A finite element formulation for clustered cables with sliding-induced friction

Abstract

Clustered (continuous) cables reflect an advantageous solution for reducing the number of tensile elements in engineering systems. During the tensioning or activation of tensile structures, such as cable structures, membranes and tensegrity structures, the deficiency of having to control too many elements can be overcome by employing clustered cables. The use of clustered cables has been shown to alter the structural behavior of tensile systems by modifying the force distribution in the systems. This effect has been showcased under the assumption of frictionless sliding of the cable elements across nodes or pulleys. However, friction can have also impact on the behavior of the system. In this paper, a new Finite Element formulation is proposed for the static analysis of tensile structures with clustered cables. The proposed formulation accommodates sliding-induced friction by the consideration of the Euler-Eytelwein equation as well as geometric nonlinearities. It is found that the sliding-induced friction can significantly modify the force distribution in the system. The applicability and importance of the proposed formulation is demonstrated through the analysis of two examples from the literature.

Keywords

Finite element clustered cable continuous cable sliding friction pulley

Introduction

Cable structures are lightweight tensile systems that have been used for a large variety of applications such as suspension and cable-stayed bridges, long-span roof structures, as well as membrane, tensegrity and hybrid structures. In many of these cases, cable lengths are adjusted in order to establish a required pre-stress state in the system¹ defined by design criteria such as safety and serviceability. However, the pre-tensioning process often becomes challenging especially when the number of cables to adjust is large. The use of clustered cables, continuous cables that slide over one or multiple nodes or pulleys,² allows to reduce the number of cables requiring pre-tensions. Clustered cables are also a potential solution for the control of active and deployable cable systems.³

Clustered cables have always been an important topic of investigation for the analysis of tensile structures.^2,4,5 Aufaure⁶ pioneered the analysis of continuous cables and developed a Finite Element formulation for cable sliding over pulleys that was used to model the cable-hanging process of multi-span electric transmission lines. Kwan and Pellegrino⁷ proposed a matrix formulation for a macro-element of an active cable composed of two or more straight segments deriving the equilibrium and flexibility matrices of active cable and pantographic elements employed in deployable structures. Zhou et al.⁸ developed a three-node sliding-cable element that was used to model parachute folding. Based on the assumption of uniform strain, Chen et al.⁹ presented a formulation of a multi-node sliding-cable element that was applied in the static and nonlinear stability analysis of a Suspen-Dome structure.

Continuous cables are also applied in tensegrity structures as tensegrity systems often include a large number of cables. Genovese¹⁰ studied the form-finding and analysis of tensegrity structures with sliding cables. Moored and Bart-Smith² presented a novel formulation for the potential energy, equilibrium equations, and stiffness matrix of tensegrity structures with clustered cables revealing that cable continuity changes the mechanical behavior of the systems as internal forces in continuous elements are assumed to be identical and kinematic constraints are reduced. Clustered cables were implemented in the dynamic relaxation method, a form-finding and analysis method for tensile structures,^11,12 by Bel Hadj Ali et al.¹³ The method was also employed by Hincz¹⁴ and Pauletti et al.⁴ for nonlinear analysis of arch-supported cable-net structures and membrane structures with continuous cables, respectively. Zhang et al.^5,15 employed a co-rotational formulation to derive a Finite Element formulation for the nonlinear analysis of clustered tensegrities demonstrating that the proposed formulation is equivalent to those proposed by Moored and Bart-Smith² and Bel Hadj Ali et al.¹³ More recently, Kan et al¹⁶ presented a sliding cable element using multibody dynamic methodology with application to nonlinear dynamic deployment analysis of clustered tensegrity.

All aforementioned studies are based on the assumption of frictionless sliding of the cable elements across nodes or pulleys. However, experimental investigations have revealed that friction has an important effect on the equilibrium of tensile structures with continuous elements. Liu et al¹⁷ experimentally studied the pre-stressing of Suspen-Dome structures. Experimental tests performed on a 10.8 m span scaled dome prototype revealed that the friction coefficient at sliding nodes varies between 0.1 and 0.5 depending on material properties and the stress level in cable elements. Results also showed significant deviations in the magnitude of internal forces in the structure with respect to their predicted design values exceeding them by 30% for some elements. Rhode-Barbarigos et al.^18,19 studied numerically and validated experimentally in a near-full scale physical model the deployment and control of a tensegrity footbridge system with clustered cables. However, recent tests conducted on the same physical model revealed that structural behavior cannot be modeled satisfactorily using the assumption of friction-free cable sliding across joints.^20–22

Taking into account sliding-induced friction in the analysis of tensile structures with clustered cables is important for correctly modeling their behavior. However, only few studies have focused on this issue. An attempt to take frictional sliding into consideration can be found in Veuve et al.²³ where the geometrically exact formulation of a cable Finite Element allowing sliding with friction is presented. Lee et al.²⁴ and Ju and Choo²⁵ derived the tangent stiffness matrix of a continuous cable including frictional effects. However, the geometric stiffness of cable elements was not considered in the formulation limiting thus its applicability to only geometrically linear structures. Based on thermal expansion, Chen et al²⁶ derived cable-sliding criterion equations including frictional effects. They implemented the derived equations into ANSYS^® for the static analysis of cable-pulley systems. A cable-pulley system with friction was also included in¹⁴ where dynamic relaxation was employed for the analysis of arch-supported structures. A more general formulation for the static analysis of tensile structures with friction using dynamic relaxation was proposed by Bel Hadj Ali et al.²⁷ However, no related work using the Finite Element method was found to provide a general formulation for clustered cables with sliding-induced friction.

In this paper, a general Finite Element formulation for clustered cables with sliding-induced friction is proposed. The proposed approach builds upon the work presented by Lee et al.²⁴ and Ju and Choo²⁵ to include sliding frictional effects to the derived tangent stiffness matrix. The derived Finite Element is then enriched with the consideration of sliding friction at the related nodes. The remainder of the paper is organized as follows: Section 2 introduces the governing equations of a two sub-element clustered cable considering frictionless sliding as well as sliding with friction. In Section 3, the stiffness matrix formulation of a frictionless clustered cable is presented. Section 4 introduces the development of a tangent stiffness matrix of a clustered cable taking into account sliding-induced friction. The proposed formulation is validated through a series of numerical examples from literature in Section 5. Finally, conclusions are discussed in Section 6.

Governing equations of a two sub-element clustered cable

Frictionless sliding

Consider a clustered cable system of three nodes where the middle node is assumed to be a pulley (Figure 1). The unconstrained reference node 2 is connected to nodes 1 and 3 by a clustered cable composed of sub-elements e₁₂ and e₂₃, respectively. The clustered cable is assumed to be tensioned and to run over a frictionless pulley at node 2.

Figure 1.

A two sub-elements sliding/clustered cable.

In any tension state, the length of the clustered cable is the sum of the length of the two sub-elements:

l = l_{1} + l_{2}

(1)

where l₁ and l₂ are the lengths of the sub-elements e₁₂ and e₂₃, respectively. Equivalently, the rest length of the clustered element, marked with the subscript zero, may be defined in a similar way as:

l_{0} = l_{0, 1} + l_{0, 2}

(2)

where l_0,1 and l_0,2 are the lengths of the sub-elements e₁₂ and e₂₃, respectively. With the assumption that the cluster cable runs over a frictionless pulley, its two sub-elements carry the same tensile force t given by:

t = E A \frac{l - l_{0}}{l_{0}}

(3)

where E and A are Young modulus and cross-section area of the clustered cable, respectively.

The coordinate vectors of the three nodes are given by:

x_{n}^{T} = {x_{n i}, x_{n j}, x_{n k}} n = 1, 2, 3

(4)

The direction cosines for sub-elements e₁₂ and e₂₃ are:

c_{1}^{T} = (x_{2} - x_{1}) / l_{1}; c_{2}^{T} = (x_{3} - x_{2}) / l_{2}

(5)

The equilibrium equations of node 1 are given by equation (6), where f_1i, f_1j and f_1k are the components of the external force F₁ applied on node 1:

\begin{array}{l} f_{1, i} = (x_{2 i} - x_{1 i}) t / l_{1} \\ f_{1, j} = (x_{2 j} - x_{1 j}) t / l_{1}; F_{1}^{T} = {f_{1 i} f_{1 j} f_{1 k}} \\ f_{1, k} = (x_{2 k} - x_{1 k}) t / l_{1} \end{array}

(6)

Equilibrium at nodes 1, 2, and 3 can thus be written in terms of the cable tension, t, in component form:

\begin{array}{l} F_{1} = - c_{1} t \\ F_{2} = (c_{1} - c_{2}) t \\ F_{3} = c_{2} t \end{array}

(7)

Based on the above governing equations, a clustered cable can be easily incorporated into vector-based numerical analysis schemes, such as dynamic relaxation, or differentiated with respect to nodal coordinates to define the tangent stiffness matrix for a Finite Element implementation.

Sliding with friction

To more accurately model the behavior of the clustered cable system of three nodes presented in the previous section, friction at the contact area between the clustered cable and the pulley at node 2 (Figure 2) is considered in the governing equation of the continuous cable elements.

Figure 2.

A two sub-elements clustered cable with slip at node 2.

When friction is small, cable sliding is possible. However, at a certain level, friction lock will occur and the frictional resistance at contact area will be large enough to hinder the cable from sliding. At this stage the internal forces t₁ and t₂ in the two sub-elements of the continuous cable can be expressed as:

\begin{array}{l} t_{1} = E A \frac{l_{1} - {\bar{l}}_{0, 1}}{{\bar{l}}_{0, 1}}; {\bar{l}}_{0, 1} = l_{0, 1} - s_{2} \\ t_{2} = E A \frac{l_{2} - {\bar{l}}_{0, 2}}{{\bar{l}}_{0, 2}}; {\bar{l}}_{0, 2} = l_{0, 2} + s_{2} \end{array}

(8)

where E and A are Young modulus and cross-section area of the clustered cable, respectively. l₁ and l₂ are the deformed lengths of the two sub-elements, while ${\bar{l}}_{0, 1}$ and ${\bar{l}}_{0, 2}$ are their corresponding initial lengths. s₂ is the sliding magnitudes occurring at node 2 (Figure 2). When friction locking occurs, the sliding magnitude at node 2, s₂, directly affects rest-lengths of the two sub-elements. In the absence of friction, modification of sub-element rest-lengths does not affect the distribution of axial forces in the sub-elements since only the rest-length of the entire cable is taken into account. However, assuming sliding-induced friction, rest-length modification directly affects the tension forces in the two sub-elements. Consequently, in order to correctly model the force distribution in the clustered cable, the sliding magnitude should also be considered.

Equilibrium at nodes 1, 2, and 3 can be written in terms of the sub-element tension (t₁ and t₂) in component form:

\begin{array}{l} F_{1} = - c_{1} t_{1} \\ F_{2} = c_{1} t_{1} - c_{2} t_{2} \\ F_{3} = c_{2} t_{2} \end{array}

(9)

where c ₁ and c ₂ are the direction cosines for sub-elements e₁₂ and e₂₃. The system of equilibrium equations (equation (9)) is no longer sufficient for the analysis of the clustered cable when taking into account sliding-induced friction. As formulated for the simple structure of Figure 2, the system of equilibrium equations has 10 unknowns, that is, the sliding magnitude s₂, and the 9 nodal displacements with only 9 equations available. The additional equation required for the solution of the system can be obtained through the development of a relation between tension forces around the sliding node.

Consider the forces at the pulley edges of node 2 (Figure 3(a)). These forces result from axial forces in the two cable sub-elements arriving at the pulley. The relationship between forces at both sides of sliding edge is given by the Euler-Eytelwein equation as:

f_{23} = α_{2} f_{21}

(10)

Figure 3.

(a) Cable-induced forces acting on a pulley; (b) representation of the contact angle.

where the force ratio α₂ is given by:

α_{2} = e^{s i g n (s_{2}) μ θ}

(11)

The force ratio α₂ is a function of the angle of contact, θ, defined in Figure 3(b) and the coefficient of friction, μ, between the pulley and the cable. The ratio of the two forces at pulley edges depends also on sliding direction defined by the sign of the sliding coordinate (sign(s₂)).

Stiffness matrix formulation of a frictionless clustered cable

The equilibrium equations formulated in the previous section are used for the development of the tangent stiffness matrix of a clustered cable. The example of the clustered cable system of three nodes is presented first (Section 3.1) followed by the generalization of the formulation (Section 3.2).

Two sub-element clustered cable (no friction)

In order to define the tangent stiffness, the component equilibrium equations (equation (7)) are differentiated with respect to the j-coordinate of node 1:

\begin{array}{l} \frac{\partial f_{1 i}}{\partial x_{1 j}} = - (x_{2 i} - x_{1 i}) \frac{\partial q_{1}}{\partial x_{1 j}} + δ_{i j} q_{1} \\ \frac{\partial f_{2 i}}{\partial x_{1 j}} = (x_{2 i} - x_{1 i}) \frac{\partial q_{1}}{\partial x_{1 j}} - δ_{i j} q_{1} - (x_{3 i} - x_{2 i}) \frac{\partial q_{2}}{\partial x_{1 j}} \\ \frac{\partial f_{3 i}}{\partial x_{1 j}} = (x_{3 i} - x_{2 i}) \frac{\partial q_{2}}{\partial x_{1 j}} \end{array}

(12)

where $δ_{i j} = 1$ if $i = j$ , $δ_{i j} = 0$ otherwise; q₁ and q₂ are the force densities in the two sub-elements of the sliding cable defined by the ratio of the force in the sub-element over its length:

q_{1} = \frac{t}{l_{1}}; q_{2} = \frac{t}{l_{2}}

(13)

The rate of change of force densities q₁ and q₂ with respect to the position of the nodes is given by:

\frac{\partial q_{1}}{\partial x_{i j}} = \frac{\partial q_{1}}{\partial l_{1}} \frac{\partial l_{1}}{\partial x_{i j}}; \frac{\partial q_{2}}{\partial x_{i j}} = \frac{\partial q_{2}}{\partial l_{2}} \frac{\partial l_{2}}{\partial x_{i j}}

(14)

where geometry shows that:

\begin{array}{l} \frac{\partial l_{1}}{\partial x_{1 j}} = - c_{1 j}; \frac{\partial l_{1}}{\partial x_{2 j}} = c_{1 j}; \frac{\partial l_{1}}{\partial x_{3 j}} = 0 \\ \frac{\partial l_{2}}{\partial x_{1 j}} = 0; \frac{\partial l_{2}}{\partial x_{2 j}} = - c_{2 j}; \frac{\partial l_{2}}{\partial x_{3 j}} = c_{2 j} \end{array}

(15)

and hence:

\begin{array}{l} \frac{\partial q_{1}}{\partial x_{1 j}} = - c_{1 j} \frac{\partial q_{1}}{\partial l_{1}}; \frac{\partial q_{1}}{\partial x_{2 j}} = c_{1 j} \frac{\partial q_{1}}{\partial l_{1}}; \frac{\partial q_{1}}{\partial x_{3 j}} = 0 \\ \frac{\partial q_{2}}{\partial x_{1 j}} = 0; \frac{\partial q_{2}}{\partial x_{2 j}} = - c_{2 j} \frac{\partial q_{2}}{\partial l_{2}}; \frac{\partial q_{2}}{\partial x_{3 j}} = c_{2 j} \frac{\partial q_{2}}{\partial l_{2}} \end{array}

(16)

The rate of change of force density q₁ with respect to length can be written as:

\frac{\partial q_{1}}{\partial l_{1}} = \frac{\partial (t / l_{1})}{\partial l_{1}} = \frac{1}{l_{1}} \frac{\partial t}{\partial l_{1}} - \frac{t}{l_{1}^{2}} = \frac{1}{l_{1}} (\frac{\partial t}{\partial l_{1}} - q_{1})

(17)

Based on equation (3), the rate of change of the cable tension t with respect to element length can be written as:

\frac{\partial t}{\partial l_{1}} = \frac{E A}{l_{0}} = k

(18)

Substituting equation (18) into equation (17) yields:

\frac{\partial q_{1}}{\partial l_{1}} = \frac{1}{l_{1}} (k - \frac{t}{l_{1}}) = \frac{1}{l_{1}} {\bar{k}}_{1}

(19)

Similarly, for force density q₂:

\frac{\partial q_{2}}{\partial l_{2}} = \frac{1}{l_{2}} (k - \frac{t}{l_{2}}) = \frac{1}{l_{2}} {\bar{k}}_{2}

(20)

Equation (12) can thus be written as:

\begin{array}{l} \frac{\partial f_{1 i}}{\partial x_{1 j}} = c_{1 i} {\bar{k}}_{1} c_{1 j} + δ_{i j} q_{1} \\ \frac{\partial f_{2 i}}{\partial x_{1 j}} = - c_{1 i} {\bar{k}}_{1} c_{1 j} - δ_{i j} q_{1} + c_{2 i} {\bar{k}}_{2} c_{1 j} \\ \frac{\partial f_{3 i}}{\partial x_{1 j}} = - c_{2 i} {\bar{k}}_{2} c_{1 j} \end{array}

(21)

or, in a matrix form:

\begin{array}{l} \frac{\partial F_{1}}{\partial x_{1}} = c_{1} {\bar{k}}_{1} c_{1}^{T} + q_{1} I_{3} \\ \frac{\partial F_{2}}{\partial x_{1}} = - c_{1} {\bar{k}}_{1} c_{1}^{T} + q_{1} I_{3} + c_{2} {\bar{k}}_{2} c_{1}^{T} \\ \frac{\partial F_{3}}{\partial x_{1}} = - c_{2} {\bar{k}}_{2} c_{1}^{T} \end{array}

(22)

where I₃ is the identity matrix. Following the same process, the component equilibrium equations are differentiated with respect to the coordinates of nodes 2 and 3 yielding the remaining components of the tangent stiffness matrix (K_T) of the clustered cable. K_T is a 9 × 9 symmetric matrix that can also be written as the sum of an elastic stiffness matrix, K_E, and a geometric stiffness matrix, K_G.

K_{T} = K_{E} + K_{G}

(23)

The elastic part of the tangent stiffness matrix is given by:

K_{E} = \frac{E A}{l_{0}} [\begin{matrix} c_{1} c_{1}^{T} & - c_{1} c_{1}^{T} + c_{1} c_{2}^{T} & - c_{1} c_{2}^{T} \\ - c_{1} c_{1}^{T} + c_{2} c_{1}^{T} & (c_{1} - c_{2}) (c_{1}^{T} - c_{2}^{T}) & c_{1} c_{2}^{T} - c_{2} c_{2}^{T} \\ - c_{2} c_{1}^{T} & c_{2} c_{1}^{T} - c_{2} c_{2}^{T} & c_{2} c_{2}^{T} \end{matrix}]

(24)

which can also be written as:

K_{E} = K_{E 1} + K_{E 2} = [\begin{matrix} K_{11} & - K_{11} & 0 \\ - K_{11} & K_{11} + K_{22} & - K_{22} \\ 0 & - K_{22} & K_{22} \end{matrix}] + [\begin{matrix} 0 & K_{12} & - K_{12} \\ K_{21} & - K_{12} - K_{21} & K_{12} \\ - K_{21} & K_{21} & 0 \end{matrix}]

(25)

where:

k_{i j} = c_{I} k c_{j}^{T} = c_{I} (\frac{E A}{l_{0}}) c_{j}^{T}

(26)

The geometric stiffness matrix is given by:

K_{G} = [\begin{matrix} K_{g 1} & - K_{g 1} & 0 \\ - K_{g 1} & K_{g 1} + K_{g 2} & - K_{g 2} \\ 0 & - K_{g 2} & K_{g 2} \end{matrix}]

(27)

where sub-matrices k_g1 and k_g2 are defined as follow:

k_{g 1} = \frac{t}{l_{1}} . (I_{3} - c_{1} c_{1}^{T}); K_{g 2} = \frac{t}{l_{2}} . (I_{3} - c_{2} c_{2}^{T})

(28)

It should be noted that with the removal of the pulley in node 2, the proposed tangent stiffness matrix results in the well-established tangent stiffness matrix of a cable element:

K_{T} = (\frac{E A}{l_{0}} - \frac{t}{l}) [\begin{matrix} c c^{T} & - c c^{T} \\ - c c^{T} & - c c^{T} \end{matrix}] + \frac{t}{l} [\begin{matrix} I & - I \\ - I & I \end{matrix}]

(29)

Multi-node clustered cable (no friction)

For a clustered cable system where the cable slides over multiple pulleys the tangent stiffness matrix can be obtained by assembling the elementary stiffness matrices corresponding to its constitutive elements.

Considering a multi-node clustered cable with N sub-elements, the tangent stiffness matrix is given by:

K_{T} = K_{E 1} + K_{E 2} + K_{G}

(30)

The elastic parts of the tangent stiffness matrix are given by:

K_{E 1} = \sum_{e = 1}^{N} c_{0} k_{e} c_{0}^{T}; k_{e} = c_{E} . k . c_{E}^{T}; c_{0}^{T} = {\begin{matrix} 1 & - 1 \end{matrix}}

(31)

\begin{array}{l} K_{E 2} = \sum_{e i = 1}^{N} \sum_{e j = 1}^{N} c_{0} k_{e i e j} c_{0}^{T} (e i \neq e j); \\ K_{e i e j} = c_{E i} . k . c_{E j}^{T}; c_{0}^{T} = {\begin{matrix} 1 & - 1 \end{matrix}} \end{array}

(32)

where c is the direction cosine, k is the axial stiffness and k the stiffness matrix of a sub-element.

The geometric part of the tangent stiffness matrix is given by:

K_{G} = \sum_{e = 1}^{N} c_{0} k_{g e} c_{0}^{T}; k_{g e} = \frac{t}{l_{e}} . (I_{3} - c_{E} c_{E}^{T}); c_{0}^{T} = {\begin{matrix} 1 & - 1 \end{matrix}}

(33)

where I₃ is the identity matrix. As noticed by,⁵ assembly of K_E₁ and K_G can be achieved by the standard Finite Element Method assembly procedure. In contrast, for K_E₂ a special assembly procedure is needed. Table 1 provides a pseudo-code for the assembly of the geometric part of the tangent stiffness K_G.

Table 1.

Pseudo-code for the assembly of the geometric part of the tangent stiffness.

For each clustered cable
N : the number of cable sub-elements A : Cross-section area of the clustered cable E : Young modulus of the clustered cable l ₀ : rest-length of the clustered cable Compute: $k = \frac{E A}{l_{0}}$
For i=1 to N a_i, b_i: Sub-elements end nodes Compute the member, i, direction cosines: c_i Compute location vector: $l o c_{i} = [{3 a}_{i} {- 2, 3 a}_{i} {- 1, 3 a}_{i} {, 3 b}_{i} {- 2, 3 b}_{i} {- 1, 3 b}_{i}]$
For j=i+1 to N a_j, b_j: Sub-elements end nodes Compute the member, j, direction cosines: c_j Compute location vector: $l o c_{j} = [{3 a}_{j} {- 2, 3 a}_{j} {- 1, 3 a}_{j} {, 3 b}_{j} {- 2, 3 b}_{j} {- 1, 3 b}_{j}]$ Compute: $k_{i j} = c_{i} . k . c_{j}^{T}$ Update the stiffness matrix of the clustered cable: End for j
End for i

Stiffness matrix formulation of a clustered cable with sliding-induced friction

Two sub-element clustered cable with sliding-induced friction

In this section, sliding-induced friction is added in the development of a modified tangent stiffness matrix for clustered cables. Similarly to the previous section, the clustered cable system of three nodes with sliding-friction at the central node is presented first (Section 4.1) followed by the generalization of the formulation for a multi-node clustered cable (Section 4.2).

Differentiating the component equilibrium equations with respect to the j-coordinate of node 1 gives:

\begin{array}{l} \frac{\partial f_{1 i}}{\partial x_{1 j}} = - (x_{2 i} - x_{1 i}) \frac{\partial q_{1}}{\partial x_{1 j}} + δ_{i j} q_{1} \\ \frac{\partial f_{2 i}}{\partial x_{1 j}} = (x_{2 i} - x_{1 i}) \frac{\partial q_{1}}{\partial x_{1 j}} - δ_{i j} q_{1} - (x_{3 i} - x_{2 i}) \frac{\partial q_{2}}{\partial x_{1 j}} \\ \frac{\partial f_{3 i}}{\partial x_{1 j}} = (x_{3 i} - x_{2 i}) \frac{\partial q_{2}}{\partial x_{1 j}} \end{array}

(34)

where q₁ and q₂ are the force densities in the two sub-elements of the sliding cable:

q_{1} = \frac{t_{1}}{l_{1}}; q_{2} = \frac{t_{2}}{l_{2}}

(35)

The rates of change of the force densities q₁ and q₂ with respect to the position of the nodes are given by:

\frac{\partial q_{1}}{\partial x_{i j}} = \frac{\partial q_{1}}{\partial l_{1}} \frac{\partial l_{1}}{\partial x_{i j}}; \frac{\partial q_{2}}{\partial x_{i j}} = \frac{\partial q_{2}}{\partial l_{2}} \frac{\partial l_{2}}{\partial x_{i j}}

(36)

where geometry shows that:

\frac{\partial l_{1}}{\partial x_{1 j}} = - c_{1 j}; \frac{\partial l_{1}}{\partial x_{2 j}} = c_{1 j}; \frac{\partial l_{2}}{\partial x_{2 j}} = - c_{2 j}; \frac{\partial l_{2}}{\partial x_{3 j}} = c_{2 j}

(37)

and hence:

\begin{array}{l} \frac{\partial q_{1}}{\partial x_{1 j}} = - c_{1 j} \frac{\partial q_{1}}{\partial l_{1}}; \frac{\partial q_{1}}{\partial x_{2 j}} = c_{1 j} \frac{\partial q_{1}}{\partial l_{1}}; \frac{\partial q_{1}}{\partial x_{3 j}} = 0 \\ \frac{\partial q_{2}}{\partial x_{1 j}} = 0; \frac{\partial q_{2}}{\partial x_{2 j}} = - c_{2 j} \frac{\partial q_{2}}{\partial l_{2}}; \frac{\partial q_{2}}{\partial x_{3 j}} = c_{2 j} \frac{\partial q_{2}}{\partial l_{2}} \end{array}

(38)

For each cable sub-element i, the rate of change of the force density with respect to the sub-element length can be written as:

\frac{\partial q_{i}}{\partial l_{i}} = \frac{\partial (t_{i} / l_{i})}{\partial l_{i}} = \frac{1}{l_{i}} \frac{\partial t_{i}}{\partial l_{i}} - \frac{t_{i}}{l_{i}^{2}} = \frac{1}{l_{i}} (\frac{\partial t_{i}}{\partial l_{i}} - q_{i})

(39)

Based on equation (8), the rate of change of the cable sub-element tension t_i with respect to sub-element length can be written as:

\frac{\partial t_{i}}{\partial l_{i}} = \frac{E A}{{\bar{l}}_{0, i}} = k_{i}^{*}

(40)

Substituting equation (41) into equation (40) yields for the force density q₁:

\frac{\partial q_{1}}{\partial l_{1}} = \frac{1}{l_{1}} (k_{1}^{*} - \frac{t_{1}}{l_{1}}) = \frac{1}{l_{1}} {\bar{k}}_{1}^{*}

(41)

and similarly for the force density q₂:

\frac{\partial q_{2}}{\partial l_{2}} = \frac{1}{l_{2}} (k_{2}^{*} - \frac{t_{2}}{l_{2}}) = \frac{1}{l_{2}} {\bar{k}}_{2}^{*}

(42)

equation (35) can then be written as:

\begin{array}{l} \frac{\partial f_{1 i}}{\partial x_{1 j}} = c_{1 i} {\bar{k}}_{1}^{*} c_{1 j} + δ_{i j} q_{1} \\ \frac{\partial f_{2 i}}{\partial x_{1 j}} = - c_{1 i} {\bar{k}}_{1}^{*} c_{1 j} - δ_{i j} q_{1} \\ \frac{\partial f_{3 i}}{\partial x_{1 j}} = 0 \end{array}

(43)

or, in a matrix form:

\begin{array}{l} \frac{\partial F_{1}}{\partial x_{1}} = c_{1} {\bar{k}}_{1}^{*} c_{1}^{T} + q_{1} I_{3} \\ \frac{\partial F_{2}}{\partial x_{1}} = - c_{1} {\bar{k}}_{1}^{*} c_{1}^{T} + q_{1} I_{3} \\ \frac{\partial F_{3}}{\partial x_{1}} = 0 \end{array}

(44)

Similarly, differentiating the component equilibrium expressions with respect to the coordinates of nodes 2 and 3 gives:

\begin{array}{l} \frac{\partial F_{1}}{\partial x_{2}} = - c_{1} {\bar{k}}_{1}^{*} c_{1}^{T} - q_{1} I_{3} \\ \frac{\partial F_{2}}{\partial x_{2}} = c_{1} {\bar{k}}_{1}^{*} c_{1}^{T} - q_{1} I_{3} + c_{2} {\bar{k}}_{2}^{*} c_{2}^{T} + q_{2} I_{3} \\ \frac{\partial F_{3}}{\partial x_{2}} = - c_{2} {\bar{k}}_{2}^{*} c_{2}^{T} - q_{2} I_{3} \end{array}

(45)

\begin{array}{l} \frac{\partial F_{1}}{\partial x_{3}} = 0 \\ \frac{\partial F_{2}}{\partial x_{3}} = - c_{2} {\bar{k}}_{2}^{*} c_{2}^{T} - q_{2} I_{3} \\ \frac{\partial F_{3}}{\partial x_{3}} = c_{2} {\bar{k}}_{2}^{*} c_{2}^{T} + q_{2} I_{3} \end{array}

(46)

In the clustered cable sub-elements, internal forces depend on node coordinates as well as the sliding magnitude occurring at the various nodes. Therefore, the component equilibrium equations are differentiated with respect to sliding magnitude in node 2:

\begin{array}{l} \frac{\partial f_{1 i}}{\partial s_{2}} = - \frac{1}{l_{1}} (x_{2 i} - x_{1 i}) \frac{\partial t_{1}}{\partial s_{2}} \\ \frac{\partial f_{2 i}}{\partial s_{2}} = \frac{1}{l_{1}} (x_{2 i} - x_{1 i}) \frac{\partial t_{1}}{\partial s_{2}} - \frac{1}{l_{2}} (x_{3 i} - x_{2 i}) \frac{\partial t_{2}}{\partial s_{2}} \\ \frac{\partial f_{3 i}}{\partial s_{2}} = \frac{1}{l_{2}} (x_{3 i} - x_{2 i}) \frac{\partial t_{2}}{\partial s_{2}} \end{array}

(47)

For each cable sub-element, the rate of change of the internal forces t₁ and t₂ with respect to the slip component s₂ can be written as:

\frac{\partial t_{1}}{\partial s_{2}} = \frac{E A}{{\bar{l}}_{0, 1}} . (\frac{l_{1}}{{\bar{l}}_{0, 1}}) = {\hat{k}}_{1}; \frac{\partial t_{2}}{\partial s_{2}} = - \frac{E A}{{\bar{l}}_{0, 2}} . (\frac{l_{2}}{{\bar{l}}_{0, 2}}) = - {\hat{k}}_{2}

(48)

Equation (48) can thus be written as:

\frac{\partial F_{1}}{\partial s_{2}} = - c_{1} {\hat{k}}_{1}; \frac{\partial F_{2}}{\partial s_{2}} = c_{1} {\hat{k}}_{1} + c_{2} {\hat{k}}_{2}; \frac{\partial F_{3}}{\partial s_{2}} = - c_{2} {\hat{k}}_{2}

(49)

For a clustered cable, nodal forces are related to an augmented displacement vector that includes both nodal displacements and sliding magnitudes:

[F] = [K_{x}, K_{s}] [\begin{matrix} u \\ s \end{matrix}]

(50)

The matrix K_x is obtained through equations (45)–(47) and can be formulated as follows:

\begin{array}{l} K_{x} = [\begin{matrix} c_{1} k_{1}^{*} c_{1}^{T} & - c_{1} k_{1}^{*} c_{1}^{T} & 0 \\ - c_{1} k_{1}^{*} c_{1}^{T} & c_{1} k_{1}^{*} c_{1}^{T} + c_{2} k_{2}^{*} c_{2}^{T} & - c_{2} k_{2}^{*} c_{2}^{T} \\ 0 & - c_{2} k_{2}^{*} c_{2}^{T} & c_{2} k_{2}^{*} c_{2}^{T} \end{matrix}] \\ + [\begin{matrix} k_{1} & - k_{1} & 0 \\ - k_{1} & k_{1} + k_{2} & - k_{2} \\ 0 & - k_{2} & k_{2} \end{matrix}] \end{array}

(51)

where sub-matrix k₁ and k₂ are defined by:

k_{1} = \frac{t_{1}}{l_{1}} . (I_{3} - c_{1} c_{1}^{T}); k_{2} = \frac{t_{2}}{l_{2}} . (I_{3} - c_{2} c_{2}^{T})

(52)

whereas, the matrix K_s is obtained through equation (50) and can be formulated as follows:

K_{s} = {[\begin{matrix} - c_{1} {\hat{k}}_{1} & c_{1} {\hat{k}}_{1} + c_{2} {\hat{k}}_{2} & - c_{2} {\hat{k}}_{2} \end{matrix}]}^{T}

(53)

Equation (51) is not sufficient for the analysis of a clustered-cable element when sliding-induced friction is considered. As formulated here the problem yields 10 unknown displacements and only 9 equations. As described in Section 2.2, the 10th equation is obtained through the development of the relation between sliding magnitude and nodal displacements.

The relationship between the forces at two sides of the pulley at node 2 is given by:

f_{23} = α_{2} f_{21}

(54)

Cable-induced forces acting at the two edges of the pulley at node 2 can be related to node displacements and slip magnitude through:

f_{21} = \frac{E A}{{\bar{l}}_{01}} [c_{1}^{T} - c_{1}^{T} - 1] [\begin{matrix} u_{1} \\ u_{2} \\ s_{2} \end{matrix}]

(55)

and

f_{23} = \frac{E A}{{\bar{l}}_{02}} [c_{1}^{T} - c_{2}^{T} 1] [\begin{matrix} u_{2} \\ u_{3} \\ s_{2} \end{matrix}]

(56)

Thus, replacing f₂₁ and f₂₃ by their expressions in equation (55) yields a new matrix expression between slip magnitude and nodal displacements:

[\begin{matrix} - α_{2} c_{1}^{T} k_{1}^{*} \\ α_{2} c_{1}^{T} k_{1}^{*} + c_{2}^{T} k_{2}^{*} \\ - c_{2}^{T} k_{2}^{*} \end{matrix}] [\begin{matrix} u_{1} \\ u_{2} \\ u_{3} \end{matrix}] + (α k_{1}^{*} + k_{2}^{*}) s_{2} = 0

(57)

Static analysis of clustered cable structures can thus be performed by iteratively solving the system of nonlinear equations formed by adding equation (58) to the system of equations of equation (51). For each iteration, it is possible to update the geometry of the structure in two steps: first, calculating the sliding magnitude using current nodal displacements (equation (58)) and second, evaluating the geometry through the resolution of equation (51).

Multi-node clustered cable with sliding-induced friction

For a clustered cable system with sliding-induced friction where the cable slides over multiple pulleys the tangent stiffness matrix can be obtained by assembling the elementary stiffness matrices corresponding to its constitutive elements.

For a multi-node clustered cable with N sub-elements, the vector of external nodal forces F can be expressed in terms of the vector of nodal displacements, u, and the vector of sliding magnitudes, s, through the following general equation

[F] = [K_{x}, K_{s}] [\begin{matrix} u \\ s \end{matrix}]

(58)

The matrix K_x is given by:

K_{x} = \sum_{e = 1}^{N} c_{0} (k_{1 e} + k_{2 e}) c_{0}^{T}; c_{0}^{T} = {\begin{matrix} 1 & - 1 \end{matrix}}

(59)

where sub-matrix k_1e and k_2e are defined as follow:

k_{1 e} = c_{e} k^{*} c_{e}^{T}; k_{2 e} = \frac{t}{l_{e}} . (I_{3} - c_{e} c_{e}^{T})

(60)

The matrix K_s is defined as:

K_{s} = \sum_{e = 1}^{N} c_{0} (c_{e} {\hat{k}}_{e}) c_{0}^{T}; c_{0}^{T} = {\begin{matrix} 1 & - 1 \end{matrix}}

(61)

As for the relation between nodal displacements and sliding magnitudes, combining all equations related to the sliding nodes yields the following matrix equation:

0 = [{\bar{K}}_{x}, {\bar{K}}_{s}] [\begin{matrix} u \\ s \end{matrix}]

(62)

Each line of the matrices ${\bar{K}}_{x}$ and ${\bar{K}}_{s}$ corresponds to the relation between the forces acting at two edges of a sliding node. For a generic sliding node n relating cable sub-elements e-1 and e, the corresponding lines in ${\bar{K}}_{x}$ and ${\bar{K}}_{s}$ are given by:

\begin{array}{l} {\bar{K}}_{x} (n, :) = [\begin{matrix} \begin{matrix} 0 & - α_{n} c_{e - 1}^{T} k_{e - 1}^{*} \end{matrix} & α_{n} c_{e - 1}^{T} k_{e - 1}^{*} + c_{e}^{T} k_{e}^{*} & \begin{matrix} - c_{e}^{T} k_{e}^{*} & 0 \end{matrix} \end{matrix}] \\ {\bar{K}}_{s} (n, :) = [\begin{matrix} \begin{matrix} 0 & - α_{n} k_{e - 1}^{*} \end{matrix} & α_{n} k_{e - 1}^{*} + k_{e}^{*} & \begin{matrix} - k_{e}^{*} & 0 \end{matrix} \end{matrix}] \end{array}

(63)

The slip magnitude at sliding nodes can thus be expressed in terms of nodal coordinates:

s = - {\bar{K}}_{s}^{- 1} {\bar{K}}_{x} u

(64)

Substituting the slip magnitude vector s in equation (60) by its expression based on equation (66) yields the matrix relationship between nodal forces and nodal displacements.

F = (K_{x} - K_{s} {\bar{K}}_{s}^{- 1} {\bar{K}}_{x}) u

(65)

Consequently, the tangent stiffness matrix, K_T, of a clustered cable with the consideration of sliding-induced friction at nodes is given by:

K_{T} = K_{x} - K_{s} {\bar{K}}_{s}^{- 1} {\bar{K}}_{x}

(66)

With the tangent stiffness matrix fully defined, the most effective way to solve the geometrically non-linear problem is to use an incremental-iterative technique. For the examples included in this paper (Section 5), a Newton-Raphson scheme was used. As described previously, the nonlinear system of equations from equation (64) is solved first to calculate the sliding magnitudes. Then, equation (60) is solved to find the nodal displacement due to the external load. These two steps are repeated iteratively until convergence to an equilibrium state.

Numerical examples

In this section, two numerical examples originating from the literature are presented and discussed. The first example is a clustered cable system where the continuous cable passes through two fixed pulleys without and with the consideration of geometric nonlinearities, while the second is a clustered tensegrity beam with continuous cables that go through multiple unconstrained pulleys/nodes. For both systems, the static equilibrium under a given set of loads is investigated.

Continuous cable through fixed pulleys (no geometric linearities)

The first example originates from Ju and Choo²⁵ and consists of the static analysis of a continuous cable passing through two pulleys, as shown in Figure 4. To study the equilibrium of the structure under a known load, Ju and Choo²⁵ employed a Finite Element formulation based on linear elastic assumptions. The continuous cable (Figure 4) starts at the fixed node 1 sliding over fixed pulleys at nodes 2 and 3 under the effect of load P = 30 kN applied at the end of the cable (node 4). The cable has a total length of 240 cm and a cross section area of 0.6 cm². It is assumed made of stainless-steel with a Young modulus of 11500 kN/cm². The continuous cable is divided into three sub-elements: L₁, L₂, and L₃ having 100 cm, 40 cm and 100 cm length, respectively. The contact angle at sliding nodes 2 and 3 are π and π/2, respectively.

Figure 4.

A continuous cable passing through two fixed pulleys (taken from Ju and Choo²⁵).

The structure is first analyzed assuming frictionless sliding and then considering frictional sliding. Table 2 includes the numerical results for sub-element internal forces, sliding magnitudes and horizontal displacement at node 4 with and without sliding-induced friction. FE₀ denotes the formulation of a clustered cable presented in Section 2 (clustered cable with no friction), while FE_fs denotes the formulation of a clustered cable with frictional sliding. FE_JC corresponds to the reference results given by Ju and Choo.²⁵ Results show agreement between the proposed formulations (FE₀ and FE_fs) when friction is not considered. Results obtained using the proposed formulation of a clustered cable with frictional sliding show also agreement with the values reported in the literature (FE_fs, given by Ju and Choo²⁵). The two formulations can thus be considered as equivalent in the absence of geometric nonlinearities (fixed nodes).

Table 2.

Analysis results for the continuous cable case study.

	Frictionless sliding		Frictional sliding (μ = 0.1)
	FE₀	FE_fs	FE_fs	FE_JC
Tension in L₁ (kN)	30.0	30.0	23.7	23.7
Tension in L₂ (kN)	30.0	30.0	27.7	27.7
Tension in L₃ (kN)	30.0	30.0	30.0	30.0
Sliding magnitude s₂ (cm)	–	0.43	0.34	0.34
Sliding magnitude s₃ (cm)	–	0.60	0.50	0.50
Displacement δ at node 4 (cm)	1.05	1.05	0.93	0.93

Actuated tensegrity beam

The second example is the analysis of the clustered tensegrity beam structure studied by Moored and Bart-Smith² under loading. The beam is composed of an assembly of three four-strut prismatic structures (quadruplex modules) with no strut-to-strut connection and with every top node of a module having two cables that connect to the bottom nodes. Having two cables per top node is a necessary condition for the existence of a feasible pre-stress basis when clustered cables are employed. A perspective view of the tensegrity beam is given in Figure 5 where grayed lines denote bars and thin lines denote cables. End nodes 1, 3, and 6 are pinned so that the system acts as a cantilever beam. Ten cable elements of the top surface and ten cable elements of the bottom surface of the structure are grouped into four cable-element clusters. The topology of the four sliding cables is given in Table 3. Clustered cable elements are also shown in dashed lines in a top view of the tensegrity beam in Figure 6. Each cable cluster is attached to two end nodes and runs over four intermediate unconstrained pulleys/nodes.

Figure 5.

A perspective view of the tensegrity beam.

Table 3.

Details about clustered cables.

Cable	Position	End nodes	Intermediate nodes
1	Top surface	6 and 21	5, 14, 13, and 22
2	Top surface	8 and 23	7, 16, 15, and 24
3	Bottom surface	1 and 18	2, 9, 10, and 17
4	Bottom surface	3 and 20	4, 11, 12, and 19

Figure 6.

Clustered elements of the tensegrity beam.

The tensegrity beam used in this study has a length of 212 cm, a width of 80 cm and a height of 30 cm. Struts are assumed made of aluminum hollow tubes with a length of 85 cm. Depending on their topology (saddle, vertical and reinforcing [2]) cables have a length of 60, 48, or 40 cm, respectively. All cable members are assumed made of stainless-steel. Material and geometrical characteristics of the elements are summarized in Table 4.

Table 4.

Material characteristics for the tensegrity structure.

Member	Material	Cross-section area (cm²)	Young modulus (kN/cm²)	Specific weight (kN/cm³)
Struts	Aluminum	2.50	7000	2.7 10⁻⁵
Cables	Stainless-steel	0.50	11500	7.85 10⁻⁵

The response of the beam under the application of two vertical concentrated forces on nodes 18 and 23 is investigated using the proposed Finite Element formulation. Since the deformed shape of the structure is very similar to its initial shape, the friction angle, θ, can be assumed constant and equal to π/2 for all sliding nodes. However, in order to investigate the influence of sliding-induced friction on the behavior of the structure, the structure is analyzed considering different values of the friction coefficient, μ. Figures 7 and 8 illustrate the evolution of tension and compression in cable 2 and strut 62, respectively, when the applied load is progressively increased from 5 to 30 daN. Cable 2 is a sub-element of the continuous cable connecting nodes 2 and 9, while strut 62 connects nodes 18 and 22 (Figure 6). As expected, the effect of sliding-induced friction increases with loading. Moreover, increasing sliding-induced friction is found to reduce the tension in cable 2, while increasing the force in strut 62.

Figure 7.

Evolution of tension magnitude at cable no. 2 with respect to the applied load.

Figure 8.

Evolution of compression magnitude at strut no. 62 with respect to the applied load.

Figure 9 illustrates the force distribution in the structure under a load of 30 daN with and without sliding-induced friction (μ = 0.3). Elements with zero force correspond to elements between support nodes. The difference between the values of axial forces with and without friction in both cables and struts emphasizes the importance of taking sliding-induced friction into consideration when studying the equilibrium of structures with continuous cables as force were found to vary from 1% to 43%.

Figure 9.

Internal forces in elements of the tensegrity beam under loading with and without sliding-induced friction.

Conclusions

Clustered or continuous cables represent a valuable solution for reducing the number of elements in tensile systems employed in engineering applications. However, most available formulations for the analysis of systems with clustered cables are developed based on the assumption of frictionless sliding. In this paper, a Finite Element formulation for clustered cable elements with sliding-induced friction is developed through the derivation of the tangent stiffness matrix at sliding nodes. Equilibrium equations are differentiated with respect to the sliding magnitude occurring on the nodes as internal forces in sub-elements depend on node coordinates as well as the sliding magnitude occurring at the various nodes. The proposed formulation shows agreement with existing formulations while also accommodating geometric nonlinearities; a key feature for the analysis of many cable systems. Two examples from literature are revisited with the consideration of sliding-induced friction. Force distribution in the structures studied is found to be significantly altered by friction highlighting the importance of a more realistic formulation for continuous cable action.

Footnotes

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: The first author gratefully acknowledges the financial support of the Fulbright Visiting Scholar Program for the academic year 2018-2019.

ORCID iD

Nizar Bel Hadj Ali

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