# Nonlinear Modeling of Cables with

Authors: Walter Lacarbonara; Arnaud Pacitti
Journal: Mathematical Problems in Engineering
 (2008)

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## Abstract

A geometrically exact formulation of cables suffering axis stretching and flexural curvature is
presented. The dynamical formulation is based on nonlinearly viscoelastic constitutive laws
for the tension and bending moment with the additional constitutive nonlinearity accounting
for the no-compression condition. A continuation method, combined with a mixed finite-difference
spatial discretization, is then employed to path-follow the static responses of cables subject to
forces or support displacements. These computations, conducted in the quasistatic regime, are
based on cables with linearly elastic material behaviors, whereas the nonlinearity is in the geometric
stiffness terms and the no-compression behavior. The finite-difference results have been
confirmed employing a weak formulation based on quadratic Lagrangian finite elements.
The influence of the flexural stiffness on the nonlinear static responses is assessed comparing the
results with those obtained for purely extensible cables. The properties of the frequencies of the
linear normal modes of cables with flexural stiffness are also investigated and compared with those
of purely extensible cables.


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## Body

## 1. Introduction

Cables are used in a variety of engineering
applications such as in suspension or cable-stayed bridges, power transmission
lines, moorings in ocean engineering, or in aerospace deployable structures.
Cables are effectively employed in long-span structures because they can be
easily engineered and are light-weight structural elements with an outstanding
stiffness in the axial direction and a significantly high strength. However,
they do possess limitations due to the lack of out-of-plane stiffness and very
light damping that make them often prone to large-amplitude vibrations. Another
serious limitation is that they cannot resist compression, therefore, they are
always prestressed to resist actions that may produce compression although
under severe dynamic excitations, the low-tension regime is barely unavoidable.Although a number of works have addressed the problem
of modeling the flexural stiffness in cables, only a few works deal with
nonlinear vibrations of general cables in low-tension regimes. Most of them
consider nonlinear vibrations of shallow/taut cables, including also multiple
resonances [1–4]. In [5, 6], a geometrically exact nonlinear model of linearly
elastic nonshallow cables was proposed, and the nonlinear modal characteristics
of the free planar motions were investigated neglecting flexural stiffness as
it is commonly done in the literature. In fact, flexural stiffness appears in
modeling cables only when the loosening effect is considered. When the excitation
levels are high, the loss of tension can be such that the cable suffers local
loosening in those segments where the overall tension vanishes and, as a
result, the cable cannot locally sustain loads unless its flexural
load-carrying capability is considered. Most of the times, the flexural
rigidity and damping are considered in a cable model to overcome numerical
divergence problems when loosening appears. For instance, when dealing with the
dynamics of submerged cables, some works focus on a general mechanical
formulation for the cable equations (e.g., [7–9]); such equations of motion are then solved by direct
integration schemes. Other works focus on the derivation or modification of
finite-element solution models (e.g., [10]), or even in simplified solution models such as the
lumped mass approach of Chai et al. [11]. Similarly, Wu et al. [12–15] used a model of cables that includes the linear
flexural stiffness contribution, and an approximate strain-displacement
relationship for the elongation, to describe nonlinear vibrations of cables
suffering loosening. Moreover, with the proposed model, the effects of
loosening on the nonlinear parametric responses of taut cables subject to
periodic horizontal displacements of the supports were investigated.However, the effects of flexural rigidity and
viscoelasticity on the nonlinear dynamics of cables have not been thoroughly
investigated, especially when loosening appears.Furthermore, the flexural
stiffness has a critical role in stock-bridge dampers used to dissipate energy.
There is a need to have a reliable and accurate mechanical model of cables to
assess the global load-carrying capability under these conditions.
There are only a few studies addressing this challenging modeling problem and
they are often based on ad hoc approximations that limit the physically
meaningful dynamic regimes.In this paper, a nonlinear geometrically exact
formulation of cables undergoing axis stretching and flexural curvature,
incorporating a nonlinearly viscoelastic constitutive law and the
no-compression condition, is first presented. The derivation of the mechanical
model is extensively inspired by the seminal work of Antman [16]. Thereafter, the
linearization and the ensuing eigenvalue problem are addressed. Then, the
finite-difference computational scheme is illustrated. Meaningful static
responses of cables subject to various loading paths—support displacements or
forces—are investigated in both cable models, here considered, the model with
flexural stiffness and that without flexural resistance.


## 2. Mechanical Formulation for Cables with Flexural Stiffness

In this section, we illustrate the geometrically exact
formulation of the equations of motion of cables suffering axis stretching and
bending curvature [16].
Shear deformations are not considered. Moreover, a planar kinematic model is
discussed neglecting out-of-plane bending and torsion. The prestressed
equilibrium is first discussed, then the dynamics around it are described in
the kinematic, dynamic, and constitutive aspects.


### 2.1. The Prestressed Cable

The cable is
stress-free in the configurationℬ (Figure 1(a)). This configuration can be any
arbitrary configuration assumed by the cable, for example, on a frictionless
horizontal plane when it is not subject to forces besides the gravity force. In
this state, the internal stresses are expected to be practically vanishing. In
particular, in Figure 1(a), ℬ represents any reference line along the cable
in the stress-free configuration (e.g., the centroidal line) and L is its undeformed length. Further, we let the arc length, denoted σ, be the coordinate identifying the material
sections of the cable. We consider the fixed Cartesian reference frame (O,e1,e3,e3) shown in Figure 1(a). When the cable ends are
fixed to two points, A and B, and the cable is let free to hang under the
action of gravity, it occupies an equilibrium configuration, here denoted ℬ0, and considered as the reference configuration
for the subsequent dynamic problem. The arc length in the configuration ℬ0 is denoted s. Let the position vector of the material point P0 of the cable in the reference configuration ℬ0 be p0. Among the different parameterizations for p0, we employ σ, and let the Cartesian representation of p0 be p0(σ):=x(σ)e1+y(σ)e2.Stress-free configurationℬ, prestressed configuration ℬ0,
and actual configuration ℬ⌣ (top). Unit vectors and rotations of the plane
kinematics (bottom left) and internal/external forces in the cable (bottom
right).
(a)(b)(c)The cable stretch is then simply defined
as(2.1)pσ0(σ)=ν0(σ)a0(σ),where a0 is the unit vector in the tangential direction
to ℬ0; the subscript here and henceforth will denote
partial differentiation with respect to the indicated variable. Consequently,
the stretch is(2.2)ν0(σ)=|pσ(σ)|=dsdσ.We suppose that the cable flexural rigidity is
negligible in the equilibriumℬ0 under its own weight, hence the stress vector
representing the contact force that the cable segment σ>σ1 exerts on the cable segment σ<σ1 through the material section at σ1 has a resultant (integrated on the domain
representing the deformed cable material section) referred to as contact force and denoted n0. On the other hand, the resultant moment is
vanishing. This assumption holds true because under its own weight (uniformly
distributed), the cable will assume a funicular configuration having the
tension carrying the weight.By denotingf0,
the force per unit reference length σ, the local form of the balance of linear and
angular momentum can be written as(2.3)nσ0(σ)+f0(σ)=o,pσ0(σ)×n0(σ)=o.
The second part of equation (2.3) implies that n0 is in the same direction as pσ0=ν0a0, that is, in the direction of the tangent to
the deformed configuration ℬ0; hence, n0(σ)=N0(σ)a0(σ),
where N0 denotes the magnitude of the contact force,
commonly referred to as the tension. Moreover, the unit vector a0 can be expressed as(2.4)a0=pσ0|pσ0|=pσ0ν0.The final balance equation is
then rewritten as(2.5)[N0(σ)a0(σ)]σ+f0=o.The constitutive law relating the tensionN0 to the stretch ν0 is introduced in the form(2.6)N0(σ)=N^0(ν0,σ).A few restrictions are imposed
on N^0 as N^0(ν0=1,σ)=0, N^ν>0.
Moreover, for materials strong in resisting tension, we may require N^0→∞ as ν0→∞.The governing equilibrium equation is obtained
substituting (2.6) into (2.5). Theequilibrium equation can be projected into the
local basis {a0,b0,c0} (with c0≡e3) or the fixed basis (e1,e2,e3). Using the local basis, accounting for (a0)σ=ν0μ0b0, with μ0=θs0 (the geometric curvature of ℬ0), and letting f0=f10a0+f20b0 yield(2.7)N^σ0+f10=0,ν0μ0N^0+f20=0.On the other hand, projecting
(2.5) into the fixed basis (e1,e2,e3) and accounting for a0=cosθ0e1+sinθ0e2 and b0=−sinθ0e1+cosθ0e2 yield(2.8)[N^0cosθ0]σ+b10=0,[N^0sinθ0]σ+b20=0,where f0=b10e1+b20e2.The relationships between(cosθ0,sinθ0) and (xσ,yσ,ν0) can be obtained considering pσ0=xσe1+yσe2 and a0=cosθ0e1+sinθ0e2, that is,(2.9)cosθ0=xσν0,sinθ0=yσν0.From the fundamental
trigonometric identity (sin2θ0+cos2θ0=1), the stretch becomes ν0=xσ2+yσ2. The solution for the static configuration ℬ0 under the dead loads and, more specifically,
under the action of gravity, is described in the Appendix. We will discuss
cables suspended from two points at the same level (horizontal cables) or at
different levels (inclined cables).


### 2.2. The Dynamic Incremental Problem

The transformation from the static configurationℬ0 to the current configuration ℬ⌣ is illustrated next. We let p(s,t)=p0(s)+u(s,t) be the position vector of the point of the
cable in ℬ⌣ and let the arc length in ℬ⌣ be denoted s⌣. We will treat, to start with, the planar
problem (see Figure 1(b)). Hence, let b(s,t) be the unit vector giving the orientation of
the cable cross-section in ℬ⌣ so that the cable-fixed basis in ℬ⌣ is (a(s,t),b(s,t)) and θ denotes the rotation angle from (a0, b0) to (a,b). Therefore, the angle that a makes with e1 is θ⌣=θ0+θ.The cable total stretch is obtained, by enforcing the
shear strain to vanish, aspσ=ν⌣a,
where ν⌣=ds⌣/dσ=νν0, that is, the product of the initial stretch
and the incremental stretch ν. To calculate the cable incremental stretch ν that arises in the motion from ℬ0 to ℬ⌣, we consider ps=νa. Then, by letting u(s,t)=u(s,t)a0(s,t)+v(s,t)b0 represent the displacement vector from ℬ0 to ℬ⌣, the position vector is p=p0+u and its gradient becomes ps=ps0+us=(1+us−μ0v)a0+(vs+μ0u)b0. Consequently,(2.10)ν=(1+us−μ0v)2+(vs+μ0u)2,(2.11)a=(1+us−μ0v)a0+(vs+μ0u)b0ν.Moreover,(2.12)sinθ=ps⋅b0ν=vs+μ0uν,cosθ=ps⋅a0ν=1+us−μ0vν.Dividing the left- and
right-hand sides of the two preceding equations and considering the inverse
tangent function yield the incremental angle θ as(2.13)θ=tan−1(vs+μ0u1+us−μ0v).Therefore, the flexural
curvature μ is(2.14)μ=dθds=1ν2[(vss+(μ0u)s)(1+us−μ0v)−(uss−(μ0v)s)(vs+μ0u)].The presentation of the kinematic model is complete
once the boundary conditions are prescribed. For generality, we consider the cable supports lying in the(e1,e2)-plane and being placed at different levels and
immovable under the dead loads, x(0)=p0(0)=o, x(L)=p0(L)=ℓe1+he2. On the contrary, the supports prescribe some smooth motions during the dynamic
change of configuration from ℬ0 to ℬ⌣. Hence, the time-dependent boundary conditions
become(2.15)p(0,t)=uA(t)=uA(t)e2,p(L,t)=p0(L)+uB(t)=(l+uB(t))e1+he2.Next, the dynamical aspects of the problem are
discussed. We letn⌣=N⌣a+H⌣b be the contact force in the cable at s⌣, let f be the incremental external forces such that f⌣(s,t)=f0(s)+f(s,t) indicates the total external force density
acting on the cable (per unit reference length s), and let m⌣ be the total flexural moment of the cable at s⌣. Please note that here f0(s) denotes the force acting in ℬ0,
however, referred, for convenience, to the unit reference length s instead of σ. In the current configuration ℬ⌣, the local form of the balance of linear and
angular momentum requires(2.16)n⌣s(s,t)+f⌣(s,t)=ρA0ptt(s,t),m⌣s(s,t)+ps(s,t)×n⌣(s,t)=o.By accounting for ps(s,t)=νa, the balance of angular momentum
yields(2.17)m⌣s+νH⌣e3=o.Further, let N and H be the incremental axial and shear forces such
that N⌣=N0+N and H⌣≡H, since the reactive shear force H0 in ℬ0 is zero. Hence, the total contact force is n⌣=(N0+N)a+Hb. Similarly, m is the incremental flexural moment coinciding
with the total flexural moment since the cable flexural stiffness is neglected
in the initial static configuration, therefore, m⌣=m. Since m=Me3, (2.17), solved for H, yields the shear force as H=−Ms/ν. We can rewrite the equation of motion (2.16)
as(2.18)(N⌣a−Msνb)s+f0+f=ρA0ptt.By considering the equilibrium
in ℬ0,(N0a0)s+f0=o, and by letting N⌣a=N0a0+N0(a−a0)+Na, the equation of motion (2.18)
becomes(2.19)(Na)s+[N0(a−a0)]s−(Msνb)s+f=ρA0ptt.The componential form of (2.19) in
the cable-fixed basis (a0,b0) becomes(2.20)[(N0+N)s+Msν(μ0+μ)]cosθ−[(N0+N)(μ0+μ)−(Msν)s]sinθ−Ns0+f1=ρA0utt,(2.21)[(N0+N)s+Msν(μ0+μ)]sinθ+[(N0+N)(μ0+μ)−(Msν)s]cosθ−μ0N0+f2=ρA0vtt,where f=f1a0+f2b0, ρA0ptt=ρA0utta0+ρA0vttb0.Suitable nonlinearly viscoelastic constitutive laws
for the tensionN and the bending moment M are given in the form(2.22)N(s,t)=N^(ν,νt,s),iffν⌣=ν0ν>1,M(s,t)=M^(μ,μt,s).For materials whose constitutive behavior is linearly
viscoelastic also at large strains and strain rates, the linearized version of
the nonlinear constitutive laws can be expressed in the form(2.23)N^=EA(ν−1)+(N^ν˙)νt,iffν⌣=ν0ν>1,M^=EJμ+(M^μ˙)μt.The constitutive law for N⌣ is nonlinear in the sense that, because the
cable cannot resist compression, in those segments of the cable where ν⌣≤1, the total tension vanishes, that is, N⌣=N0+N^=0. When a total loss of tension occurs, the cable
does not undergo any local length changes, hence letting ν⌣=ν0ν=1 yields the incremental stretching in terms of
the prestretching, that is, ν=1/ν0. In the corresponding cable subdomains, the
balance equations are accordingly modified putting N⌣=0 and ν⌣=1.
Hence,(2.24)−[N0a0]s−(Msνb)s+f=ρA0ptt.The strain-displacement relationships (2.10) and (2.14) are
substituted into the constitutive equations (2.22) and these, in turn, are
substituted, along with (2.12), into the equations of (2.20) and (2.21) delivering the final
governing equations of motion in the unknown displacements u and v.We finally nondimensionalize the equations of motion
introducing the followinnondimensional quantities:(2.25)s*=sℓ,u*=uℓ,v*=vℓ,t*=H0ρA0ℓ2t,μ0*=ℓμ0,μ*=lμ,f*=ℓfH0,k=EA0H0,N0*=N0H0,N*=NH0,M*=MℓH0,Λ=EJ0ℓ2H0.By dropping the star for notational simplicity and by
neglecting the dissipative parts of the incremental dynamic tension and moment,
the nondimensional equations of motion, for linearly elastic cables,
become(2.26)Ns0[1+us−μ0vν−1]−N0(μ0+μ)(vs+μ0uν)+k[νs1+us−μ0vν−(ν−1)(μ0+μ)(vs+μ0uν)]+Λ[(μ0+μ)(μsν)(1+us−μ0vν)+(μssν−νsμsν2)(vs+μ0uν)]+f1=utt,(2.27)Ns0[vs+μ0uν]+N0[(μ0+μ)1+us−μ0vν2−μ0]+k[(ν−1)(μ0+μ)(1+us−μ0vν+νsvs+μ0uν)]+Λ[(μ0+μ)(μsν)(vs+μ0uν)+(μsνsν2−μssν)(1+us−μ0vν)]+f2=vtt,where the stretch ν and the curvature μ are expressed by (2.10) and (2.14). On the other
hand, the equations of motion in the case of a total decompression, with a
linearly elastic material, become(2.28)ν02(μ0+μ)(1+us−μ0v)(Λμ)s+ν0(vs+μ0u)[ν0(Λμ)s]s−Ns0+f1=utt,ν02(μ0+μ)(vs+μ0u)(Λμ)s−ν0(1+us−μ0v)[ν0(Λμ)s]s−μ0N0+f2=vtt.


## 2.1. The Prestressed Cable

The cable is
stress-free in the configurationℬ (Figure 1(a)). This configuration can be any
arbitrary configuration assumed by the cable, for example, on a frictionless
horizontal plane when it is not subject to forces besides the gravity force. In
this state, the internal stresses are expected to be practically vanishing. In
particular, in Figure 1(a), ℬ represents any reference line along the cable
in the stress-free configuration (e.g., the centroidal line) and L is its undeformed length. Further, we let the arc length, denoted σ, be the coordinate identifying the material
sections of the cable. We consider the fixed Cartesian reference frame (O,e1,e3,e3) shown in Figure 1(a). When the cable ends are
fixed to two points, A and B, and the cable is let free to hang under the
action of gravity, it occupies an equilibrium configuration, here denoted ℬ0, and considered as the reference configuration
for the subsequent dynamic problem. The arc length in the configuration ℬ0 is denoted s. Let the position vector of the material point P0 of the cable in the reference configuration ℬ0 be p0. Among the different parameterizations for p0, we employ σ, and let the Cartesian representation of p0 be p0(σ):=x(σ)e1+y(σ)e2.Stress-free configurationℬ, prestressed configuration ℬ0,
and actual configuration ℬ⌣ (top). Unit vectors and rotations of the plane
kinematics (bottom left) and internal/external forces in the cable (bottom
right).
(a)(b)(c)The cable stretch is then simply defined
as(2.1)pσ0(σ)=ν0(σ)a0(σ),where a0 is the unit vector in the tangential direction
to ℬ0; the subscript here and henceforth will denote
partial differentiation with respect to the indicated variable. Consequently,
the stretch is(2.2)ν0(σ)=|pσ(σ)|=dsdσ.We suppose that the cable flexural rigidity is
negligible in the equilibriumℬ0 under its own weight, hence the stress vector
representing the contact force that the cable segment σ>σ1 exerts on the cable segment σ<σ1 through the material section at σ1 has a resultant (integrated on the domain
representing the deformed cable material section) referred to as contact force and denoted n0. On the other hand, the resultant moment is
vanishing. This assumption holds true because under its own weight (uniformly
distributed), the cable will assume a funicular configuration having the
tension carrying the weight.By denotingf0,
the force per unit reference length σ, the local form of the balance of linear and
angular momentum can be written as(2.3)nσ0(σ)+f0(σ)=o,pσ0(σ)×n0(σ)=o.
The second part of equation (2.3) implies that n0 is in the same direction as pσ0=ν0a0, that is, in the direction of the tangent to
the deformed configuration ℬ0; hence, n0(σ)=N0(σ)a0(σ),
where N0 denotes the magnitude of the contact force,
commonly referred to as the tension. Moreover, the unit vector a0 can be expressed as(2.4)a0=pσ0|pσ0|=pσ0ν0.The final balance equation is
then rewritten as(2.5)[N0(σ)a0(σ)]σ+f0=o.The constitutive law relating the tensionN0 to the stretch ν0 is introduced in the form(2.6)N0(σ)=N^0(ν0,σ).A few restrictions are imposed
on N^0 as N^0(ν0=1,σ)=0, N^ν>0.
Moreover, for materials strong in resisting tension, we may require N^0→∞ as ν0→∞.The governing equilibrium equation is obtained
substituting (2.6) into (2.5). Theequilibrium equation can be projected into the
local basis {a0,b0,c0} (with c0≡e3) or the fixed basis (e1,e2,e3). Using the local basis, accounting for (a0)σ=ν0μ0b0, with μ0=θs0 (the geometric curvature of ℬ0), and letting f0=f10a0+f20b0 yield(2.7)N^σ0+f10=0,ν0μ0N^0+f20=0.On the other hand, projecting
(2.5) into the fixed basis (e1,e2,e3) and accounting for a0=cosθ0e1+sinθ0e2 and b0=−sinθ0e1+cosθ0e2 yield(2.8)[N^0cosθ0]σ+b10=0,[N^0sinθ0]σ+b20=0,where f0=b10e1+b20e2.The relationships between(cosθ0,sinθ0) and (xσ,yσ,ν0) can be obtained considering pσ0=xσe1+yσe2 and a0=cosθ0e1+sinθ0e2, that is,(2.9)cosθ0=xσν0,sinθ0=yσν0.From the fundamental
trigonometric identity (sin2θ0+cos2θ0=1), the stretch becomes ν0=xσ2+yσ2. The solution for the static configuration ℬ0 under the dead loads and, more specifically,
under the action of gravity, is described in the Appendix. We will discuss
cables suspended from two points at the same level (horizontal cables) or at
different levels (inclined cables).


## 2.2. The Dynamic Incremental Problem

The transformation from the static configurationℬ0 to the current configuration ℬ⌣ is illustrated next. We let p(s,t)=p0(s)+u(s,t) be the position vector of the point of the
cable in ℬ⌣ and let the arc length in ℬ⌣ be denoted s⌣. We will treat, to start with, the planar
problem (see Figure 1(b)). Hence, let b(s,t) be the unit vector giving the orientation of
the cable cross-section in ℬ⌣ so that the cable-fixed basis in ℬ⌣ is (a(s,t),b(s,t)) and θ denotes the rotation angle from (a0, b0) to (a,b). Therefore, the angle that a makes with e1 is θ⌣=θ0+θ.The cable total stretch is obtained, by enforcing the
shear strain to vanish, aspσ=ν⌣a,
where ν⌣=ds⌣/dσ=νν0, that is, the product of the initial stretch
and the incremental stretch ν. To calculate the cable incremental stretch ν that arises in the motion from ℬ0 to ℬ⌣, we consider ps=νa. Then, by letting u(s,t)=u(s,t)a0(s,t)+v(s,t)b0 represent the displacement vector from ℬ0 to ℬ⌣, the position vector is p=p0+u and its gradient becomes ps=ps0+us=(1+us−μ0v)a0+(vs+μ0u)b0. Consequently,(2.10)ν=(1+us−μ0v)2+(vs+μ0u)2,(2.11)a=(1+us−μ0v)a0+(vs+μ0u)b0ν.Moreover,(2.12)sinθ=ps⋅b0ν=vs+μ0uν,cosθ=ps⋅a0ν=1+us−μ0vν.Dividing the left- and
right-hand sides of the two preceding equations and considering the inverse
tangent function yield the incremental angle θ as(2.13)θ=tan−1(vs+μ0u1+us−μ0v).Therefore, the flexural
curvature μ is(2.14)μ=dθds=1ν2[(vss+(μ0u)s)(1+us−μ0v)−(uss−(μ0v)s)(vs+μ0u)].The presentation of the kinematic model is complete
once the boundary conditions are prescribed. For generality, we consider the cable supports lying in the(e1,e2)-plane and being placed at different levels and
immovable under the dead loads, x(0)=p0(0)=o, x(L)=p0(L)=ℓe1+he2. On the contrary, the supports prescribe some smooth motions during the dynamic
change of configuration from ℬ0 to ℬ⌣. Hence, the time-dependent boundary conditions
become(2.15)p(0,t)=uA(t)=uA(t)e2,p(L,t)=p0(L)+uB(t)=(l+uB(t))e1+he2.Next, the dynamical aspects of the problem are
discussed. We letn⌣=N⌣a+H⌣b be the contact force in the cable at s⌣, let f be the incremental external forces such that f⌣(s,t)=f0(s)+f(s,t) indicates the total external force density
acting on the cable (per unit reference length s), and let m⌣ be the total flexural moment of the cable at s⌣. Please note that here f0(s) denotes the force acting in ℬ0,
however, referred, for convenience, to the unit reference length s instead of σ. In the current configuration ℬ⌣, the local form of the balance of linear and
angular momentum requires(2.16)n⌣s(s,t)+f⌣(s,t)=ρA0ptt(s,t),m⌣s(s,t)+ps(s,t)×n⌣(s,t)=o.By accounting for ps(s,t)=νa, the balance of angular momentum
yields(2.17)m⌣s+νH⌣e3=o.Further, let N and H be the incremental axial and shear forces such
that N⌣=N0+N and H⌣≡H, since the reactive shear force H0 in ℬ0 is zero. Hence, the total contact force is n⌣=(N0+N)a+Hb. Similarly, m is the incremental flexural moment coinciding
with the total flexural moment since the cable flexural stiffness is neglected
in the initial static configuration, therefore, m⌣=m. Since m=Me3, (2.17), solved for H, yields the shear force as H=−Ms/ν. We can rewrite the equation of motion (2.16)
as(2.18)(N⌣a−Msνb)s+f0+f=ρA0ptt.By considering the equilibrium
in ℬ0,(N0a0)s+f0=o, and by letting N⌣a=N0a0+N0(a−a0)+Na, the equation of motion (2.18)
becomes(2.19)(Na)s+[N0(a−a0)]s−(Msνb)s+f=ρA0ptt.The componential form of (2.19) in
the cable-fixed basis (a0,b0) becomes(2.20)[(N0+N)s+Msν(μ0+μ)]cosθ−[(N0+N)(μ0+μ)−(Msν)s]sinθ−Ns0+f1=ρA0utt,(2.21)[(N0+N)s+Msν(μ0+μ)]sinθ+[(N0+N)(μ0+μ)−(Msν)s]cosθ−μ0N0+f2=ρA0vtt,where f=f1a0+f2b0, ρA0ptt=ρA0utta0+ρA0vttb0.Suitable nonlinearly viscoelastic constitutive laws
for the tensionN and the bending moment M are given in the form(2.22)N(s,t)=N^(ν,νt,s),iffν⌣=ν0ν>1,M(s,t)=M^(μ,μt,s).For materials whose constitutive behavior is linearly
viscoelastic also at large strains and strain rates, the linearized version of
the nonlinear constitutive laws can be expressed in the form(2.23)N^=EA(ν−1)+(N^ν˙)νt,iffν⌣=ν0ν>1,M^=EJμ+(M^μ˙)μt.The constitutive law for N⌣ is nonlinear in the sense that, because the
cable cannot resist compression, in those segments of the cable where ν⌣≤1, the total tension vanishes, that is, N⌣=N0+N^=0. When a total loss of tension occurs, the cable
does not undergo any local length changes, hence letting ν⌣=ν0ν=1 yields the incremental stretching in terms of
the prestretching, that is, ν=1/ν0. In the corresponding cable subdomains, the
balance equations are accordingly modified putting N⌣=0 and ν⌣=1.
Hence,(2.24)−[N0a0]s−(Msνb)s+f=ρA0ptt.The strain-displacement relationships (2.10) and (2.14) are
substituted into the constitutive equations (2.22) and these, in turn, are
substituted, along with (2.12), into the equations of (2.20) and (2.21) delivering the final
governing equations of motion in the unknown displacements u and v.We finally nondimensionalize the equations of motion
introducing the followinnondimensional quantities:(2.25)s*=sℓ,u*=uℓ,v*=vℓ,t*=H0ρA0ℓ2t,μ0*=ℓμ0,μ*=lμ,f*=ℓfH0,k=EA0H0,N0*=N0H0,N*=NH0,M*=MℓH0,Λ=EJ0ℓ2H0.By dropping the star for notational simplicity and by
neglecting the dissipative parts of the incremental dynamic tension and moment,
the nondimensional equations of motion, for linearly elastic cables,
become(2.26)Ns0[1+us−μ0vν−1]−N0(μ0+μ)(vs+μ0uν)+k[νs1+us−μ0vν−(ν−1)(μ0+μ)(vs+μ0uν)]+Λ[(μ0+μ)(μsν)(1+us−μ0vν)+(μssν−νsμsν2)(vs+μ0uν)]+f1=utt,(2.27)Ns0[vs+μ0uν]+N0[(μ0+μ)1+us−μ0vν2−μ0]+k[(ν−1)(μ0+μ)(1+us−μ0vν+νsvs+μ0uν)]+Λ[(μ0+μ)(μsν)(vs+μ0uν)+(μsνsν2−μssν)(1+us−μ0vν)]+f2=vtt,where the stretch ν and the curvature μ are expressed by (2.10) and (2.14). On the other
hand, the equations of motion in the case of a total decompression, with a
linearly elastic material, become(2.28)ν02(μ0+μ)(1+us−μ0v)(Λμ)s+ν0(vs+μ0u)[ν0(Λμ)s]s−Ns0+f1=utt,ν02(μ0+μ)(vs+μ0u)(Λμ)s−ν0(1+us−μ0v)[ν0(Λμ)s]s−μ0N0+f2=vtt.


## 3. The Linearization and Vibration Eigenvalue Problem

The linearization of (2.19) can be systematically obtained once we introduce a small parameter, denoted ϵ, which suitably quantifies the deviations from
the prestressed configuration ℬ0. We neglect the dissipative parts of the
tension and bending moment and the forcing. We assume ν=ν(ϵ),μ=μ(ϵ),N^=N^(ν(ϵ)),M^=M^(μ(ϵ)),us=us(ϵ),a=a(ϵ) as continuously differentiable functions of ϵ. Further,(3.1)ν(0)=1,μ(0)=0,N^(ν(0))=0,M^(μ(0))=0,a(0)=a0,b(0)=b0,us(0)=o.Therefore, the elastic parts of
the tension and flexural moment can be expanded in series of ϵ as(3.2)N^(ν(ϵ))a(ϵ)=ϵEA0ν1a0+O(ϵ2),M^(μ(ϵ))sν(ϵ)b(ϵ)=ϵ(EJ0μ1)sb0+O(ϵ2),where ν1=νϵ(0) and μ1=μϵ(0) denote the first-order part of the stretch and
the curvature (i.e., containing only linear terms in us). On the other hand, the geometric part of
the internal force, expanded in series of ϵ, yields N0(a(ϵ)−a0)=N0a1+O(ϵ2),
where a1=aϵ(0) is the first-order deviation of a from a0. By retaining only first-order terms in the
expansion of the equations of motion, we obtain the linearized equations of
motion(3.3)(EA0ν1a0)s+(N0a1)s−((EJ0μ1)sb0)s=ρA0utt.To calculate ν1, μ1,
and a1, we consider (2.10), (2.14), and (2.11), and by
differentiating them with respect to ϵ and by setting ϵ=0, we obtain(3.4)ν1=us−μ0v,μ1=vss+μs0u+μ0us,a1=(vs+μ0u)b0.The first-order variation of a can be alternatively obtained from a=cosθ(ϵ)a0+sinθ(ϵ)b0=a0+ϵθ1b0+O(ϵ2),
where the first-order variation of θ, given by (2.13), is θ1=vs+μ0u. Therefore,(3.5)[EA0(us−μ0v)]s+μ0[EJ0(vss+μs0u+μ0us)]s−μ0N0(vs+μ0u)=ρA0utt,μ0[EA0(us−μ0v)]−[EJ0(vss+μs0u+μ0us)]ss+Ns0(vs+μ0u)+N0(vs+μ0u)s=ρA0vtt.In nondimensional
form,(3.6)[k(us−μ0v)]s+μ0[Λ(vss+μs0u+μ0us)]s−μ0N0(vs+μ0u)=utt,μ0k(us−μ0v)−[Λ(vss+μs0u+μ0us)]ss+Ns0(vs+μ0u)+N0(vss+μs0u+μ0us)=vtt.


### 3.1. The Cable Elastogeometric Parameters

The linear as well as nonlinear free motions of
linearly elastic nonshallow cables without flexural stiffness depend on two
parameters [5] related
to their geometric and elasticstiffnesses, namely, γ and k,
contrary to shallow cables [1] whose linear motions depend solely on Irvine's
elastogeometric parameter λ. Irvine's parameter combines the two
characteristic parameters of nonshallow cables according to λ2:=γ2k/ηe with ηe=∫0η0sec2θ0ds, η0=L/ℓ. For cables suffering stretching and bending in
the plane and made of a linearly elastic material, the independent governing
parameters are three, namely, (γ,k,Λ) or, equivalently,  (γ,λ,Λ).We discuss in more detail the cable parameters
pointing out their mechanical significance and pertinent ranges of variation.
The region of admissible elastic stiffnessk in the (γ,λ)-plane was discussed in [5] considering the isostiffness
curves according to the definition of Irvine's parameter. The boundaries of the
admissible region correspond to k1=5⋅102 and k3=5⋅104, respectively. These values were determined
considering that k=E/S0 and S0=H0/A0 denotes the engineering tensile stress at the
mid-span section. Hence, the minimum k1 is attained when S0 is maximum, here taken as the yielding tensile
strength Sy. Because typical working tensile stresses are
around 5–10% of Sy, a reasonable value of k is of the order of 103. However, by considering lower tensile stresses
in the static configuration, values of k of the order of 104 can be reasonably reached.By assuming a typical circular cross-section,J0=(A0)2/(4π), hence the nondimensional flexural stiffness
becomes Λ=kA0/(4πℓ2). Further, it is γ=ρgℓ/S0 and k=E/S0. Hence, by considering the following as design
data: (i) the sag ℓ, (ii) the material properties (ρ,E), and (iii) the working tensile stress S0, the parameters γ and k (or λ) are accordingly determined and are
independent of A0 whereas Λ depends on k, ℓ, and A0. Therefore, considering iso-k curves, the nondimensional flexural stiffness
parameter Λ remains constant only if the cable
cross-sectional areas do not vary.


## 3.1. The Cable Elastogeometric Parameters

The linear as well as nonlinear free motions of
linearly elastic nonshallow cables without flexural stiffness depend on two
parameters [5] related
to their geometric and elasticstiffnesses, namely, γ and k,
contrary to shallow cables [1] whose linear motions depend solely on Irvine's
elastogeometric parameter λ. Irvine's parameter combines the two
characteristic parameters of nonshallow cables according to λ2:=γ2k/ηe with ηe=∫0η0sec2θ0ds, η0=L/ℓ. For cables suffering stretching and bending in
the plane and made of a linearly elastic material, the independent governing
parameters are three, namely, (γ,k,Λ) or, equivalently,  (γ,λ,Λ).We discuss in more detail the cable parameters
pointing out their mechanical significance and pertinent ranges of variation.
The region of admissible elastic stiffnessk in the (γ,λ)-plane was discussed in [5] considering the isostiffness
curves according to the definition of Irvine's parameter. The boundaries of the
admissible region correspond to k1=5⋅102 and k3=5⋅104, respectively. These values were determined
considering that k=E/S0 and S0=H0/A0 denotes the engineering tensile stress at the
mid-span section. Hence, the minimum k1 is attained when S0 is maximum, here taken as the yielding tensile
strength Sy. Because typical working tensile stresses are
around 5–10% of Sy, a reasonable value of k is of the order of 103. However, by considering lower tensile stresses
in the static configuration, values of k of the order of 104 can be reasonably reached.By assuming a typical circular cross-section,J0=(A0)2/(4π), hence the nondimensional flexural stiffness
becomes Λ=kA0/(4πℓ2). Further, it is γ=ρgℓ/S0 and k=E/S0. Hence, by considering the following as design
data: (i) the sag ℓ, (ii) the material properties (ρ,E), and (iii) the working tensile stress S0, the parameters γ and k (or λ) are accordingly determined and are
independent of A0 whereas Λ depends on k, ℓ, and A0. Therefore, considering iso-k curves, the nondimensional flexural stiffness
parameter Λ remains constant only if the cable
cross-sectional areas do not vary.


## 4. Computational Scheme: Finite-Difference versus Finite Elements

We employ a
finite-difference discretization method to path-follow the nonlinear static
solutions of (2.26) and (2.27) with f1 and f2 being time-independent and being applied
through a loading device with a sufficiently slow rate, that is, in a
quasistatic fashion. The boundary conditions at A and B prescribe the displacements and the bending
moments. In particular, the six boundary conditions are as follows:(4.1)u(0)=uA,v(0)=vA,u(η0)=uB,v(η0)=vB,M(0)=M(η0)=0.By considering a grid withn points including the boundary points A and B, the interior points are n−2. At the interior points, we enforce the two
balance equations, with an ensuing number of 2(n−2) field equations. Overall, the number of
equations is 2(n+1). If the problem is fully formulated in the
displacement components (u,v),
then the unknowns would be 2n leading to an overconstrained system of
nonlinear equations. The problem is circumvented by employing a standard mixed
approach (in the sense that it is neither the
displacement nor the force method) consisting in taking as unknowns the
displacement (u,v) and the bending moment M resulting into 3n unknowns. The bending moment, treated as
unknown in the balance equations, has to satisfy the constitutive equation
which is added explicitly as an independent equation. Therefore, the overall
number of equations comprises the 3(n−2) field equations to which the six boundary
conditions are to be added, resulting into a system of 3n equations in 3n unknowns.Therefore, the field equations read(4.2)[Ns0+(k(ν−1))s+Msν(μ0+μ)]cosθ−[(N0+k(ν−1))(μ0+μ)−(Msν)s]sinθ−Ns0+f1=0,[Ns0+(k(ν−1))s+Msν(μ0+μ)]sinθ+[(N0+k(ν−1))(μ0+μ)−(Msν)s]cosθ−μ0N0+f2=0,M−Λ1ν2[(vss+(μ0u)s)(1+us−μ0v)−(uss−(μ0v)s)(vs+μ0u)]=0.To enhance the accuracy of the finite-difference
scheme, we employed a five-point scheme [17] based on an equidistant point grid which can be
written, considering the first-order space derivative, as follows:(4.3)112×Δ[−2548−3616−3−3−1018−611−808−1−16−181033−1636−4825],where Δ=si−si−1 denotes the distance between two adjacent
points of the grid.For the chosenn-point grid, we have 3n equations, each expressed as a function of(4.4)[(ui−2,ui−1,ui,ui+1,ui+2),(vi−2,vi−1,vi,vi+1,vi+2),(Mi−2,Mi−1,Mi,Mi+1,Mi+2)]and the external force term fi,
where i is the index associated to the grid point. Out
of the six boundary conditions, four equations are kinematic, namely, u0=uA,un=uB,v0=vA,vn=vB, two of them are mechanical, M0=Mn=0. Further, mention must be made of the fact that
the discretization at the two points adjacent to the boundary points is not
clearly centered.The problem is solved step-by-step employing a
zeroth-order path-following scheme where the Newton-Raphson iterative scheme is
exploited at each load step to find the new solution point. The procedure was
implemented in MATHEMATICA [18]. At each load step, the external force is increased
by Δfi and the solution point of the preceding step
is used as the initial guess in the updated load step; in this sense, the
continuation procedure is based on a zeroth-order predictor. At the end of each
load step, the determined solution is expected to satisfy the balance equations
ensuring that the pointwise remainders are below a prescribed numerical
tolerance.The same analyses have been conducted employing COMSOL
[19]. COMSOL Multiphysics
allows to approximate partial-differential equations of various kinds via a
finite element procedure. The number of quadratic Lagrangian finite elements
was set to 7680 in all calculations, for a total number of 76 805 degrees of
freedom, and the tolerance was fixed to 10−6. The high number of finite elements was not
strictly needed, it was chosen in all calculations for accuracy reasons. A
close agreement between the finite difference-based (with number of grid points
greater than or equal to 30) and finite element-based results has been found
and it is such that only the outcomes of COMSOL are reported next.


## 5. Illustrative Examples of Nonlinear Static Responses

It is of interest to investigate into the differences
exhibited by the nonlinear structural responses of cables possessing flexural
stiffness and those of cables whose flexural stiffness is neglected. The
objective is to assess the extent of the bending stiffness contribution within
the context of static loading processes. It is clear, however, that the most
significant effects are expected to be exhibited in dynamic regimes, especially near resonances and instabilities.In this section, we present some illustrative examples
of nonlinear responses to three different loading scenarios. All loading cases
are selected so as to induce a gradual loss of tension that enhances the
bending moment contribution. The first case features a horizontal cable, lying
under its own weight between two points at the same level. A constant uniformly
distributed vertical upward load is applied over a central region of the cable.
In the other two cases, the loading path is a prescribed incremental
displacement of the right support to the left so as to loosen the cable. We
consider both a horizontal and an inclined cable.Two different cables are considered in the numerical
computations and they have the following nondimensional governing parameters:η0=1.09615, γ=1.5, k=1.10⋅104, Λ=4.43⋅10−4, for the horizontal cable, and γ=0.47, k=3.70⋅103, Λ=1.40⋅10−4, for the inclined cable. These parameters
correspond to a steel cable whose initial length is L=142.5 m for the first configuration whose span is l=130 m, Young's effective modulus is E=100 GPa; its cross-sectional area and moment of
inertia are A0=8⋅10−3 m2 and J0=5.1⋅10−6 m4, respectively. The height of the inclined cable is h=30 m and its initial length is L=136.4 m.In all loading scenarios, we determine the cable
response curves, depicting variations of the vertical displacement of a control
point (pointC whose arc length coordinate is s=1/3L) with the magnitude of the force or support
displacement. In particular, the loading path is discretized into NL steps so that, by indicating with fNL the load magnitude at the end of the loading
path and with fj the magnitude at the jth step, we let αj=fj/fNL be the load multiplier. We monitored the
configurations and state of stress at three given load steps, namely, α=1/3,2/3,1. For each of those three states, the tension,
shear force, and flexural moment distributions along the cable are analyzed so
as to point out the influence of the flexural rigidity throughout a comparison
of the results with those obtained using the standard model that neglects the
flexural stiffness.


### 5.1. Horizontal Cable Subject to an Upward Vertical Load

The first case is that of a horizontal cable, shown in
Figure2, subject to an upward vertical load, distributed over a small region
centered about the midspan whose length is Δs=0.1873η0.
The nondimensional load amplitude is varied in the range [0.03,12]. Because the force per unit reference length
has been nondimensionalized with respect to mg/γ, and γ=1.5; the maximum load is fNL=8mg and the resultant load becomes F=8mg(Δsℓ), which is about 3/2W,
where W=mgL0 is the total weight of the cable. The loading
process is discretized into NL=400 load steps with a resulting load step Δf=0.02993.Figure 2
Response curves of the horizontal cable: variation of
the vertical displacement of the control pointC with the load magnitude obtained with the
flexural stiffness (solid line) and without (dashed line).The response curve in Figure2 shows a softening
behavior of the cable control point in both models. We further note that the
curve representing the model with flexural stiffness is globally above the
curve obtained with the standard cable model as it is to be expected since the
cable with flexural stiffness is clearly stiffer than the purely extensible
cable. This difference in behaviors is exhibited neither at the beginning nor
at the end of the loading path whereas the most prominent difference is
appreciated when α=2/3, that is, when the upward load is nearly equal
the weight of the cable.Figure3 shows the cable configurations at various
load magnitudes, the initial static configuration ℬ0 (catenary configuration), two intermediate
configurations ℬ⌣ when α=1/3 (F≃1/2W) and α=2/3 (F≃W) and the final configuration at the maximum
load (F≃3/2W). A region of negative curvatures is
localized around the subdomain where the load is applied upward. In particular,
when the load is between W and 3/2W, the cable crosses the horizontal line passing
through the supports and the extent of the central segment of the cable lifted
above the horizontal line gradually increases up to the end of the loading path
when it is nearly one third of the cable. At two thirds of the maximum load,
the difference introduced by the flexural stiffness is quite remarkable
especially if we consider the displacement of the midspan point around which we
note appreciably different curvatures, sharper in the purely extensible cable.
In Figure 4 (top), we show the tension along the cable whose evolution has two
distinct phases. First, the application of the load tends to decrease uniformly
the total tension in the cable until the load reaches a sufficient value to
overcome the weight of the cable in the central region. At this stage, the
curvature is reversed at the midspan and the tension gradually starts to exhibit
sharp decreasing variations thus introducing a lack of uniformity of
distribution. While at the midspan section the curvature (and the bending
moment) increases, the tension tends to vanish; on the other hand, around the
midspan, the tension has to balance part of the total weight of the cable.
Concurrently, the shear load exhibits a boundary layer within the central
loaded region, and the magnitude of the jump increases with the load amplitude.
At the peak of the loading path, the bending moment is clearly localized within
the central boundary layer with the peak moment being one order of magnitude
higher than elsewhere in the cable.Figure 3
Cable configurations under an upward vertical load
obtained with the flexural stiffness (solid lines) and without (dashed
lines).Tension (top), shear force (middle), and bending
moments (bottom) of the cable under an upward vertical load obtained with the flexural stiffness (solid lines) and without (dashed lines).
(a)(b)(c)


### 5.2. Horizontal and Inclined Cables Subject to Support Displacements

The right end boundary of the horizontal cable is moved
horizontally to the left up to a nondimensional value ofuB=0.808 (i.e., a dimensional displacement of the
considered cable of 105 m) in NL=1050 steps. Figure 5 shows the response curve with
the prescribeddisplacement uB on the vertical axis and the displacement of
the control point C on the abscissa axis. Besides, Figure 6
presents the configurations of both cables, with and without flexural
stiffness, at the three load steps (α=1/3,2/3,1) and Figure 7 reports the associated tension,
shear force, and bending moment.Figure 5
Response curves of the horizontal cable: variation of
the vertical displacement of the control pointC with the support displacement obtained with
the flexural stiffness (solid line) and without (dashed line).Figure 6
Cable configurations under a prescribed displacement
of the support.Tension (top), shear force (middle), and bending
moments (bottom) of the cable subject to displacements of the support obtained
with the flexural stiffness (solid lines) and without (dashed lines).
(a)(b)(c)During the whole loading path, a decrease of tension
and an increase of shear force and bending moment are observed in the central
region as the two boundaries are brought closer; further, as in the first
loading scenario, the decrease becomes even more remarkable when the curvature
is increasing. At the end of the loading path, the central region presents a
tension that is almost close to zero bringing the cable close to a total loss
of tension. Further, the shear force at the boundaries is appreciable. We also
note that, as it has already been pointed out in previous studies, the standard
model of purely extensible cables generates numerical instabilities when the
tension levels are too low while the consideration of flexural stiffness
overcomes this problem. Moreover, the close observation of the differences in
the configurations exhibited by the two cables reveals that the flexural
stiffness has an appreciable influence on the equilibrium configuration even in
the static regime, and points out the fact that it cannot be neglected in those
segments of the cable where the tension is very low.To quantify the differences in the state of stress,
let us now consider the midspan section whenα=1,
and let S=N/A0 denote the maximum tensile stress for the
cable without flexural stiffness, and let S¯=N¯/A0+M¯/Wf0  be the tensile
stress of the cable with flexural stiffness (where Wf0=J0/A/π is the cable bending modulus). Calculating the relative percent
difference between S¯ and S yields a value about 53% which indicates that we would be led to
underestimate the maximum tensile stress by the same amount with the standard
cable model.In the last loading scenario, the right support of the
inclined cable is moved horizontally to the left up to a nondimensional value
ofuB=0.769 (i.e., a dimensional displacement of the
considered cable of 100 m) in NL=1000 steps. Figure 8 shows the response curves
whereas Figure 9 presents the configurations of both cables, with and without
flexural stiffness, at the three load steps (α=1/3,2/3,1), and Figure 10 reports the associated
tension, shear force, and bending moment. This case is very similar to that of
the horizontal cable although this cable is shallow and quite taut. The final
maximum curvature is smaller than in the other case while the final lowest
tension is bigger. It has to be noticed that, in this case, the most stressed
region, on consideration of the shear force and flexural moment, is more
shifted to the left due to the evident asymmetry of the problem.Figure 8
Response curves of the inclined cable: variation of
the vertical displacement of the control pointC with the support displacement obtained with
the flexural stiffness (solid line) and without (dashed line).Figure 9
Cable configurations under a prescribed displacement
of the support.Tension (top), shear force (middle), and bending
moments (bottom) of the cable subject to displacements of the support obtained
with the flexural stiffness (solid lines) and without (dashed lines).
(a)(b)(c)


## 5.1. Horizontal Cable Subject to an Upward Vertical Load

The first case is that of a horizontal cable, shown in
Figure2, subject to an upward vertical load, distributed over a small region
centered about the midspan whose length is Δs=0.1873η0.
The nondimensional load amplitude is varied in the range [0.03,12]. Because the force per unit reference length
has been nondimensionalized with respect to mg/γ, and γ=1.5; the maximum load is fNL=8mg and the resultant load becomes F=8mg(Δsℓ), which is about 3/2W,
where W=mgL0 is the total weight of the cable. The loading
process is discretized into NL=400 load steps with a resulting load step Δf=0.02993.Figure 2
Response curves of the horizontal cable: variation of
the vertical displacement of the control pointC with the load magnitude obtained with the
flexural stiffness (solid line) and without (dashed line).The response curve in Figure2 shows a softening
behavior of the cable control point in both models. We further note that the
curve representing the model with flexural stiffness is globally above the
curve obtained with the standard cable model as it is to be expected since the
cable with flexural stiffness is clearly stiffer than the purely extensible
cable. This difference in behaviors is exhibited neither at the beginning nor
at the end of the loading path whereas the most prominent difference is
appreciated when α=2/3, that is, when the upward load is nearly equal
the weight of the cable.Figure3 shows the cable configurations at various
load magnitudes, the initial static configuration ℬ0 (catenary configuration), two intermediate
configurations ℬ⌣ when α=1/3 (F≃1/2W) and α=2/3 (F≃W) and the final configuration at the maximum
load (F≃3/2W). A region of negative curvatures is
localized around the subdomain where the load is applied upward. In particular,
when the load is between W and 3/2W, the cable crosses the horizontal line passing
through the supports and the extent of the central segment of the cable lifted
above the horizontal line gradually increases up to the end of the loading path
when it is nearly one third of the cable. At two thirds of the maximum load,
the difference introduced by the flexural stiffness is quite remarkable
especially if we consider the displacement of the midspan point around which we
note appreciably different curvatures, sharper in the purely extensible cable.
In Figure 4 (top), we show the tension along the cable whose evolution has two
distinct phases. First, the application of the load tends to decrease uniformly
the total tension in the cable until the load reaches a sufficient value to
overcome the weight of the cable in the central region. At this stage, the
curvature is reversed at the midspan and the tension gradually starts to exhibit
sharp decreasing variations thus introducing a lack of uniformity of
distribution. While at the midspan section the curvature (and the bending
moment) increases, the tension tends to vanish; on the other hand, around the
midspan, the tension has to balance part of the total weight of the cable.
Concurrently, the shear load exhibits a boundary layer within the central
loaded region, and the magnitude of the jump increases with the load amplitude.
At the peak of the loading path, the bending moment is clearly localized within
the central boundary layer with the peak moment being one order of magnitude
higher than elsewhere in the cable.Figure 3
Cable configurations under an upward vertical load
obtained with the flexural stiffness (solid lines) and without (dashed
lines).Tension (top), shear force (middle), and bending
moments (bottom) of the cable under an upward vertical load obtained with the flexural stiffness (solid lines) and without (dashed lines).
(a)(b)(c)


## 5.2. Horizontal and Inclined Cables Subject to Support Displacements

The right end boundary of the horizontal cable is moved
horizontally to the left up to a nondimensional value ofuB=0.808 (i.e., a dimensional displacement of the
considered cable of 105 m) in NL=1050 steps. Figure 5 shows the response curve with
the prescribeddisplacement uB on the vertical axis and the displacement of
the control point C on the abscissa axis. Besides, Figure 6
presents the configurations of both cables, with and without flexural
stiffness, at the three load steps (α=1/3,2/3,1) and Figure 7 reports the associated tension,
shear force, and bending moment.Figure 5
Response curves of the horizontal cable: variation of
the vertical displacement of the control pointC with the support displacement obtained with
the flexural stiffness (solid line) and without (dashed line).Figure 6
Cable configurations under a prescribed displacement
of the support.Tension (top), shear force (middle), and bending
moments (bottom) of the cable subject to displacements of the support obtained
with the flexural stiffness (solid lines) and without (dashed lines).
(a)(b)(c)During the whole loading path, a decrease of tension
and an increase of shear force and bending moment are observed in the central
region as the two boundaries are brought closer; further, as in the first
loading scenario, the decrease becomes even more remarkable when the curvature
is increasing. At the end of the loading path, the central region presents a
tension that is almost close to zero bringing the cable close to a total loss
of tension. Further, the shear force at the boundaries is appreciable. We also
note that, as it has already been pointed out in previous studies, the standard
model of purely extensible cables generates numerical instabilities when the
tension levels are too low while the consideration of flexural stiffness
overcomes this problem. Moreover, the close observation of the differences in
the configurations exhibited by the two cables reveals that the flexural
stiffness has an appreciable influence on the equilibrium configuration even in
the static regime, and points out the fact that it cannot be neglected in those
segments of the cable where the tension is very low.To quantify the differences in the state of stress,
let us now consider the midspan section whenα=1,
and let S=N/A0 denote the maximum tensile stress for the
cable without flexural stiffness, and let S¯=N¯/A0+M¯/Wf0  be the tensile
stress of the cable with flexural stiffness (where Wf0=J0/A/π is the cable bending modulus). Calculating the relative percent
difference between S¯ and S yields a value about 53% which indicates that we would be led to
underestimate the maximum tensile stress by the same amount with the standard
cable model.In the last loading scenario, the right support of the
inclined cable is moved horizontally to the left up to a nondimensional value
ofuB=0.769 (i.e., a dimensional displacement of the
considered cable of 100 m) in NL=1000 steps. Figure 8 shows the response curves
whereas Figure 9 presents the configurations of both cables, with and without
flexural stiffness, at the three load steps (α=1/3,2/3,1), and Figure 10 reports the associated
tension, shear force, and bending moment. This case is very similar to that of
the horizontal cable although this cable is shallow and quite taut. The final
maximum curvature is smaller than in the other case while the final lowest
tension is bigger. It has to be noticed that, in this case, the most stressed
region, on consideration of the shear force and flexural moment, is more
shifted to the left due to the evident asymmetry of the problem.Figure 8
Response curves of the inclined cable: variation of
the vertical displacement of the control pointC with the support displacement obtained with
the flexural stiffness (solid line) and without (dashed line).Figure 9
Cable configurations under a prescribed displacement
of the support.Tension (top), shear force (middle), and bending
moments (bottom) of the cable subject to displacements of the support obtained
with the flexural stiffness (solid lines) and without (dashed lines).
(a)(b)(c)


## 6. Natural Frequencies of Cables with Flexural Stiffness

This section discusses concisely the vibration
behavior of cables without flexural stiffness, in shallow and nonshallow
regimes, with respect to the behavior of cables with flexural stiffness. To
this end, variations of the lowest natural frequencies with Irvine's parameterλ are here reported. In Irvine's theory of
shallow cables and in the generalized theory described in [5], the free motions of cables
only depend on the two parameters (γ,λ) which completely characterize the geometric
and elastic properties of linearly elastic cables. Taking into account the
flexural stiffness introduces a new elastogeometric parameter, Λ.
In Figure 11, variations of the lowest natural frequencies with λ/π are shown as obtained with the two relevant
models, with incorporation of the flexural stiffness (solid lines) and without
flexural stiffness (dashed lines). We consider three different regimes: shallow
cables with γ=0.1, transition cables with γ=0.75,
and nonshallow cables with γ=1.5. In Figure 11 (top), in the case of shallow cables
(γ=0.1), differences between the two models are not
easily detectable except mild differences for λ>0.7π. On the other hand, for nonshallow cables (γ=1.5), we clearly observe a deviation of the loci
of the eigenfrequencies to higher values with increasing λ. As a matter of a fact, flexural stiffness
effects are expected to be more significant for cables with larger
cross-section areas, that is, in our case, those corresponding to larger values
of λ. Here, the assumed data are γ (that defines the geometric stiffness) and E,
Young's modulus. At the same time, higher modes present a number of curvature
variations, hence a number of nodes, greater than the lower modes so that the
flexural rigidity is expected to impact the frequencies of the higher modes as
it can be seen in Figure 11. Furthermore, the increase of the natural
frequencies due to the flexural stiffness effects do not seem to generate new
crossovers.Variation of the lowest natural frequencies withλ/π obtained with the flexural stiffness (solid
lines) and without (dashed lines) when γ=0.1,0.75,1.5, respectively.
(a)(b)(c)


## 7. Concluding Remarks

A geometrically exact formulation of cables undergoing
axis stretching and flexural curvature has been proposed. The model, in its
general form, is suitable to treat more general cable regimes, such as the
loosening regime whereby a local loss of tension occurs.The equations of motion have been formulated for
cables with nonlinearly viscoelastic constitutive laws and general loading
conditions. The particular case of cables with a linearly elastic constitutive
law has been then considered in the numerical calculations conducted on
horizontal and inclined cables by employing a path-following scheme with two
different discretizations: a finite-difference approach and a finite element formulation.
In particular, low-tension regimes have been investigated, and the flexural
stiffness influence on shallow and nonshallow cable behaviors has been studied
outlining its importance in the structural response in the mentioned
low-tension regime.The most remarkable loading case, presented in this
paper, is relative to a displacement of the right support moved toward the left
support. It has been shown that an underestimation of the maximum tensile
stress of about50% would be reached were we to employ the crude
model of purely extensible cables. Although the probability of such a
phenomenon is low in civil engineering applications since the displacements are
expected to be small, the extreme scenarios here reproduced can certainly
concern mooring cables or cables in tethered space applications. In addition,
consideration of the flexural stiffness is important to correctly evaluate the
fatigue life of cables.Modeling linearly elastic cables with flexural
stiffness has led to a new independent parameter, denotedΛ, which represents the ratio between the
flexural and the geometric stiffnesses. The presence of flexural stiffness
modifies the loci of the higher frequencies at higher stiffnesses, especially
in nonshallow cables.More in-depth investigations are needed to correctly
unfold the cables behavior near  instabilities or in the fully developed
post-critical scenarios (galloping, parametric reso-nance,…, etc.) whereby
loosening phenomena and nonlinear viscoelasticity within the boundary layers
are expected to play a critical role on the response.



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*Source: 370767-2008-06-04.xml*