Theoretical and numerical studies of the slip resistance of main cable clamp composed of an upper and a lower part

Abstract

Cable clamps are important connection members widely used in suspension bridges and cable structure buildings. The clamps are usually tightened on the cable through pre-stressed bolts and resist cable axial component of the external load by friction. Current relevant standards provide slip resistance formulas for anti-slip design of the clamps but are conceptive and simple, lacking explicit and quantitative mechanical derivation. This paper develops an analytical model and proposes a novel slip failure criterion based on the slippage amount, aiming at understanding the force state and estimating the slip resistance of the main cable clamp composed of an upper and a lower part. Finite element analyses then validate that the analytical model can correctly reveal the influences of the multiple factors including hanger tensile force and orthotropic friction on the force state of the cable-clamp system. Moreover, the original Coulomb-friction-law-based slip resistance formula is briefly revised by introducing a partial factor in order to take the nonlinearity of the connection system into account. The revised slip resistance formula implies its promising applicability in obtaining reliable and flexible solution to anti-slip problem of the clamp with different level of safety redundancy.

Keywords

suspension bridge cable clamp frictional resistance coefficient of friction analytical model finite element modeling

Introduction

In suspension bridges, the connection between the main cable and the hangers is through a cable clamp. The clamp normally consists of two semi-cylindrical halves pre-tightened by high-strength steel bolts to initiate necessary friction on the cable-clamp interface (Gimsing and Georgakis, 2013). The clamp transmits the concentrated vertical hanger force to the main cable. Due to the inclination between the hanger and the main cable, the component of hanger force parallel to the cable axis will induce great slipping trend, which only be prevented by the interface friction. Ensuring enough frictional resistance is therefore critical to the reliable connection of the clamp as well as the structural safety.

Several structural design codes involve the anti-slip requirements of clamps. JGJ 257-2012 (2012) outlines various types of the cable joints used in building structures, advises styles of clip/clamp fittings for specific situation, and emphasizes that cables should avoid slipping in such fittings. ASCE/SEI 19-96 (2016) claims the actual coefficient of friction (COF) is a variable, and recommends the maximum design COF between clamps and strands or ropes shall to be 0.1 to prevent slipping and reserve some safety redundancy. JTG/T D65-05-2015 (2015) provides the slip resistance formula of main cable clamps for suspension bridges based on the Coulomb friction law. However, in the formula, the radial clamping force, which is extremely sensitive to the frictional resistance, is only related to the bolts preload, without considering the influence of the hanger force. In the slip resistance formula given by the Eurocode 3: BS EN 1993-1-11 (2006), the total radial clamping force is simply assumed to be equal to the sum of the bolts preload and the component of external tension force perpendicular to the cable axis. The reasonability of this simplification still lacks prudent investigation.

In order to inspect the anti-slip capacity of the tightening main cable clamps designed for suspension bridges, a series of laboratory push-slip experimental tests with diameter of full-scaled cable specimens have been conducted (Ma et al., 2014; Si et al., 2018; Zhou et al., 2015). In these tests, the clamp was first tightened on a cable segment with the measured bolts preload, and then was gradually pushed along the cable axis direction by a jack. In general, the ultimate frictional resistance between the tested clamp and the cable was defined as the pushing load of the jack where the rapid increase of relative slippage occurs. According to the frictional resistance formula given in the JTG/T D65-05-2015 (2015), the COF between the clamps and the main cable composed of zinc-coating wires was tested in range of 0.213∼0.349. In fact, the slip behavior of solids is complicated and the COF is a variable depending on a number of factors according to the adhesion friction theory (Bowden and Tabor, 2001). For convenience, the COF is usually defined as a ratio between the frictional resistance force and the normal compression force according to the classical Coulomb friction law. However, the two forces are not easy to judge or measure accurately in the experiments (Zhang and Li, 2013). This is also one of the key reasons for the obvious discretization of the tested COF. On the other hand, the push-slip experiments cannot reproduce real mechanical condition of the clamp due to the absence of the hanger tension action.

Considering the limitations of the push-slip experiments, theoretical and numerical methods are needed to study the cable-clamp slip resistance issue. In recent years, substantial efforts have been exerted to investigate the behavior and the inter-wire contact conditions of steel strands or wire ropes (Chen et al., 2017; Judge et al., 2012; Liang et al., 2017; Montoya et al., 2012; Yu et al., 2014, 2017). Many methodologies such as semi-analytical approaches, simplified contact and full contact numerical models have been devised, which may offer references to deeply study the contact and frictional resistance problem for the cable-clamp connection. However, thus far, these methods were just applied to the object of multiple discrete wires in single-strand or single-rope scale. Unaffordable intensive computational cost and tremendous convergence difficulty limit their applicability in cables of larger size like the main cables of suspension bridges, which comprises thousands of steel wires. Instead, the main cable was modeled as homogeneous elastic solid in relevant studies. Chen et al. (2013) performed finite element analysis to mirror the push-slip experiment of the cable clamps designed for arch-cable-structure buildings. Ruan et al. (2019) conducted the push-slip tests of the clamps tightened on a fully-locked-coil-rope. As the push load increases, the clamp successively experienced nonslip stage, stable slip stage and slip failure stage. Castro-Fresno et al. (2012) introduced the design of a new metallic cable-joining clip used for slope stabilization nets. Results of the load-displacement curve from the experiment and the finite element simulation presented similar trend.

Analogous issues of the slip resistance between saddle and wire cable also drew much attention. Zhang et al. (2016) and Cheng et al. (2018) proposed an analytical model to estimate the cable-saddle slip resistance. In this model, the cable in the saddle groove was regarded as an integral and homogeneous elastomer. Contrasting to the integral elastomer model, Shen et al. (2017) and Wang et al. (2017) proposed an analytical model for analyzing the contact force of the cable-saddle system based on the discrete cable wires.

However, for the cable-clamp connection system, currently, little literature exists to deeply investigate issues about interface force condition and its influential factors, which are sensitive to determine the slip resistance of the clamp over the cable. Restricted to simple design methods disregarding these issues, large anti-slip safety redundancy is required in design and needs to be realized either through underestimating real COF between clamps and cables (ASCE/SEI 19-96, 2016) or through adopting high level of the clamp bolt preload (JTG/T D65-05-2015, 2015), both are not quite reasonable. On the other hand, with rapid development of the modern communication and the infrastructure construction, more long-span heavy-loading railway and multi-function combined cable stayed bridges (Qin and Gao, 2017) and pre-stressed spatial structures (Chen and Dong, 2013; Chen and Feng, 2016; Guo and Jiang, 2016; Quagliaroli et al., 2015) are demanded, thus requiring larger scaled cable-clamp connection system with higher anti-slip capacity. Current experimental schemes and the simplified calculation methods by the design codes are unable to provide economic and reliable solution to the slip resistance issue.

In view of these problems, preliminary research on the contact condition and the slip behavior between cable and clamp through theoretical and numerical methodologies is necessary and important to provide more reliable and economic anti-slip design solution for the structures. This paper focuses on the issues of the contact force, the ultimate frictional resistance and the slip behavior on the interface between the main cable of suspension bridge and the clamp composed of an upper and a lower part. First, the slip resistance formula provided by codes (Eurocode 3: BS EN 1993-1-11, 2006; JTG/T D65-05-2015, 2015) are discussed. Secondly, a novel analytical model to solve the contact force and the frictional resistance of the clamp is developed. Thirdly, the finite element (FE) models are developed to verify the analytical model accompanied with systematic parametric studies. Finally, based on the parametric analyses, a new slip failure criterion is proposed and the slip resistance formula from the proposed analytical model is further revised by the criterion. These studies are favorable to improve the anti-slip design and verification for the clamps used in both cable structurers of suspension bridges and buildings. Future performed experiments concerning the slip resistance of clamps will also be inspired by the work.

Analytical model

Description of the problem

JTG/T D65-05-2015 (2015) gives a slip resistance formula of main cable clamp designed for suspension bridges

F_{fc} = μ k P_{tot}

(1)

where $F_{fc}$ is the slip resistance force; $μ$ is the COF between main cable and clamp; and $k$ is the compression factor defined as the ratio of the normal compression force on the cable-clamp interface to the total preload of bolts, which is denoted by $P_{tot}$ . The value of $k$ takes 2.8. (According to equation (22) established in the following section, this value is actually on the assumption that the value of the circumferential COF of the cable-clamp interface equals to 0.15 and the hanger tensile load is neglected.) From equation (1), the slip resistance is evaluated based on the classical Coulomb friction law; the normal compression force on the cable-clamp interface is determined only by the preload of bolts. However, the influence of the hanger tensile load applied to the clamp is omitted. In addition, the value of $k$ is advised to be constant without any explanation in the code. Therefore the analytical meaning of $k$ is unclear as well as its influence to the solution of the slip resistance.

In the Eurocode 3: BS EN 1993-1-11, 2006, the slipping between cable and clamp is prevented by verifying

F_{E d_{| |}} \leq \frac{μ (F_{E d_{⊥}} + F_{r})}{γ_{M, fr}}

(2)

Where $F_{E d_{| |}}$ is the component of the tension force applied to clamp parallel to cable axis; $F_{E d_{⊥}}$ is the component of the tension force applied to clamp perpendicular to cable axis; $F_{r}$ is the radial clamping force from the preload of bolts; and $γ_{M, fr}$ is the partial factor for friction, recommended as 1.65. From equation (2), the total normal compression force on the cable-clamp interface is assumed to be the sum of $F_{r}$ and $F_{E d_{⊥}}$ . Nevertheless, the quantitative relationship between $F_{r}$ and the total preload of bolts remains unclear. Besides, the accuracy to express the total radial (normal) compression interface force as the sum of $F_{r}$ and $F_{E d_{⊥}}$ is also worth prudent investigation. The relevant studies concerning these problems are covered in the following sections.

Assumptions of the analytical model

Figure1 illustrates the force state of the cable clamp composed of an upper and a lower part under actions of the bolt tightening force and the hanger tensile force. In order to predict the slip resistance of the clamp under the two kinds of actions, the functions of the pressure and the radial compression on the cable-clamp interface are first developed, in which, the following assumptions are used:

The main cable composed of a large amount of parallel steel wires is regarded as a homogeneous elastic cylinder.

All the forces distribute uniformly along the cable axis direction. Therefore, the plane stress state can represent the real spatial force state.

The small deformation hypothesis is valid. Therefore, the superposition principle is appropriate for the stress analysis.

The ratio between the circumferential friction and the radial pressure on the cable-clamp interface always keeps constant and equals to the value of the circumferential COF of the interface.

The lower part of clamp always contacts with the surface of the cable, that is, no separation appears.

The friction of the cable-clamp interface is orthotropic: the value of COF of the circumferential direction is independent of that of the axial direction.

Figure 1.

Force condition for the clamp composed of an upper and a lower part: (a) purely under the action of the bolts preload, (b) infinitesimal element analysis for the upper part of clamp in condition of (a), (c) infinitesimal element analysis for the lower part of clamp in condition of (a), (d) purely under the action of the hanger tensile load, (e) infinitesimal element analysis for the upper part of clamp in condition of (d), and (f) infinitesimal element analysis for the lower part of clamp in condition of (d).

Derivation of analytical model for the slip resistance

In this part, the analytical model for the slip resistance of the clamp is developed through the following three steps.

Step1: Consider the condition of the clamp purely under the action of bolts preload.

As illustrated in Figure 1(a), the bolts are arranged in the two sides of the clamp. The total preload of bolts of the single side is denoted $P$ . The interface forces between cable and clamp include the normal pressure $q$ and the circumferential friction $f$ . Two representative infinitesimal elements of the upper part and the lower part of the clamp wall are shown in Figure 1(b) and (c), respectively. The polar coordinate systems with the origin at the center of section of main cable are established. In the radial direction, the equilibrium equation for the element can be given by

(F_{N} + d F_{N}) \sin \frac{d θ}{2} + F_{N} \sin \frac{d θ}{2} = qR d θ

(3)

where $F_{N}$ is the internal tensile force supported on the section of the infinitesimal element where the circumferential angle is $θ$ , and $d F_{N}$ is its differentiation with respect to $d θ$ , $R$ is the radius of the cable section. By ignoring the second-order term, equation (3) then becomes

F_{N} = qR

(4)

In the circumferential direction, the equilibrium equation for the element can be given by

\begin{matrix} F_{N} + d F_{N} + d f = F_{N} \\ \Rightarrow d F_{N} = - d f \end{matrix}

(5)

where $d f$ is the circumferential friction and can be expressed as

d f = μ_{θ} qR d θ

(6)

following the assumption 4 in the section “Assumptions of the analytical model.” $μ_{θ}$ is the COF of the circumferential direction.

Substituting equations (4) and (5) into equation (6), gives

\frac{d F_{N}}{F_{N}} = - μ_{θ} d θ

(7)

After the indefinite integral operation on equation (7), yields

\ln F_{N} = - μ_{θ} θ + C

(8)

where $C$ is the integration constant, which can be determined by force boundary condition. At the boundary where $θ = 0$ , there is $F_{N} = P$ , thus $C = \ln P$ , equation (8) then can be arranged as

F_{N} = P e^{- μ_{θ} θ}

(9)

By substituting equation (4) into equation (9), the function of the radial pressure on the cable-clamp interface can be given by

q (θ) = \frac{P e^{- μ_{θ} θ}}{R}

(10)

$q (θ)$ is symmetric about x-axis and y-axis. When integral $q (θ)$ in complete circle of the clamp, the total radial compression force $N_{r}$ on the interface can be expressed as

N_{r} = \frac{2 P_{tot}}{μ_{θ}} (1 - e^{- μ_{θ} π / 2})

(11)

where $P_{tot} = 2 P$ , is the total preload of all the clamp bolts.

Step 2: Consider the variation of the force state due to the hanger tensile load.

The vertical concentrated hanger tensile load is supported by the lower part of clamp and is denoted by $T$ . The inclination of the clamp is denoted by $φ$ , which is the intersection angle between the cable axis and the horizontal direction. Thus the component of $T$ perpendicular to the cable axis equals to $T \cos φ$ , which causes the variation of force state of the connection system, as illustrated in Figure 1(d). What needs to be agreed here is all the variations illustrated in Figure 1(d) to (f), are positive and possess definite direction. Suppose $T \cos φ$ causes the decrease of the pressure $q$ as well as the circumferential stress $f$ on the inner face of the lower part of clamp with respect to the reference force state depicted in Figure 1(a). Therefore $Δ q$ is toward the center of cable section and $Δ f$ is upward. The infinitesimal element of the lower part of clamp wall is shown in Figure 1(f). According to the equilibrium of the element in the radial direction, there is

Δ F_{N} = Δ qR

(12)

According to the equilibrium of the element in the circumferential direction, there is

d Δ F_{N} = - d Δ f

(13)

Following the assumption 4 in the section “Assumptions of the analytical model,” $d Δ f$ can also be expressed as

d Δ f = - μ_{θ} Δ qR d θ

(14)

Note $Δ q$ is toward the center of the cable section, thus inducing decrease of the friction $Δ f$ , a negative sign is added on the term at the right of the equation. Substituting equations (12) and (13) into equation (14), and given the force boundary condition of $Δ F_{N} = Δ P$ at position of $θ = 0$ , $Δ F_{N}$ then can be expressed as

Δ F_{N} = Δ P e^{μ_{θ} θ}

(15)

Substituting equation (12) into equation (15), the function of $Δ q (θ)$ can be given by

Δ q (θ) = \frac{Δ P e^{μ_{θ} θ}}{R}

(16)

Next, the equilibrium equation of the half of the lower clamp in the vertical direction of the cable section plane is established by

\begin{matrix} \int_{0}^{π / 2} q (θ) \sin θ R d θ + \int_{0}^{π / 2} μ_{θ} q (θ) \cos θ R d θ + Δ P \\ = T \cos φ / 2 \end{matrix}

(17)

By substituting equation (16) into equation (17), the variation of the total preload of the bolts at single side, which is denoted by $Δ P$ , can be arranged as

Δ P = \frac{T \cos φ}{4} \frac{1 + {μ_{θ}}^{2}}{(1 + μ_{θ} e^{μ_{θ} π / 2})}

(18)

Being similar to equation (10), the function of $Δ q (θ)$ of the upper part of clamp can be given by:

Δ q (θ) = \frac{Δ P e^{- μ_{θ} θ}}{R}

(19)

Step 3: Stress superposition and integration.

By using the superposition principle and integration operations, the total radial compression force on the upper half of the cable-clamp interface and the lower half of the cable-clamp interface, under both actions of the bolts preload and the hanger tensile load, can be given by equations (20) and (21), respectively,

N_{r}^{u} = \frac{2 (P + Δ P)}{μ_{θ}} (1 - e^{- μ_{θ} π / 2})

(20)

N_{r}^{l} = \frac{2 P}{μ_{θ}} (1 - e^{- μ_{θ} π / 2}) - \frac{2 Δ P}{μ_{θ}} (e^{μ_{θ} π / 2} - 1)

(21)

where the superscripts $u$ and $l$ represent the values of the upper part and the lower part of the clamp, respectively.

Finally, the slip resistance of the cable clamp, which is denoted by $F_{fr}$ , can be obtained based on the classical Coulomb friction law:

F_{fr} = μ_{a} (N_{r}^{u} + N_{r}^{l})

(22)

where $μ_{a}$ is the COF of the cable axial direction.

From the previous theoretical derivation, an analytical model of the slip resistance of the clamp composed of an upper part and a lower part, which comprehensively considers the influences of bolts preload, the hanger tensile load and the orthotropic friction of the cable-clamp interface, is established. Each factor is involved as an explicit and independent parameter in the functions, thus providing a possibility for systematic parametric analyses. In the following section, finite element analyses are performed to further investigate the slip behavior of the clamp over the cable, and to verify the proposed analytical model.

Finite element model

Engineering case

Three-dimensional FE models are developed in this section using ANSYS software (ANSYS Inc., 2009). A suspension bridge across the Yangtze River in China is selected as the engineering case for this study, and the overall layout of the bridge is shown in Figure 2. In this bridge, the clamps are the pinned connection composed of an upper and a lower part. The lower part of each hanger clamp is shaped as a vertical gusset plate with two pin holes. Then two relevant hanger cables each has a fork socket end at the top are pinned connected to the pin holes, and transmit the concentrated vertical load directly to the gusset plate. Figure 3 shows the clamp selected as the case for this study. The clamp is fabricated of alloy casting steel, connected by 16 high-strength alloy steel stud bolts of 48 mm diameter, which are arranged at two sides symmetrically. The length of clamp is 1.6 m. The diameter of the inner groove of the clamp is 0.84 m. The inclination angle $φ$ of the clamp is 23.5°.

Figure 2.

Overall layout of the suspension bridge selected as the engineering case for the study (unit: m).

Figure 3.

Structure diagram of the case clamp (unit: mm).

Modeling strategy

Figure 4(a) shows the meshed FE model including a segment of main cable, the clamp and the clamping stud bolts. The clamp is modeled using tetrahedral-shaped SOLID185 elements with the mesh size of 40 mm. The main cable segment is modeled as a mean-field transversely isotropic elastic cylinder based on a cylindrical coordinate system, where z-axis represents the axil direction of the cable, $r$ -axis and $θ$ -axis represent the radial direction and the circumferential direction of the cable section, respectively. The cable segment model is meshed with hexahedral-shaped SOLID185 elements. The stud bolts are modeled using BEAM188 elements.

Figure 4.

FE model development: (a) cable-clamp assembly, (b) bolt-clamp connection, and (c) contact elements on the embedding rabbets interface.

The fasteners of the bolts include nuts and washers, which are used to screw the bolt shank and transmit its axial tension force on the bearing face of the clamp. Because the local stress distributed in the area of the washer-bearing surface is not the focus in this study, the model of nuts and washers are neglected. Instead, the bolt is connected to the clamp through establishing a rigid area as seen in Figure 4(b). In the rigid area, the node on the end of the bolt shank is selected as the master node, and the nodes on the washer-bearing surface of the clamp are selected as the slave nodes. In this way, the ends of the bolt are tied to the clamp, thus being able to transmit the bolt tension load to the washer-bearing face of the clamp. PRETS179 elements are embedded in the meshed bolts to generate the pretension sections, thereby the initial preload of the bolts and the following adaptive tension variation of the bolts due to the deformation of the connector system can be simulated.

The axial length of the cable cylinder is 0.8 + 1.6 + 0.8 = 3.2 m. The middle segment of 1.6 m is the area where the clamp of 1.6 m axial length is installed on. In this contact area, the mesh size of the cable is set as 47 mm. The radial of the cable cylinder is equal to the radial of the clamp groove, so that the two parts are just contact in initial state. CONT174 and TARGE170 elements are used to generate standard surface-to-surface contact pairs in this area. Among them, the cable surface is selected as contact surface, and the inner surface of the clamp is selected as target surface. Besides, the contact pairs are also generated on the rabbet interfaces as seen in Figure 4(c), so the upper part and the lower part of the clamp model can move as a complete part over the cable. The augmented Lagrangian contact algorithm is adopted for the contact solutions. The orthotropic friction is assigned to the cable-clamp interface, where the COF of the $z$ -axis direction ( $μ_{a}$ ) adopts 0.15, and the COF of the $θ$ -axis direction ( $μ_{θ}$ ) adopts 0.15∼0.4.

As previously interprets, the real main cable which contains thousands of parallel steel wires is modeled as the transversely isotropic elastic cylinder based on the cylindrical coordinate system. Although lacking necessary investigation to estimate the quantitative equivalent material properties of the mean-field cable model, what is definite is that the radial elastic modulus ( $E_{r}$ ) of it is much less than that of steel solid due to the void existing in the cable section. Thereby the radial elastic modulus assumes to be 5000 MPa in this model. The rest material properties constants including the circumferential elastic modulus ( $E_{θ}$ ), the axil elastic modulus ( $E_{z}$ ), the shear moduli ( $G_{zr}$ , $G_{z θ}$ , and $G_{r θ}$ ), and the Poisson’s ratios ( $ν_{zr}$ , $ν_{z θ}$ , and $ν_{r θ}$ ) are also defined based on assumption. The basic information of the FE model is summarized in Table 1.

Table 1.

Information of the FE model.

Items	Element types	Material properties
Clamp	SOLID185	$E = 210 GPa, ν = 0.3$
Stud bolts	BEAM188, PRES179	$E = 210 GPa, ν = 0.3$
Cable cylinder	SOLID185	$\begin{matrix} E_{r} = E_{θ} = 5000 MPa, E_{z} = 195 GPa; \\ G_{zr} = G_{z θ} = 7900 MPa, G_{r θ} = 1923 MPa; \\ ν_{zr} = ν_{z θ} = ν_{r θ} = 0.3 \end{matrix}$
Cable-clamp contact surface	CONT174, TARGE170	$μ_{θ} = 0.15 ~ 0.4, μ_{a} = 0.15$
Rabbets contact surfaces	CONT174, TARGE170	$μ_{R} = 0.15$

The pylon side (upper side) of the end of the cable cylinder is constrained at all nodes of the face in the axial direction and at the node of the section center in the lateral direction. In the first load step, 500 MPa tensile stress is applied to the midspan side (lower side) of the end of the cable cylinder to provide the cable model with tension. Meanwhile, the clamp is constrained to prevent rigid body movement. Then the constraints of the clamp are removed and the clamp contacts with the cable being tightened by applying 650 kN of pretension to each bolt. Finally, the incremental concentrated vertical load, representing the hanger load, is applied on the bearing face of the two pin holes.

Finite element results

Radial interface force

The radial interface force is the normal compression force on the cable-clamp interface, which critically determines the slip resistance of the clamp. Figure 5 shows typical pressure distribution development on the interface with respect to the increase of the hanger tensile force. The pressure distribution on the lower half of the interface is observed to be generally identical with that on the upper half when only the bolts tightening force is applied. However, as the hanger load increases, the pressure on the bottom area of the lower part decreases gradually. When the hanger load reaches 5400 kN (double design load), an obvious zero-pressure area appears at the bottom of the lower part, indicating that large out-of-plane bending of the clamp cylinder wall occurs and leads to the separation between the surface of cable and inner groove surface of the clamp. When the hanger load grows to triple design load of 8100 kN, the zero-pressure area extends to the whole bottom area of the lower part. While the pressure on the lateral area (near the rabbets) of the lower part increases. On the other hand, the pressure on the upper part always increases as the hanger load increases.

Figure 5.

Development of the pressure distribution on the cable-clamp interface subjects to the hanger load (unit: MPa).

The development of the radial compression force accompanied with the hanger load is revealed further in Figure 6. In general, the radial compression force of the upper clamp increases as the hanger load increases. The radial compression force of the lower clamp decreases as the hanger load increases until a critical point, and then increases. That is because after the whole bottom area of the lower part of the clamp separates from the surface of the main cable, the radial force of this area contracts to zero hence no longer decreases, whereas the radial force on the lateral area increases evidently. The trend of the total radial force of the two halves of the clamp is analogous to that of the lower half of the clamp. What shall to be noticed is that within double design hanger load of 5400 kN, the total radial force of the clamp is consistently reduced by the hanger load. Therefore, the hanger load is validated to be unfavorable to the slip resistance of the pinned connection of the clamp with the upper and the lower parts. In addition, the circumferential COF also affects the results of radial interface force as seen in Figure 6. Larger $μ_{θ}$ consistently induces smaller radial interface force within triple design hanger load.

Figure 6.

Radial compression on the cable-clamp interface subjects to the hanger load (unit: kN): (a) compression of the upper clamp, (b) compression of the lower clamp, and (c) total compression of the upper clamp and the lower clamp.

Bolt tension

According to the above analytical model, the hanger load will induce an added bolt tension ( $Δ P$ ). The Finite element method (FEM) results of the average added tension of 16 clamping bolts are shown in Figure 7. The results show that, in the range of 6000 kN (2.2 times of the design load) of the hanger tensile force, the added bolt tension increases linearly with the hanger tensile force; even though the difference made by $μ_{θ}$ is slight, larger $μ_{θ}$ consistently induces smaller added bolt tension. As hanger load exceeds 6000 kN, difference of the added bolt tension with different $μ_{θ}$ enlarges until a critical point, and then tends to be small again. In this stage, nonlinear deformation of the clamp and the contact state develops abundantly, as well as evident nonlinearity of the added bolt tension.

Figure 7.

Average added bolt tension subjects to the hanger load (unit: kN).

Axial friction

The axial friction on the interface is shown in Figure 8. It can be seen from Figure 8(a) that, the total axial friction in case of the different $μ_{θ}$ keeps consistent with the axial component of the hanger force, which verifies the reliability of the FEM for revealing the equilibrium relationship of the connection system. The axial friction of the lower clamp and the upper clamp is depicted in Figure 8(b) and (c), respectively, and are compared in Figure 8(d). The lower clamp is observed to bear lager component of the hanger load by friction than the upper clamp until the hanger load reaches about 7000 kN (2.6 times of the design load). In this stage, the rising rate of the friction of the lower clamp gradually drops to zero until reaching ultimate value of the friction. The friction of the upper clamp always increases with the increase of the hanger load, and eventually exceeds the friction of the lower clamp. This curve-intersection phenomenon of the friction could be explained as the lower clamp prefer to bear lager sliding load by friction due to the vertical hanger force is supported on the gusset directly; simultaneously, the radial compression on the lower half of the interface is constantly reduced by the rising hanger load, thus eventually reaching the ultimate friction. However, the radial interface force of the upper clamp keeps rising with the hanger load, as well as the frictional resistance, therefore the friction eventually exceeds that of the lower clamp. In addition, it is observed that $μ_{θ}$ can hardly affect the resultant axial friction on the upper part and the lower part of the clamp.

Figure 8.

Axial friction subjects to the hanger load (unit: kN): (a) total friction of the entire clamp, (b) friction of the lower clamp, (c) friction of the upper clamp, and (d) comparison between the friction of the upper clamp and the lower clamp.

Slip behavior

From the beginning of contact, accumulated slippage is recorded for every contact element generated on the surface of the cable model. Owing to the wedging action of the rabbets, the upper part and the lower part of the clamp is observed to slip as an integral part. In general, the trend of axial slippage distributed on every contact element is similar. Therefore, the results of a contact element located on the upper half interface near the end of midspan side of the clamp is selected to represent the axial slip behavior of the clamp. The slippage-load curve is depicted in Figure 9(a). It can be seen in the initial loading stage, the hanger load increases but the clamp dose not slip. This stage can be defined as the non-slip stage. Once the load rises to a critical value, the slippage occurs and increases rapidly, indicating the slip failure of the clamp. Define term the slip acceleration as second derivative of slippage to hanger load. The term expresses the rate of the slippage increase, and is more sensitive to present occurrence of the slip failure as shown in Figure 9(b). In addition, lager $μ_{θ}$ is observed to consistently lead to earlier onset of the slip failure as expected according to the Coulomb friction law. Because lager $μ_{θ}$ always induces smaller radial interface compression as known from the previous analyses.

Figure 9.

Slippage and slip acceleration subject to the hanger load: (a) slippage, and (b) slip acceleration.

Analytical model validation and results discussion

From comprehensive analyses of the previous FEM results, it can be found easily that in the initial stage where the hanger load in range of about double design load, all categories of force keep the linear relationship with the hanger load. Then, nonlinearity appears due to the large deformation of the clamp, as well as the change of the contact condition with the main cable.

First, the liner part of the FEM results is observed to generally support the analytical model from qualitative perspective, including the following aspects: (1) As the hanger load increases, the radial interface compression of the upper clamp increases, the compression of the lower clamp decreases, and the total compression decreases according to equations (20)∼(22). (2) The action of the hanger load will induce the added bolt tension and keeps linear relationship with it according to equation (18). (3) Larger $μ_{θ}$ consistently leads to smaller pressure and compression force on the connection interface, and leads to smaller value of the added bolt tension, according to equations (10), (11), and (18), respectively.

Then, quantitative comparisons between the analytical solutions and the FEM results are made. Analytical model overestimates the radial interface force of the upper clamp, and underestimates that of the lower clamp, yet yields accurate solutions for total value of the radial interface force for the entire clamp, as shown in Figure 10. As shown in Figure 10(d), in the range of 4000 kN (1.3 times of the design load) of the hanger load, the discrepancy of the total radial interface force between the two methods is less than 3%; in the range of 7000 kN (2.3 times of the design load) of the hanger load, that discrepancy is less than 7%.

Figure 10.

Comparison between the analytical solutions and the FEM results of the radial interface force: (a) in case of $μ_{θ} = 0.4$ , (b) error rate of the radial force of the upper clamp, (c) error rate of the radial force of the lower clamp, and (d) error rate of the total radial force.

Restricted to the assumptions on which the analytical model based, there is difference between the analytical solutions and the FEM solutions, especially considering the gradually rising nonlinearity accompanied with increase of the hanger load. However, the analytical model is verified to correctly reveal effects of the hanger load and $μ_{θ}$ on the force state of the cable-clamp connection system. It is important that that the analytical model can give the solution of the total radial interface compression with highly accuracy, which is crucial to correctly predict the slip resistance of the clamp in the theoretical frame of the Coulomb friction law.

Revised slip resistance model based on parametric slip analysis

The FEM of the cable-clamp system developed above can simulate nonlinear contact and slip behavior of the clamp over the surface of the main cable under actions of bolts preload and hanger load. As shown in Figure 9, the slip failure of the clamp is represented by the rapid increase of the slippage. The slip accelerations corresponding to the slippage limits of 0.2 and 0.5 mm are analyzed from the data statistics for the contact elements generated on the surface of the cable cylinder, which are located at circumferential angles (as illustrated in Figure 11) of $\pm 10 °$ , $\pm 30 °$ , $\pm 60 °$ , and $\pm 90 °$ , near the midspan side of end of clamp. Generally, the slip accelerations are positive as listed in Table 2, indicating unstable contact state. Therefore, the slippage limits just like 0.2 and 0.5 mm can be regarded as the slip failure criterion aiming at determining the slip resistance for cable clamp.

Figure 11.

Circumferential angels of the cable-clamp interface.

Table 2.

Slip accelerations determined by the slippage limits of 0.2 and 0.5 mm (unit: $μ m / M N^{2}$ ).

Circumferential COF	Slippage limit of 0.2 mm			Slippage limit of 0.5 mm
Circumferential COF	Hanger load(10³ kN)	Upper halfclamp	Lower half clamp	Hanger load(10³ kN)	Upper halfclamp	Lower halfclamp
0.15	7.5	40	53	9.3	133	115
0.20	7.2	48	67	9.0	233	224
0.25	6.9	47	69	8.7	331	334
0.30	6.6	26	44	8.4	425	434
0.35	6.3	16	35	8.1	491	508
0.40	6.0	−28	−11	7.8	578	596

In order to further investigate reasonability of the slippage-based failure criterion, systematic parametric FEM studies collaborating with the Coulomb-friction-law-based analytical model are performed. Three sets of variables including the inclination angle of the clamp, the circumferential COF and the average pressure on the contact interface due to the bolts preload are studied. Among them, the inclination angle adopts $30 °$ , $23.5 °$ _, and $17 °$ ; the circumferential COF adopts 0.2, 0.3, and 0.4; the average contact pressure adopts 2, 3, 4, and 5 MPa, converting to the uniform bolts preload calculated by equation (11). The axial circumferential COF still adopts 0.15. The hanger load increment adopts 200 kN for the finite element solutions.

The ultimate anti-slip load determined by the analytical model and by the slippage limits of 0.2 and 0.5 mm, from the FEM solutions, are summarized in Table 3. Then the original Coulomb-friction-law-based analytical model for the slip resistance of the clamp can be briefly revised by introducing a partial factor as

F_{fr, s} = γ_{s} μ_{a} (N_{r}^{u} + N_{r}^{l})

(23)

where the partial factor $γ_{s}$ is defined as ratio between the ultimate anti-slip load ruled by the slippage-based failure criterion and that ruled by the Coulomb friction law. Thus, the revised model can take nonlinearity of the cable-clamp system into account. The partial factors corresponding to the slippage limits of 0.2 and 0.5 mm, based on the parametric analysis cases, are shown in Figure 12(a) and (b), respectively. Note that when the anti-slip design or verification for the clamp adopts the 0.2 mm slippage-based failure criterion, more conservative result than that is based on the Coulomb friction law will be obtained (i.e. $γ_{s} < 1.0$ ). While, in most cases, the 0.5 mm slippage-based failure criterion will obtain generally the same result with that is based on the Coulomb friction law (i.e. $γ_{s} \approx 1.0$ ). That is, the slip failure criterion based on the slippage can offer more flexible solution to the anti-slip capacity when different level of safety redundancy is required.

Table 3.

Contrast of the ultimate anti-slip load based on the criterions of the Comlomb friction law and the slippage limits of 0.2 and 0.5 mm.

Inclination angles (deg)	Circumferential COF	Average pressure by the bolts preload (MPa)
		2	3	4	5	2	3	4	5
		Analytical solutions (10³ kN)				FEM solutions (10³ kN)
30	0.2	2.41	3.61	4.82	6.02	2.0/2.4	3.2/3.6	4.2/4.8	5.0/5.8
	0.3	2.36	3.55	4.73	5.92	2.0/2.4	3.2/3.6	4.0/4.6	5.0/5.8
	0.4	2.33	3.50	4.66	5.83	2.0/2.4	3.0/3.4	3.8/4.6	4.6/5.6
23.5	0.2	2.97	4.45	5.94	7.42	2.6/3.2	3.8/4.6	4.8/5.8	6.0/7.0
	0.3	2.90	4.36	5.80	7.26	2.6/3.0	3.8/4.4	4.8/5.8	5.8/7.0
	0.4	2.85	4.27	5.69	7.12	2.6/3.0	3.6/4.4	4.6/5.6	5.6/7.0
17	0.2	3.94	5.90	7.88	9.86	3.4/4.6	4.6/6.2	5.8/7.6	6.8/9.0
	0.3	3.82	5.74	7.64	9.56	3.4/4.2	4.6/5.8	5.8/7.4	6.8/8.8
	0.4	3.72	5.58	7.44	9.30	3.4/4.0	4.6/5.8	5.8/7.4	6.6/9.0

Ultimate anti-slip load determined by the slippage limits of 0.2 and 0.5 mm are listed before and behind the slash sign “/”, respectively.

Figure 12.

Partial factors based on the slippage limits of: (a) 0.2 mm, and (b) 0.5 mm.

Conclusion

This paper studied the slip resistance of the main cable clamp composed of upper and lower parts. The main findings and conclusions are summarized as follows:

The bolts preload, the hanger load, and the orthotropic friction are concerned in the proposed analytical model comprehensively, which is validated by the finite element studies. The analytical model is valuable to help deeply understand the pinned connection system of the cable and the clamp in view of the forces equilibria and the multiple factors influences, providing a reliable solution of the slip resistance for the clamp in practice.

Larger circumferential COF consistently leads to smaller radial (normal) pressure on the cable-clamp interface, which is dominated by the equilibrium principle. Therefore, the orthotropic friction should be taken into account to predict the slip resistance of the clamp.

For the pinned connection system of the clamp composed of upper and lower parts, the perpendicular component of the hanger tensile load will cause the reduction of the total radial compression on the cable-clamp interface, which is unfavorable to the slip resistance. This effect should be properly concerned in the anti-slip design or verification of the clamp.

The original Coulomb-friction-law-based slip resistance formula can be revised by introducing a partial factor to convert to the slip resistance ruled by the slip failure criterion based on the slippage, thus taking the nonlinearity of the cable-clamp system into account. The revised model offers more flexibility in practice when different safety redundancy is required.

Footnotes

Acknowledgements

The financial support from these grants is gratefully acknowledged.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study was sponsored by the Science and Technology Research and Development Project of China Railway Corporation (grant no. 2015G002-A) and the National Natural Science Foundation of China (grant no. 51178396).

ORCID iD

Rusong Miao

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