Finite element modeling of cable sliding and its effect on dynamic response of cable-supported truss

Abstract

The cable system of cable-supported structures usually bears high tension forces, and clip joints may fail to resist cable sliding in cases of earthquake excitations or sudden cable breaks. A analytical method that considers the dynamic cable sliding effect is proposed in this paper. Cable sliding behaviors and the resultant dynamic responses are solved by combining the vector form intrinsic finite element framework with cable force redistribution calculations that consider joint frictions. The cable sliding effect and the frictional tension loss are solved with the original length method that uses cable length and the original length relations. Then, the balanced tension distributions are calculated after frictional sliding. The proposed analytical method is achieved within MATLAB and applied to simulate the dynamic response of a cable-supported plane truss under seismic excitation and sudden cable break. During seismic excitations, the cable sliding behavior in the cable-supported truss have an averaging effect on the oscillation magnitudes, but it also magnifies the internal force response in the upper truss structure. The slidable cable joints can greatly reduce the ability of a cable system to resist sudden cable breaks, while strong friction resistances at the cable-strut joints can help retain internal forces.

Keywords

sliding cable / explicit solution framework / original length method / seismic response / cable rupture

Introduction

Cables, such as strands, wires, and wire cables, are widely used in tension structures and bridges because of their high strengths and flexible features. These characteristics help ensure a long span or achieve aesthetic structural designs, but they also bring many geometric nonlinear properties that cause highly dynamic problems [¹,²]. Cable-supported structures are generally the interactive types, and their structural studies often involve static or dynamic analyses between rigid parts and their flexible cable system. Numerous investigations on design and construction problems have been conducted, but in-depth studies on the working mechanism of cable joints and the influence of sliding cables during construction and service periods are lacking. In general, cables are assumed to be discontinued by cable-strut joints and their tensile forces are determined with the static balance principle [³–⁵]. However, this assumption is not always reasonable. At certain periods, such as the cable stretching process, the continuous cables slide across the cable-strut joints and transfer the pretension forces [⁶]. Such cable-strut joints are fastened at the completion of construction; meanwhile, the joint frictional resistance will likely to be designed to resist the maximum horizontal force difference under static loading conditions [⁷,⁸]. However, due to the high tension forces preserved in the cable system, the cable-strut joint may fail to clamp the continuous cable under dynamic perturbations like cable ruptures or seismic excitations, and cable slips are bound to occur. To ensure the safety of cable structures, special attention should be accorded to the frictional sliding of a continuous cable and its influence on static and dynamic behaviors.

Many studies have been conducted on the simulation of sliding cable elements. Zhou et al. [⁹] developed a three-node sliding cable element based on the uniform strain assumption along the entire cable range and applied the element to the modeling of an airdrop system. Based on the same idea, Chen et al. [¹⁰] developed a multi-node sliding cable element and implemented it to study the static behaviors of suspen-dome structures with sliding cables. However, the uniform strain assumption did not consider the friction resistance at the cable-strut joints and the relative simulations were limited to the frictionless slipping condition. Tang and shen [¹¹] proposed a five-node curved cable element and introduced sliding stiffness to define the relation between internal force and cable length variation. Then, this relation was incorporated into an iteration process to determine the appropriate sliding amounts to reach the force balancing state at the cable joints. This iteration idea was then extended by Nie et al. [¹²] and Cai et al. [¹³], and were in combined with the elastic catenary model to consider the effect of self-weight. Furthermore, friction resistance was added into the balance state iteration [¹⁴,¹⁵] and different adjustment methods were adopted in the calculation [¹⁶]. However, the method involved complex matrix deductions, and iterations were mainly used in the static calculations.

Given that the frictionless cable sliding is unrealistic, then subsequent studies have focused on friction joint problems. Guo and Cui [¹⁷] proposed a frozen-heated method that utilized virtual temperature-increased and temperature-decreased loads to study the influence of gliding cable joints. Then Liu and Chen [¹⁸] and Zhao et al. [¹⁹] adopted the same method in their finite element (FE) studies of cable-supported structures. Chen et al. [²⁰] established cable-sliding criteria-based equations to conduct the static analysis of cable–pulley systems. Yan et al. [²¹] proposed an equivalent friction element by using an additional virtual friction node and by configuring virtual equivalent stiffness matrixes.

Although various efforts were conducted to solve the cable sliding problems, those studies mainly focused on static or quasi-static (construction process) conditions. Frictional sliding and its influence on mechanical responses under dynamic excitations were rarely reported. In general cable supported structures, the magnitude of cable forces is quite high while the tension difference at each joint often remains at a comparatively low level to ensure a self-balancing state. And the resisting capacities of cable-strut joints are often designed to resist the maximum cable force difference in all static conditions [²²,²³]. However, this clamping effect may not be fully effective when the system has large pulling forces. During the service period, structures may inevitably experience dynamic disturbances, such as earthquake or cable clamp and joint failures, which may induce considerable impact and violent internal force changes. Once the oscillations surpass the frictional resistance at the clip joint, micro-slip occurs and the cable tension is redistributed. Moreover, cable sliding often involves dynamic variations in cable length and inconsecutive force changes against friction directions. Hence, the structural stiffness matrix is also a time-varying factor that often causes the traditional FE analyses to fail with singularity and non-convergence. The motivation of this work is to establish a finite element solution to simulate the frictional sliding behaviors of continuous cables and use this modeling method to investigate the dynamic responses of a cable-supported truss considering the cable sliding effects.

Finite element modeling of frictional cable sliding and basic solving framework

Basic explicit solution framework with vector mechanic

The cable sliding modeling method proposed in this study can be effectively used within the explicit analytical framework. The basic solving process adopts the vector form intrinsic finite element (VFIFE) method proposed by Ting et al. [²⁴,²⁵]. Subsequently, the calculation method for the cable sliding problem is incorporated into the VFIFE solving framework. The VFIFE method assumes that the structure is a system composed of a group of mass particles formed by mass lumping and connected by intermediate elements. Figure 1 shows an example of a beam discretion and the resultant internal force systems. For a single beam element, all the beam mass is concentrated to the mass particles at two end nodes. The mass particles comply with Newton’s second law, and the motion is described by global position vectors. The intermediate beam element has no mass but bears internal forces that react to the mass particles in converse directions. Hence, the problem of structural deformation and motion is converted into the moving pattern of nodes or finite mass particles.

As shown in Fig. 2, the position of an arbitrary particle i can be expressed by its global coordinatesX_i(t). Furthermore, based on Newton’s second law, the motion equation of particle i can be written as

(1)m_i{\ddot{\mathit{X}}}_i={\mathit{F}}_i^{\mathrm{e}\mathrm{x}\mathrm{t}}-{\mathit{F}}_i^{\mathrm{int}}-{\mathit{F}}_i^{\mathrm{d}\mathrm{m}\mathrm{p}},

where M_iis the mass of particle i; {\ddot{\mathit{X}}}_i is the acceleration vector; and {\mathit{F}}_i^{\mathrm{e}\mathrm{x}\mathrm{t}}, {\mathit{F}}_i^{\mathrm{int}}, and {\mathit{F}}_i^{\mathrm{d}\mathrm{m}\mathrm{p}} are the external force, internal force, and damping force vectors at particle i, respectively. Each particle at any moment is in a dynamic equilibrium state.

All the position vectors and mechanical factors, such as stresses and strains, in the dynamic analysis are time-varying functions. Then, the total time-duration in the explicit solution framework is rendered to be discrete in small time intervals. The motion equation can be solved at each time interval through explicit time integration. Here, we utilize the central difference method as the principle algorithm. Given position vectors of any node i at subsequent moment {\mathit{T}}_{k-\mathrm{1}}, {{\displaystyle \mathit{T}}}_k, and {\mathit{T}}_{k+\mathrm{1}} as {\mathit{X}}_i(k-\mathrm{1}), {\mathit{X}}_i(k), and {\mathit{X}}_i(k+\mathrm{1}), the relations are as follows:

(2){\dot{\mathit{X}}}_i(k)={\displaystyle \frac{\mathrm{1}}{\mathrm{2}\Delta \mathit{T}}}({\mathit{X}}_i(k+\mathrm{1})-{\mathit{X}}_i(k-\mathrm{1})),

(3){\ddot{\mathit{X}}}_i(k)={\displaystyle \frac{\mathrm{1}}{\mathrm{2}\mathrm{\Delta }{\mathit{T}}^{\mathrm{2}}}}({\mathit{X}}_i(k+\mathrm{1})-\mathrm{2}{\mathit{X}}_i(k)+{\mathit{X}}_i(k-\mathrm{1})),

where \Delta \mathit{T} is the constant time interval, while {\dot{\mathit{X}}}_i(k)and {\ddot{\mathit{X}}}_i(k)are the velocity and acceleration of mass node i at {{\displaystyle \mathit{T}}}_k, respectively. Combined with Eq. (1), the position vector {\mathit{X}}_i(k+\mathrm{1})can be expressed or calculated by {\mathit{X}}_i(k-\mathrm{1})and {\mathit{X}}_i(k). Here, we adopt the damping factor \xi and consider damping force as a proportional function of moving speed.

(4){\mathit{F}}_i^{\mathrm{d}\mathrm{m}\mathrm{p}}(k)=\xi m_i{\dot{\mathit{X}}}_i(k)

The explicit formula for nodal position calculation therefore yields the following:

(5){\mathit{X}}_i(k+\mathrm{1})=\left({\displaystyle \frac{\mathrm{2}}{\mathrm{2}+\xi \Delta t}}\right){\displaystyle \frac{\Delta {\mathit{T}}^{\mathrm{2}}}{m_i}}({\mathit{F}}_i^{\mathrm{e}\mathrm{x}\mathrm{t}}(k)-{\mathit{F}}_i^{\mathrm{int}}(k))+\left({\displaystyle \frac{\mathrm{4}}{\mathrm{2}+\xi \mathrm{\Delta }\mathit{T}}}\right){\mathit{X}}_i(k)-\left({\displaystyle \frac{\mathrm{2}-\xi \mathrm{\Delta }\mathit{T}}{\mathrm{2}+\xi \mathrm{\Delta }\mathit{T}}}\right){\mathit{X}}_i(k-\mathrm{1})\hspace{1em}k=\mathrm{1},\mathrm{2....}

In the calculation of mechanical factors at the initial time step {\dot{\mathit{X}}}_i(\mathrm{0}), {\ddot{\mathit{X}}}_i(\mathrm{0}), a necessary parameter {\mathit{X}}_i(-\mathrm{1}) is non-existent but can be obtained by

(6){\mathit{X}}_i(-\mathrm{1})={\mathit{X}}_i(\mathrm{0})-{\displaystyle \frac{\mathrm{2}+\xi \mathrm{\Delta }\mathit{T}}{\mathrm{2}}}\mathrm{\Delta }\mathit{T}{\dot{\mathit{X}}}_i(\mathrm{0})+{\displaystyle \frac{\mathrm{\Delta }{\mathit{T}}^{\mathrm{2}}}{\mathrm{2}m}}({\mathit{F}}_i^{\mathrm{e}\mathrm{x}\mathrm{t}}(\mathrm{0})-{\mathit{F}}_i^{\mathrm{int}}(\mathrm{0})).

Finite element modeling of frictional cable sliding behavior

Cable-supported structures are generally composed of cables, rigid beams and struts, and continuous cables that span across several cable-strut joints. In the proposed modeling method, the continuous cable is assumed to be a sliding cable element with two end nodes as the anchored supports and intermediate nodes as the cable-strut joints. Figure 3 shows an example of a sliding cable element. The cable is divided into N−1 segments by N nodes. The following basic assumptions are adopted:

1) No sliding is allowed at the two end nodes, and the position, moving path, and sliding state of those inside nodes are decided by the relation between unbalanced cable tension and nodal frictional resistance.

2) The cable segments remain elastic and in straight state throughout the entire analysis period. The cable length and tension conditions can therefore be calculated with comparative locations of cable nodes by using Hook’s relation.

3) A node cannot slide across its adjacent nodes along the cable. The node order remains unchanged during the entire analysis period.

The frictional sliding behavior and the tension force redistributing patterns are solved with the original cable length method under exlicit solution framework. The calculating procedure can be summarized as follows:

Step 1: Calculate the nodal positions at the next time step by using explicit time integration. At time spot k, given the current cable forces as \mathit{T}(k)=\{{\mathit{T}}_{\mathrm{1}}(k),{\mathit{T}}_{\mathrm{2}}(k),\mathrm{...},{\mathit{T}}_i(k),\mathrm{...},{\mathit{T}}_{N-\mathrm{1}}(k)\} and the nodal positions as \mathit{X}(k)=\{{\mathit{X}}_{\mathrm{1}}(k),{\mathit{X}}_{\mathrm{2}}(k),{\mathit{X}}_{\mathrm{3}}(k),\mathrm{...},{\mathit{X}}_i(k),\mathrm{...},{\mathit{X}}_{N-\mathrm{1}}(k),{\mathit{X}}_N(k)\} respectively, and assume that no slippage has occurred and the cable forces are unchanged in the next time interval, then, by using the central difference explicit integration and Eqs. (2)-(5), the nodal positions at time step k+1 can be obtained as\mathit{X}(k+\mathrm{1})=\{X_{\mathrm{1}}(k+\mathrm{1}),{\mathit{X}}_{\mathrm{2}}(k+\mathrm{1}),{\mathit{X}}_{\mathrm{3}}(k+\mathrm{1}),\mathrm{...},{\mathit{X}}_i(k+\mathrm{1}),\mathrm{...},{\mathit{X}}_{N-\mathrm{1}}(k+\mathrm{1}),{\mathit{X}}_N(k+\mathrm{1})\}. Based on the two subsequent nodal positions \mathit{X}(k) and\mathit{X}(k+\mathrm{1}), cable force \mathit{T}(k+\mathrm{1}) can be calculated with Hook’s law as follows:

(7){{\displaystyle l}}_i^{\mathrm{0}}(k)={\displaystyle \frac{{{\displaystyle l}}_i(k)EA}{{{\displaystyle \mathit{T}}}_i(k)+EA}},

(8){{\displaystyle \mathit{T}}}_i(k+\mathrm{1})={\displaystyle \frac{{{\displaystyle l}}_i(k+\mathrm{1})-{{\displaystyle l}}_i^{\mathrm{0}}(k)}{{{\displaystyle l}}_i^{\mathrm{0}}(k)}}EA,\hspace{1em}k=\mathrm{0},\mathrm{1},2,\mathrm{...},

where E gives the elastic modulus of cable material, and A is the section area of cable. {{\displaystyle l}}_i^{\mathrm{0}}(k) is the original length of cable piece iat time spot k, and due to the no slippage assumption at this step, then the origin length at time spot k+ 1 was equals {{\displaystyle l}}_i^{\mathrm{0}}(k) at this period. Moreover, the l_i(k) and l_i(k+\mathrm{1}) are the cable lengths that calculated as \left|{\mathit{X}}_{i+\mathrm{1}}(k)-{\mathit{X}}_i(k)\right|and\left|{\mathit{X}}_{i+\mathrm{1}}(k+\mathrm{1})-{\mathit{X}}_i(k+\mathrm{1})\right|respectively. Owing to the no-compression capacity characteristic of the cable, if , then{\mathit{T}}_i(k+\mathrm{1})=\mathrm{0}. However, due to the non-slip assumption, the obtained \mathit{T}(k+\mathrm{1}) may not really reflect the real cable force conditions and the tension difference between adjacent cable segments may surpass the friction resistance at the intermediate joint.

Step 2: Calculate the frictional resistance at each joint and determine the sliding state at time spot k+1. In consideration of the joint sliding effect, the tension difference and friction resistance need to be initially determined for each cable joint. Figure 4 shows the force diagram at turning node i. Here, {\mathit{T}}_{i+\mathrm{1}}(k+\mathrm{1}) and {\mathit{T}}_i(k+\mathrm{1}) represent the tension forces at the two connected cables, while n represents the middle line direction between vector direction ({\mathit{X}}_{i-\mathrm{1}}-{\mathit{X}}_i) and ({\mathit{X}}_{i+\mathrm{1}}-{\mathit{X}}_i). Then, we have

(9)\{\begin{array}{l}\mathrm{\Delta }{\mathit{T}}_i(k+\mathrm{1})=\left|{\mathit{T}}_i(k+\mathrm{1})-{\mathit{T}}_{i-\mathrm{1}}(k+\mathrm{1})\right|\\ R_i(k+\mathrm{1})={\mathit{F}}_i^{\mathrm{0}}+\mu ({\mathit{T}}_i(k+\mathrm{1})+{\mathit{T}}_{i-\mathrm{1}}(k+\mathrm{1}))\cos {\theta }_i(k+\mathrm{1})\end{array},

where \Delta {\mathit{T}}_i(k+\mathrm{1}) is the tension difference at node i at the moment of k+ 1, \mu is the Coulomb coefficient; and R_i(k+\mathrm{1}) is the joint friction resistance that consists of the inherent friction capacity {{\displaystyle \mathit{F}}}_i^{\mathrm{0}} and the sliding friction from contact compression \mu ({\mathit{T}}_i(k+\mathrm{1})+{\mathit{T}}_{i-\mathrm{1}}(k+\mathrm{1}))\cos {\theta }_i(k+\mathrm{1}).

The sliding state can be determined by comparing the values of \Delta {\mathit{T}}_i(k+\mathrm{1})andR_i(k+\mathrm{1}). If , then no sliding is induced and cable tension adjustments are not needed for this joint; otherwise, slips happen and the original length of the connected cable will be changed. For those sliding joints, we need to determine the slippage amount and the ultimate balancing state at the current nodal locations. To facilitate the calculation, a vector indicator expressed as \delta (k)={({\delta }_{\mathrm{1}},{\delta }_{\mathrm{2}},{\delta }_{\mathrm{3}},,,{\delta }_i,,,{\delta }_{N-\mathrm{2}},{\delta }_{N-\mathrm{1}},{\delta }_N)}_k is used to record the sliding state at the cable nodes. As in Fig. 4, if or no slippage happen, then {\delta }_i=\mathrm{0}. If \Delta {\mathit{T}}_i(k+\mathrm{1})>R_i(k+\mathrm{1}), then slippages are initiated. If T_i(k+\mathrm{1})>T_{i-\mathrm{1}}(k+\mathrm{1}), then {\delta }_i(k+\mathrm{1})=\mathrm{1}; otherwise{\delta }_i(k+\mathrm{1})=-\mathrm{1}. Once the joint sliding state \delta (k+\mathrm{1}) is determined, the cable force redistribution can be calculated based on the frictional drag relations.

The slippage on one cable-strut joint can influence the tension state of two adjacent cable segments and subsequently affect the sliding state and frictional balancing relations at the adjacent joints. Here, a sliding region is defined as a continuous range where slippage occurs on all the inside nodes and the constituent cable segments have tension forces interacting with one another. As in Fig. 3, if a cable slips at nodes 2, 3, and 4, then all the cable pieces between nodes 1 and 6 form a sliding region. Subsequently, the tension force will be redistributed across all those cable pieces. The joint sliding state indicator \delta (k+\mathrm{1}) is introduced to solve the problems of those sliding regions, and the tension adjustments for each continuous sliding region are simultaneously calculated.

Step 3: Perform the cable force redistributing calculation for each sliding region. Until now, the cable node positions at time spot k+ 1, the internal tension at continent cable pieces under the provisional non-slip condition, and the possible sliding state at each node were all obtained. Then during this step, the cable force redistribution and balancing process will be performed. The nodal positions are still fixed as \mathit{X}(k+\mathrm{1}) and only the internal forces of the cable segments are redistributed in each sliding region. Therefore, for any sliding region(a,a+\mathrm{1},\mathrm{...},b-\mathrm{1},b), the sum of the tensioned-cable lengths and the original lengths are unchanged throughout the force redistributing process because the slip only occurs at inside nodes. Thus,

(10)\underset{i=a}{\overset{i=b}{\mathrm{\Sigma }}}{l}_i^{\mathrm{0}}(k+\mathrm{1})=\underset{i=a}{\overset{i=b}{\mathrm{\Sigma }}}{l}_i^{\mathrm{0}}{(k+\mathrm{1})}^{*},

(11)\underset{i=a}{\overset{i=b}{\mathrm{\Sigma }}}{l}_i^{\mathrm{0}}(k+\mathrm{1})+\underset{i=a}{\overset{i=b}{\mathrm{\Sigma }}}{\displaystyle \frac{{\mathit{T}}_i(k+\mathrm{1})l_i^{\mathrm{0}}(k+\mathrm{1})}{EA}}=\underset{i=a}{\overset{i=b}{\mathrm{\Sigma }}}{l}_i^{\mathrm{0}}{(k+\mathrm{1})}^{*}+\underset{i=a}{\overset{i=b}{\mathrm{\Sigma }}}{\displaystyle \frac{{\mathit{T}}_i{(k+\mathrm{1})}^{*}{l}_i^{\mathrm{0}}{(k+\mathrm{1})}^{*}}{EA}},

where l_i^{\mathrm{0}}(k+\mathrm{1}) and l_i^{\mathrm{0}}{(k+\mathrm{1})}^{*} are the original cable lengths while{\mathit{T}}_i(k+\mathrm{1}) and {\mathit{T}}_i{(k+\mathrm{1})}^{*} are the tension forces before and after frictional sliding. Here during the first cable force redistribution process, the l_i^{\mathrm{0}}(k+\mathrm{1}) equals l_i^{\mathrm{0}}(k)due to the non-slip assumption in Step 2. Combined with Eqs. (10) and (11), the following relation can be obtained:

(12)\underset{i=a}{\overset{i=b}{\mathrm{\Sigma }}}{\displaystyle \frac{{\mathit{T}}_i(k+\mathrm{1})l_i^{\mathrm{0}}(k+\mathrm{1})}{EA}}=\underset{i=a}{\overset{i=b}{\mathrm{\Sigma }}}{\displaystyle \frac{{\mathit{T}}_i{(k+\mathrm{1})}^{\ast }{l}_i^{\mathrm{0}}{(k+\mathrm{1})}^{*}}{EA}}\hspace{1em}i=a,a+\mathrm{1...}b-\mathrm{1},b.

At any sliding node (Fig. 4), equilibrium equations can be derived as follows:

(13){\mathit{T}}_i{(k+\mathrm{1})}^{\ast }-{\mathit{T}}_{i-\mathrm{1}}{(k+\mathrm{1})}^{\ast }={\delta }_i(k+\mathrm{1})[{\mathit{F}}_i^{\mathrm{0}}+\mu ({\mathit{T}}_i{(k+\mathrm{1})}^{\ast }+{\mathit{T}}_{i-\mathrm{1}}{(k+\mathrm{1})}^{\ast })\cos {\theta }_i],

(14){\mathit{T}}_i{(k+\mathrm{1})}^{\ast }={\displaystyle \frac{(\mathrm{1}+{\delta }_i(k+\mathrm{1})\mu \cos {\theta }_i)}{(\mathrm{1}-{\delta }_i(k+\mathrm{1})\mu \cos {\theta }_i)}}{\mathit{T}}_{i-\mathrm{1}}{(k+\mathrm{1})}^{\ast }+{\displaystyle \frac{{\delta }_i(k+\mathrm{1}){\mathit{F}}_i^{\mathrm{0}}}{(\mathrm{1}-{\delta }_i(k+\mathrm{1})\mu \cos {\theta }_i)}}.

All of the tension forces at the constitute cable segments can be expressed by tension force at the first (or the last) cable segment within the same sliding region. Two influencing factors, c_i(k+\mathrm{1})={\displaystyle \frac{(\mathrm{1}+{\delta }_i(k+\mathrm{1})\mu \cos {\theta }_i)}{(\mathrm{1}-{\delta }_i(k+\mathrm{1})\mu \cos {\theta }_i)}} and b_i(k+\mathrm{1})={\displaystyle \frac{{\delta }_i(k+\mathrm{1}){\mathit{F}}_i^{\mathrm{0}}}{(\mathrm{1}-{\delta }_i(k+\mathrm{1})\mu \cos {\theta }_i)}}, are defined to aid the calculation. Then, {\mathit{T}}_i{(k+\mathrm{1})}^{\ast }can be calculated as

(15){\mathit{T}}_i{(k+\mathrm{1})}^{\ast }=C_i(k+\mathrm{1}){\mathit{T}}_a{(k+\mathrm{1})}^{\ast }+B_i(k+\mathrm{1}),

(16)\{\begin{array}{l}{C}_i(k+\mathrm{1})=\underset{j=a+\mathrm{1}}{\overset{j=i}{\mathrm{\Pi }}}{{\displaystyle c}}_j(k+\mathrm{1}),C_a=\mathrm{1},i=a+\mathrm{1},a+\mathrm{2},\mathrm{...},b-\mathrm{1}\\ B_i(k+\mathrm{1})=\underset{j=a+\mathrm{1}}{\overset{j=i-\mathrm{1}}{\mathrm{\Sigma }}}(b_j(k+\mathrm{1})\underset{m=j+\mathrm{1}}{\overset{m=i}{\mathrm{\Pi }}}{{\displaystyle c}}_m(k+\mathrm{1})+b_i(k+\mathrm{1}),B_a=\mathrm{0},i=a+\mathrm{1},a+\mathrm{2},\mathrm{...},b-\mathrm{1}\end{array}.

Substituting Eqs. (15) and (16) into Eq. (12) yields {\mathit{T}}_a{(k+\mathrm{1})}^{\ast }as the only unknown variable, and the subsequent nonlinear equations can be solved by the Newton method.

As previously discussed, the explicit solution framework calculates time-dependent problems by discretizing the time duration into small intervals. The time interval is somewhat small, which renders the slippage amounts during each time step to be negligible as opposed to the cable segment lengths. Therefore, an approximation relation, that the original length of each cable segment is unchanged and only the cable forces are redistributed, is adopted here. Then Eq. (12) can be simplified as

(17)\underset{i=a}{\overset{i=b}{\mathrm{\Sigma }}}{\mathit{T}}_i(k+\mathrm{1})l_i^{\mathrm{0}}(k+\mathrm{1})=\underset{i=a}{\overset{i=b}{\mathrm{\Sigma }}}{\mathit{T}}_i{(k+\mathrm{1})}^{\ast }{l}_i^{\mathrm{0}}(k+\mathrm{1}).

Equations (15), (16), and (17) can collectively solve {\mathit{T}}_a{(k+\mathrm{1})}^{\ast }. Subsequently, {\mathit{T}}_{a+\mathrm{1}}{(k+\mathrm{1})}^{\ast },{\mathit{T}}_{a+\mathrm{2}}{(k+\mathrm{1})}^{\ast }...{\mathit{T}}_{b-\mathrm{1}}{(k+\mathrm{1})}^{\ast } within the sliding region can be solved.

Step 4: Re-determine the joint sliding state and the sliding regions, and repeat the cable force redistribution calculations. Once the cable forces are updated, the frictional resistance at each cable joint will be changed. Therefore, the sliding state and continuous sliding regions need to be re-determined, and the entire process of steps (2) and (3) are repeated until no slip occurs at any node within the cable. Then the final {\mathit{T}}_i(k+\mathrm{1}) at cable piece i can be obtained. And the overall \mathit{T}(k+\mathrm{1})denotes the ideal redistributed cable tension at time step k+\mathrm{1}. With these balanced cable forces and current nodal locations, the internal force {\mathit{F}}^{\mathrm{int}}(k+\mathrm{1}) can be obtained.

The above calculations are established based on tensioned cable conditions in which all cable segments are always in tensioned state and their axial forces can be calculated with Eqs. (7) and (8). However, under dramatic dynamic disturbances, some cable pieces may be slackened, which may influence the cable force determination and redistribution calculations during dynamic analyses. Therefore, several additional calculations are needed in consideration of these factors.

At the end of Step1, the tension and slackness states need to be checked. For cable piece i, if in step (1), which means that and the current segment length is smaller than the original length at time i, then the present cable is slackened, and set{\mathit{T}}_i(k+\mathrm{1})=\mathrm{0} and l_i^{\mathrm{0}}(k+\mathrm{1})=l_i^{}(k+\mathrm{1}). Calculate the length difference\mathrm{\Delta }{l}_i(k+\mathrm{1})=l_i^{\mathrm{0}}(k)-l_i^{}(k+\mathrm{1}). Conduct steps 2 to 3 to derive a balanced \mathit{T}(k+\mathrm{1}) at the end of step 3.

Prior to step 4, calculate the tensionless length l_i^{\mathrm{0}}{(k+\mathrm{1})}^{*} with Eq. (7) and re-determine the tension state. If l_i^{\mathrm{0}}{(k+\mathrm{1})}^{*}+\mathrm{\Delta }{l}_i(k+\mathrm{1})>l_i(k+\mathrm{1}), which means that segment i continues to be slackened, then set {\mathit{T}}_i{(k+\mathrm{1})}^{\ast }=\mathrm{0}, and update the \mathrm{\Delta }{l}_i(k+\mathrm{1})=\mathrm{\Delta }{l}_i(k+\mathrm{1})-(l_i(k+\mathrm{1})-l_i^{\mathrm{0}}{(k+\mathrm{1})}^{*}), and l_i^{\mathrm{0}}{(k+\mathrm{1})}^{*}=l_i(k+\mathrm{1}); Otherwise, if l_i^{\mathrm{0}}{(k+\mathrm{1})}^{*}+\Delta l_i(k+\mathrm{1})\le l_i(k+\mathrm{1}), which means that the cable segment has reverted to the tension state. Then, set\mathrm{\Delta }{l}_i(k+\mathrm{1})=\mathrm{0}. The current cable force can be calculated as

(18){\mathit{T}}_i{(k+\mathrm{1})}^{\ast }={\displaystyle \frac{l_i(k+\mathrm{1})-(l_i^{\mathrm{0}}{(k+\mathrm{1})}^{*}+\mathrm{\Delta }{l}_i(k+\mathrm{1}))}{l_i^{\mathrm{0}}{(k+\mathrm{1})}^{*}+\mathrm{\Delta }{l}_i(k+\mathrm{1})}}EA.

The entire calculation is performed during each time interval. The mechanical factors at each time spot can altogether reflect the dynamic response and the varying trends.

Element force calculation and global coordinate transformation

Similar to the cable force determinations, the element forces of cables, struts, and beams, as well as their variations, can be calculated by referring to the comparative change of the nodal positions between two subsequent steps within the nodal coordinate system. Given that the global coordinate system is adopted in the explicit solution framework, a coordinate transforming process is thus needed to decompose the element forces into component forces along the global system axes. The detailed internal force formulation and decomposition of the link and beam elements can be found in Refs. [²⁴] and [²⁵], and only a simple description is provided here.

As shown in Fig. 1, if the discrete element is a link or a strut, then the internal force only has a tension or a compression force along its longitudinal direction, which can be expressed as follows (i.e., at time k+\mathrm{1}):

(19){\mathit{F}}_{AB}^{\mathrm{int}}=[{\mathit{F}}_A^{\mathrm{int}},{\mathit{F}}_B^{\mathrm{int}}]=EA{\displaystyle \frac{l(k+\mathrm{1})-l^{\mathrm{0}}(k+\mathrm{1})}{l^{\mathrm{0}}(k+\mathrm{1})}}[-e_{AB},e_{AB}],

where {\mathit{F}}_A^{\mathrm{int}} and {\mathit{F}}_B^{\mathrm{int}} are the internal force vectors at the two end nodes A and B within the global coordinate system; l(k+\mathrm{1}) and l^{\mathrm{0}}(k+\mathrm{1}) denote the cable length and original length at time k+\mathrm{1}; and e_{AB} is the directional vector of element AB.

For a beam element, each end node has a tension force and a twisting moment along the longitudinal direction and a pair of orthogonal transverse force and bending moment right to the axis. The position vector of the beam element at the two adjacent time steps k,k+\mathrm{1} is

(20){{\displaystyle \mathit{X}}}_{AB}^t=[{{\displaystyle \mathit{X}}}_A^{\mathit{T}},{{\displaystyle \mathit{X}}}_B^t]=[({{\displaystyle \mathit{X}}}_A^{\mathit{T}},{{\displaystyle y}}_A^{\mathit{T}},{{\displaystyle z}}_A^{\mathit{T}},{{\displaystyle R\mathit{X}}}_A^{\mathit{T}},{{\displaystyle Ry}}_A^{\mathit{T}},{{\displaystyle Rz}}_A^{\mathit{T}}),({{\displaystyle \mathit{X}}}_B^{\mathit{T}},{{\displaystyle y}}_B^{\mathit{T}},{{\displaystyle z}}_B^{\mathit{T}},{{\displaystyle R\mathit{X}}}_B^{\mathit{T}},{{\displaystyle Ry}}_B^{\mathit{T}},{{\displaystyle Rz}}_B^{\mathit{T}})]\hspace{1em}\mathit{T}=k,k+\mathrm{1.}

As shown in Fig. 5, both the element rotation {\beta }_{k-(k+\mathrm{1})}and the twist angle variation \mathrm{\Delta }R\mathit{X} can be calculated and expressed with nodal position vectors at timek,k+\mathrm{1}. Here an intermediate state is involved that that the magnitude of internal forces and node forces are the same as those at time spot k+ 1, but with the force directions remaining unchanged as at time k. In Fig. 5, node forces at the intermediate state are given as ({\mathit{F}}^{\prime }(k+\mathrm{1}),M^{\prime }(k+\mathrm{1})). Then according to beam theory, the internal forces at time k+\mathrm{1} can be calculated as follows:

(21){\mathit{F}}^{\text{'}}(k+\mathrm{1})=\{{{\displaystyle \mathit{F}}}_A^{\text{'}}(k+\mathrm{1}),{{\displaystyle \mathit{F}}}_B^{\text{'}}(k+\mathrm{1})\}=\left\{\left(\begin{array}{l}{\mathit{F}}_{A-x}^k-{\displaystyle \frac{EA}{l(k)}}(l(k+\mathrm{1})-l(k))\\ {\mathit{F}}_{A-y}^k+{\displaystyle \frac{\mathrm{6}E{I}_z}{{(l(k))}^{\mathrm{2}}}}(\mathrm{\Delta }{\varphi }_z^A+\mathrm{\Delta }{\varphi }_z^B)\\ {\mathit{F}}_{A-z}^k-{\displaystyle \frac{\mathrm{6}E{I}_y}{{(l(k))}^{\mathrm{2}}}}(\mathrm{\Delta }{\varphi }_y^A+\mathrm{\Delta }{\varphi }_y^B)\end{array}\right),\left(\begin{array}{l}{F}_{B-x}^k-{\displaystyle \frac{EA}{l(k)}}(l(k+\mathrm{1})-l(k))\\ {\mathit{F}}_{B-y}^k-{\displaystyle \frac{\mathrm{6}E{I}_z}{{(l(k))}^{\mathrm{2}}}}(\mathrm{\Delta }{\varphi }_z^A+\mathrm{\Delta }{\varphi }_z^B)\\ {\mathit{F}}_{B-z}^k+{\displaystyle \frac{\mathrm{6}E{I}_y}{{(l(k))}^{\mathrm{2}}}}(\mathrm{\Delta }{\varphi }_y^A+\mathrm{\Delta }{\varphi }_y^B)\end{array}\right)\right\},

(22)M^{\text{'}}(k+\mathrm{1})=\{{{\displaystyle M}}_B^{\text{'}}(k+\mathrm{1}),{{\displaystyle M}}_B^{\text{'}}(k+\mathrm{1})\}=\left\{\left(\begin{array}{l}{M}_{A-x}^k-{\displaystyle \frac{G{J}_x}{l(k)}}(\mathrm{\Delta }{\varphi }_{\mathit{X}}^B)\\ M_{A-y}^k+{\displaystyle \frac{E{I}_y}{l(k)}}(\mathrm{4}\mathrm{\Delta }{\varphi }_y^A+\mathrm{2}\mathrm{\Delta }{\varphi }_y^B)\\ M_{A-z}^k+{\displaystyle \frac{E{I}_z}{l(k)}}(\mathrm{4}\mathrm{\Delta }{\varphi }_z^A+\mathrm{2}\mathrm{\Delta }{\varphi }_z^B)\end{array}\right),\left(\begin{array}{l}{M}_{B-\mathit{X}}^k+{\displaystyle \frac{G{J}_{\mathit{X}}}{l(k)}}(\mathrm{\Delta }{\varphi }_{\mathit{X}}^B)\\ M_{B-y}^k+{\displaystyle \frac{E{I}_y}{l(k)}}(\mathrm{2}\mathrm{\Delta }{\varphi }_y^A+\mathrm{4}\mathrm{\Delta }{\varphi }_y^B)\\ M_{B-z}^k+{\displaystyle \frac{E{I}_z}{l(k)}}(\mathrm{2}\mathrm{\Delta }{\varphi }_z^A+\mathrm{4}\mathrm{\Delta }{\varphi }_z^B)\end{array}\right)\right\}\mathrm{.}

Here ({\mathit{F}}_{A-\mathit{X}}^k,{\mathit{F}}_{A-y}^k,{\mathit{F}}_{A-z}^k,M_{A-\mathit{X}}^k,M_{A-y}^k,M_{A-z}^k) and (F_{B-\mathit{X}}^k,F_{B-y}^k,F_{B-z}^k,M_{B-\mathit{X}}^k,M_{B-y}^k,M_{B-z}^k) are the node forces and moments along the element coordinate system at time spot k. And G{J}_{\mathit{X}},E{I}_y, E{I}_zare the longitudinal twist and bending module along two bending directions within the element system of the calculated element. Then (\mathrm{\Delta }{\varphi }_{\mathit{X}}^A,\mathrm{\Delta }{\varphi }_y^A,\mathrm{\Delta }{\varphi }_z^A)and (\mathrm{\Delta }{\varphi }_{\mathit{X}}^B,\mathrm{\Delta }{\varphi }_y^B,\mathrm{\Delta }{\varphi }_z^B) are the longitudinal twist and nodal rotation extent at node A and B from time spot k to k+ 1. On this basis, the calculated ({\mathit{F}}^{\prime }(k+\mathrm{1}),M^{\prime }(k+\mathrm{1})) follow the element system of time k (i.e., virtual locationA^{\prime }(k+\mathrm{1})B^{\prime }(k+\mathrm{1})). The internal beam force directions need to be shifted to the element system at time k+\mathrm{1} (i.e., real location, A(k+\mathrm{1})B(k+\mathrm{1})) with the element-rotating vector {\beta }_{k-(k+\mathrm{1})} until they are decomposed to the global system.

The original lengths of the beam and link elements are unchanged during calculation. However, for the multi-node sliding cable element, the tensionless length of each cable segment is a time-dependent value due to the sliding behaviors at the internal joints. Thus, internal force updating and mass lumping calculations are needed at each time increment. Figure 6 shows an example of a cable element at time k and time k+\mathrm{1} . Owing to the slippage at clip joints, the original lengths of the constituent segments are changed between two successive time spots. The original length at time k+\mathrm{1} differs from that at time k and needs to be determined from the balanced cable force by using Eq. (7). The particle mass can be calculated as follows:

(23)m_{i-A}=m_{i-B}={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho A{\displaystyle \frac{l_i(k+\mathrm{1})EA}{{\mathit{T}}_i(k+\mathrm{1})+EA}},

where {{\displaystyle m}}_{i-\mathrm{1}} and {{\displaystyle m}}_{i-\mathrm{2}} are the discrete masses at the two end nodes (nodes A and B) of cable piece i. The internal force decomposition process is similar to that of the link element,

(24){\mathit{F}}_i^{\mathrm{int}}(k+\mathrm{1})=[{\mathit{F}}_{i-A}^{\mathrm{int}}(k+\mathrm{1}),{\mathit{F}}_{i-B}^{\mathrm{int}}(k+\mathrm{1})]=EA{\displaystyle \frac{l_i(k+\mathrm{1})-l_i^{\mathrm{0}}(k+\mathrm{1})}{l_i^{\mathrm{0}}(k+\mathrm{1})}}[-e_i(k+\mathrm{1}),e_i(k+\mathrm{1})],

where {\mathit{F}}_i^{\mathrm{int}}(k+\mathrm{1}) represents the internal force of cable piece i at time spot k+ 1; {\mathit{F}}_{i-A}^{\mathrm{int}}(k+\mathrm{1})and{\mathit{F}}_{i-B}^{\mathrm{int}}(k+\mathrm{1})denote the nodal forces at the two end nodes A and B; l_i(k+\mathrm{1})and l_i^{\mathrm{0}}(k+\mathrm{1})are the cable length and original cable length at time k+ 1; and e_i(k+\mathrm{1})is the direction vector of cable piece i at time k+ 1.

Frictional cable sliding calculation and analytical process

The analytical process at each time step can be described as follows. First, based on the nodal positions at the last time step, the element forces are decomposed in global coordinate system and explicit time integration calculations are performed to obtain the current nodal position vectors. Then, the current element forces at link and beam elements are calculated. Next, the sliding states at the cable joints are determined and the tension force redistributing conditions are calculated with the original length method. During this step, the cable force redistribution process is repeated until the joint sliding state indicator {\delta }_k revealing no sliding happened. Then the mass lumping for the cable elements at the current time spot are performed. The above processes are repeated to obtain the displacement vectors and the mechanical factors for the next time step.

The entire analytical process is conducted in MATLAB with the flowchart given in Fig. 7. The central difference method is a conditionally convergent method. Thus, the integrating time intervals need to be sufficiently small to ensure a reasonable result. The time step for the structural components is commonly between 10⁻⁵ and 10⁻⁴ s.

Quasi-static analysis and validation

Continuous sliding cable

The effectiveness of the proposed numerical method is initially validated through a sliding cable test in Ref [²¹]. Figure 8 shows a continuous wire rope that passes through several rigid nails. The cable is fixed on one end and gradually pulled from the other end until a balanced state is reached. Each cable piece is connected to a force measurer that records the tension force, as shown in Fig. 8(b). According to Ref. [²¹], the inherent friction resistance at each turning joint is assumed to be zero and the Coulomb friction factor is 0.26. The calculation diagram is shown in Fig. 9. The cable characteristics, friction balance relation, and boundary settings are the same as those in Ref [²¹]. The damping factor is 0.5 for this example. Figure 10 shows the cable force changing process and the comparison with experimental data. Owing to the joint friction, the cable forces experience tension loss when crossing the turning joint. The consistent results validate the accuracy of the proposed dynamic sliding method in solving the frictional sliding problem under quasi-static loading conditions.

Cable-supported truss structure

A cable-supported truss is a new hybrid spatial structure that involves a tensioned cable-strut system below the truss to form a self-balanced structural system. During the construction of a cable-supported truss, the continuous cable is usually stretched from one side by applying the cable sliding effect to transfer the pretension force. The cable-strut joints will inevitably undergo friction loss, which in turn may influence the distribution of internal force. However, the commonly used discontinuous cable model neglects this tension loss, which then may result in improper evaluations. A sliding cable analysis for the cable-supported truss is performed (Fig. 11). The structural parameters are the same as those in Ref. [²¹], where elastic modulus is E= 2 × 10⁵ N/mm², member area is A= 100 mm², and loading force is P = 1 KN. The simulation investigates the one-side stretching process, in which the clip joints are not clamped thus enables the cable to slide through (Fig. 11 (b). Assumes that the inherent friction resistance is zero and the sliding friction coefficient is 0.2.

To validate this advanced explicit method, the extreme conditions in which cable-strut joints are frictionless (i.e., without inherent friction and Coulomb friction resistance) are established in ANSYS to validate the accuracy of the MATLAB solutions (Fig. 12). The results of the no-friction case in MATLAB are highly consistent with the ANSYS results. The frictionless sliding case presents symmetric internal force distribution, while the frictional sliding model has a different performance. The cable force has a gradient reduction from one stretching end to the other end. Furthermore, the uneven tension distribution induces asymmetric internal force distributions in the upper truss part, especially the member force of top chords (TT) and bottom chords (TB) and V-shaped cable struts (CS) (Fig. 12).

Dynamic analysis with frictional sliding effect

Seismic performance of cable-supported truss

The proposed method shows its effectiveness in solving quasi-static sliding problems and is then used to study the cable sliding effect on seismic performance. Earthquake excitation often induces violent transient force changes that may surpass the resisting ability at cable-strut joints, further leading to cable sliding. Here, the same cable-supported truss in Fig. 11 is used and the balanced model in Fig. 11(f) is adopted as the initial state of dynamic analysis. To better compare the frictional sliding effect, a regular artificial seismic excitation is adopted.

(25)acc=\mathrm{500}(\sin \mathrm{100}\mathit{T}+\sin \mathrm{120}t)\cdot \sin \mathrm{10}\mathit{T}\cdot (\mathrm{1}-\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{2}\mathit{T}),

where acc is the applied acceleration. No additional joints or attached masses are considered in the dynamic simulation except for the member qualities, but this renders the original earthquake responses to be somewhat weak. Then, as described in Eq. (25), the dynamic excitation is amplified to reach marked performance. Four friction conditions with different combinations of inherent friction resistance and friction coefficient (Table 1) are considered. The first is the no-sliding or discontinuous cable condition in which a large value is given to the inherent friction., Whereas the three others are sliding cases that consider two inherent friction-resisting capacities and two Coulomb coefficients. The discontinuous cable case is also performed in ANSYS-DYNA, and the internal force responses of the representative members are compared in Fig. 12. MATLAB obtained somewhat smaller response than that ANSYS-DYNA, but the difference is acceptable and the two solutions are consistent with each other. This finding proves the ability and effectiveness of the advanced method to solve earthquake problems.

Figures 13–14 show the responses of the member forces under different joint friction conditions. To fully reflect the influencing trends, the structural members are numbered and several representatives are selected, as in Fig. 13(b). When seismic excitations occur, the cable forces undergo violent oscillations, such as cable CC1 in Fig. 13(a). Once the tension gap between adjacent cables surpasses the friction resistance at the intermediate joint, the sliding occurs and the internal forces are redistributed. Thus, tension plateaus are observed in cases 3 and 4. The threshold of cable force increases with inherent friction resistance. For instance, case 2 has nearly the same seismic response as the discontinuous cable condition while case 4 has a higher friction-resisting ability than case 3. As shown in Fig. 13(c), the varying amplitude of the cable tension and the plateau force all decrease from two ends of the truss to the center.

Figure 14 shows the seismic response of the internal force in the upper truss and supporting struts. The sliding behavior can obviously magnify the seismic responses at the top and bottom chords. Furthermore, the amplification becomes intensified when the inherent friction of the cable-strut joints is decreased. In addition, the Coulomb coefficient has a comparatively weak effect on oscillation amplitude but can reduce the density of the high-force response. The oscillating movement in the discontinuous cable case is large in the one-fourth span region, and it is relatively weak at the end regions and the center region of the truss. However, the vibrating amplitudes in the sliding cable cases are limited, which leads to the amplified oscillation at the members in the two ends and the middle span region (TT2 and TT6). By contrast, the seismic response of TT4 is similar in all cases, and the bottom chords have nearly the same varying trend as the upper ones.

Figure 14(b) shows the internal force response at vertical chords during dynamic excitations. The side bracing chords (diagonal chords TD1–TD3 and vertical chords TV2–TV4) of the cable-supported plane truss present obvious oscillation response, but the cable sliding behaviors do not increase the seismic response and they even alleviate the vibrating amplitudes (TV4 in Fig. 14(b)). The internal force vibration is weak in the no-sliding cable case for the supporting struts at the mid-span region (diagonal chords TD4–TD6 and vertical chords TV4–TV6). When the sliding resisting ability at the cable-strut joints is decreased, the oscillation amplitudes are increased, and they reach nearly thrice of that of the no-sliding case. The results indicate that when the seismic evaluation or earthquake design does not consider the sliding cable effect, the safety of the mid-span bracing chords will be overestimated. The cable sliding behavior can increase dynamic response at the bracing struts, and this magnifying effect increases from the two end regions to the center part, as shown in Fig. 14(c). In all, the clip joint sliding induces an averaging effect on force damping but magnifies the internal force response of the upper truss structure.

Dynamic response of cable loss

Cables are exposed to corrosion, fatigue, improper loads, or poorly constructed measurements during the service life of cable-supported structures, and these scenarios may reduce joint friction resistance or cause cable breaks. Previous studies on sudden cable loss usually adopt discontinuous cable simplification and seldom consider the frictional sliding effect. Here, the proposed explicit solving method is applied to the cable rupture or sudden tension loss problems to explore the influence of cable sliding on structure responses. In accordance with the structure type and practical failure probability, only the end cable break case is considered in this study (Fig. 15(a)). Three inherent friction levels (5, 3, and 1 kN) and two Coulomb friction coefficients (0.4 and 0.2) are selected. A base model with an inherent friction resistance of 50 kN and a coefficient of 0.4 is pre-calculated to simulate the discontinuous cable condition and then compared with ANSYS for validation. The analyzed models are labeled as Cxy (C refers to the case, x is the inherent friction, and y is the coefficient, e.g., C5k04 refers to the case with 5 kN inherent friction and 0.4 coefficient) while the base condition is numbered as C0.

Figure 15 shows the internal force response of a top truss chord TT6 and a cable piece CC2 under different inherent friction conditions. The MATLAB and ANSYS results of the base model C0 are compared in the same plot. The internal forces in MATLAB vibrate around the ANSYS values, both for the upper truss members and sub-tensioned cables, which proves the effectiveness of the proposed method. Cable sliding has an obvious loosening effect on the top chords, and this varying trend aggravates when the inherent friction resistance is reduced. Moreover, the lower the friction resistance is at the cable-strut joints, the weaker the ability of the sub-cable system to retain tension levels and the more violent the oscillation occurs during rupture impact. Figure 15(c) shows the partially enlarged illustration of CC2 tension during the rupture moment. The sudden rupture of cable CC1 has resulted in an immediate tension drop to 0. A sliding occurs once the transient tension change surpasses the resisting capacity at the clip joints, and this phenomenon manifests as strength platforms after the cable break scenario in the tension history (Fig. 15 (c)). The tension of CC2 is limited within the friction-resisting capacity at the intermediate joint between CC1 and CC2 owing to the sliding effect. When the sliding amount is increased, the cables are slackened and the average value of cable tension is gradually decreased. If the frictional resisting strength is high, then the cable forces can retain their comparatively high levels and undergo damping with large amplitudes.

The tension histories of the remaining cables (Fig. 16) are compared to fully understand the cable slackening effect. The force retaining ability and average of cable tension all decrease with the inherent friction level. The CC5 and CC6 cables in model C5k04 have consistent vibrations at comparatively high levels, which indicate minimal tension loss. By contrast, most of the cable forces in C1k04 have dropped to low levels with nearly arithmetic gradients between adjacent cable pieces.

Apart from the sudden break conditions, the corresponding modeling of gradual cable loss is also performed. The tension force of CC1 gradually decreases to zero. Figure 16(b) illustrates the cable force response in the gradual fail case of C3k04, in which the cable tensions are all retained at the higher level relative to those of the sudden break cases. The differences indicate that sudden tension loss and the effect of induced impact can aggravate cable sliding and slackening behaviors.

After the cable break, the internal force oscillations gradually decay to steady (As in Figs. 15(b) and 15(c)). Then the stabilized values are calculated, and some representative members are selected (Fig. 17) and compared for all friction conditions and between the two failure processes. Given the comparatively large intersection angle between adjacent cables, the Coulomb coefficient has a minimal effect on member force under both failure modes. This scenario reflects as having nearly the same internal forces between different Coulomb coefficient cases, such as C3k04 and C3k02. In a rupture fail condition, the tension loss extent is larger than the corresponding gradual fail case, which result in a large tension change in the former before and after the cable break. The member force of the top chord TT6 has a dropping trend when sliding is allowed, and this dropping trend aggravates with the decrease of inherent friction capacity. Moreover, the extent of the decrease is more intensified in the rupture failure condition than the gradual fail ones. The tension drop magnitude at the top chords increases from the two ends to the center. Meanwhile, the internal forces of the bottom chords decrease at the two end regions, but they appear with increase trends at the mid-span region. The member force at the diagonal or vertical chords undergoes a decreasing trend, and the extent of the drop is more obvious for the end struts. Therefore, the cable sliding behaviors in the cable-supported truss can greatly reduce the resisting ability for sudden cable breaks and a strong friction resistance at the cable-strut joints can help retain the internal forces.

Conclusions

An FE analytical method for frictional cable sliding behaviors is proposed in this paper. The cable sliding behaviors and the resultant dynamic responses are solved by combining the explicit solution framework with the cable force redistribution calculations that consider joint friction. By adopting the explicit solution framework, the nodal coordinates at the next time step can be obtained through explicit time integration. Then the cable sliding process in each time increment is solved with the original length method that uses cable length and original length relations, after which the balanced tension distributions after frictional sliding are calculated.

The effectiveness of the proposed method is validated by the quasi-static simulations of a continuous sliding cable and the stretching process of a cable-supported truss. The continuous sliding cable exhibits gradual tension loss when the turning point is crossed. In the cable-supported truss modeling, a gradient tension reduction due to joint frictional loss induces uneven internal force changes and asymmetric inner force distribution to the upper truss parts.

When a cable-supported truss is under seismic excitations, the cable sliding behaviors have an averaging effect on the force damping of the sub-cable system. The frictional sliding behaviors can magnify the seismic response to the internal force of the upper truss structure. For some members, the amplification is somewhat dramatic. Meanwhile, with regard to the cable break problems, cable sliding has an obvious loosening effect on the tension state of the cable-supported truss, and the force changes aggravate with the reduction of inherent friction resistance at the cable-strut joints. The smaller the friction resistances of the cable-strut joints are, the lower the ability of the sub-cables to retain their tension levels. Thus, more violent damping can occur during rupture impact.

All the simulations and comparisons indicate that the discontinuous cable assumptions can induce large deviations or even lead to false evaluations of the dynamic response of cable-supported structures. By integrating the failure mechanism or plasticity description of the material properties, the advanced method is able to simulate highly complicated problems, such as degradation failure or plastic deformation propagation processes under dynamic excitations.

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