Analytical Method and Analysis of Cold-Joint Interface

Abstract

The behaviour of the interface between two concrete layers, subjected to shear, is a complex process that is influenced by many different parameters. Knowledge of concrete interface performance is insufficient to this day. Most of the existing analytical methods are only suitable for determining the highest interface shear resistance and do not consider the interface behaviour at other stages. This article focuses on smooth concrete interfaces, which have their layers cast at different times (cold-joint interface). By analysing the results of different experimental push-off tests, presented in the literature, a novel analytical method was developed for the previously described concrete interface. Furthermore, numerical models of push-off tests were developed. A parametric numerical analysis was employed to determine the influence of various physical parameters that might affect interface shear behaviour. Most importantly, the results acquired using the proposed analytical method were compared with the findings of numerical analysis and experimental results acquired by other authors. The values of shear stress and slippage were found to be relatively close. Therefore, the proposed analytical method is suitable for the analysis of smooth concrete interfaces cast at different times.

1. Introduction

Layered reinforced concrete structures are widely used in modern construction. This might include a precast, hollow core slab with a layer of concrete topping [1], a semi-precast (filigree) slab which incorporates void-forming inserts [2], a precast concrete girder and cast-in-place concrete bridge deck [3] or simply two concrete layers that are cast at different times and many others. Layered concrete structures provide a great many advantages. During the installation of a filigree slab, the precast concrete layer acts as a residual formwork, which increases the construction rate [4]. Additionally, the usage of filigree slabs provides flexibility in the choice of structural systems. By reinforcing the cast-in-place layer, a balcony slab might be installed more conveniently and, by reinforcing the layer over the supports, a continuous slab can be constructed. A layered concrete construction can be chosen for repairing and strengthening existing concrete members [5], which is also a suitable option for a voided slab, which can reduce material quantities, reduce weight and provide openings for the installation of different building services [6].

The major characteristic which determines the advantages of layered concrete structures is the mutual performance of concrete layers. The behaviour and properties of a layered concrete member subjected to bending can be considered to be equivalent to the behaviour and properties of a monolithic concrete member, as long as the interface between the layers is completely stiff. However, in the case of a partially stiff layer interface, the advantage of mutual layer performance is lost and each layer behaves, performing individually. In this situation, the properties and behaviour of the individual layers must be considered [7,8]. Layered slab behaviour is shown in Figure 1.

In the case of insufficient interface stiffness and large shear stress, an interface might be damaged and layers might slip relative to each other [9,10,11,12,13,14]. The performance of a concrete interface which is crossed by connectors is a complex process that depends on many different parameters. The foundation for the study of shear interface behaviour was first established by Birkeland and Birkeland (1966) [15], which was named the “shear friction theory”. Nowadays, most design norms base their models on this theory to describe shear interface behaviour. As significant as it was, some assumptions made by the theory, such as the reinforcement being the main contributor to the total shear capacity of the interface, might not be true in all cases. Studies by other authors have found that the calculation methods presented in the design norms are not fully accurate and the different assumptions adopted by the calculations are not correct in all cases [16,17,18]. Liu et al. (2021) [19] found out that, in their case, concrete was the most important contributor to the ultimate interface shear capacity, whereas the influence of the connectors was less noticeable.

The goal of this research was to further extend the knowledge of shear interface behaviour and the influence of different contributors to interface shear resistance. A statistical analysis of the experimental shear tests performed by other authors [16,18,19,20,21,22,23,24,25,26,27,28] was carried out and curves of the shear stress–slippage relationship at different stages were proposed. This analytical model can be employed not only up to the ultimate interface shear capacity (which is the case for most calculation methods in the design norms), but also until the stage of severe layer slippage. This analytical method is mostly suitable for determining the behaviour of the interface between the layers of normal concrete with a smooth interface surface with and without transverse connectors, cast at different times (cold joint interface). The analytical method can also be employed in engineering practice. By using it, it can be determined whether the layers of the structural member are susceptible to separation. Calculations could show if the materials used (such as concrete or connectors steel) are of sufficient strength. Additionally, the analysis could indicate whether enough connectors are used for the layered structure to remain intact. Additionally, the ratio of shear stress and slippage provides a characteristic called the interface slip modulus, which is the main characteristic defining interface stiffness [29]. The shape of the curve itself is an indicator of whether the deterioration of the interface would be sudden and uncontrollable or slower and less dangerous.

Finally, a numerical analysis is performed to determine the influence of interface friction and interface connector diameter on interface behaviour and, most importantly, a validation of the analytical method is conducted using the results of the numerical analysis and experimental results.

2. Shear Interface Behaviour

Interface strength and stiffness depend on the following mechanisms: cohesive bonds, friction (due to an external compressive force or connectors subjected to tension) and connector dowel action [5,21,30]. Shear mechanisms are depicted in Figure 2. It is important to emphasize that the influence of each shear mechanism on shear transfer changes during different interface behaviour stages. For instance, when shear forces are mainly resisted by interface cohesive bonds, dowel action does not have a significant impact on interface behaviour, and on the contrary, when shear forces are resisted by dowel action, chemical bonds between the layers are lost and cohesive contribution disappears [16,21].

Researchers who have studied concrete interfaces subjected to shear describe the process of failure of interfaces crossed by connectors and the performance of individual shear mechanisms [5,16,21]. When the interface has not yet shown any signs of cracking or damage, cohesion is the main shear transfer mechanism. For the dependence of shear–layer displacement at this stage, rigid behaviour is usually observed. When cracks start to appear and when the displacement reaches a value that should not be ignored, shear force transfer is gradually taken over by other shear mechanisms. The second mechanism, interface friction, is caused by compressive forces that occur as a result of an external load perpendicular to the interface and the connectors crossing the interface. As cracks develop throughout the interface and concrete layers slip, connectors join the shear transfer and resist the slippage. This causes clamping (compression) in the interface. This phenomenon significantly increases shear stress at the interface in the case of considerable surface roughness (e.g., when concrete aggregates protrude through the surface plane). As an uneven concrete surface slips, cracks in the interface open in a direction perpendicular to the interface and connectors are subjected to even greater tension. In the literature, this phenomenon is called dilation [20]. As a result, even higher compression is achieved. The last shear mechanism is dowel action. During this mechanism, bars which cross the interface are subjected to shear and local bending [31]. Walraven and Reinhardt (1981) [32] stated that dowel action does not significantly influence the interface behaviour as long as the interface is not noticeably cracked. This confirms that, in the earlier stages of interface behaviour, shear force is mostly resisted by cohesive bonds. Observing the dependence of shear stress–layer displacement, it can be noticed that the influence of dowel action increases with increasing displacement of layers.

An Interface which is not crossed by any type of transverse bar behaves slightly differently to an interface with connectors. It can be assumed that an interface starts cracking and fails at the same time. This means that the interface behaves rigidly up to the highest shear-bearing capacity. When the interface fails, the only shear mechanism that governs its behaviour is the surface friction. At this stage, shear stress decreases to a value that is close to 0. This can be seen in the experimental studies of interfaces without connectors that are subjected to shear [22,23,24,25,26,27,28].

3. Analytical Method

In order to accurately consider the subtleties of concrete interfaces, which are subjected to shear, the authors of this study propose two curves: one for an interface that is crossed by connectors (Figure 3) and another for an interface without connectors (Figure 4). The curve of an interface with connectors consists of 4 interface behaviour stages and the curve of an interface without connectors consists of 3 stages. Each of these stages begins and ends with certain values of interface shear stress $\tau$ and slip $s$.

3.1. Curve of Concrete Interface with Connectors

• At stage 1, shear at the concrete interface is only transferred by cohesive bonds between concrete layers. The authors of experimental shear tests noted that shear stress ${\tau }_A$ (Equation (1)) is the point at which the first cracks develop at the interface:

  $${\tau }_A=0.5f_{ctm}$$

  (1)

  where $f_{ctm}$ is the tensile strength of concrete. The measurement of slip $s_A$ is a difficult task, as shear stress ${\tau }_A$ occurs at particularly low slippage. Model Code 2010 [5] suggests that this occurs at values lower than 0.05 mm. After analysing different experimental interface shear tests, it was determined that the value of $s_A$ should not be lower than 0.01 mm. Ultimately, the authors of this research propose that the value of $s_A$ chosen should be in the interval of 0.01 mm–0.05 mm (Equation (2)):

  $$s_A=a=0.01÷0.05 \mathrm{m}\mathrm{m}$$

  (2)
• At the start of stage 2, cohesive forces at the interface begin to decrease and the other shear transfer component, which consists of the connectors, joins the shear resistance. However, the analysis of the experimental results indicates that, at this stage, cohesion is still the main contributor to shear transfer. When this analytical model was built, it was assumed that, at the end of stage 2, the highest shear stress of ${\tau }_{max}$ (Equation (3)) was reached at the concrete interface. The analysis of experimental tests led to the conclusion that this value depended on coefficient $\kappa$ (Equation (4)), which is influenced by the ratio of $\frac{f_y·\rho }{f_c}$. $f_c$ is the compressive strength of the concrete, $f_y$ is the strength of shear connectors and $\rho$ is the ratio of connectors which cross the interface. When the ratio of $\frac{f_y·\rho }{f_c}$ is equal to an approximate value of 0.18, the highest shear stress ${\tau }_{max}$ is equal to the following shear stress point ${\tau }_C$. If the ratio is lower than 0.18, ${\tau }_{max}$ is higher than ${\tau }_C$. Additionally, the value of $\kappa$ must meet the condition of $\kappa \ge 1.$ The physical meaning of this ratio is that the higher the concrete strength, the more that ${\tau }_{max}$ is influenced by cohesion bonds. The lower the concrete strength, the more it is assumed that cohesion had already been lost and that ${\tau }_{max}$ had been reached only by the shear resistance of connectors. Shear stress ${\tau }_{max}$ can be determined by:

  $${\tau }_{MAX}=\kappa \left({\tau }_{fr}+0.5{\tau }_{dw}\right)$$

  (3)

  where ${\tau }_{fr}$ is the interface shear stress due to friction and ${\tau }_{dw}$ is the interface shear stress due to dowel action. Coefficient $\kappa$ determines the influence of cohesion for the highest interface shear capacity:

  $$\left\{\begin{array}{c}\kappa =0.35(\frac{f_y\rho }{f_c}{)}^{-0.62}\\ \kappa \ge 1.0\end{array}\right.$$

  (4)
  Shear slip $s_{max}$ (Equation (5)) is composed of two components. The first component defines the magnitude of the slip based on the value of concrete strength $f_c$. The greater the value, the lower the slippage. The second component defines the influence of connectors on the value of the slip. It was noticed that the lower the ratio of $\frac{f_y·\rho }{f_c}$, the stiffer the shear stress–slip behaviour and the smaller the slip $s_{max}$. On the contrary, the higher the ratio, the more ductile the behaviour and the larger the slip. Shear slip can be determined by:

  $$s_{MAX}=106{{·f}_c}^{-2.08}+18·{\left(\frac{f_y\rho }{f_c}\right)}^{2.08}$$

  (5)
• At stage 3, it is assumed that all cohesive bonds between the concrete layers are lost and that the friction and dowel action are the only shear mechanisms that transfer shear stress along the interface. This is indicated by the shear stress ${\tau }_C$ (Equation (6)). The equation for this stress was acquired from Model Code 2010 [5] and modified with reference to the experimental results to suit the behaviour of this type of interface. It is presumed that, at the start of the stage (${\tau }_C)$, friction is a predominant shear mechanism and dowel action is a secondary mechanism. This is indicated by coefficients of 1 and 0.5 before the friction and dowel action components, respectively. Shear stress ${\tau }_C$ can be determined by:

  $${\tau }_C={\tau }_{fr}+0.5{\tau }_{dw}$$

  (6)
  Friction stress is composed of three friction stress values that are multiplied by the friction coefficient $\mu$ (Equation (7)). ${\sigma }_n$ is the value of external compressive stress that is imposed perpendicular to the interface plane. $f_y\rho$ determines the friction stress value due to connector clamping force and $0.005f_c$ defines the minimal value of friction stress. This equation shows how much friction stress remains when there is no external compression (${\sigma }_n$) and no connectors crossing the interface. Friction stress ${\tau }_{fr}$ can be determined by:

  $${\tau }_{fr}=\mu \left({\sigma }_n+f_y\rho +0.005f_c\right)$$

  (7)
  The magnitude of dowel action stress depends on the strength of concrete and connector steel and the connector ratio (Equation (8)). The more the interface is crossed by connectors and the stronger the materials that are used, the more dowel action partakes in shear resistance. Dowel action stress ${\tau }_{dw}$ can be determined by:

  $${\tau }_{dw}=\rho \sqrt{f_c{f}_y}$$

  (8)
  Stage 3 ends with a slip $s_C$ (Equation (9)). Shear slip is composed of two components. Just like the previously defined shear slip ($s_{max}$), the value of $s_C$ decreases with increasing concrete strength $f_c$. The second component determines how much the slippage is suppressed by the presence of connectors. The larger the total connectors’ cross-sectional area $A_s$, the smaller the total value of $s_C$. Slip $s_C$ can be determined:

  $$s_C=2.3·f_c^{-0.28}-0.00003·{A_s}^{1.4}$$

  (9)
• At stage 4, concrete interface behaviour is governed by friction resistance and dowel action. At the end of stage (${\tau }_D$) (Equation (10)), dowel action becomes a predominant shear mechanism and friction becomes a secondary mechanism. This is indicated by the coefficients of 0.5 and 1 before the friction and dowel action components, respectively. Connectors yield later at this stage. Shear stress ${\tau }_D$ can be determined by:

  $${\tau }_D=0.5{\tau }_{fr}+{\tau }_{dw}$$

  (10)
  The final component of this shear stress–slippage relationship curve is the slip $s_D$ (Equation (11)). The value of this component was chosen to be 6 mm. The analysis of experimental results led to the conclusion that 6 mm was an optimal value to end the estimation of concrete interface behaviour. On average, the value of 6 mm was the point at which many experimental tests were stopped. Additionally, the authors of this paper decided that concrete interface subjected to slip larger than 6 mm could be considered severely broken and no further evaluation would be necessary. Slip $s_D$ can be taken as:

  $$s_D=6 \mathrm{m}\mathrm{m}$$

  (11)

3.2. Curve of Interface without Connectors

• During the analysis of the literature describing the experimental results of interfaces without connectors, it was noticed that, from the start of the shear loading to the maximum interface shear strength (${\tau }_{max}$), interface behaviour was more rigid than that for interfaces with shear connectors. The experimental curves displayed a linear stress–slip relationship from the start of the loading to the maximum shear stress. This means that point ${\tau }_A$ does not exist. It can be assumed that the interface starts cracking and fails at the same time, which is at point ${\tau }_{max}$. The elimination of the shear point ${\tau }_A$ results in stage 1 starting from the start of the loading and ending at point ${\tau }_{max}$ (Equation (12)). The calculation of shear stress ${\tau }_{max}$ is similar to that for concrete interfaces with connectors; however, coefficient $\theta$ is used instead of coefficient $\kappa$ (Equation (13)). The physical meaning of $\theta$ is that the larger the interface area $A_c$ and the higher the concrete’s strength, the larger the value of $\theta$. This value has to meet the condition of $\theta \ge 1$. The higher the value of $\theta$, the further ${\tau }_{max}$ is from the next shear stress point ${\tau }_C$. Shear stress ${\tau }_{max}$ can be determined by:

  $${\tau }_{MAX}=\theta ·{\tau }_{fr}$$

  (12)
  Coefficient $\theta$ can be determined by:

  $$\left\{\begin{array}{c}\theta =1.71f_c{A}_c+2.95\\ \theta \ge 1.0\end{array}\right.$$

  (13)
  The calculation of slip $s_{max}$ is quite similar to that for an interface with connectors; however, in order to determine the slip value for a concrete interface with no connectors, all equation components that describe connector properties, such as $f_y$ and $\rho$, have to be equal to 0 (Equation (14)). This leads to the equation having only one component, which describes how the slip is influenced by concrete’s compressive strength $f_c$. Slip $s_{max}$ is determined by:

  $$s_{MAX}=106{{·f}_c}^{-2.08}$$

  (14)
• Stage 2 of the behaviour of the interface without connectors ends with shear stress ${\tau }_C$. At this stage, interface behaviour is only governed by friction stress. That is why dowel action should be eliminated (Equation (15)). Shear stress ${\tau }_C$ is determined by:

  $${\tau }_C={\tau }_{fr}$$

  (15)
  All factors that cause friction stress related to connectors should be eliminated (Equation (16)). Friction stress ${\tau }_{fr}$ can be determined by:

  $${\tau }_{fr}=\mu \left({\sigma }_n+0.005f_c\right)$$

  (16)

  when calculating slip $s_C$, the connector area $A_s$ must be taken to be equal to 0. By doing so, the equation is only left with a part describing how the slip is influenced by concrete’s compressive strength $f_c$. Slip $s_C$ is determined by:

  $$s_C=2.3·f_c^{-0.28}$$

  (17)
• At stage 3, the interface is slipping until the final point ${\tau }_D$. Here, dowel action also has to be eliminated (Equation (18)). Shear stress ${\tau }_D$ is determined by:

  $${\tau }_D=0.5{\tau }_{fr}$$

  (18)

Interface slips to the final point $s_D$. Slip $s_D$ can be taken as:

$$s_D=6 \mathrm{m}\mathrm{m}$$

(19)

The authors of this paper also developed shear stress functions at different interface behaviour stages, with shear slip $s$ being the unknown variable. Functions for interface with connectors (Equations (20)–(23)) were constructed using interface shear stress and slippage equations. Shear stress functions of interface with connectors:

$${\tau }_I\left(s\right)=0.5\frac{f_{ctm}·s}{a}$$

(20)

$${\tau }_{II}\left(s\right)=\frac{(s-a)(\kappa ·\left({\tau }_{fr}+0.5{\tau }_{dw}\right)-0.5f_{ctm})}{106·{f_c}^{-2.08}+18·{\left(\frac{f_y\rho }{f_c}\right)}^{2.08}-a}+0.5f_{ctm}$$

(21)

$${\tau }_{III}\left(s\right)=\frac{(s-106·{f_c}^{-2.08}-18·{\left(\frac{f_y\rho }{f_c}\right)}^{2.08})\left({\tau }_{fr}+0.5{\tau }_{dw}\right)(1-\kappa )}{2.3·f_c^{-0.28}-0.0001·{A_s}^{1.4}-106·{f_c}^{-2.08}-18·{\left(\frac{f_y\rho }{f_c}\right)}^{2.08}}+\kappa ·\left({\tau }_{fr}+0.5{\tau }_{dw}\right)$$

(22)

$${\tau }_{IV}(s)=\frac{(s-2.3·f_c^{-0.28}+0.0001·{A_s}^{1.4})\left(0.5{\tau }_{dw}-0.5{\tau }_{fr}\right)}{6-2.3·f_c^{-0.28}+0.0001·{A_s}^{1.4}}+{\tau }_{fr}+0.5{\tau }_{dw}$$

(23)

The same was performed for the functions of interfaces without connectors (Equations (24)–(26)). The shear stress functions for this type of interface are as follows:

$${\tau }_I\left(s\right)=\frac{\theta {·\tau }_{fr}·s}{106{{·f}_c}^{-2.08}}$$

(24)

$${\tau }_{II}\left(s\right)=\frac{{\tau }_{fr}·(s-106·{f_c}^{-2.08})(1-\theta )}{2.3·f_c^{-0.28}-106·{f_c}^{-2.08}}+\theta ·{\tau }_{fr}$$

(25)

$${\tau }_{III}\left(s\right)={\tau }_{fr}-\frac{0.5{\tau }_{fr}·\left(s-2.3·f_c^{-0.28}\right)}{6-2.3·f_c^{-0.28}}$$

(26)

All of the stress values (shear stress and normal stress) in the model should be calculated in [mPa] and slippage in [mm]. Material strengths, such as $f_c$ and $f_y$, must also be provided in [mPa] values. The connector cross-sectional area in Equations (9), (22) and (23) is calculated in [mm²]. The area of the interface in Equation (13) is calculated in [m²].

The limitations of this proposed model may include the fact that this model was built for interfaces with connectors that are perpendicular to the interface plane. On many occasions, connectors might not be perpendicular to the interface plane, which is how filigree slabs are constructed. Additionally, the part of the model which evaluates the behaviour of the interface without connectors could be further updated, as experimental data for this type of interface were limited.

4. Validation of Analytical Method

Validation of the proposed analytical method was carried out by comparing the analytical calculations to experimental results and by performing a numerical simulation of the experimental test in the FEA software DIANA (Figure 5). The experimental results and data for the push-off tests were taken from the study of Fang et al. (2020) [33].

Three different experimental specimens were chosen as a reference: S-C65L30-6D8-Ba, S-C65L40-6D10-Ba and S-C65L50-6D12-Ba. The numerical model for S-C65L30-6D8-Ba was assigned the following physical parameters: for an older concrete layer, the elastic modulus was $E_c=38.57 \mathrm{G}\mathrm{P}\mathrm{a}$ and Poisson’s ratio $\nu =0.2$; for compressive concrete behaviour, a parabolic compression curve based on fracture energy was applied [34]. Concrete’s compressive strength was $f_c=65 \mathrm{M}\mathrm{P}\mathrm{a}$ and the fracture energy was $G_F=27.74 \mathrm{N}/\mathrm{m}\mathrm{m}$. The compressive fracture energy was calculated according to the study by Lourenço (1996) [35]. Concrete’s tension is described by an exponential curve that depends on the concrete’s tensile fracture energy [36]. The tensile strength was equal to $f_t=4.272 \mathrm{M}\mathrm{P}\mathrm{a}$; the fracture energy was determined according to Model Code 1990 [37] and was equal to $G_F=0.111 \mathrm{N}/\mathrm{m}\mathrm{m}$. The properties for a younger concrete layer were as follows: elastic modulus $E_c=30.59 GPa$; Poisson’s ratio $\nu =0.2$; compressive strength $f_c=30 \mathrm{M}\mathrm{P}\mathrm{a}$; compressive fracture energy $G_F=24.66 \mathrm{N}/\mathrm{m}\mathrm{m}$; tensile strength $f_t=2.355 \mathrm{M}\mathrm{P}\mathrm{a}$; tensile fracture energy $G_F=0.065 \mathrm{N}/\mathrm{m}\mathrm{m}$. The connector yield stress was $f_y=285.13 \mathrm{M}\mathrm{P}\mathrm{a}$, ultimate tensile strength $f_u=429.41 \mathrm{M}\mathrm{P}\mathrm{a}$ and the connector’s elastic modulus $E_s=197 \mathrm{G}\mathrm{P}\mathrm{a}$. The Von Mises plasticity model was assigned for the connectors. The number of shear connector bars crossing the interface was 6 bars, diameter of connector bars $\varnothing =8 \mathrm{m}\mathrm{m}$ and concrete interface area $A_c=0.0375 {\mathrm{m}}^2$. The 3D model was meshed into 15 mm quadrilateral finite elements. The interface between the layers was modelled by applying specific parameters. It was assumed that the interface properties were mainly influenced by the properties of the weaker concrete layer because that is where the first cracks were most likely to appear. Therefore, interface characteristics were calculated in reference to a weaker concrete layer. The interface’s linear parameters were a normal stiffness modulus, which was equal to concrete elastic modulus $E_c=30.59 \mathrm{G}\mathrm{P}\mathrm{a}$, and shear stiffness modulus, which was equal to $K_s=0.4·E_c=12.24 \mathrm{G}\mathrm{P}\mathrm{a}$. Interface plasticity is governed by the Coulomb friction model [38] (Figure 6).

The friction model requires the values of concrete cohesion, friction angle and dilation angle. Cohesion stress was determined according to Eurocode 2 [39]. For a smooth-surface interface, the suggested cohesion coefficient was $c=0.35$. The cohesion stress was equal to $C=c·f_t=0.82 \mathrm{M}\mathrm{P}\mathrm{a}$. The interface friction angle was taken as $\phi =$ 30°, which is a recommended value for a smooth interface [40]. The dilation angle, which determines the width of an interface crack in the perpendicular direction, was assumed to be equal to $\psi =0$. By choosing this value, it was considered that the surface was smooth enough that layers slipped without significant displacement in the perpendicular direction. Additionally, the interface model was supplemented by a curve, which, in the model, was called “cohesion hardening”. This indicates the magnitude of the concrete layer bonding stress. However, in this case, the first stress value represented a value of cohesion stress, which was already presented, and the second value specified a value of the concrete layer clamping stress, which was caused by tensioned connectors (after the loss of a cohesive bond) (Figure 7). The relative displacement values were chosen to be small enough to be automatically adjusted while performing a non-linear analysis and, at the same time, realistic. Clamping stress was determined according to Eurocode 2 [39]. All the physical parameters for three numerical models are presented in

Table 1. Concrete characteristics for numerical models.

Properties	S-C65L30-6D8-Ba		S-C65L40-6D10-Ba		S-C65L50-6D12-Ba
	Older Layer	Newer Layer	Older Layer	Newer Layer	Older Layer	Newer Layer
Elastic modulus	38.57 GPa	30.59 GPa	38.57 GPa	33.35 GPa	38.57 GPa	35.65 GPa
Compressive strength	65 MPa	30 MPa	65 MPa	40 MPa	65 MPa	50 MPa
Compressive fracture energy	27.74 N/mm	24.66 N/mm	27.74 N/mm	26.44 N/mm	27.74 N/mm	27.50 N/mm
Tensile strength	4.27 MPa	2.36 MPa	4.27 MPa	3.02 MPa	4.27 MPa	3.63 MPa
Tensile fracture energy	0.111 N/mm	0.065 N/mm	0.111 N/mm	0.079 N/mm	0.111 N/mm	0.093 N/mm

and

Table 2. Connectors and interface characteristics for numerical models.

Properties	S-C65L30-6D8-Ba	S-C65L40-6D10-Ba	S-C65L50-6D12-Ba
Connectors’ cross-sectional diameter	8 mm	10 mm	12 mm
Number of connectors	6	6	6
Connectors’ steel elastic modulus	197 GPa	203 GPa	205 GPa
Connectors’ steel yield stress	285.13 MPa	411.88 MPa	418.98 MPa
Connectors’ steel ultimate strength	429.41 MPa	559.07 MPa	535.87 MPa
Interface normal stiffness modulus	30.59 GPa	33.35 GPa	35.65 GPa
Interface shear stiffness modulus	12.24 GPa	13.34 GPa	14.26 GPa
Interface cohesion	0.82 MPa	1.06 MPa	1.27 MPa
Interface friction angle	30°	30°	30°
Interface clamping stress	1.38 MPa	3.10 MPa	4.55 MPa

.

Shear stress–relative slip curves were acquired by numerical analysis and compared with experimental and analytical curves that were calculated using the same physical parameters as those in the numerical model. Three figures (Figure 8, Figure 9 and Figure 10) are presented which display a shear stress–slip relationship. Each figure consists of a curve acquired by numerical analysis, an experimental curve and a curve calculated using the proposed analytical method.

The interface of an experimental specimen (Figure 8) behaved plastically up to the highest shear resistance. This implies that the highest shear resistance was reached with the help of interface friction and connectors, as the interface began to crack early. The experimental analysis curve reached a maximum shear stress value of $\tau =3.55 \mathrm{M}\mathrm{P}\mathrm{a}$ at a shear slip of $s=0.59 \mathrm{m}\mathrm{m}$. Then, the shear stress slightly decreased and the push-off test was stopped at a shear stress value of $\tau =3.46 \mathrm{M}\mathrm{P}\mathrm{a}$, which corresponded to a slip of $s=4.3 \mathrm{m}\mathrm{m}$. The analytical curve displays a different behaviour. However, the highest shear stress values were similar. The highest shear stress for the analytical curve was $\tau =3.50 \mathrm{M}\mathrm{P}\mathrm{a}$, which was 1.4% lower than that of the experimental curve. In addition, the corresponding slip was determined to be $s=0.29 \mathrm{M}\mathrm{P}\mathrm{a}$, which was lower than the experimental slip by 103%. After the highest interface shear capacity, the analytical curve displayed a decrease in shear stress to $\tau =2.62 \mathrm{M}\mathrm{P}\mathrm{a}$, which meant that cohesive stress was lost and interface shear resistance consisted only of friction and the connectors’ resistance. From this point, shear stress gradually decreased. By comparing the experimental and analytical curves, their shear stress and slippage values, it can be stated that the proposed analytical method was able to predict a similar shear interface behaviour, especially for the highest interface shear capacity. However, the values at the stage of friction and connectors’ action should be much closer to the experimental values. By analysing this case and by taking into account the results presented in Figure 9 and Figure 10, it can be stated that the proposed analytical model should be improved to predict the cases more accurately when weaker concrete and smaller interface connector ratios are used. The numerical analysis curve was similar to the experimental curve by its shape. Nonetheless, the determined shear stress values were smaller. The loss of cohesion was visible at a shear stress of $\tau =1.63 \mathrm{M}\mathrm{P}\mathrm{a}$ and slip of $s=0.08 \mathrm{m}\mathrm{m}$. After this point, it was assumed that concrete-to-concrete (cohesion) bonding should be neglected and that the shear resistance of the interface was mainly achieved by the interface’s friction and connectors’ resistance. The numerical analysis curve reached the highest shear value of $\tau =2.28 \mathrm{M}\mathrm{P}\mathrm{a}$, which was 56% lower than the experimental value. This shear stress was determined at a slippage of $s=0.37 \mathrm{m}\mathrm{m}$, which was 59% lower than the experimental slip.

In the case of specimen S-C65L40-6D10-Ba (Figure 9), the experimental curve displayed a very similar shape and values to the analytical one. From the start of the push-off test to the highest interface shear capacity, the experimental data indicated a plastic interface behaviour. The highest shear stress was $\tau =5.53 \mathrm{M}\mathrm{P}\mathrm{a}$, which was reached at a slip of $s=0.58 \mathrm{m}\mathrm{m}$. The corresponding analytical values were $\tau =5.43 \mathrm{M}\mathrm{P}\mathrm{a}$ and $s=0.53 \mathrm{m}\mathrm{m}$. The differences between the experimental and analytically determined values were 2% for shear stress and 9% for shear slip. It can be considered that the values were significantly similar, bearing in mind how unpredictable shear interface behaviour can be. From the highest shear point, both curves displayed a decrease in shear stress. The experimental stress stopped when slip $s=4 \mathrm{m}\mathrm{m}$ was reached. At this point, the experimentally determined shear stress value was $\tau =4.15 \mathrm{M}\mathrm{P}\mathrm{a}$. At slip $s=4 \mathrm{m}\mathrm{m}$, the analytical curve displayed a shear stress value of approximately $\tau =4.5 \mathrm{M}\mathrm{P}\mathrm{a}$. The difference between the two shear stresses was 8%. As explained earlier, the analytical model seemed to depict a better correspondence when a higher interface concrete strength and connector ratio were used. The numerical curve portrays a similar behaviour from the start of the loading to the point of complete cohesion loss to the specimen in Figure 8. This time, the loss of cohesion was visible at a shear stress of $\tau =2.21 \mathrm{M}\mathrm{P}\mathrm{a}$ and slip of $s=0.12 \mathrm{m}\mathrm{m}$. After the previous point, the shear stress reached a value of $\tau =4.36 \mathrm{M}\mathrm{P}\mathrm{a}$, which was achieved at slip $s=0.49 \mathrm{m}\mathrm{m}$. The value of shear stress was lower than the experimental value by 27%. At the same time, the disparity of slip values was lower, with a difference of 18%. In the case of the numerical analysis of S-C65L40-6D10-Ba, this shear stress point was not the highest shear stress that was acquired at the interface, because, after this point, the shear stress–slip curve decreased to a value of $\tau =4.05 \mathrm{M}\mathrm{P}\mathrm{a}$ and then increased to a shear stress pf $\tau =4.58 \mathrm{M}\mathrm{P}\mathrm{a}$. The increase in shear stress is the result of the larger involvement of interface connectors in interface shear transfer.

The diagrams for specimen S-C65L50-6D12-Ba (Figure 10) showed a considerable similarity between the analytical and experimental curves. This time, a more rigid experimental interface behaviour from the start of the loading to the highest shear resistance could be noticed. The reason for this might be the higher concrete strength used. The maximum shear stress for the experimental specimen was $\tau =7.14 \mathrm{M}\mathrm{P}\mathrm{a}$ and $\tau =7.45 \mathrm{M}\mathrm{P}\mathrm{a}$ for the analytical curve. The difference between these values was 4%. The slippage values at this point were $s=0.85 \mathrm{m}\mathrm{m}$ for the experimental curve and $s=0.62 \mathrm{m}\mathrm{m}$ for the analytical curve. The difference between these values was 37%. After the highest interface shear resistance, the stress values began to decrease. The experimental test was ended at slip $s=4.18 \mathrm{m}\mathrm{m}$. The corresponding shear stress value was $\tau =5.1 \mathrm{M}\mathrm{P}\mathrm{a}$. At this point, the shear stress for the analytical curve appeared to be identical. After analysing the figures (Figure 8, Figure 9 and Figure 10) in which experimental and analytical interface shear stress–slippage relationships are compared, it can be concluded that the proposed analytical method is suitable for predicting the behaviour of an interface subjected to shear.

By analysing the numerical analysis curve, it can be noticed that the shape of the curve was similar to a curve in Figure 9; however, the values of interface shear stress were naturally higher. Just like in the curve in Figure 9, the highest shear stress value was not reached when the interface cracked. After the formation of a crack, a point is reached at which the shear stress value of $\tau =5.84 \mathrm{M}\mathrm{P}\mathrm{a}$ was acquired, which was 22% lower than the experimental value. The slip corresponding to the stress was $s=0.59 \mathrm{m}\mathrm{m}$, which was 44% lower than the experimentally determined value. After the point of $\tau =5.84 \mathrm{M}\mathrm{P}\mathrm{a}$, the shear stress slightly decreased and started to increase until the end of the test due to the connectors becoming tense. After three comparisons (Figure 8, Figure 9 and Figure 10) between the experimental values and the values determined by numerical analysis, it can be stated that the finite element analysis is an appropriate tool for the analysis of shear interface behaviour. However, the performance of an interface (from the start of the loading to the point of interface connectors failure) is an intricate problem and the models which are used today should be improved to predict the interface behaviour more accurately.

5. Numerical Parametric Interface Analysis

A parametric analysis was performed by using the numerical model of specimen S-C65L40-6D10-Ba. The purpose of the parametric analysis was to find out how the interface friction angle and connector bar diameter influenced the interface behaviour.

In Figure 11, four stress–slip curves are displayed.

Table 3. Shear stress and slip values of different friction angle interfaces.

	φ = 10°		φ = 20°		φ = 30°		φ = 40°
Point	τ, MPa	s, mm	τ, MPa	s, mm	τ, MPa	s, mm	τ, MPa	s, mm
A	2.39	0.18	3.15	0.36	4.36	0.49	5.39	0.59
B	1.41	6.00	2.58	6.00	4.59	6.00	6.25	1.30

shows the values of shear stress and slippage. Each curve was acquired with a different value for the interface friction angle, which is used in the Coulomb friction model [36]. As seen in the legend of the figure, each friction angle value corresponds to a certain friction coefficient value, as the friction coefficient is a tangential value of the friction angle. By changing the value of the friction angle $\phi$, the value of interface clamping stress also changes, which was considered when the analysis was performed. Four different values of friction were used φ = 10°, 20°, 30° and 40°. Different values of friction resulted in different shear stresses at point A. Comparing the results with the lowest shear stress of φ = 10°, it appears that, when the friction was φ = 20°, the shear stress value increased by 31.8%; when φ = 30°, it increased by 82.4%; when φ = 40°, it increased by 125.5%. Comparing the slip values, it can be also seen that the increase in friction caused a larger slip at this point. When the friction was φ = 10° the lowest value was determined; when the friction was φ = 20°, the value increased by 100.0%; when φ = 30°, the value increased by 172.2%; when φ = 40°, the value increased by 227.8%. From this point, with the different values of friction angle, different interface behaviours were determined. The shear stress value for φ = 10° kept decreasing until the end of the test, which was at point B. The shear stress value was 69.5% lower than the value at point A. When the friction was φ = 20°, the shear stress value decreased and at the end of the test, it reached a value that was 22.1% lower than the value at point A. When the friction was φ = 30°, the shear stress increased by 5.3%. When the friction was φ = 40°, the determined increase in shear stress was 16.0%. The difference here was that, at the end of the test, the interface failed. By increasing the friction angle, the interface became stiffer. It started to behave like a monolithic joint. Interface failure became more sudden and occurred at a relatively lower layer slippage. Comparing the shear stress values at the end of test (B) with the lowest value of φ = 10°, the shear stress value when the friction was φ = 20° increased by 83.0%; when φ = 30° the shear stress increased by 225.5%; when φ = 40°, the shear stress value increased by 343.3%. After completing the parametric analysis with friction being a variable, a few patterns became noticeable. By increasing the interface friction value, higher interface shear resistance was achieved. Additionally, with a larger value of interface friction, a larger layer slip was needed for the interface to fully crack. Lastly, with the increase in friction, a higher shear stress was achieved in the further stages of interface performance, after the interface cracked and cohesive bond was lost.

In Figure 12, the variation in the interface connector diameter is presented. Four curves display the shear stress–layer slippage relationships based on different interface connector diameters. By changing the value of the bar diameter, the value of interface clamping stress also changed, which was considered when the analysis was performed.

The same kind of comparison, which was used for Figure 11, was performed for Figure 12.

Table 4. Shear stress and slip values of interfaces with different connector diameters.

	$\mathit{\varnothing } = 6 \mathbf{mm}$		$\mathit{\varnothing } = 8 \mathbf{mm}$		$\mathit{\varnothing } = 10 \mathbf{mm}$		$\mathit{\varnothing } = 12 \mathbf{mm}$
Point	τ, MPa	s, mm	τ, MPa	s, mm	τ, MPa	s, mm	τ, MPa	s, mm
A	2.27	0.26	3.23	0.39	4.36	0.49	5.57	0.61
B	1.92	6.00	3.17	6.00	4.58	6.00	6.07	1.82

shows the values of shear stress and slippage. At point A, curve $\mathrm{\varnothing }=8 \mathrm{m}\mathrm{m}$ reached a shear stress value which was 42.3% higher than that for $\mathrm{\varnothing }=6 \mathrm{m}\mathrm{m}$. For $\mathrm{\varnothing }=10 \mathrm{m}\mathrm{m}$, this value was 92.1% higher and for $\mathrm{\varnothing }=12 \mathrm{m}\mathrm{m}$, it was 145.4% higher than the lowest value. The slip values were 50.0%, 88.5% and 134.6% larger than the lowest value for $\mathrm{\varnothing }=6 \mathrm{m}\mathrm{m}$, respectively. At the end of both tests (point B), the shear stress values for $\mathrm{\varnothing }=6 \mathrm{m}\mathrm{m}$ and $\mathrm{\varnothing }=8 \mathrm{m}\mathrm{m}$ were, respectively, 18.2% and 1.9% lower than the values at point A. For $\mathrm{\varnothing }=10 \mathrm{m}\mathrm{m}$ and $\mathrm{\varnothing }=12 \mathrm{m}\mathrm{m}$, they were, respectively, 5.0% and 9.0% larger than the values at the previous point. Comparing the shear stress values at the end of the test with the lowest value of $\mathrm{\varnothing }=6 \mathrm{m}\mathrm{m}$, the values were 65.1%, 138.5% and 216.1% higher. Comparing the behaviours of the interfaces, the curve for $\mathrm{\varnothing }=12 \mathrm{m}\mathrm{m}$ is prominent. At the end of the test, the interface failed. This occurred when the interface’s concrete was crushed by a large connector diameter. A parametric analysis with a variation in the interface connector diameter led to some conclusions. First, by increasing the interface connector diameter, a higher interface shear resistance was achieved. Additionally, with a larger connector diameter, a larger layer slip was needed for the interface to fully crack. Lastly, with an increase in connector diameter, a higher shear stress was achieved in the further stages of interface performance, after the interface cracked and the cohesive bond was lost.

6. Conclusions

• This article presents a novel analytical model for the calculation of layered concrete interface behaviour. Two shear stress–shear slippage relationship curves (for interfaces with connectors and without connectors) were developed. The proposed analytical model could be employed up to the point when the layers severely slipped. Cold-joint interface behaviour analysis was performed by predicting the shear stress and slippage values using the proposed analytical method, by building a numerical model and employing finite element analysis. It appears that the analytical values were similar to the experimental values, especially at the point of highest interface shear stress. The numerical analysis shear stress results were lower than the experimental results up to the point of highest experimental shear stress and then, in some cases, surpassed the experimental results at the stage of friction and connector action.
• The comparison of the experimental results, numerical analysis results and the results acquired by the analytical method led to the conclusion that the proposed analytical method is suitable for the analysis of a concrete interface with layers cast at different times.
• The numerical parametric analysis led to the conclusion that, by increasing the interface friction magnitude, a higher interface shear resistance could be achieved. The interface behaved more rigidly in the earlier behaviour stages. Gradually, at a certain friction angle (coefficient), the cold-joint interface started to behave like a monolithic interface.
• The parametric analysis of the influence of connector diameter led to the conclusion that a higher interface shear capacity can be obtained by choosing a larger connector diameter. As with higher friction, an interface with a larger connector bar starts to behave more rigidly. However, by increasing the diameter of the bar, the interface might fail earlier due to the crushing of the interface’s concrete.