 43%), and according to MC2010 [4] for SF (43%). Comparing the curves τ b (s) shown in Figure 15, one can see significant differences in their courses. Therefore, the use of curves described in the bond model in MC2010 [4] was determined to be the best approach, based on characteristic points specified from the experimental results.
In the definition of the cohesive surface in ABAQUS, the stiffness of the concrete-bar interface should also be defined. It was assumed that the linear relationship τ b (s) occurs up to a value of 15% of the maximum bond stress (τ b,lin ), which well reflects the range of adhesion action during the test. At the same time, the stress τ b,lin initiates the bond stiffness degradation process. Using Equations (8) and (9), the bond stiffnesses shown in Table 14 were obtained. Downloaded from mostwiedzy.pl Numerical simulations employed the same τ b (s) curve for the POT-10 and POT-12 series corresponding to pull-out failure. After pre-simulations, the curves describing the bond behavior were calibrated to best match the experimental results. Figures 16 and 17 compare the curves obtained from experiments, FEM simulations, and bond model for POT-10 and POT-12, respectively. stiffness degradation process. Using Equations (8) and (9), the bond stiffnesses shown Table 14 were obtained.

Comparative Analysis of Experimental and Numerical Results-POT-10 and POT-12 Ser
Numerical simulations employed the same τ b s curve for the POT-10 and POT series corresponding to pull-out failure. After pre-simulations, the curves describing bond behavior were calibrated to best match the experimental results.   In the pre-peak and peak ranges, all three curves have a remarkably similar cou This proves that the parameters of the bond model and the parameters describing bo in ABAQUS were properly chosen. The differences in the curves start in the post-p range, especially between the experimental and numerical curves. This is directly due the use of a linear bond stress-slip relationship, which is an oversimplification of the τ b curve description proposed in MC2010 [4] for this range. For the range of residual bo stress, there are also noticeable discrepancies between the experimental and numer results. Nevertheless, the overall course of the curves adequately describes the trend the τ b s curves. stiffness degradation process. Using Equations (8) and (9), the bond stiffnesses shown Table 14 were obtained.

Comparative Analysis of Experimental and Numerical Results-POT-10 and POT-12 Ser
Numerical simulations employed the same τ b s curve for the POT-10 and POT series corresponding to pull-out failure. After pre-simulations, the curves describing bond behavior were calibrated to best match the experimental results.   In the pre-peak and peak ranges, all three curves have a remarkably similar cou This proves that the parameters of the bond model and the parameters describing bo in ABAQUS were properly chosen. The differences in the curves start in the post-p range, especially between the experimental and numerical curves. This is directly du the use of a linear bond stress-slip relationship, which is an oversimplification of the τ curve description proposed in MC2010 [4] for this range. For the range of residual bo stress, there are also noticeable discrepancies between the experimental and numer results. Nevertheless, the overall course of the curves adequately describes the trend the τ b s curves. In the pre-peak and peak ranges, all three curves have a remarkably similar course. This proves that the parameters of the bond model and the parameters describing bond in ABAQUS were properly chosen. The differences in the curves start in the post-peak range, especially between the experimental and numerical curves. This is directly due to the use of a linear bond stress-slip relationship, which is an oversimplification of the τ b (s) curve description proposed in MC2010 [4] for this range. For the range of residual bond stress, there are also noticeable discrepancies between the experimental and numerical results. Nevertheless, the overall course of the curves adequately describes the trend of the τ b (s) curves.

Comparative Analysis of Experimental and Numerical Results-POT-16 Series
Numerical simulations employed τ b (s) curve for the POT-16 series corresponding to splitting failure. Analogously to the POT-10 and POT-12 series, pre-simulations were performed to calibrate the model. Figure 18 summarizes the curves as for the previously described analyses.
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Comparative Analysis of Experimental and Numerical Results-POT-16 Series
Numerical simulations employed τ b s curve for the POT-16 series correspond to splitting failure. Analogously to the POT-10 and POT-12 series, pre-simulations w performed to calibrate the model. Figure 18 summarizes the curves as for the previou described analyses. The numerical simulation of the POT-16 series was not completed due to conv gence problems. The dashed line for the numerical curve is an extension of the cu where the calculations stopped. Based on Figure 18, there is good agreement between numerical curve and the curve from the bond model in the pre-peak range. The exp mental curve differs from the other two. This is because it is a curve averaging the resu of the six specimens of the POT-16 series, which had the largest scatters (see Table 10) Figures 19-21 show the maximum principal stresses corresponding to the maxim bond stresses.  The numerical simulation of the POT-16 series was not completed due to convergence problems. The dashed line for the numerical curve is an extension of the curve where the calculations stopped. Based on Figure 18, there is good agreement between the numerical curve and the curve from the bond model in the pre-peak range. The experimental curve differs from the other two. This is because it is a curve averaging the results of the six specimens of the POT-16 series, which had the largest scatters (see Table 10). Figures 19-21 show the maximum principal stresses corresponding to the maximum bond stresses.

Comparative Analysis of Experimental and Numerical Results-POT-16 Series
Numerical simulations employed τ b s curve for the POT-16 series corresponding to splitting failure. Analogously to the POT-10 and POT-12 series, pre-simulations were performed to calibrate the model. Figure 18 summarizes the curves as for the previously described analyses. The numerical simulation of the POT-16 series was not completed due to convergence problems. The dashed line for the numerical curve is an extension of the curve where the calculations stopped. Based on Figure 18, there is good agreement between the numerical curve and the curve from the bond model in the pre-peak range. The experimental curve differs from the other two. This is because it is a curve averaging the results of the six specimens of the POT-16 series, which had the largest scatters (see Table 10). Figures 19-21 show the maximum principal stresses corresponding to the maximum bond stresses.     During the pull-out tests, the POT-16 series specimens were damaged by splitting, which is evident from the curve shown in Figure 14. This means that the vertical component of the bond force, causing radial compressive stresses in the cracked bar cover and circumferential tensile stresses in the uncracked bar cover [52], was dominant due to insufficient bar confinement in the concrete. This should lead to exceeding the concrete tensile strength and splitting the concrete. From the map of maximum principal stresses (maximum tensile stresses, Figure 21), the maximum tensile stress occurring in the specimen had a value of 2.83 MPa, and at the edge of the block, about 2.40 MPa. Equal parameter values for the concrete material model were used for the numerical analysis of the POT. Table 15 shows the mean concrete tensile strengths for each series of pull-out test. As can be seen, the mean value of f ctm was the lowest for the POT-16 series. Therefore, the FEM simulation of the POT-16 series was carried out for the modified concrete tensile strength equal to 2.71 MPa. Figure 22 presents the result of this simulation, namely the map of the maximum principal stresses in the concrete block corresponding to the maximum bond stress. During the pull-out tests, the POT-16 series specimens were damaged by splitting, which is evident from the curve shown in Figure 14. This means that the vertical component of the bond force, causing radial compressive stresses in the cracked bar cover and circumferential tensile stresses in the uncracked bar cover [52], was dominant due to insufficient bar confinement in the concrete. This should lead to exceeding the concrete tensile strength and splitting the concrete. From the map of maximum principal stresses (maximum tensile stresses, Figure 21), the maximum tensile stress occurring in the specimen had a value of 2.83 MPa, and at the edge of the block, about 2.40 MPa. Equal parameter values for the concrete material model were used for the numerical analysis of the POT. Table 15 shows the mean concrete tensile strengths for each series of pull-out test. As can be seen, the mean value of f ctm was the lowest for the POT-16 series. Therefore, the FEM simulation of the POT-16 series was carried out for the modified concrete tensile strength equal to 2.71 MPa. Figure 22 presents the result of this simulation, namely the map of the maximum principal stresses in the concrete block corresponding to the maximum bond stress. From Figure 22, it can be seen that the introduced modification of the concrete material model affected the tensile stresses in the concrete. This view of tensile stresses prompts the conclusion that the concrete block of the specimen would be damaged by splitting  Figure 22, it can be seen that the introduced modification of the concrete material model affected the tensile stresses in the concrete. This view of tensile stresses prompts the conclusion that the concrete block of the specimen would be damaged by splitting failure, which is what happened in the experimental study.

Conclusions
Based on the conducted analysis, the following conclusions can be drawn:

•
The diameter of the bar is a quantity that determines the dimensions of the concrete block and bond length in the pull-out test. It affects the bond behavior in the test, that is, the course of the local bond stress-slip relationship, as well as the pull-out force, which increases as the bar diameter increases. • For test pieces in which the bond length does not exceed five times the bar diameter, the bond stress is treated as a pull-out force uniformly distributed on the lateral surface of the bar, cooperating with the concrete. Thus, the maximum stress τ b has similar values for different bar diameters and for the same type of bond failure. This is an important conclusion in the context of the division of bars into groups in terms of bar diameter, which was presented in the standard [18].

•
If a concrete block of the same size is used, the bar confinement by the concrete cover depends on the diameter d b . The conducted tests confirmed the notation in MC2010 [4] regarding the ratio of the bar cover thickness to its diameter. Specimens from the POT-10 and POT-12 series, in which the condition c/d b ≥ 5 was met, were damaged by pull-out.

•
The Contact Cohesive Behavior (CCB) method effectively reproduces the effect of bond at the concrete-bar interface in the pull-out test. The use of this method in FEA simulations performed in ABAQUS allowed for the obtainment of τ b (s) curves consistent with the experimental results.

•
The numerical simulations performed showed how important it is to properly define material models that correctly reflect the operation of the POT specimen. If the material models are properly defined, it is possible to observe and analyze results that are unknown during experiments (e.g., stress analysis in concrete).

•
The correct determination of parameters specifying bond behavior, such as stiffness and damaged initiation and evolution, is crucial for the simulation of concrete-bar interface performance. In the presented numerical analysis of the POT, the local bond stress-slip relationship contained in MC2010 [4] was used. The characteristic points were determined using representative curves obtained from experiments. This solution was chosen because the τ b (s) curves according to MC2010 [4] correctly reflected the trend of the experimental curves, while the characteristic points of the graph, which are mainly determined by the maximum bond stress, were characterized by significant discrepancies from the POT results, as shown by the parametric analysis performed. This conclusion indicates that the topic of research on improving models of the local bond stress-slip relationship is still an open issue, which will form the basis for further research by the authors of this paper.