A nonlinear model of a curved beam for t

The incremental equilibrium equations around the prestressed reference configuration are derived, in which shear forces are condensed via a perturbation procedure.. The nonlinear, nontri

A nonlinear model of curved, prestressed, no-shear, elastic beam, loaded by wind forces, is formulated The beam is assumed to be planar in its reference configura-tion under its own weight and static wind forces The unit extension, two bending curvatures and the torsional curvature are taken as strains and then expanded up-to second-order terms in the three displacement components and in the angle of twist The incremental equilibrium equations around the prestressed reference configuration are derived, in which shear forces are condensed via a perturbation procedure By using a linear elastic constitutive law and accounting for inertial effects, the com-plete equations of motion are obtained They are successively strongly simplified by estimating the order of magnitude of all their terms, under the hypotheses of small sag-to-span ratio, high slenderness, compact section and by neglecting tangential inertia forces and inertia torsional couples A system of two integro-differential equations in the two transversal displacements only is drawn A simplified model of aerodynamic forces is then developed according to the quasi-static theory The nonlinear, nontrivial equilibrium path of the cable subjected to increasing static wind forces is successively evaluated, and the influence of the angle of twist on the equilibrium is discussed Then stability is studied by discretizing the equations of motion via a Galerkin approach and analyzing the small oscillations around the nontrivial equilibrium Analysis of the limit cycle (galloping) arising at the Hopf bifurcation is left for future investigation Finally, the role of the angle of twist on the dynamic stability of the cable is discussed

Keywords: Cables, galloping, twist in cables, aeroelasticity, instability, bifurcation.

1 Introduction

The analysis of galloping oscillations of iced cables requests a careful formulation both of the mechanical model and of the aeroelastic forces, especially concerning non-linear regimes (Luongo & Piccardo [1]) The forces are usually modelled referring to the quasi-steady theory, and they depend on the mean wind speed and on the angle of attack, which in its turn descends from the velocity of the structure and from its flow exposition The structure is generally modelled as a perfectly flexible cable, that is as

a one-dimensional continuum capable of translational displacements only (Luongo et

al [2], Lee & Perkins [3]) This assumption is reliable since the torsion stiffness of the cable is usually very high and the bending stiffness is negligible, respect to the ge-ometric one, because of the slenderness of the structure However, simplified models

of cables have highlighted the importance of the twist angle on the determination of aerodynamic forces and, therefore, on the dynamical behavior of the system In partic-ular, Yu et al [4] have introduced the torsion ignoring the initial curvature of the cable, and neglecting all the mechanical nonlinearities Luongo & Piccardo [5] have tried to correct the classic model, adding to the elastic potential energy of the flexible cable an energy of pure torsion ignoring, still, every term of mechanical coupling Therefore, the formulation of a consistent cable-beam model is a matter of great interest, able to take into account all the stiffnesses involved in the problem To the best of authors’ knowledge, similar models are usually employed in fully numerical approaches (for instance, Diana et al [6]) but they are confined to the linear range and are not yet employed in semi-analytical analysis, like the one here proposed A first approach to the subject, devoted to the linear problem, has been presented by the authors in [7]

In this paper a nonlinear model of curved elastic prestressed beam, subjected to aerodynamic forces induced by wind, is formulated By taking into account the high slenderness of the body, the model is remarkably simplified via an analysis of the magnitude orders af all the terms in the equations of motion As major result, it is shown that, at the leading order, the dynamic behavior of the cable is governed by the same equations of the perfectly flexible model, in which, however, the positional and velocity-dependent forces also depend on the angle of twist, which is an integral

function of the transversal displacements In other words, the twist is a passive

vari-able, slave of the normal and binormal translations The reduced model thus obtained

permits to investigate the critical and postcritical aeroelastic behavior of the cable,

by highlighting the role of torsion on the stability of the structure Numerical results

so far obtained are limited to the bifurcation analysis; the postcritical behavior will

be investigated in the next future, along the lines of previous works by the authors [1, 5, 8]

The paper is organized as follows The complete equations of motion are derived

in Section 2 under the hypotesis of no-shear deformation of the beam In Section 3 a reduced model is drawn from the complete one by neglecting small terms and stati-cally condensing the tangent translation and the angle of twist; therefore, two integro-differential equations, only in the transversal displacements, are obtained In Section 4

an approximate model for the aerodynamic forces acting on the cable, consistent with

the appoximations introduced, is developed In Section 5 the nontrivial equilibrium path of the cable subjected to static wind forces is evaluated, and bifurcations causing galloping (Hopf bifurcations) are detected Numerical results are discussed in Section

6, and some conclusions are drawn in Section 7

Let us consider the cable as a beam constituted by a flexible centerline and rigid cross-sections The actual configurationC of the body is described by the curve x = x(s, t) and by the field of basis β := {a1(s, t), a2(s, t), a3(s, t)}, where x is the position occupied by the material point P at the (undeformed) abscissa s at the time t, and β

is the inertial principal triad, with a1normal to the cross section (Fig 1a)

Let us assume as reference configuration ¯C of the body at the time t = 0 the planar

curve x¯ = ¯x(s) and the field of basis ¯β := {¯a1(s, t), ¯a2(s, t), ¯a3(s, t)} coincident with the Frenet triad, with a¯1 ≡ ¯x′ the tangent, ¯a2 the normal and a¯3 the binormal

to the curve at s (where ( )′ = d/ds) Therefore, according to the Frenet formulas,

¯

a′

1 = ¯κ¯a2,¯a′

2 = −¯κ¯a1,¯a′

3 = 0, where ¯κ = ¯κ(s) is the modulus of the curvature vector

¯

κ= ¯κ¯a3in ¯C

The referential description of C respect to ¯C requires the assignment of the dis-placement vector field u(s, t) and of the rotation tensorial field R(s, t), such that:

x= ¯x+ u , ai = R¯ai , i = 1, 2, 3 (1) where the independent variables s and t have been omitted In particular, the orthog-onal tensor R describes a rotation that leads the triad ¯β to match the triad β; it admits the following scalar representation (both in ¯β and β):

[R] =hcoscosϑϑ22cossinϑϑ33 sincosϑϑ11sincosϑϑ23cos+sinϑϑ3−cos ϑ1 sin ϑ12sinsinϑϑ33 coscosϑϑ11sinsinϑϑ22cossin ϑϑ33−sin ϑ+sinϑ11cossinϑϑ33

i (2)

where ϑ3, ϑ2 and ϑ1 are three elementary rotations ordinately assigned around the (updated) homonymous axes

The slenderness of the beam suggests to neglect the shear deformation; therefore the cross-sections are assumed to remain orthogonal to the centerline in any config-uration This internal constraint is expressed by the condition x′ = (1 + ε)a1, a1

being the normal to the section and dx/dS = x′/(1 + ε) the unit vector tangent to the strained centerline at the actual abscissaS = S(s), with ε := dS/ds − 1 the unit

extension By accounting for Eqs (1) and (2)

′

[(1 + u′− ¯κv)2+ (v′+ ¯κu)2]1 , tan ϑ3 =

v′ + ¯κu

1 + u′ − ¯κv (3) are derived, together with:

ε =(1 + u′− ¯κv)2

+ (v′+ ¯κu)2

+ w′212

−1 ≃ u′− ¯κv +1

2(v′+ ¯κu)2

+ w′2

(4)

whereu, v, w are the components of u in ¯β Due to the constraints (3), the configu-ration variablesu, v, w, ϑ1, ϑ2, ϑ3are reduced to the three translation components and the unique rotation componentϑ := ϑ1, referred as the twist angle.

In addition to the unit extension, bending and torsion must be introduced as

fur-ther measures of strain They are defined as the components in ¯β of the incremental curvature vector:

ˆ

i.e as the difference between the components of κ and κ in the bases β and ¯¯ β, re-spectively By using:

a′

i = κ × ai , ¯a′

which define the curvature vectors in the two bases, together with Eq (1)b,κ turns outˆ

to be the axial vector of the skew-simmetric tensor (see Appendix A):

ˆ

By expanding the components of κ in a two-terms Taylor series and accounting forˆ the constraints conditions (3), it follows:

ˆ

κ1 = ϑ′ + ¯κw′+ ¯κ2

vw′+ w′v′′+ ¯κ′uw′

ˆ

κ2 = −w′′+ ¯κϑ + [(u′− ¯κv)w′]′ + ϑ [(¯κu)′+ v′′]

ˆ

κ3 = v′′+ (¯κu)′ + ϑw′′− 12κ(ϑ¯ 2

+ w′2) − [(¯κu + v′)(u′ − ¯κv)]′

(8)

The equilibrium equations are then derived By considering an infinitesimal element

of cable in the actual configuration (Fig 1b), and denoting by t(s, t) and m(s, t) the in-ternal contact force and couple, respectively, acting at abscissas at time t, the balance equations, in Lagrangian form, read:

where b := b(s, t) and c := c(s, t) are body forces and couples densities per unde-formed arc-length, including inertial effects It is assumed that in the planar reference configuration ¯C the cable is loaded by body forces ¯b(s) and no couples: ¯c(s) ≡ 0 By neglecting flexural effects in its own plane, the cable is stressed in ¯C exclusively by axial forces, namely ¯t = ¯T ¯a1 andm¯ = 0, with ¯t′+ ¯b= 0 By subtracting this latter from Eq (9)a, the incremental equilibrium equations are obtained:

(t′− ¯t′) + ˆb= 0 , m′+ x′ × t + ˆc = 0 (10)

with ˆb := b − ¯b and ˆc := c − ¯c By letting t = ( ¯T + ˆT1)a1 + ˆT2a2 + ˆT3a3,

m = ˆM1a1+ ˆM2a2 + ˆM3a3, using Eq (6)a and (1)b to express the derivatives of a′

i

in ¯β, and by recalling that, from Eqs (1)band (6)b, x′ = ¯a1+ u′ = (1 + u′− ¯κv)¯a1+

(v′ + ¯κu)¯a2+ w′¯a3, Eq (10), projected onto ¯β, read:

T′

1− ¯T ¯κ(v′+ ¯κu) − T2¯κ + b1+ h.o.t = 0

T′

2+T (v¯ ′+ ¯κu)′

+ T1κ + b¯ 2+ h.o.t = 0

T′

3+ ( ¯T w′)′+ b3 + h.o.t = 0

M′

1− M2κ + c¯ 1+ h.o.t = 0

M′

2+ M1¯κ − T3+ c2+ h.o.t = 0

M′

3+ T2+ c3+ h.o.t = 0

(11)

where hats have been omitted on ˆTiand ˆMiandh.o.t stands for “higher order terms”

In Eqs (11),T2 andT3are reactive shear forces associated with the internal constrains

of zero-shear In order to express the equilibrium equations in terms of configuration variables, they must be condensed To this end, Eqs (11)e,fare solved for the unknows

T2 and T3 by applying a perturbation method First, all the kinematic quantities are rescaled to introduce a small perturbation parameter ǫ; then T2 and T3 are expanded

inǫ-series up-to ǫ3

-order, and the relevant perturbation equations are solved in chain Finally, the shear forces are replaced in the remaining four equations and only terms up-to ǫ3

-order retained The whole procedure has been performed by the symbolic manipulation software Mathematicar[9], and the relevant equations have found to be quite cumbersome; therefore, only their linear part is explicitely given here:

T′

1 − ¯T ¯κ(v′ + ¯κu) + M′

3¯κ + ˜b1+ h.o.t = 0

−M′′

3 +T (v¯ ′+ ¯κu)′

+ T1¯κ + ˜b2+ h.o.t = 0

M′′

2 + (M1κ)¯ ′+ ( ¯T w′)′+ ˜b3+ h.o.t = 0

M′

1 − M2¯κ + c1+ h.o.t = 0

(12)

where ˜bi are modified forcesbi including the effects ofc2 andc3couples

¯

a1

¯

a2

¯

a2

a3

s

x= ¯x+ u

u

¯ x

¯ P

P ϑ

¯ C

A

B

0

C S

(a)

dS C

−t

−m

t+ dt

m+ dm b

Figure 1: (a) configurations of the cable; (b) forces and couples on an infinitesimal element

A linear, uncoupled, elastic law is then assumed among the stress components in the actual base β and the strain components in the reference base ¯β, namely:

T1 = EAε , M1 = GJκ1 , M2 = EI2κ2 , M3 = EI3κ3 (13)

where hats have been omitted onκˆi, andEA, GJ, EI2andEI3are axial, torsional and flexural stiffnesses of the cable With Eqs (13), the equations of motion (12) read: EA(u′ − ¯κv)′+ EI3κ [v¯ ′′+ (¯κu)′]′

− ¯T ¯κ(v′+ ¯κu) + ˜b1− m¨u + h.o.t = 0 EA¯κ(u′− ¯κv) − EI3[v′′+ (¯κu)′]′′+T (v¯ ′+ ¯κu)′

+ ˜b2− m¨v + h.o.t = 0

EI2(−w′′+ ¯κϑ)′′+ GJ [¯κ(¯κw′+ ϑ′)]′+ ( ¯T w′)′+ ˜b3− m ¨w + h.o.t = 0

GJ(¯κw′+ ϑ′)′− EI2¯κ(−w′′+ ¯κϑ) + c1 − J1ϑ + h.o.t = 0¨

(14)

where inertia forces have been made explicit,m being the mass linear density and J1

the inertia polar moment of the section It is worth nothing that Eqs (14) are block uncoupled in their linear part, i.e in-plane small oscillations of the cable are inde-pendent of out-of-plane small oscillations, these latter involving torsion However, nonlinearities couple all these equations; moreover, if the forces ˜bidepend on the con-figuration variables, as it occurs for the aerodynamic forces, even the linear equations are coupled

Equations (14) must be sided by suitable boundary conditions If the cable is restrained at both ends by sferical hinges, displacements and moments must vanish there, namely:

u = 0,

v = 0,

w = 0,

GJ(ϑ′ + ¯κw′) + h.o.t = 0

EI2(−w′′+ ¯κϑ) + h.o.t = 0

EI3[v′′+ (¯κu)′] + h.o.t = 0

(15)

ats = 0, l

The problem is completed by the initial conditions; here it is assumed that the body

is at rest att = 0

The equations of motion previously obtained are too complicated to be treated analyt-ically; therefore, a simplified model is developed in this Section First, the classical hypothesis of small sag-to-span ratiod/l is introduced [2, 3, 10], commonly accepted for cables falling into the technical range Then, advantage is obtained from the fact the cable is a very slender body; hence the flexural-torsional effects are expected to be smaller than the funicular effects, perhaps except close to the boundaries On the other hand, due to the (small but finite) initial curvature, the bending moment contributes to the moment equilibrium around the tangent to the cable, so that bending and torsion couple

3.1 Magnitude order analysis

According to the previous ideas, an analysis of the magnitude orders of all the terms

in the equilibrium Eqs (14) and boundary conditions (15) is performed along the following lines

By assuming that the cable is flat, and forces in the reference configuration are uniformly distributed, the static profile is well described by a parabola Consistenly:

 d l

2

≪ 1, T (s) ≃ ¯¯ T = const, ¯κ(s) ≃ 8d

By taking into account the slenderness of the cable, and the smallness of ¯T /A in comparison with the elastic modulusE, it follows:

EI2 ,3

EAl2 = O r2

l2



≪ 1, O(EI2) = O(EI3) = O(GJ),

¯ T

EA ≪ 1, EIT l¯2,32 ≪ 1

(17)

wherer is a characteristic dimension of the (compact) cross-section Inequality (17)d

holds for almost all real cables

Then, it needs to estimate the magnitude order of the displacement component ratios It is assumed that:

u = O d

lv

 , ϑ = Ow

d



together with:

∂nu

∂sn = Ou

ln



nv

∂sn = Ov

ln



nw

∂sn = Ow

ln

 , n = 1, 2, (19)

and :

∂ϑ

∂s = O

 d

l3w



2

ϑ

∂s2 = O d

l4w



(20)

Equation (18)ais suggested by the results of the linear theory of small vibrations [10], and on the fact that u → 0 in the (prevalently) transversal motions (v ≫ u) when d/l → 0 Equations (19) also follow from the linear theory, when the trigonometric nature of the eigenfunctions is recognized Estimates (18)band (20) are instead drawn

by inspection of the solution of the linearized Eq (14)d and the relevant boundary conditions (see Appendix B) Equation (18)c is self-explaining It is worth noting that, due to the different boundary conditions, the translationsu, v and w must vanish

at the ends, and therefore they are fastly varying functions, whereas the twist angleϑ, being different from zero at the ends, can vary in a much slower manner As illustrated

in detail in Appendix B,ϑ is indeed a slowly varying function in symmetrical modes

(in whichϑ = O (w/d)), and again a fastly varying function in antisymmetrical modes (in whichϑ = O (wd/l2

)) The upper estimate of ϑ has been adopted for any motions,

in order to account also for nonsymmetrical modes of cables supported at different levels

By using previous estimations in Eqs (14) and retaining only dominant terms

among linear, quadratic and cubic terms, separately, the following reduced equations

are obtained, in theEI2 = EI3(=: EI) case (circular cross-section):

EA



u′− ¯κv +12v′2+1

2w

′2

′

+ b1− m¨u = 0 EA



¯

κ(u′− ¯κv +12v′2+1

2w

′2) + [(u′− ¯κv + 12v′2+ 1

2w

′2)v′]′

 + + ¯T v′′+ b2− m¨v = 0 EA



(u′− ¯κv +1

2v

′2+ 1

2w

′2)w′

′

+ ¯T w′′+ b3− m ¨w = 0 GJϑ′′− EI ¯κ2

ϑ + (EI + GJ)¯κw′′− EI ¯κϑv′′+ GJ(v′′w′)′+ c1− J1ϑ = 0¨

(21)

As major result of the analysis, Eqs (21)a,b,c are identical to that of the flexible ca-ble [3], while Eq (21)drepresents an additional nonlinear equation in the twist angle

If the body forces are independent of ϑ or even zero (as it happens in the free vi-brations), the translational motion is independent of ϑ, which is therefore a passive

variable, slave of the translations In contrast, if the body forces depend on ϑ, as in the aerodynamic case, the twist angle does affect the dynamics of the body

It is interesting to observe that the reduced equation (21)d, stating the moment equilibrium around the tangent, can be obtained in a much simpler way by substituting reduced expressions of the strains (4) and (8), based on the estimates performed here,

in the linear part of the equilibrium equation (12)d However, this is a posteriori observation, emerging from the analysis, but unpredictable a priori.

Equations (21) must be integrated with the reduced boundary conditions:

u = v = w = 0, GJ(ϑ′+ ¯κw′+ w′v′′) = 0, ats = 0, l (22) Equations expressing the vanishing of the bending moments must instead be ignored, consistently with the approximation adopted, that does not permit to describe the boundary layers

3.2 Static condensation

It is well known that, in the framework of the parabolic cable theory, the tangential inertia force −m¨u can be neglected in the prevalently transversal motions, since the longitudinal natural frequencies are much higher than the transversal ones This cir-cumstance permits to statically condense the tangent displacementu, by expressing it

as an integral of the transverse displacementsv and w:

u(s, t) = e(t)s +

Z s

0

[¯κv(ξ, t) −1

2v

′(ξ, t)2

−1

2w

′(ξ, t)2

where

e(t) = −1

l

Z l

0

[¯κv − 1

2v

′2− 1

2w

An analogous procedure is here applied to the equation (21)d governing the twist Since the torsional frequencies of a single cable turn out to be much higher than the transversal ones, the inertia couple−J1ϑ is neglected and ϑ obtained in integral form.¨

To solve (21)dand the boundary conditions (22)din the unknownϑ, first v and w are scaled by a perturbaton parameterǫ ≪ 1, then ϑ is expanded in ǫ-series:

v → ǫv, w → ǫw, ϑ = ǫϑ1+ ǫ2ϑ2+ (25) The following perturbation equation are obtained:

ǫ : GJϑ′′

1 − EI ¯κ2

ϑ1 = −¯κ (GJ + EI) w′′

ǫ2

: GJϑ′′

2 − EI ¯κ2

ϑ2 = EI ¯κv′′ϑ1 − GJ(v′′w′)′ (26) wherec1 = 0 has been taken for semplicity The relevant boundary conditions read:

ǫ : GJ (ϑ′

1+ ¯κw′) = 0

ǫ2

: GJ (ϑ′

By solving in chain Eqs (26), it follows:

ϑ1(s, t) = −GJ + EI√

GJ EI

Z s

0

w′′(ξ, t) sinh [k(s − ξ)] dξ + A1cosh ks + B1sinh ks

ϑ2(s, t) =r EI

GJ

Z s

0

v′′(ζ, t)ϑ1(ζ, t) sinh [k(s − ζ)] dζ+

−k1

Z s

0

(v′′(ξ, t)w′(ξ, t))′sinh [k(s − ξ)] dξ + A2cosh ks + B2sinh ks

(28)

wherek := ¯κpEI/GJ has been set and where the arbitrary constants A1, B1, A2,B2

are determined by Eqs (27)

Modelling aerodynamic loads is a very difficult task, often handled in literature under strong hypoteses Moreover, the iced cable problem adds further difficulties to the popular problem of indefinite cylinder, due to the curvature of its centerline and the random variation of the section To tentatively tackle the problem, a simple model is

adopted here, by introducing the following assumptions: (a) the quasi-static theory

[11] is believed applicable, according to which the loads acting on the moving body at

a certain instant are identical to those exerted on the body at rest in the same position;

(b) the curvature of the cable, due to its smallness, is negligible; (c) loads are evaluated

in the current configuration C, by accounting for the twist angle ϑ, but neglecting the smaller flexural rotationsϑ2 ,3 = O (ϑd/l) (remember Eqs (3) and (18)), which,

according to the so-called cosine rule [12], have small influence; (d) the ice is assumed

¯

ai

ax

az

ay

C0

¯

C

−y(s)

U

U

ϕ

ϕ

ϕ s

a2 0

¯

a2

a2

a3 0 ≡ az

¯

a3

a3

bd

bl (a)

V

˙u

(b)

ϑ

ϑ γ

G

Figure 2: Aerodynamic forces: (a) cable configuration; (b) transversal section, mean wind velocity U, relative wind velocity V, angle of attack γ, drag force bd and lift force bl

to be uniformly distributed along the cable, consistently with the hypotesis of planar

reference configuration; (e) the aerodynamic couples are neglected.

Let us now consider a wind flow of mean velocity U= Uaz, blowing horizontally and normally to the initial (no wind) planar configuration of the cable (Fig 2a) Three different attitudes of the cross-section in its own plane are considered (Fig 2b): (a)

the initial configurationC0 (axes a2 0, a3 0 ≡ az), in which the cable is only subjected

to gravity; (b) the reference configuration ¯C (axes ¯a2, ¯a3), in which the cable is loaded

also by (uniform) static wind forces; (c) the actual configuration C (axes a2, a3), in which the cable is loaded also by (non uniform) dynamic wind forces The twist angle caused by the static forces coincides, to within small quantities of order O(d2

/l2

), with the angle of rotation ϕ experienced by the cable in passing from C0 to ¯C (see Fig 2a,b), which depends only on the mean wind velocityU The twist angle ϑ caused

by the dynamic forces depends, in addition toU, on the abscissa s and on time t The angleϕ is assumed to be large; the angle ϑ is assumed to be small but finite

According to the quasi-static theory, the flow exerts on the section the following aerodynamic force:

ba= 1

2ρV r(cd(γ)V + cl(γ)a1× V) (29) whereρ is the air density, V is the relative velocity of the wind respect to the section,

V = ||V|| its modulus, and cdandcltwo aerodynamic coefficients, called of drag and

lift, respectively These latter depend on the shape of the section and on the angle of attack,

γ := − arcsin VV · a2



(30) i.e on the angle between V and a reference material axis, here taken as a3 The two components of ba, along-wind bdand cross-wind bl, are usually known as drag and lift forces, respectively (Fig 2b)