Nonlinear static and dynamic analysis of mixed cable elements

Abstract

This paper presents a family of finite elements for the nonlinear static and dynamic analysis of cables based on a mixed variational formulation in curvilinear coordinates and finite deformations. This formulation identifies stress measures, in the form of axial forces, and conjugate deformation measures for the nonlinear catenary problem. The continuity requirements lead to two distinct implementations: one with a continuous axial force distribution and one with a discontinuous. Two examples from the literature on nonlinear cable analysis are used to validate the proposed formulation for St Venant-Kirchhoff elastic materials. These studies show that displacements and axial forces are captured with high accuracy for both the static and the dynamic case.

1 INTRODUCTION

Cable structures are of great interest in many engineering applications because they offer numerous advantages, such as high ultimate strength, light weight or prestressing capabilities, among others. Nonetheless, a highly nonlinear behavior arises in this type of structures because of their high flexibility. For analyzing cable structures, two families of elements have traditionally been considered: truss elements and catenary elements.

For truss elements, the cable is discretized in a series of straight 2-node elements. In this case, the geometric nonlinearity is often accounted for by a corotational formulation, involving the transformation of the node kinematic variables under large displacements. Truss elements suffer from excessive mesh refinement to obtain accurate results, especially when assuming a constant axial force distribution in the element. Moreover, they may exhibit snap-through instabilities at states of nearly singular stiffness.

Catenary elements use linear kinematics to discretize the cable into a series of curved elements that satisfy the catenary equation. These elements solve the global balance of linear momentum by explicit integration and assuming linear elasticity [1]. As a result, loads are not adjusted with the cable elongation, so that these elements cannot be extended to nonlinear elasticity or inelasticity. Recently, the authors have proposed a general formulation for a catenary element in finite deformations and curvilinear coordinates [2] that overcomes these limitations.

2 MIXED FORMULATION OF THE CATENARY PROBLEM

2.1 Kinematics

Fig. 1 shows the motion of a cable from a reference configuration Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{P}_0}

to a current configuration Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{P}}

. Define an orthogonal frame Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \{ \mathbf{G}_i\} _{i=1}^3}

with associated coordinates Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \{ \xi ^i\} _{i=1}^3}
at any point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle P\in \mathcal{P}_0}

, such that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathbf{G}_1 = \frac{d\mathbf{X}}{d\xi ^1} \; \; ; \; \; \mathbf{G}_1\cdot \mathbf{G}_2 = 0 \; \; ; \; \; \| \mathbf{G}_2\|=1 \; \; ; \; \; \mathbf{G}_3=\frac{\mathbf{G}_1 \times \mathbf{G}_2}{\| \mathbf{G}_1 \times \mathbf{G}_2\| }

(1)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \xi ^1}

is the parameter describing the curve. Under the motion Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{x}=\boldsymbol{\chi }(\mathbf{X})}

, this frame is convected to the orthogonal frame Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \{ \mathbf{g}_i\} _{i=1}^3} . Let upper case letters denote variables in the reference configuration and lower case letters, variables in the current configuration.

$Motion \mathbfx=χ(\mathbfX(ξ¹)) of the cable \mathcalC.$

Figure 1: Motion Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathbf{x}=\boldsymbol{\chi }(\mathbf{X}(\xi ^1))

of the cable Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathcal{C}

.

The relevant stretch and Green-Lagrange strain of the problem, in the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{g}_1}

direction, are

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \lambda = \frac{\| \mathbf{g}_1\| }{\| \mathbf{G}_1\| } \;\;\;\;\;\; ; \;\;\;\;\;\; \mathrm{E}=\frac{1}{2}(\lambda ^2-1)\| \mathbf{G}_1\| ^2

(2)

The displacement vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{u}}

depends only on the curvilinear coordinate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \xi ^1}

,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathbf{u}(\mathbf{X}(\xi ^1))=\mathbf{x}(\mathbf{X}(\xi ^1))-\mathbf{X}(\xi ^1) = u_A(\xi ^1) \mathbf{E}_A

(3)

Therefore, the relationship between the displacement field Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{u}}

and the relevant Green-Lagrange strain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathrm{E}}
can be computed [2] as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathrm{E}=\frac{d \mathbf{u}}{d \xi ^1}\cdot \mathbf{G}_1+\frac{1}{2}\left|\frac{d \mathbf{u}}{d \xi ^1}\right|^2=\frac{d \mathbf{u}}{d \xi ^1} \cdot \left(\mathbf{G}_1 +\frac{1}{2}\frac{d \mathbf{u}}{d \xi ^1} \right)=\frac{1}{2}\frac{d \mathbf{u}}{d \xi ^1}\cdot \left(\mathbf{G}_1+\mathbf{g}_1\right)

(4)

It is relevant to observe that the frame Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \{ \mathbf{G}_i\} _{i=1}^3}

is orthogonal, but not orthornomal in general. Indeed, the metric tensor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle G_{ij}=\mathbf{G}_i \cdot \mathbf{G}_j}
is not necessarily the identity operator and the Green-Lagrange strain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathrm{E}}
may not be physical. Nevertheless, one can construct an orthonormal basis Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \{ \mathbf{\hat{G}}_A\} _{A=1}^3=\{ \mathbf{G}_i/\| \mathbf{G}_i\| \} _{i=1}^3}
such that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathbf{E}=\mathrm{E}_{ij} \mathbf{G}^i \otimes \mathbf{G}^j = \mathrm{\hat{E}}_{AB}\mathbf{\hat{G}}_A \otimes \mathbf{\hat{G}}_B

(5)

Hence, the components

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathrm{\hat{E}}_{AB}=\mathrm{E}_{ij}(\mathbf{\hat{G}}_A \cdot \mathbf{G}^i)(\mathbf{\hat{G}}_B \cdot \mathbf{G}^j)

(6)

are physical quantities.

2.2 Equilibrium and principle of virtual work

For expressing the equilibrium equation of the cable in finite deformations, let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{n}}

denote the axial force in the current configuration, thus a Cauchy representation. Observe that the first Piola-Kirchhoff  and the Cauchy representations of the axial force coincide for the problem in hand, which does not account for changes in the cross section dimensions.

The Cauchy axial force can be pulled back to the reference configuration to obtain a second Piola-Kirchhoff representation of the axial force, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{N}} . It can be shown [2] that, using the orthonormal basis in Eq. 5, namely with components Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{n}=\mathrm{\hat{n}}\mathbf{\hat{g}}_1}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{N}=\mathrm{\hat{N}}\mathbf{\hat{G}}_1}

,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathrm{\hat{n}}=\lambda \mathrm{\hat{N}}

(7)

Denoting by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle s}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle S}
the arc-length coordinates in the current and reference configurations, respectively, the cable distributed load can be described as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathbf{w}ds = \mathbf{\hat{W}}dS = \mathbf{W}d\xi ^1

(8)

Then, global equilibrium for the cable in the current configuration Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{P}}

states

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathbf{n}(s)-\mathbf{n}(0)+\int _0^s\mathbf{w}\, ds = \int _0^s \rho \mathbf{a}\, ds

(9)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \rho }

is the material density in the current configuration and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{a}}

, the total acceleration. The corresponding local statement in the current configuration can be obtained with the fundamental theorem of calculus and the localization theorem

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{d}{ds}(\mathrm{n}\mathbf{g}_1)+\mathbf{w}=\rho \mathbf{a}

(10)

or, in the reference configuration,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{d}{dS}(\mathrm{\hat{N}}\sqrt{G^{11}}\mathbf{g}_1)+\mathbf{\hat{W}}=\rho _0\mathbf{a}

(11)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle G^{ij}=\mathbf{G}^i \cdot \mathbf{G}^j}

represents the dual metric tensor and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \rho _0 = \lambda \rho }

.

In summary, if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathrm{\hat{N}}=\Psi (\mathrm{\hat{E}})}

is a frame-indifferent constitutive relation between the phy-sical Green-Lagrange strain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathrm{\hat{E}}}
and the physical 2nd Piola-Kirchhoff axial force Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathrm{\hat{N}}}

, the pair of fields (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{u}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathrm{\hat{N}}} ) will be the solution of the cable problem, if and only if, they satisfy

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left\{ \begin{array}{rl}\displaystyle G^{11}\frac{d\mathbf{u}}{d\xi ^1} \cdot \left(\mathbf{G}_1+\frac{1}{2}\frac{d\mathbf{u}}{d\xi ^1}\right)-\mathrm{\hat{E}}=0 & \mathrm{in} \; \Omega=(0,L) \\[3.5mm] \displaystyle \frac{d}{dS}\left(\sqrt{G^{11}}\mathrm{\hat{N}}\mathbf{g}_1 \right)+\mathbf{\hat{W}}=\rho _0 \mathbf{a} & \mathrm{in} \; \Omega=(0,L)\\[] \mathrm{\hat{N}}-\Psi (\mathrm{\hat{E}})=0 & \mathrm{in} \; \Omega=(0,L) \\[] \mathbf{u} = \mathbf{\bar{u}} & \mathrm{on} \; \Gamma _u \\[1mm] \displaystyle \sqrt{G^{11}}\mathrm{\hat{N}}\mathbf{g}_1= \mathbf{\bar{T}} & \mathrm{on} \; \Gamma _q \end{array} \right.

(12)

for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 0<S<L}

equivalent to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \xi ^1_1<\xi ^1<\xi ^1_2}

.

The corresponding two-field weak statement of Eq. 12 can be obtained by considering any variation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \delta \mathbf{u}\in \mathcal{V}} , the space of displacement test functions, any variation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \delta \mathrm{\hat{N}}\in \mathcal{W}} , the space of axial force test functions, and integrating the equilibrium equation by parts,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left\{ \begin{array}{r}\displaystyle \int _0^L \delta \mathrm{\hat{N}}\left\{G^{11}\frac{d\mathbf{u}}{d\xi ^1} \cdot \left(\mathbf{G}_1+\frac{1}{2}\frac{d\mathbf{u}}{d\xi ^1}\right)-\mathrm{\hat{E}}\right\}dS=0 \\[3.6mm] \displaystyle \int _0^L \frac{d(\delta \mathbf{u})}{dS} \cdot \mathrm{\hat{N}}\sqrt{G^{11}}\mathbf{g}_1 \, dS + \int _0^L \delta \mathbf{u}\cdot \rho _0 \mathbf{a}\, dS =\left[\delta \mathbf{u}\cdot \mathbf{\bar{T}} \right]_{\Gamma _q}+\int _0^L \delta \mathbf{u}\cdot \mathbf{\hat{W}}\, dS \end{array} \right.

(13)

where the constitutive relation is imposed strongly. The spaces for the trial solutions of the displacements and axial forces, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{S}}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{N}}

, respectively, are

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \begin{array}{c}\mathcal{S}&=\{ \mathbf{u}\in H^1(0,L) \, | \, \mathbf{u}=\mathbf{\bar{u}} \; \mathrm{on} \; \Gamma _u\} \\ \mathcal{N}&= \left\{\mathrm{\hat{N}}\in H^0(0,L) \, | \, \mathrm{\hat{N}}>0, \; \mathrm{and} \; \mathrm{\hat{N}}=g^{11}\sqrt{G_{11}}\mathbf{\bar{T}}\cdot \mathbf{g}_1 \; \mathrm{on} \; \Gamma _q \right\} \end{array}

(15)

Similarly, the spaces for the test functions of the displacements and the axial forces, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{V}}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{W}}

, respectively, become

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \begin{array}{c}\mathcal{V}&=\{ \delta \mathbf{u}\in H^1(0,L) \, | \, \delta \mathbf{u}=0 \; \mathrm{on} \; \Gamma _u\} \\ \mathcal{W}&=\{ \delta \mathrm{\hat{N}} \in H^0(0,L) \, | \, \delta \mathrm{\hat{N}}=0 \; \mathrm{on} \; \Gamma _q\} \end{array}

(17)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle H^k(\Omega )}

is the Sobolev space for the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k-}

th weak derivative in the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle L^2(\Omega )}

norm.

As a result, there are no continuity requirements for the axial force field Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathrm{\hat{N}}} . This implies the possibility of exploring cable finite elements with continuous or discontinuous axial force distribution.

3 FINITE-ELEMENT IMPLEMENTATION

3.1 Discretization

The discretization of the governing equations requires interpolations for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathrm{\hat{N}}(\xi ^1)} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{u}(\xi ^1)}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{a}(\xi ^1)}

. Assume a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k} -th order Galerkin interpolation for the axial forces

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathrm{\hat{N}}=\boldsymbol{\varphi }^t\mathbf{\hat{N}}=\mathbf{\hat{N}}^t \boldsymbol{\varphi }

(18)

and an Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle l} -th order Galerkin interpolation for the displacement and acceleration fields

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathbf{u}=\boldsymbol{\phi }^t \mathbf{\hat{u}} \;\;\;\;\;\; ; \;\;\;\;\;\; \mathbf{a}=\boldsymbol{\phi }^t \mathbf{\hat{a}}

(19)

Then, using the same shape functions for the reference configuration, the current configu-ration is obtained as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathbf{x}=\mathbf{X}+\mathbf{u}=\boldsymbol{\phi }^t (\mathbf{\hat{X}}+\mathbf{\hat{u}})=\boldsymbol{\phi }^t \mathbf{\hat{x}}

(20)

With these interpolation functions, one can discretize the weak statement in Eq. 13 for a finite element Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Omega _e}

and a time step Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n}
as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left\{ \begin{array}{r}\displaystyle \int _{\Omega _e} \delta \mathbf{\hat{N}}^t \boldsymbol{\varphi } \left\{G^{11}\mathbf{\hat{u}}^t_n \boldsymbol{\phi }' \left(\mathbf{G}_1+\frac{1}{2}(\boldsymbol{\phi }')^t\mathbf{\hat{u}}_n\right)-\mathrm{\hat{E}}(\mathbf{\hat{N}}_n)\right\}dS=0 \\[3.6mm] \displaystyle \int _{\Omega _e} \delta \mathbf{\hat{u}}^t \boldsymbol{\phi }' G^{11} \boldsymbol{\varphi }^t \mathbf{\hat{N}}_n \mathbf{\hat{g}}_n \, dS + \int _{\Omega _e} \delta \mathbf{\hat{u}}^t \rho _0 \boldsymbol{\phi } \boldsymbol{\phi }^t\mathbf{\hat{a}}_n\, dS =\left[\delta \mathbf{\hat{u}}^t\boldsymbol{\phi } \mathbf{\bar{T}}_n \right]_{\partial \Omega _e}+\int _{\Omega _e} \delta \mathbf{\hat{u}}^t \boldsymbol{\phi } \mathbf{\hat{W}}_n\, dS \end{array} \right.

(21)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (\cdot )'}

represents the derivative with respect to the curvilinear coordinate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \xi ^1}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{\hat{g}}_n=\mathbf{G}_1+(\boldsymbol{\phi }')^t\mathbf{\hat{u}}_n}
is the numerical counterpart to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{g}_1}

.

3.2 Time integration and consistent linearization

Once the discretization of the problem has been performed, the corresponding time-dependent equations need to be solved. As stated before, one can consider cable finite elements with a continuous or a discontinuous axial force field.

3.2.1 Mixed cable element with continuous axial force

For the element with continuous axial force distribution, the cable is subdivided into Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle e}

elements of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k}

-th order in axial forces and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle l} -th order in displacements. By defining the expanded stress divergence term Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{R}=(\mathbf{R}_1,\mathbf{R}_2)}

with components

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \begin{array}{c}\mathbf{R}_1 (\mathbf{\hat{N}}_n, \mathbf{\hat{u}}_n) &= \int _{\Omega _e} \boldsymbol{\varphi } \left(G^{11} \mathbf{\hat{u}}^t_n\boldsymbol{\phi }' \left(\mathbf{G}_1+\frac{1}{2}(\boldsymbol{\phi }')^t\mathbf{\hat{u}}_n\right)-\mathrm{\hat{E}}(\mathbf{\hat{N}}_n)\right)dS \\[0.3em] \mathbf{R}_2 (\mathbf{\hat{N}}_n, \mathbf{\hat{u}}_n) &= \int _{\Omega _e} G^{11} \boldsymbol{\varphi }^t\mathbf{\hat{N}}_n \, \boldsymbol{\phi' } \mathbf{\hat{g}}_n \, dS \end{array}

(23)

and the mass matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{M}}

as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathbf{M}=\int _{\Omega _e} \rho _0 \boldsymbol{\phi }\boldsymbol{\phi }^t\, dS

(24)

one can rewrite Eq. 21 in an implicit scheme as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left[ \begin{array}{c}\mathbf{R}_1(\mathbf{\hat{N}}_{n+1},\mathbf{\hat{u}}_{n+1}) \\ \mathbf{R}_2(\mathbf{\hat{N}}_{n+1},\mathbf{\hat{u}}_{n+1}) \end{array} \right]+ \left[ \begin{array}{c}\mathbf{0} \\ \mathbf{M}\mathbf{\hat{a}}_{n+1} \end{array} \right]= \left[ \begin{array}{c}\mathbf{0} \\ \mathbf{F}_{ext,n+1} \end{array} \right]

(25)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{F}_{ext,n+1}}

refers to the external forces considered at the time step Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n+1}

.

Introducing Newmark's time integrator [3], one obtains the system of equations

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \begin{array}{c}\frac{1}{\beta \Delta t_n^2}\left[ \begin{array}{c}\mathbf{0} \\ \mathbf{M}\mathbf{\hat{u}}_{n+1} \end{array} \right]&+ \left[ \begin{array}{c}\mathbf{R}_1(\mathbf{\hat{N}}_{n+1},\mathbf{\hat{u}}_{n+1}) \\ \mathbf{R}_2(\mathbf{\hat{N}}_{n+1},\mathbf{\hat{u}}_{n+1}) \end{array} \right] \\ &= \left[ \begin{array}{c}\mathbf{0} \\ \mathbf{F}_{ext,n+1} \end{array} \right]+ \frac{1}{\beta \Delta t_n^2}\left[ \begin{array}{c}\mathbf{0} \\ \mathbf{M}(\mathbf{\hat{u}}_n + \Delta t_n \mathbf{\hat{v}}_n) \end{array} \right]+ \frac{1-2\beta }{2\beta }\left[ \begin{array}{c}\mathbf{0} \\ \mathbf{M}\mathbf{\hat{a}}_n \end{array} \right] \end{array}

(32)

Hence the consistent linearization of the former equation, namely Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{\Phi }(\mathbf{\hat{u}}_{n+1},\mathbf{\hat{N}}_{n+1})=\mathbf{0}} , around a point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{\bar{V}}_{n+1}=(\mathbf{\hat{u}}_{n+1},\mathbf{\hat{N}}_{n+1})}

and for the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k}

-th iterate establishes

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathcal{L}\boldsymbol{\Phi }=\boldsymbol{\Phi }|_{\mathbf{\bar{V}}_{n+1}}^{(k)}+\underbrace{\left. \frac{\partial \mathbf{\Phi }}{\partial \mathbf{\hat{V}}_{n+1}} \right|_{\mathbf{\bar{V}}_{n+1}^{(k)}} ( \mathbf{\hat{V}}_{n+1}^{(k+1)}-\mathbf{\bar{V}}_{n+1}^{(k)})}_{\mathrm{D}\boldsymbol{\Phi }(\mathbf{\bar{V}}_{n+1}^{(k)} ,\Delta \mathbf{V}_{n+1})}=\mathbf{0}

(33)

where the Fréchet derivative Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \partial \boldsymbol{\Phi }/\partial \mathbf{\hat{V}}_{n+1}|_{\mathbf{\hat{V}}_{n+1}^{(k)}}}

corresponds to the dynamic stiffness Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{K}}
of the problem, with components

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {2} \mathbf{K}_{\mathbf{N}\mathbf{N}} = -\int _{\Omega _e} \boldsymbol{\varphi } \frac{\partial \mathrm{\hat{E}}}{\partial \mathbf{\hat{N}}_{n+1}} \, dS = -\int _{\Omega _e} \boldsymbol{\varphi } \frac{\partial \mathrm{\hat{E}}}{\partial \mathrm{\hat{N}}} \boldsymbol{\varphi }^t \, dS

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathbf{K}_{\mathbf{N}\mathbf{u}} = \int _{\Omega _e} G^{11}\boldsymbol{\varphi }\mathbf{\hat{g}}^t_{n+1} (\boldsymbol{\phi' })^t dS = \mathbf{K}_{\mathbf{u}\mathbf{N}}^t

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathbf{K}_{\mathbf{u}\mathbf{u}}^s= \int _{\Omega _e} G^{11} \boldsymbol{\varphi }^t \mathbf{\hat{N}}_{n+1}\boldsymbol{\phi' }(\boldsymbol{\phi' })^t dS

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathbf{K}_{\mathbf{u}\mathbf{u}}^d= \frac{1}{\beta \Delta t_n^2}\mathbf{M}+ \mathbf{K}_{\mathbf{u}\mathbf{u}}^s

(34)

in the form

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathbf{K}=\left.\frac{\partial \boldsymbol{\Phi }}{\partial \mathbf{\hat{V}}_{n+1}}\right|_{\mathbf{\bar V}_{n+1}^{(k)}}=\left[\begin{array}{cc}\mathbf{K}_{\mathbf{N}\mathbf{N}} & \mathbf{K}_{\mathbf{N}\mathbf{u}} \\ \mathbf{K}_{\mathbf{u}\mathbf{N}} & \mathbf{K}_{\mathbf{u}\mathbf{u}}^d \end{array} \right]

(35)

In order to satisfy stability of the solution scheme, it is necessary [2] that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \ker \, (\mathbf{K}_{\mathbf{N}\mathbf{N}}-\mathbf{K}_{\mathbf{N}\mathbf{u}}(\mathbf{K}_{\mathbf{u}\mathbf{u}}^s)^{-1}\mathbf{K}_{\mathbf{N}\mathbf{u}}^t)=\mathbf{0}

(36)

3.2.2 Mixed cable element with discontinuous axial force

For the element with discontinuous axial force distribution, the cable is also subdivided into Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle e}

elements of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k}

-th order in axial forces and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle l} -th order in displacements. In this case, however, the axial forces are treated as internal degrees of freedom, and are consequently condensed out at the element level before assembly of the element response. This generates a discontinuity in the axial forces, which is allowed by the condition Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathrm{\hat{N}}\in H^0(0,L)} . The stress divergence term Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{R}(\mathbf{\hat{N}}_n,\mathbf{\hat{u}}_n)}

is then understood as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathbf{R}(\mathbf{\hat{N}}_n(\mathbf{\hat{u}}_n),\mathbf{\hat{u}}_n)=\int _{\Omega _e} G^{11}\boldsymbol{\varphi }^t\mathbf{\hat{N}}_n \, \boldsymbol{\phi' } \mathbf{\hat{g}}_n \, dS

(37)

and one can rewrite Eq. 21 in an implicit scheme as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathbf{R}(\mathbf{\hat{u}}_{n+1})+\mathbf{M}\mathbf{\hat{a}}_{n+1}=\mathbf{F}_{ext,n+1}

(38)

Introducing Newmark's time integrator [3], one obtains the system of equations

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{1}{\beta \Delta t_n^2}\mathbf{M}\mathbf{\hat{u}}_{n+1}+\mathbf{R}(\mathbf{\hat{u}}_{n+1})=\mathbf{F}_{ext,n+1}+\frac{1}{\beta \Delta t_n^2}\mathbf{M}(\mathbf{\hat{u}}_n+\Delta t_n \mathbf{\hat{v}}_n)+\frac{1-2\beta }{2\beta }\mathbf{M}\mathbf{\hat{a}}_n

(39)

Hence the consistent linearization of the former equation, namely Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{\Phi }(\mathbf{\hat{u}}_{n+1})=\mathbf{0}} , around a point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{\bar{u}}_{n+1}}

and for the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k}

-th iterate establishes

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathcal{L}\boldsymbol{\Phi }=\boldsymbol{\Phi }|_{\mathbf{\bar{u}}_{n+1}^{(k)}}+\underbrace{\left.\frac{\partial \boldsymbol{\Phi }}{\partial \mathbf{\hat{u}}_{n+1}}\right|_{\mathbf{\bar{u}}_{n+1}^{(k)}}(\mathbf{\hat{u}}_{n+1}^{(k+1)}-\mathbf{\bar{u}}_{n+1}^{(k)})}_{\mathrm{D}\boldsymbol{\Phi }(\mathbf{\bar{u}}_{n+1}^{(k)},\Delta \mathbf{u}_{n+1})}=\mathbf{0}

(40)

where the Fréchet derivative Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \partial \boldsymbol{\Phi }/\partial \mathbf{\hat{u}}_{n+1}|_{\mathbf{\bar{u}}_{n+1}^{(k)}}}

corresponds to the condensed dynamic stiffness Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{K}}
of the problem

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathbf{K}=\mathbf{K}_{\mathbf{u}\mathbf{u}}^d-\mathbf{K}_{\mathbf{u}\mathbf{N}}\mathbf{K}_{\mathbf{N}\mathbf{N}}^{-1}\mathbf{K}_{\mathbf{N}\mathbf{u}}

(41)

with the components defined in Eq. 34.

The stability condition of the solution scheme reads [2] in this case as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \ker \, (\mathbf{K}_{\mathbf{u}\mathbf{u}}^s-\mathbf{K}_{\mathbf{u}\mathbf{N}}\mathbf{K}_{\mathbf{N}\mathbf{N}}^{-1}\mathbf{K}_{\mathbf{N}\mathbf{u}})=\mathbf{0}

(42)

4 NUMERICAL EXAMPLES

The proposed formulation is implemented in two cable elements with continuous and discontinuous axial force distributions. The elements, deployed in the general purpose finite element program FEAP [4] and Matlab toolbox FEDEASLab [5], use a linear approximation for the axial forces (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k=1} ) and a quadratic approximation for the displacements Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (l=2)} . The two-dimensional element results in eight degrees of freedom (DOFs), six displacement DOFs and two axial force DOFs, while the three-dimensional element results in eleven DOFs, nine displacement DOFs and two axial force DOFs.

Both elements are implemented with a St Venant - Kirchhoff elastic material model with stored energy Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{U}}

in terms of the stretch Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \lambda }
and the generalized Young's modulus Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle E}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathcal{U}=\frac{E}{8}(\lambda ^2-1)^2

(43)

Thus, if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle A}

is the area of the cross section and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathrm{\hat{N}}_0}
the prestressing force,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathrm{\hat{N}}-\mathrm{\hat{N}}_0=(EA)\mathrm{\hat{E}}

(44)

with constant material stiffness Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \partial \mathrm{\hat{N}}/\partial \mathrm{\hat{E}}=EA} .

4.1 Example 1: Stability of a 3d pulley system

The first example investigates the stability of a 3d cable supported by a pulley that was previously studied by Impollonia et al [6]. The structural model, whose geometric and material properties are shown in Table 1, consists of a cable anchored at both ends and supported by an intermediate roller. In this case, the inertia forces in Eq. 21 and the mass term of the stiffness are not considered as the problem is analyzed in a static manner.

Table. 1 Geometry and Material Properties for Example 1.
Property	Value
Cross-sectional area	805 mmFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle ^2}
Elastic modulus	16.0 kN/mmFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle ^2}
Cable self-weight	62.0679 N/m
Cable length	500 m

The objective of this example is to determine the equilibrium configurations of the cable under the assumptions that the pulley is free to move horizontally and that the pulley radius and friction are negligible. For the nonlinear analysis, the cable is subdivided into two segments, one for each span, with the reference curvilinear coordinate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \xi ^1}

of the pulley as problem unknown. This curvilinear coordinate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \xi ^1}
is used to construct the finite element mesh in each iteration.

Following the form finding procedure by Argyris et al [2,7], the analysis starts from a straight reference configuration, and imposes a displacement Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{u}=(-200,0,50)}

m at the right support and a pair of displacements Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle u_2 = 50}
m and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle u_3= 100}
m at the intermediate roller. Because friction is not considered, the jump in the Cauchy axial force at the roller support must be zero. As a result, the problem is solved by iterating over the curvilinear coordinate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \xi ^1}
so that the jump in the Cauchy axial force at the pulley becomes zero.

Table. 2 Results for Example 1 from different studies.
	Impollonia et al [6]	Present work (continuous)	Present work (discontinuous)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \xi ^1_1} (m)	126.12	126.26	126.25
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N_1} (kN)	14.12	8.31-13.99	8.31-13.99
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \xi ^1_2} (m)	219.98	219.46	219.47
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N_2} (kN)	10.79	4.02-10.68	4.02-10.68
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \xi ^1_3} (m)	424.76	424.70	424.70
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N_3} (kN)	17.42	10.25-17.28	10.25-17.28

Table 2 summarizes the results for the equilibrium configurations with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \xi ^1_i}

refering to the curvilinear coordinate of the pulley and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N_i}

, to the axial force. Because the study by Impollonia et al [6] does not consider finite deformations, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \xi ^1_i}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N_i}
correspond to infinitesimal deformations. For the present study, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \xi ^1_i}
corresponds to the reference confi-guration and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N_i}

, to the Cauchy axial force. While the values of the present study agree well with those by Impollonia et al [6], it is worth noting the variation of the Cauchy axial force that the current formulation captures, as indicated by the range of axial force values in Table 2. In contrast, the model in [6] overestimates the axial force by reporting a value corresponding to the maximum of the current formulation.

Three equilibrium states result from the analysis, as Fig. 2 shows: three stable confi-gurations denoted with solid lines (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle C_1}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle C_3}

), and one unstable configuration, denoted with a dashed line (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle C_2} ), as reflected in the change of direction for the horizontal component of the reaction at the pulley. The Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle x_1}

positions of the pulley for these equilibrium states in Fig. 2 are Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle x_1^1 = 56.54 / 56.53}
m, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle x_1^2=134.00/134.01}
m and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle x_1^3=274.31/274.31}
m for the continuous and the discontinuous formulations, respectively.

Figure 2: Deformed shape (30 elements) of equilibrium states for Example 1.

4.2 Example 2: Free vibration in finite deformations

The second example studies the large-amplitude free vibration of two cables with diffe-rent sag/span ratio that were investigated by Srinil et al [8]. The structural model consists of a cable anchored at both ends and spanning 850 m in both cases. Table 3 summarizes the geometric and material properties of the cables denoted by C1 and C2.

Table. 3 Geometry and Material Properties for Example 1.
Property	C1	C2
Cross-sectional area	0.1159 mFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle ^2}	0.1159 mFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle ^2}
Elastic modulus	17.94 GPa	17.94 GPa
Density	8337.9 kg/mFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle ^3}	8337.9 kg/mFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle ^3}
Cable length	840.48 m	870.51 m
Prestressing	-	345 kN

First, following the shape finding procedure by Argyris et al [7], the equilibrium confi-guration and the first two natural modes of vibration around this configuration are obtained for both cables by solving the standard eigenvalue problem

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \det [\mathbf{K}(\mathbf{u}_{eq})-w^2\mathbf{M}]=0

(45)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle w}

is the angular frequency, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{u}_{eq}}
refers to the displacement field at the equilibrium state, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{K}}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{M}}
correspond to the static stiffness and mass matrices of the formulation in Sec. 3.2. Both cables are discretized with a mesh of 14 elements. Results are presented in Table 4, where the end tension is given in the Cauchy representation, and "S" and "A" refer to the symmetric and antisymmetric modes, respectively. Fig. 3 shows the normalized first symmetric and antisymmetric modes of both cables. From this figure, it is interesting to note that, when the sag/span ratio increases, the single extremum for the symmetric mode divides into three because of increasing horizontal displacements.


Figure 3: Normalized vertical eigenvectors for Example 2.

Table. 4 Results for equilibrium configurations and natural vibration in Example 2.
	C1		C2
	Present	Srinil et al [8]	Present	Srinil et al [8]
Sag [m]	28.01	28.39	89.28	89.57
Sag/span [-]	1/30	1/30	1/9.5	1/9.5
End tension [kN]	30432	30000	10500	10500
Frequency (1st S) [Hz]	0.124	0.123	0.158	0.158
Frequency (1st A) [Hz]	0.208	0.206	0.112	0.112

To evaluate the large-amplitude free vibration, an initial displacement field is imposed corresponding to an amplified first symmetric mode, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{u}_0=\alpha \mathbf{u}_{m1}} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{u}_{m1}}

is the normalized first symmetric mode. The parameters for Newmark's method are Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \beta=0.25}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \gamma=0.5}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta t=0.05}
s. No differences are observed between the continuous and the discontinuous formulations as the problem in hand is symmetric in geometry and loads.

Fig. 4(a) shows the normalized vertical displacements and Cauchy axial forces for cable C1 and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \alpha=15} . The evolution of the energy for this case is presented in Fig. 5(a). Likewise, Fig. 4(b) and Fig. 5(b) present the normalized vertical displacements and Cauchy axial forces, and energy evolution, respectively, for cable C2 and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \alpha = 15} . While cable C1 behaves linearly in displacements, cable C2 shows a high dependence on high-frequency modes. Also, high-frequency contributions are observed in both cases for the axial force, becoming more relevant for the large sag/span ratio, as observed by Srinil et al [6]. The total energy is conserved for cable C1, whereas it shows minor oscillations for cable C2.


Figure 4: Normalized vertical displacements and Cauchy axial forces for Example 2.


Figure 5: Energy evolution for Example 2.

5 CONCLUSIONS

The paper presents a general formulation of catenary elements based on finite deformations and curvilinear coordinates for the nonlinear static and dynamic analysis of cables. From the weak statement of the problem, two implementations are derived: one with a continuous axial force distribution and one with a discontinuous.

As demonstrated by the first example, the formulation is capable of determining equilibrium configurations of three-dimensional cable arrangements with high accuracy, especially in axial forces, compared to other elements in the literature which do not distinguish between Cauchy and 2nd PK axial forces. Furthermore, the second example shows that the natural modes of vibration around equilibrium configurations can also be obtained by the proposed formulation. Because the energy is conserved in the analyzed range of sag/span ratios, Newmark's implicit method can be used to solve the nonlinear dynamic problem. Nevertheless, as observed in the literature, high-frequency contributions in the axial force appear in the analysis, with their amplitude increasing with the sag/span ratio.

In conclusion, because of their consistency and versatility, the proposed catenary elements seem well suited for the nonlinear static and dynamic analysis of nonlinear elastic cables under general loading.

BIBLIOGRAPHY

[1] Andreu, A., Gil, L. and Roca, P. A new deformable catenary element for the analysis of cable net structures. Comp. Struct. (2006) 84:882–1890.

[2] Crusells-Girona, M., Filippou, F.C. and Taylor, R.L. A mixed formuation for nonlinear analysis of cable structures. Comp. Struct. (2017) 186:50–61.

[3] Newmark, N. A method of computation for structural dynamics. J. Eng. Mech.-ASCE (1959) 85:67–94.

[4] Taylor, R.L. FEAP - Finite Element Analysis Program. Univ. of California, Berkeley (2014). http://www.ce.berkeley/feap.

[5] Filippou, F. C. FEDEASLab - Finite Elements in Design, Evaluation and Analysis of Structures. Univ. of California, Berkeley (2007). http://www.ce.berkeley.edu/filippou/Courses/FEDEASLab.htm.

[6] Impollonia, N., Ricciardi, G. and Saitta, F. Statics of elastic cables under 3D point forces. Int. J. Solids Struct. (2011) 48:1268–1276.

[7] Argyris, J. H., Angelopoulos, T. and Bichat, B. A general method for the shape finding of lightweight tension structures. Comp. Meth. Appl. Mech. Eng. (1974) 3:135–149.

[8] Srinil, N., Rega, G. and Chucheepsakul, S. Three-dimensional non-linear coupling and dynamic tension in large-amplitude free vibrations of arbitrarily sagged cables. J. Sound Vib. (2004) 269:823–852.