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This paper presents an elastic shear stress distribution theoretical model at the carbon fiber-reinforced polymer (CFRP)-adhesive interface of a single-rod and a multi-rod straight-pipe bonding anchor. A comparison between theoretical and finite element analysis results reveals that the accuracy of the theory can be used to guide the preliminary design of CFRP rod bonding anchors. The mechanical performance of the inner cone bonding anchor for multi-rods are evaluated within different coefficients of friction and inner inclined angles. Numerical results indicate that the straight-parabolic inner cone bonding anchor has a significant effect on reducing the shear force at the loading end.

Carbon fiber-reinforced polymer (CFRP) has a high strength, is light weight, with excellent fatigue performance and non-corrosive properties and is, thus, widely used in the structures of buildings and bridges in various forms. As an effective form of CFRP, pultruded CFRP rods or cables can better utilize the unique advantages mentioned above. A cable composed of CFRP rods, as shown in

(

The specific strength of the CFRP cable is fairly high compared to that of a steel cable, while the density of CFRP is one-fourth that of steel. This makes it more suitable for large-span structures, such as large-span roof systems [

However, the anchorage system is the key factor of whether CFRP cables can reach tensile strength in service. Compared to steel, the anisotropy and brittleness of CFRP make it more difficult to anchor. Currently, anchorage systems can be roughly divided into the clamp anchor, bonding anchor and barrel-wedge anchor [

Several CFRP rod anchorage systems were studied. Parametric analysis of the barrel-wedge-type anchor has been reported. A four-wedge type anchor used to grip a single CFRP rod was proposed [

There is some research on multi-rod anchors, but not as much as single-rod ones, because it is more difficult to grip a collection of rods as opposed to only one. To anchor multiple rods, the bonding anchor is preferred over the barrel-wedge-type anchor. The CFCC (Carbon Fiber Composite Cable) strand in Japan already has a mature post-tensioning anchorage system using highly expansive cementing material used in the Penobscot Narrows Cable Stayed Bridge [

Based on the existing research, the bonding stress distribution of CFRP single-rod and multi-rod bonding anchors is analyzed by a theoretical model and a finite element program in this paper.

The ultimate loading capacity of a CFRP rod anchor generally depends on the shear stress distribution along the rod. The more uniform the stress distribution, the higher the anchorage efficiency.

A theory was proposed to calculate the ultimate capacity of a straight-pipe bonding anchor [

An elastic theoretical model is proposed in this paper to calculate shear stress referring to [

All the materials are elastic: the stress state of the anchorage system remains elastic.

Deformation of the metal barrel is not considered, because the stiffness of the barrel is much higher than that of the adhesive. Therefore, the internal surface of the barrel is considered fixed.

Interface slippage is ignored, so that the displacement compatibility can be founded at the interface between the barrel and adhesive (hereinafter, called the first interface) and the interface between the adhesive and CFRP rods (hereinafter, called the second interface).

Thus, the theory is suitable for preliminary shear stress analysis in an elastic state.

The cross-section of a single-rod anchor is shown in

Tensile load (_{0}) is applied at the loading end of the CFRP rod. We assume that the axial displacement of the rod has a uniform distribution in one section that is set as _{b}(

(

_{c} is the longitudinal elastic modulus of the CFRP rod, and _{b}(

Assuming that the shear strain has a linear distribution in the radial direction, the average shear strain of the adhesive can be gained with Equation (3):

Here,

With Equations (1), (2) and (5) and setting ^{2}

In Equation (6), _{1}^{x}_{2}^{−x}_{0}, the value of _{1} and _{2} can be calculated as below:

Therefore, the shear stress distribution can be gained:

The stress distribution in multiple rods can also be derived from theory. A common arrangement is multiple rods symmetrically anchored with one rod in the center and other rods arranged in a ring (_{1}, while it is _{2} between side rods and the barrel. The other parameters are the same as the single-rod anchor system described in

(

In contrast to the single rod, the shear stress distribution on the surface of the side rods is not uniform along the perimeter. To simplify the theory, the adhesive is considered to be two layers of concentric rings, and the shear stress on the side rods is divided into the inner stress distribution, τ_{2i}(_{2o}(_{1}(_{1}(_{2}(_{1}(_{2}(

According to the equilibrium and interaction relationships from

Through simplification, we get:

The shear force for each layer of adhesive is also considered to be approximately equal:_{1}(_{1})τ_{2i}(_{1})τ_{2o}(_{1} + 2_{2})τ_{b}(

It can be seen from _{1}(_{2}(_{2}(

With Equations (10)–(15), the displacements, _{1}(_{2}(

In the equations above, the coefficients are listed respectively:

With Equations (16) and (17), the displacement can be solved. By eliminating _{2}(

The boundary conditions at the free end and the loading end are _{1} (0) = 0, _{2} (0) = 0, _{1} (0) = _{0} and _{2} (0) = _{0}, which can be converted into conditions that only contain _{1}, as listed in Equation (20):

With Equations (19) and (20), _{1} (

The finite element method was used to study the stress distribution in the CFRP rod bonding anchor and to test the accuracy of the theoretical model. The anchor was modeled by the Solid45 element in ANSYS software, and nodes on the interface were merged, because slippage is not considered.

The CFRP rod, adhesive grout and steel tube were modeled, respectively (

(

The material properties of CFRP and steel are presented in

Material properties (GPa).

Material |
_{x} |
_{y}_{z} |
_{xy}_{xz} |
_{yz} |
_{xy}_{xz} |
_{yz} |
---|---|---|---|---|---|---|

CFRP rod | 181 | 10.05 | 0.28 | 0.3 | 7.17 | 7.17 |

Steel | 200 | / | 0.27 | / | 78.74 | / |

Numerical cases for the single-rod anchor.

No. | 1 | 2 | 3 |
---|---|---|---|

Modulus of adhesive (GPa) | 5 | 5 | 2 |

Thickness of adhesive (mm) | 2 | 5 | 5 |

In the FE (Finite Element) model, the outer surface of the steel tube is restrained in all degrees. A tensile stress of 1500 MPa is then applied to the loading end of the CFRP rod. To compare the numerical and theoretical datasets, each case is also analyzed by theory in

From

(

Two kinds of parameter combinations are selected (

(

Numerical cases for the seven-rod anchor.

No. | Thickness of adhesive Layer 1 (mm) | Thickness of adhesive Layer 2 (mm) |
---|---|---|

1 | 5 | 5 |

2 | 8 | 2 |

Here, the boundary conditions are the same as the single-rod anchor, and the tensile stress is still set as 1500 MPa on each rod. We used FE to obtain the axial displacement, the shear stress on the surface of the center rod, as well as the shear stress on the inner and outer surface of the side rods. Mathmetica 6.0 solved differential Equation (19). For each case, the displacement distribution gained by theory and the finite element method is shown in

(

(

As can be seen, the rod in the center has a shear stress peak with a relatively low value at the loading end. The six rods on the side have a complex shear stress distribution. The shear stress in different positions on the same section of side rods may have opposite directions that are negative inside the rod and positive outside the rod. The peak value is also much bigger than the center rod. As the space between the center rod and side rods increases, the stiffness of the anchor and the shear stress peak increase.

The theoretical and finite element results are generally the same, except for some local positions. Some deviation is seen at the peak point of shear stress close to the end cross-section of the anchorage, and the shear stress on the inner surface of the side rod changes its direction in the FE results. This may be caused by stress discontinuity or approximation error in the application of the theory, which has been explained in

Zhang

A straight-pipe bonding anchor for multiple CFRP rods has some disadvantages (

Additional finite element analysis was performed to investigate the mechanical properties of the inner cone multi-rod bonding anchor. The seven-rod bonding anchor is designed by an anchorage manufacturer in China (

(

The ANSYS Solid185 element is applied to model the inner cone barrel, adhesive and CFRP rods, and the 1/6 model is shown in

The coefficient of friction (COF) at the first interface and the inner inclined angle were varied to study their influence on the stiffness, radial stress value and shear stress distribution along the rod. The material properties of the rod are the same as in

Numerical cases of the inner cone bonding anchor.

Case No. | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|

COF at the first interface | 0 | 0.05 | 0.1 | 0.2 | 0.05 | 0.05 | 0.05 |

Inner inclined angle | 3% | 3% | 3% | 3% | 2% | 4% | Straight-parabolic |

The convergence of the FEM (Finite Element Method) is studied through mesh-refinement of the FE model. The meshing of the straight-parabolic inner cone anchor model is refined with the element number from 39,200 to 148,140. After mesh-refinement, the tensile load value is 50.8 kN when the displacement reaches 15 mm, while it is 50.5 kN in the original model. Under the same stress state of 2000 MPa, the maximum radial compressive stress at the first interface is 111.5 MPa compared to the original value 104.3 MPa. The relative error of the load and stress value is 0.6% and 7%, which is acceptable. Therefore, the convergence of the FEM results is reliable.

The loading curves of the bonding anchor with different first-interface coefficients of friction are illustrated in

Loading curves with different inner inclined angles are displayed in

(

Meanwhile, the radial stress of the straight-parabolic anchor has a larger value at the free end

(

(

Campell

Based on the research of this paper, the following conclusions can be made:

With fundamental assumptions of elasticity and non-slip, a theoretical model for single and multiple rod bonding anchors is proposed by calculating the shear stress distribution on the rods. The theoretical equations are derived from the equilibrium and geometric conditions of the CFRP rods and adhesive. The theoretical model is suitable for preliminary shear stress analysis in the elastic state, and it can provide support for further study on decreasing the shear stress at the loading end cross-section of the anchorage.

Finite element results show good agreement with the theoretical results, thus confirming the accuracy of the theory. For the single rod bonding anchor, as the modulus of the adhesive increases or the thickness of the adhesive decreases, the stiffness of the single rod and the peak value of the shear stress will increase. For multi-rod anchors, the stiffness and shear stress peak value will also increase as the space between the center rod and side rods increases. Anyway, the side rods have a complex stress state under tensile force, which may lead to premature failure. Therefore, the modulus and thickness of the adhesive, as well as the space between the rods need to be considered in the anchorage design. The modulus of the adhesive should be high enough to ensure the stiffness of the anchor, but not too high to cause shear or sliding failure. A thin adhesive layer can help to improve the anchorage performance on the condition that the bond strength is guaranteed. Small spacing between the rods is preferred in a multi-rod anchor, because the longitudinal radial stress is more uniform, and the influence on the side rods caused by asymmetry becomes smaller. The size of the anchorage can also be smaller.

In the parametric analysis of the in-cone bonding anchor, as the coefficient of friction at the barrel-adhesive interface increases, the stiffness will increase and the radial compression will decrease, such that it will be more difficult for the steel barrel to yield under a certain load. The stiffness of the anchor also increases when the inner inclined angle increases, with the side effect that the radial pressure is more uneven. In the anchorage design, the inner inclined angle should be appropriate under the overall consideration of the stiffness and radial pressure.

Straight-parabolic in-cone anchoring is shown to be an efficient way to reduce the concentrated shear stress and to improve the anchorage efficiency by moving the stress peak from the loading end to the free end. It can be a feasible type of anchorage for CFRP cable made up of parallel CFRP rods in engineering.

This work is supported by the China 863 Project (No. 2012AA03A204) and China 973 Project (2012CB026200), and the first author acknowledges the support of Beijing Higher Education Young Elite Teacher Project (YETP0078).

The work presented here was carried out in collaboration between all authors. Peng Feng, Pan Zhang and Lieping Ye defined the research theme. Peng Feng and Pan Zhang designed the research methods, carried out the finite element analysis and theoretical analysis, analyzed the data and wrote the paper. Xinmiao Meng co-worked on the data collection and their interpretation and presentation. All authors have contributed to, seen and approved the manuscript.

The authors declare no conflict of interest.