Liu Yun Qian Zhendong Xia Kaiquan

(1College of Civil and Transportation Engineering, Hohai University, Nanjing 210098, China)(2Intelligent Transportation System Institute, Southeast University, Nanjing 210096, China)(3China Electric Power Research Institute, Beijing 100192, China)

High-voltage overhead transmission lines can be idealized as cable-rod-beam coupling systems[1-6]. The cable element is always used to simulate the transmission line, and the truss element is always used to simulate the insulator and the components of the tower. When the transmission lines are connected with other rods, the slippage may occur at a clamp or a joint. Therefore, a new cable element should be developed to consider the influence of the sliding characteristics on the mechanical response of transmission lines.

Some scholars put forward the analysis method considering sliding characteristics. Tang and Shen[7]presented a new finite element model with five-node curved cable elements using quartic polynomial interpolation. Guo and Cui[8]calculated the cable tension on both two sides of the sliding point by applying different temperature loads representing either heating or cooling to each side of the sliding point. Zhang and Dong[9]presented an algorithm for the analysis of the continuous cable in tension structures based on a two-node catenary cable element. Wei and Liu[10]developed a numerical method by the finite element method (FEM) dealing with the cable-sliding problem in cable structures. Aufaure et al.[11-12]presented the three-node finite element formulation of a length of cable passing through a pulley and clamp respectively, i.e. the expressions of the internal forces and of the stiffness matrix. Nie et al.[13]put forward a nonlinear method for calculating the continuous cables sliding at the middle support. Wei[14]developed an effective numerical method for the cable sliding problem in cable structures, and a two-node catenary cable element was built to model the cables based on the analytical solution of elastic catenary. McDonald et al.[15]developed a pulley element which can model a finite length of cable supported somewhere along its length by a pulley. Zhou et al.[16]used the principle of virtual work and the total Lagrange(TL) formulation to derive the element internal force vector and the tangent stiffness matrix. Chen et al.[17]presented the multi-node sliding cable element for the analysis of cable structures with cables threading through a number of joints and being able to slide inside them.

In the previous studies, the formulation of the catenary element used in the group of sliding cables is too complicated to solve in the finite element analysis, and the total Lagrangian formulation is used to derive the tangent stiffness matrix of the active sliding cable. In this paper, the slippage between cables and joint structures in the transmission lines is considered. The geometric nonlinear stiffness matrix of the three-node straight sliding cable element is deduced based on the updated Lagrangian (UL) formulation. Finally, two examples are given to verify the proposed sliding cable element.

1 Finite Element Formulation of Linear Sliding Cable Element

Fig.1 shows a string of the sliding cable element (SCE) consisting of one active three-node SCE passing through the sliding point and multiple inactive two-node SCEs. A special geometrically nonlinear three-node cable element is developed to model the active sliding cables, as shown in Fig.2. Standard geometrically nonlinear two-node cable elements are used to model the inactive sliding cables. The primary assumption used in this paper to develop the active sliding cable element is that the strain is uniform along the entire element. This assumption implies that there is no resistance, such as friction, at the sliding point. The cross-sectional area of the cable element does not vary with loading, and there is the axial strain in the cable element with no bending moment.

Fig.1 A group of sliding cables

Fig.2 An active sliding cable element

Fig.2 shows a definition sketch of an active sliding cable element in its initial, current and unknown configurations. The fundamental kinematic assumption of the sliding cable element states that the strain is uniform along the element; i.e., the strain in both parts is the same at any time. Applying the principle of virtual work and an updated Lagrangian formulation[18], the incremental virtual work done by the internal force is

(1)

whereε11is the Green-Lagrange strain;S11is the second Piola-Kirchhoff stress; andA0is the initial cross section area of the element, which is assumed to be constant over the entire element length. In the UL formulation, the integration is performed over the current configuration. Because the strain and stress are assumed to be constant along the element, the integration in Eq.(1) is performed analytically as

(2)

where

(3)

(4)

For the three-node sliding cable element, the Green-Lagrange strain is given by

(5)

(6)

Considering the linearization of the balance equation,tS11can be expressed as

tS11=D1111tε11

(7)

The second Piola-Kirchhoff stress and the Cauchy stress are given by

(8)

and

(9)

whereEis Young’s modulus. Performing the variation of Eq.(3) yields

δtε11=(l1+l2)(δl1+δl2)

(10)

The initial, current and unknown element length are determined from the respective nodal coordinates (xi,yi,zi) as

(11)

(12)

(13)

The current nodal coordinates are related to the initial coordinates (0xi,0yi,0zi) and the current nodal displacements (tu,tv,tw) by

txi=0xi+tui,tyi=0yi+tvi,tzi=0zi+twii=1,2,3

(14)

The unknown nodal coordinates are related to the current coordinates (txi,tyi,tzi) and the current nodal displacements (t+Δtu,t+Δtv,t+Δtw) by

t+Δtxi=txi+t+Δtui,t+Δtyi=tyi+t+Δtvi
t+Δtzi=tzi+t+Δtwii=1,2,3

Substituting Eq.(15) into Eq.(11) and performing the variation, we can obtain

(16)

where

Δxi=t+Δtx3-t+Δtxi, Δyi=t+Δty3-t+Δtyi
Δzi=t+Δtz3-t+Δtzii=1,2

(17)

Substituting Eq.(16) into Eq.(10), the virtual strain term can be written as

δtε11=-(l1+l2)ΔTδd

(18)

where

The incremental virtual work given by Eq.(2) can be rewritten as

(19)

and the internal force vector is

FI=-(β1+β0)ΦΔ

(20)

where

Note that theβterm is constant. Taking the partial derivative of the internal force with respect to the nodal displacement yields the following element tangent stiffness matrix:

(21)

where

(22)

(23)

and

(24)

(25)

(26)

and

(27)

It should be noted that the element equations, as written, are singular when the slider node coincides exactly with either of the end nodes (li=0). The element equations for these limit cases can be derived analytically.

2 Verification Problems

A two-span continuous cable structure and a three-span continuous cable structure are implemented to verify the sliding cable element deduced above.

2.1 Example 1

The initial configuration, without gravity, is the unstressed straight lineOP1P2dotted in Fig.3.Orepresents the anchorage of the cable on a dead end.P1is the first pulley fixed at the foot of an insulator chainCP1.P2is the second pulley fixed on the other dead end.O,P1andP2are level. We seek the profile adopted by the cableOP1P2when its unstretched length is given. The calculation parameters of this example are defined as follows:OP1=8m;P1P2=12m;q0=0.2kN/m;E=1.7×105MPa;A0=6.74×10-5m2. (Eis the modulus of elasticity;A0is the cross sectional area;q0is the cable weight per unit length.)

Fig.3 Equilibrium of a two-span cable with a given unstretched length

Tab.1 lists the tension of the two kinds of cable structures simulated by the linear space cable element without considering sliding and the linear space cable element considering sliding. As for the linear space cable element, the elastic modulus is modified and an initial strain is applied so that the geometrical non-linearity of the structure should be considered. The tension of the cable structure simulated by the linear space cable element considering sliding is observed to correlate well with the results in Ref.[13]. As shown in Tab.2, the differences are all within 1%, which indicates the effectiveness and validity of the model adopted in this study. The tension of the cable structureOP1simulated by the linear space cable element considering sliding is greater than that of the simulated by the linear space cable element without considering sliding.

Tab．1 Cable tension of example 1 kN

Tab．2 Cable tension in Ref.[13] kN

2.2 Example 2

A three-span continuous cable structure with non-uniform height supports is applied to verify the element, as shown in Fig.4. The calculation parameters of this example are defined as follows:q0=0.2kN/m;E=1.7×105MPa;A0=6.74×10-5m2. The unstressed lengths of each span are 8.26, 12.52, and 16.64m, respectively. Nodes 2 and 3 are defined as sliding nodes. The tension of the linear sliding cable element is very close to that in Ref.[13] as shown in Tab.3, which further proves the correctness of the proposed element in this paper.

Fig.4 A three-span unequal height support continuous cable(unit:m)

Tab．3 Cable tension of example 2

3 Conclusions

1) The three-node linear sliding cable element is put forward in this paper to consider the slippage in the transmission line structures.

2) The deduced linear sliding cable element is correct and can be used in the analysis of the practical transmission lines structures without considering additional effects, such as friction.

3) Because there is a large difference between the tension of the sliding cable element and that of the cable element without considering sliding, the sliding characteristics should be considered in practical engineering.

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