1. Introduction
Over the past three decades, the use of carbon fiber reinforced polymer (CFRP) composite materials in fabricating cables as a substitute for steel cables has emerged as a primary research focus in the field of high-performance cable structures [
1]. This design concept stems from the outstanding properties of CFRP itself. Compared to steel, CFRP shows higher strength and lighter weight [
2], and its excellent corrosion and fatigue resistance contribute fundamentally to extending the service life of the cables [
3,
4]. Theoretically, replacing steel with CFRP in cable fabrication addresses significant issues associated with traditional steel cables and significantly advances the development of cable structures. Currently, CFRP tendons and plates represent the primary structural forms used in the production of cables. When the strength is equivalent, CFRP plates with a rectangular cross-section exhibit a larger anchorage area and better bending performance [
5]. Therefore, CFRP plates show enormous potential for development in cable structures. A literature review indicates that cable roofs, due to their orthogonal loading characteristics, are considered ideal structural applications for CFRP plate cables [
1,
6]. However, this novel CFRP cable roof structure also poses a series of challenges for the anchorage of CFRP plates.
The influence of the new CFRP cable roof structure system on anchor design manifests mainly in two aspects. Firstly, consideration needs to be given to the dimensions of the anchors. Anchors connect the CFRP cables to the supporting framework or other components [
7], and they are tailored to suit the structural characteristics of CFRP. For CFRP plates, prestressed structures usually employ both adhesive and mechanical anchors [
8]. In cable roof structures, bulky adhesive anchors disrupt the spatial arrangement of CFRP cables and add to the burden of connections [
9], conflicting with lightweight design principles.
Additionally, cable roof structures frequently utilize modular design and prefabricated components, expediting construction completion. Conversely, the maintenance period for adhesive anchors is typically longer, thus not conducive to reducing the construction period. Therefore, mechanical anchors, such as the large-angle wedge anchors proposed by Han et al. [
10] and designed for beam string structures, are more suitable for CFRP cable roof structures due to their small size, lightweight, and easy installation [
1]. These anchors are used to anchor transversely enhanced CFRP tendons, offering convenience in disassembly and reusability. Mohee FM et al. [
11] have set clear design criteria for the dimensions and quality of the anchors in designing a small-sized mechanical anchor for anchoring a single layer of 50 mm × 1.2 mm CFRP plate in prestressed structures. Specifically, the anchor’s weight should be less than 7 kg, and the anchorage length should be less than 110 mm for ease of application. Secondly, the quantity of anchors needs to be taken into account. The complex spatial arrangement of cable members in cable roof structures necessitates a significant number of anchors to connect them to the supporting framework effectively. Liu et al. [
12] suggested using a more flexible CFRP continuous strip as the prestressing system on the roof. Wrapping CFRP continuous strips at the nodes minimizes the need for anchors, placing them only at both ends. However, this approach may lead to structural instability. Wang et al. [
9] proposed a novel integrated barrel-wedge anchorage system (NISWAS) and validated its design concept through finite element analysis (FEA). The system comprised a steel barrel, a multi-channel integrated wedge, and six CFRP tendons with a diameter of 5 mm. Compared to a traditional wedge anchorage system, NISWAS achieves the anchoring of multiple CFRP tendons, significantly improving anchoring efficiency while reducing the volume and self-weight of the anchoring end. This provides a valuable reference for the anchoring of CFRP plates. Therefore, it is necessary to design small-sized mechanical anchors that can simultaneously anchor multiple layers of CFRP plates.
The anchoring principle of mechanical anchors relies on the frictional force generated between the inner surface of the anchor and the interface of the CFRP plate to balance the tension force. Therefore, pressure must be applied perpendicularly to the surface of the CFRP plate [
13]. Based on the method of pressure generation, mechanical anchors can be subdivided into bolt-clamping and wedge anchoring, each with unique characteristics and application scenarios. Bolt clamping anchoring features precise pressure control and ease of disassembly, making it suitable for the modular and cyclical construction of cable-stayed roof structures. However, the adverse effects of bolt relaxation on the long-term service performance of anchors must be considered. This may lead to pressure loss on the CFRP plate, making bolt clamping anchoring unable to provide sufficient load-bearing capacity. In contrast, with their special design, wedge anchors exert stronger extrusion pressure during anchoring through the follow-up of the wedge, forming a self-anchor. This characteristic makes wedge anchors excel in long-term fixation scenarios, and higher structural stability requirements, but traditional wedge anchors are unable to anchor multiple-layer CFRP plates. Therefore, considering the impact of bolt relaxation [
14], the type of small mechanical anchor studied in this paper is a wedge anchor.
Currently, almost all literature on wedge anchors is focused on anchoring single-layer CFRP plates. Anchoring multiple-layer CFRP plates poses challenges without increasing the dimensions of the anchor. The fundamental issue lies in the relatively singular frictional force transmission path within the anchor. Taking traditional wedge anchors as an example, the tension force in the cable is balanced by the friction force generated between the wedge and the CFRP plate. This friction force is ultimately transmitted to the barrel through the interface (CFRP plate-wedge-barrel) and balanced by the restraining force acting on the barrel, as illustrated in
Figure 1 [
15]. Given the structural form of the CFRP plate, the positive pressure generating the friction force primarily acts on the two wide faces of the CFRP plate. Without increasing the anchoring size, the pressure required for anchoring even the double-layer CFRP plates exceeds the allowable compressive stress range of the CFRP plate. Another potential issue is the uneven distribution of compressive stress on the CFRP plate, which typically increases gradually from the free end to the loading end of the anchor, reaching its maximum at the tip of the wedge [
16]. This stress distribution results in the loading end region becoming the primary contribution area to the anchoring capacity. Consequently, the CFRP plate section at the loading end of the anchor also becomes the most susceptible to damage, bearing the maximum compressive stress. This compressive stress concentration may lead to premature failure of the CFRP plate before reaching its ultimate load-bearing capacity. Anchoring a multiple-layer CFRP plate is likely to further amplify this stress concentration phenomenon since the pressure requirements for anchoring multiple layers are much higher than those for anchoring a single layer, and the distance the wedge needs to advance will be greater. Therefore, reducing the pressure requirements for anchoring multiple-layer CFRP plates is crucial. This necessitates the release of a portion of the friction force used to balance external loads onto the barrel through new force transmission paths in advance.
Furthermore, the contribution of relatively low compressive stress in the free-end region to anchoring capacity is minimal [
17]. To overcome this issue, Mohee FM et al. [
11] proposed a circular edge wedge design concept, which employed a segment of a circular arc with a radius of 3000 mm as the longitudinal profile for the inner surface of the barrel and the outer surface of the wedge. This design ensures a relatively uniform distribution of compressive stress on the CFRP plate, resulting in satisfactory outcomes. Portnov et al. [
18] designed special profiled clamping plates with varying curvature surfaces to achieve a smooth transition of shear stress from the loading end to the free end. It is evident that stress concentration within the anchor can be alleviated by designing the contact surface profiles of the wedge, clip, and CFRP plate [
1,
13]. However, the effects of these designs need to be evaluated through theoretical analysis, as there is currently a lack of effective means to directly measure the internal pressure distribution of the anchor. Overall, these two main issues mentioned above make it difficult for traditional wedge anchors to effectively anchor multi-layer CFRP plates.
To address the current limitation of wedge anchors, which can only anchor single-layer CFRP plate cable, this study aims to achieve efficient anchoring of double-layer CFRP plate cable by proposing an innovative wedge anchor. This paper elaborates on the design concept and methodology of the “secondary force transmission path,” which effectively releases frictional force onto the barrel in advance, thus reducing the pressure requirements for anchoring double-layer CFRP plates. Additionally, this paper presents a theoretical analysis model for calculating the internal contact pressure distribution of the anchor. This model is utilized to predict the influence of parameter variations on the compressive stress of the CFRP plate. Through FEA, the internal stress distribution of the anchor is thoroughly investigated, and the results are compared with the calculations from the theoretical analysis model to validate its effectiveness. The research findings demonstrate that under a 100 mm anchorage length, the new anchor design efficiently anchors double-layer CFRP plate cables with cross-sectional dimensions of 50 mm 2 mm.
2. New Anchor Design Concept
The core idea of the new anchor lies in establishing a novel frictional force transmission path to reduce the pressure requirements for anchoring double-layer CFRP plates. Additionally, it ensures a reasonable distribution of pressure on the CFRP plate to avoid stress concentration at the loading end of the anchor. Therefore, two key design concepts for the new anchor are proposed, and their design schemes are illustrated in
Figure 2. The first concept pertains to the design of the frictional force transmission pathway. To transmit the frictional force between the double-layer CFRP plates to the barrel through alternative paths, clips need to be introduced between the CFRP plates, connecting them to the barrel, thus forming a secondary force transmission path. For this purpose, a specially shaped clip comprising a flat portion and a wedge portion was designed. It can be visualized as if a rectangular metal clip is cut from both ends along its length at a specific distance. The flat portion of the clip clamps the CFRP plate, while the wedge portion connects to the barrel. The connection between the clip and the anchor is established via a wedge-shaped channel inside the barrel, positioned on the side of the barrel’s inner wall, not under pressure. This channel runs from the loading end to the free end of the barrel, matching the angle of the wedge portion of the clip for proper positioning. After assembling the CFRP plate, clip, and barrel, a secondary frictional force transmission path is established at the interface of “CFRP plate-clip-barrel”.
The next aspect involves designing the surfaces of the barrel’s inner surface and the wedge’s outer surface. In the new design, the combined thickness of the assembled wedge, clips, and CFRP plate must be greater than the thickness of the barrel opening. This difference in dimensions is known as the interference distance [
11]. When the assembly is inserted into the barrel opening, the barrel compresses the outer surface of the wedge, creating pressure due to the interference distance. This pressure is then transmitted through the interface and ultimately distributed onto the CFRP plate. Thus, for a reasonable distribution of pressure on the CFRP plate within the anchorage length, the longitudinal profiles of the wedge and barrel were designed using a quadratic function to describe their surfaces. This function is continuous, which helps to avoid stress discontinuities inside the anchor [
19]. Additionally, to enhance the pressure in the free-end region and its contribution to anchoring, the slope and curvature of this function gradually increase from the loading end to the free end of the anchor. Lastly, at the exit position of the anchor loading end, it ensures that the first derivative of the function is equal to zero, guaranteeing that the pressure reaches its minimum at the loading end, thereby avoiding stress concentration.
Combining the two aforementioned design concepts, a variable curvature wedge anchor (VCWA) is proposed for anchoring double-layer CFRP plates (
Figure 2). The VCWA primarily consists of four components: a hollow circular-section barrel with multiple channels, two variable curvature wedges, three irregular-shaped clips, and double-layer CFRP plates.
Research by Mohee et al. [
11] indicated that wedge anchors with an anchorage length of around 100 mm could effectively anchor CFRP plates. According to the design concept of the new anchor, with the optimization of internal stress in the anchor and minimization of anchorage length as objectives, the decisive factors mainly include the magnitude of the compressive stress acting on the surface of the CFRP plate, the friction coefficient between the metal clip and the surface of the CFRP plate, and the interlaminar shear strength of the CFRP plate. Based on previous research conclusions, the in-plane interlaminar shear strength of CFRP plates can reach 60 MPa [
20]. To ensure that interlaminar shear failure of the CFRP plate does not occur due to friction in the anchoring zone, the average frictional stress is conservatively limited to no more than 30 MPa as a design objective. Additionally, previous studies have shown that the friction coefficient between steel plates treated with appropriate surface sandblasting and CFRP plates subjected to a roughening treatment can reach around 0.3 [
21]. Therefore, according to the principle of friction, the corresponding compressive stress requirement is 100 MPa. Relative to the transverse compressive strength of the CFRP plate, there is still a considerable margin [
22]. Based on the fundamental anchoring principles mentioned above, for a CFRP plate with a cross-section of 50 mm
2 mm and a strength of 2800 MPa, the minimum anchorage length can be less than 100 mm.
Therefore, aiming for a 100 mm anchorage length as our target and satisfying all the design principles mentioned above, a function is prioritized to describe the surface shape as follows:
where
x represents the anchorage length, with values ranging from 0 to 100 mm, and the parameter
c controls the thickness of the wedge. To confirm the performance and feasibility of VCWA, the following two sections will conduct theoretical analysis and finite element analysis on VCWA. These two analytical methods will help verify whether the design of VCWA meets expectations.
3. Theoretical Analysis of VCWA
Based on the principle of friction, wedge anchoring systems typically require applying significant compressive stress to the CFRP plate before tensioning to ensure sufficient frictional force to balance the tension. This process is generally achieved by setting a certain presetting distance for the wedge. The key point in anchoring double-layer CFRP plates lies in whether the presetting process can provide sufficient compressive stress while avoiding damaging the CFRP plate. Therefore, the theoretical analysis model in this paper focuses on the magnitude and distribution of contact pressure between the wedge, clip, and CFRP plate [
11].
In this section, the presetting process of the wedge is considered as being acted upon by a rigid flat punch pressing against a plane, as shown in
Figure 3. According to previous by Johnson et al. [
23], the punch features a flat bottom with a width of 2
a and sharp square corners, and it is restricted to non-tilted pressing, ensuring that the deformed contact surface remains parallel to the undeformed solid surface. While real wedges cannot be entirely rigid, the compression deformation of the wedge by the barrel serves as the source of anchoring force. Thus, to align the anchor analysis with this theoretical model, several assumptions are proposed [
17,
24]:
- (1)
The barrel is considered a rigid body;
- (2)
Assuming that the deformation of the contact surface between the barrel and the wedge is much smaller than that of the wedge pressing into the clips and CFRP plate assembly;
- (3)
Spatial issues are simplified to approximate the plane strain problems for analysis;
- (4)
The assembly of clips and CFRP plate is regarded as an elastic half-space bounded by a flat surface;
- (5)
Averse frictional forces caused by pressing are neglected.
Now, the assembly consisting of stacked clips and CFRP plates is considered an elastic half-space, and the wedge pressing into this assembly is likened to the action of a rigid flat punch. As shown in
Figure 3, the contact interface is the
x-
y plane, with the
z-axis pointing towards the interior of the elastic half-space.
P represents the pressure exerted by the barrel on the wedge, which causes normal displacement in the
z-direction.
Q denotes the tangential force transmitted by the frictional force at the contact interface. Relative to the anchor, transverse frictional forces caused by pressing are typically neglected. In summary, the pressure distribution model inside the anchor is simplified as follows: under given boundary conditions, the pressure distribution at the contact interface caused by the punch pressing into an elastic half-space under the action of pressure
P is to be determined.
3.1. Press-In Caused by Rigid Flat Punch
According to the research of Johnson et al. [
23], the pressing problem of a rigid punch can be regarded as the stress and deformation problem in an elastic half-space caused by a concentrated load of intensity
P per unit length distributed along the
x-axis within a narrow band. Combining the assumptions of the above theoretical model, the force analysis is shown in
Figure 4. At point
B, positioned at a horizontal distance
s from the origin on the contact surface, the force acting on the element of width
ds can be considered as a concentrated force acting vertically on point
B, with a value of
pds. Based on elastic theory, it becomes feasible to ascertain the stress components at any point
A beneath the elastic half-space induced by these forces, along with the displacement at any point
C on the contact surface.
For anchorage, it is only necessary to consider the pressure distribution caused by the wedge pressing into the contact surface and the magnitude of compressive stress within the CFRP plate. In the theoretical analysis model, the stress-strain relationship induced by the wedge pressing into the clip and the CFRP plate assembly is linear. This implies that as stress increases, strain increases at the same rate. In this scenario, the clamp and CFRP plate stack are treated as a single entity, and the following are collectively referred to as the assembly. Therefore, after determining the distance
z of the CFRP plate from the contact surface, the stress component in the
z-direction at any point within the CFRP plate is given [
23]:
here,
a represents half of the width of the CFRP plate.
The pressure distribution is given by the integral, as Equation (3) [
23]:
where
represents the Poisson’s ratio of the elastic half-space;
denotes the normal displacement of the contact surface;
E stands for the elastic modulus of the elastic half-space. Since the elastic half-space consists of the clip and CFRP plate, the values of
and
E are, respectively, the equivalent Poisson’s ratio and equivalent elastic modulus of the assembly [
11]:
where
and
represent the Poisson’s ratios of the CFRP plate and the clip, respectively;
represents the transverse elastic modulus of the CFRP plate;
represents the elastic modulus of the clip;
,
, and
denote the thicknesses of the CFRP plate, the clip, and the assembly, respectively.
The general solution form of Equation (3) is provided as the following Equation [
23]:
where
is a known function, and the constant
K is determined by the total normal load in the contact area, expressed as follows [
23]:
according to the given boundary conditions within the contact area, where
,
, Equation (6) can ultimately be simplified to the following [
23]:
Equation (9) allows for the calculation of the compressive stress distribution resulting from the wedge pressing into the assembly in the anchoring width direction. At the edge of the wedge (), the compressive stress theoretically can reach an infinite value. This presents difficulties when considering the compressive stress near the edge. However, the calculated results from Equation (9) remain meaningful in situations away from the edge, and the compression situation in the middle section of the wedge better reflects the average compressive stress borne by the CFRP plate. Therefore, when validating Equation (9) using FEA, only the middle part is chosen for validation. The boundary where the edge is ignored is determined by comparing FEA results with theoretical results. Subsequently, Equation (9) is substituted into Equation (2) to determine the compressive stress within the compressed object, which is then utilized to calculate the compressive stress within the CFRP plate. These equations are solved using MATLAB software (R2023a).
3.2. Contact Pressure Distribution between Barrel and Wedge
Determining the unknown pressure P is necessary to calculate the compressive stress distribution using Equation (9). This pressure arises from the squeezing of the barrel on the wedge and is distributed on their contact surface. Therefore, this problem can be treated as a contact problem between the barrel and the wedge. For simplicity, the deformation at the contact is entirely determined by the geometric shapes involved.
Figure 5 illustrates a longitudinal sectional view of the 1/2 model, showing the internal contact within the anchor along the anchorage length direction. Along the
x-axis, the presetting displacement
of the wedge will cause the compression deformation of the micro-element at
in the
z direction, with the size being as follows.
In the scenario where the barrel is considered a rigid body, the deformation
is jointly caused by the deformation of the wedge and the assembly. Thus, the micro-element at
must satisfy the following conditions:
Here,
represents the interference distance;
and
are the elastic moduli of the wedge and the assembly respectively;
and
are the elastic strains of the wedge and the assembly respectively;
and
are the thicknesses of the wedge and the assembly respectively, and since
Figure 5 represents a symmetrical structure’s 1/2 model,
is taken as half of the actual thickness of the assembly.
It is worth noting that within the contact area
, when the position of the cross-section at
x1 is different, the thickness of the wedge is also different. Therefore,
tw is a function of
x. If the interference distance is entirely controlled by the thickness of the wedge, the expression for
tw would be:
Similarly, within the contact area
,
can also be expressed as a function of
x:
According to Equations (11) and (12), the elastic strain of the wedge can be expressed by the following Equation:
Then, the compressive stress in the contact region between the barrel and the wedge is given by the following:
The load
P per unit length along the
x-axis can be expressed as follows:
It can be observed that the value of P is dependent on the position x of the section chosen. By substituting Equation (17) into Equation (9), the compressive stress distribution resulting from the wedge pressing into the assembly in the anchorage width direction can be determined.
3.3. Parameter Design Scheme
The above theoretical analysis method is used to predict the influence of key parameters on the compressive stress of the CFRP plate. These parameters mainly include the presetting distance of the wedge
, the wedge thickness control parameter
c, and the interference distance
. For ease of comparison, the section position
x = 60 is selected, and the initial values of
,
, and
are set. When one of these parameters changes and the other parameters remain at their initial values, then the maximum value given by Equation (2) is computed. Other given parameters are determined based on the dimensions and material properties of the anchor components, with specific values detailed in
Table 1. The selection of dimensions and material properties will be discussed in the next section.
As shown in
Figure 6a, as the presetting distance of the wedge increases, the compressive stress endured by the CFRP plate gradually increases. When
mm, the CFRP plate reaches its transverse compressive strength, and further pushing the wedge may cause damage to the CFRP plate. According to the discussion in
Section 2 regarding the anchorage length set to 100 mm in the VCWA and considering a maximum average frictional stress not exceeding 30 MPa with a friction coefficient of 0.3 between the steel plate and the CFRP plate, the average compressive stress requirement for a 50 mm
2 mm CFRP plate is 100 MPa. Therefore, it is recommended to control the compressive stress of the CFRP plate between 100–120 MPa. Thus, it is suggested that the presetting distance of the wedge be
mm. Similarly, considering the relationship between the interference distance and the compressive stress, as shown in
Figure 6b, it is recommended to set
mm. In the VCWA, the thickness of the wedge is the sum of the thickness control parameter
c and the interference distance
. Therefore, the value of
c is related to the thickness of the wedge. As shown in
Figure 6c, the compressive stress decreases accordingly as the parameter
c increases. This trend is consistent with the research conducted by Mohee et al. [
11]. on different wedge thicknesses of 6.05 mm, 8.05 mm, 10.05 mm, 13.05 mm, and 15.05 mm, as well as with the research conducted by Ye et al. [
26] on wedge thicknesses of 15 mm, 18 mm, and 25 mm. This indicates that as the wedge thickness increases, the maximum contact pressure inside the anchor decreases, and the shear stress on the surface of the CFRP plate decreases as well [
11,
26]. Based on compressive stress control for the CFRP plate, the value of
c is determined as
mm. Considering the possibility of increased compressive stress on the CFRP plate due to the wedge’s follow-up during load-bearing, the final optimized choice for
c is set to 12.8 mm.
Based on the analysis above, a set of proposed optimized parameter design schemes is as follows: presetting distance of the wedge mm, interference distance mm, and the wedge thickness control parameter c = 12.8 mm. In the next section, the FEA of the VCWA will be conducted under these determined parameters to validate their feasibility and further analyze the mechanical behavior within the anchor.