PLEASE SCROLL DOWN FOR ARTICLE This article was downloaded by: [Vairo, Giuseppe] On: 22 July 2009 Access details: Access Details: [subscription number 913232142] Publisher Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK Mechanics of Advanced Materials and Structures Publication details, including instructions for authors and subscription information: http://www.informaworld.com/smpp/title~content=t713773278 A Closed-Form Refined Model of the Cables' Nonlinear Response in Cable- Stayed Structures Giuseppe Vairo a a Department of Civil Engineering, University of Rome “Tor Vergata”, Rome, Italy Online Publication Date: 01 August 2009 To cite this Article Vairo, Giuseppe(2009)'A Closed-Form Refined Model of the Cables' Nonlinear Response in Cable-Stayed Structures',Mechanics of Advanced Materials and Structures,16:6,456 — 466 To link to this Article: DOI: 10.1080/15376490902781217 URL: http://dx.doi.org/10.1080/15376490902781217 Full terms and conditions of use: http://www.informaworld.com/terms-and-conditions-of-access.pdf This article may be used for research, teaching and private study purposes. Any substantial or systematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.
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PLEASE SCROLL DOWN FOR ARTICLE
This article was downloaded by: [Vairo, Giuseppe]On: 22 July 2009Access details: Access Details: [subscription number 913232142]Publisher Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK
Mechanics of Advanced Materials and StructuresPublication details, including instructions for authors and subscription information:http://www.informaworld.com/smpp/title~content=t713773278
A Closed-Form Refined Model of the Cables' Nonlinear Response in Cable-Stayed StructuresGiuseppe Vairo a
a Department of Civil Engineering, University of Rome “Tor Vergata”, Rome, Italy
Online Publication Date: 01 August 2009
To cite this Article Vairo, Giuseppe(2009)'A Closed-Form Refined Model of the Cables' Nonlinear Response in Cable-StayedStructures',Mechanics of Advanced Materials and Structures,16:6,456 — 466
To link to this Article: DOI: 10.1080/15376490902781217
URL: http://dx.doi.org/10.1080/15376490902781217
Full terms and conditions of use: http://www.informaworld.com/terms-and-conditions-of-access.pdf
This article may be used for research, teaching and private study purposes. Any substantial orsystematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply ordistribution in any form to anyone is expressly forbidden.
The publisher does not give any warranty express or implied or make any representation that the contentswill be complete or accurate or up to date. The accuracy of any instructions, formulae and drug dosesshould be independently verified with primary sources. The publisher shall not be liable for any loss,actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directlyor indirectly in connection with or arising out of the use of this material.
A Closed-Form Refined Model of the Cables’ NonlinearResponse in Cable-Stayed Structures
Giuseppe VairoDepartment of Civil Engineering, University of Rome “Tor Vergata”, Rome, Italy
In this paper a refined second-order model for the analysis ofthe elastic response of cables in cable-stayed structures is proposed.Starting from the exact catenary theory, the Dischinger-Ernst for-mulation is rationally deduced and, performing a second orderapproximation both with respect to the stress variation and theapparent chord-strain one, a refined explicit equivalent constitu-tive relationship for elastic stays is obtained. Moreover, the geo-metrical non-linearities induced on the cable by the deformationof the stay-supported structure are taken into account through asecond order displacement approach. Accordingly, a closed-formformulation describing nonlinear cable-structure interaction ef-fects is obtained which, unlike the classical secant model, does notneed iterative procedures, opening the possibility to develop re-fined analytical solutions for cable structures. Several numericalexamples and benchmarks on different stay configurations are pre-sented, showing the effectiveness and the accuracy of the proposedmodel.
Keywords elastic cables, cable-stayed structures, equivalent elasticmodulus, Dischinger-Ernst theory
1. INTRODUCTIONIn several engineering fields (civil, mechanical, aeronauti-
cal, etc.) using stays for supporting and/or stabilizing functionalstructures (e.g., bridge decks, roofs, floors, masts, guyed tow-ers, transmission lines, etc.) allows to obtain innovative andadvanced design solutions, more efficient material utilization,simple erection procedure, as well as considerable aesthetic re-sults [1–4].
In cable-stayed structures a stay works solely in tension andcarries not only the structural dead load but also live and en-vironmental (wind, snow, etc.) loads acting upon the supportedstructure. Starting from the dead-load-depending equilibriumconfiguration, live and environmental loads produce variationin stay configuration because of displacements imposed at cableend-points by the deformation of the supported structure. The
Received 25 June 2008; accepted 17 July 2008.Address correspondence to Giuseppe Vairo, Department of Civil
Engineering, University of Rome “Tor Vergata,” viale Politecnico 1,00133 Roma, Italy. E-mail: [email protected]
corresponding stay-structure interaction is highly nonlinear dueto geometrical and material nonlinearities of the stay-supportedstructure, as well as essentially because of the cable sag ef-fect induced by the stay’s own dead load. When the cable isassumed as a perfectly flexible member, catenary theory givesthe exact description of its elastic behavior. Catenary-based ap-proaches for static and dynamic analysis of cable structuresare presented, for instance, in [1–3, 5–8]. The complexity ofthe relevant governing equations does not enable closed formsolutions to be obtained and, consequently, numerical iterativetreatments are needed. A number of nonlinear finite element for-mulations and numerical methods have been proposed in spe-cialized literature and employed for modelling and analysingcable-stayed structures (e.g., [9–15]). Nevertheless, when a ca-ble undergoes high stress levels, which is the usual conditionin many cable structures, the problem can be advantageouslysimplified. In this case, a well established approach for describ-ing interaction forces between cable and structure is based onthe well-known formulations of Dischinger [16] and Ernst [17],and is referred to as the equivalent modulus approach. Accord-ingly, the relationship for a single stay between axial (i.e., alongthe cable’s chord direction) force and axial elongation is repre-sented by means of a fictitious equivalent elastic modulus (usu-ally denoted as the Dischinger’s modulus), non-linearly depend-ing on the stress level in the cable because of geometrical sageffect.
In the framework of the Dischinger’s theory, secant and tan-gent moduli can be introduced. The first one depends on bothcable’s initial stress (due to dead loads) and live-load-dependingcable stress (a-priori unknown), giving an accurate evaluationof the elastic behavior of the stay (e.g., [2, 3, 18]). Accord-ingly, displacement and stress variations in cable-stayed struc-ture acted upon live loads can be accurately determined, butiterative procedures have to be employed. On the other hand,tangent modulus depends on the cable’s initial stress (a knowndatum) only, and allows to represent stay-structure interactionin closed form. Nevertheless, equivalent tangent elastic modu-lus does not take into account the stiffening effect due to largedisplacements, results in a softer cable response and, as a con-sequence, it induces a precision lack when live-load-dependinghigh stress variations occur [2, 3, 18, 19].
456
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CABLES’ NONLINEAR RESPONSE IN CABLE-STAYED STRUCTURES 457
Dischinger’s theory has been employed in several numericalstudies and finite element formulations recently proposed forthe analysis of cable-stayed structures (e.g., [12, 20–28]). Onthe other hand, analytical models and closed-form solutionsremain unreplaceable for conception and design phases, givinga synthetic understanding of the basic behavior of the cablestructure and therefore allowing to fulfill in a better way thedesign requirements, reducing time-consuming computationsand/or risks of nonconverged results.
In order to obtain closed-form solutions able to take into ac-count non-linear stay-structure interaction and avoiding numer-ical iterative procedures, usually-employed analytical formula-tions involve first-order displacement approaches and a tangentDischinger-type treatment for cables.
Accordingly, technical guidelines employed for the initialdesign phases are usually based on the tangent Dischinger’s for-mulation (e.g., [1–3, 5–7]). De Miranda et al. [29] and Comoet al. [30] proposed the earliest continuous model for the anal-ysis of statical behavior of long-span cable-stayed bridges and,in the framework of the tangent Dischinger’s approach, ob-tained closed-form solutions which a number of researchers(e.g., [31–40]) applied for dynamical and aeroelastic analysis ofsuch structures.
However, solutions based on the tangent Dischinger’s mod-ulus can be too unrefined for cable-stayed structures with verylong stays and in which stress variations induced on the staysby the design live loads are not negligible. This is the caseof long-span cable-stayed bridges, where live-load-dependingstress variations in cables can have the same order of magnitudeof the initial stress level. Moreover, in such structures first-ordertangent models are not able to take into account coupling effectsbetween bending and torsion, resulting in an underestimation ofthe structure’s stiffness. Therefore, a poor cable modelling re-flects on a poor description of the overall cable-stayed structurebehavior.
As a consequence, feasibility and design of structures withlong cable-stayed spans and acted upon by high live loads (e.g.,railway loads in cable-stayed bridges, snow and wind loads incable-stayed roofs, etc.) need an evaluation of deformabilityeffects more accurate than the tangent one. Furthermore, it isworth observing that when the assumption of small displace-ments for both stays and supported structure is enforced, geo-metrical non-linearities due to the finite rotation of cable’s chordcannot be accounted for, resulting in an improper evaluation ofcable-structure interaction. This effect is particularly evident inthe case of modern long-span cable-stayed bridges [41], wherequasi-horizontal long stays connect girder mid-span to towerstop. Therein, conventional linear analysis based on small de-formations and displacements has been proved to be ineffective[5].
In order to furnish a refined closed-form representation of thestay-structure interaction, a quasi-secant second-order formula-tion for the analysis of elastic stays is here proposed. It doesnot need iterative procedure and is able to take into account fi-
nite displacements and deformations of the cable as well as thenonlinear variation of the cable’s stiffness during the loadingprocess. As confirmed in [42], this approach opens the pos-sibility to develop more refined closed-form solutions for theanalysis of cable-stayed structures.
Starting from some relevant aspects of the catenary theory,the exact equivalent along-the-chord formulation is discussed,rationally deducing tangent and secant Dischinger’s moduli(Section 2). From these preliminary results, the quasi-secantequivalent approach is developed (Section 3) and a second-orderdisplacement formulation is proposed, furnishing a closed-formrepresentation of the cable-structure interaction forces (Section4). Finally, several applications are discussed (Section 5), show-ing effectiveness and accuracy of the proposed model.
2. PREPARATORY RESULTS
2.1. Outline of the Catenary Equilibrium ProblemLet a homogeneous cable comprising isotropic linearly elas-
tic material be hanged by its ends at the fixed point O and atthe structure S in P (Figure 1), whose dead load q induces thecable’s traction. Let Ac be the cross-section area of the cable,γc its specific weight and Ec the Young modulus. Any flexuraleffect is assumed to be negligible.
Let π be the gravity plane containing the equilibrium config-uration � of the cable, and (O, x, y) a planar Cartesian frame onπ with {i, j} the corresponding orthonormal basis, j being the di-rection of the gravity acceleration. Moreover, let L = (P −O) · iand h = (P − O) · j be the horizontal and vertical cable’s pro-jections, respectively, and α the angle between the x-axis andthe cable’s chord. Accordingly, vector (P − O) and the chordunit vector c are defined from:
(P − O) = � c = � (cos α i + sin α j) (1)
FIG. 1. Reference stay configuration: notation.
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458 G. VAIRO
� being the chord length. The equilibrium configuration � ofthe cable is described by the elastic catenary:
y(x, τ)
L= −1
τcosh
(−τ
x
L+ C1
)+ C2 (2)
with
C(τ) = τeτ
(eτ − 1)tan α
C1(τ) = ln(C +
√C2 + eτ
)C2(τ) = 1
τcosh C1 (3)
and where τ is a dimensionless measure of the cable weightversus cable stress ratio:
τ = γcAcL
T(4)
and where T is the horizontal (along i) component of the cable’stensile force, which is constant along the cable. In detail, T =fP · i = −fO · i, where fP and fO indicate the constraint reactionsat the cable’s end-points P and O, respectively. Reactions fPand fO lie in π and can be represented through the cable tractionN (x, τ) and the unit vector t(x, τ) tangent at the elastic catenary,both evaluated at end-points P and O:
fP = NP tP , fO = −NO tO (5)
where
t(x, τ) =(
i + ∂y
∂xj)[
1 +(
∂y
∂x
)2]−1/2
(6)
N (x, τ) = T cosh(−τ
x
L+ C1
)(7)
and with NO and NP related each other through the equilibriumequation:
NP − NO + γc Ac h = 0 (8)
It is worth emphasizing that force fP describes the interactionacted from the cable upon the structure S.
Let the along-the-chord stress σ = F/Ac be introduced,where F = T/ cos α is the equivalent along-the-chord thrust.Accordingly, σ is an apparent stress giving a measure of theaverage tensile stress on the cable.
When the cable works as a stay, small values of τ (i.e.,high values of T ) are usually experienced and small sag effectsconsequently appear. In the limit of very high stress level (i.e.,when τ tends to zero), σ acquires the meaning of a measure of
the real cable’s stress and, for every x ∈ [0, L]:
limτ→0+
∂y
∂x= tan α, =⇒ lim
τ→0+t = c,
limτ→0+
N = T
cos α= F (9)
Let be L the length of the elastic line corresponding to �, Lu
the unstressed length of the cable and �L the length variationdue to the tensile stress. From compatibility it results:
L = Lu + �L (10)
The exact values for L and �L turn out to be:
L(τ) =∫ L
0
[1 +
(∂y
∂x
)2]1/2
dx = 2L
τsinh
τ
2cosh
(τ
2− C1
)(11)
�L(τ) = T
EcAc
∫ L
0
[1 +
(∂y
∂x
)2]
dx
= ω� cos2 α
2
[1 + 1
τsinh τ cosh (τ − 2C1)
](12)
where τ = γ�/σ and the dimensionless thrust ω = σ/Ec isintroduced.
Accordingly, given the chord length �, the angle α and thedimensionless ratios τ and ω, the unstressed length of the cableLu is univocally determined by Eq. (10).
In order to characterize the stay-structure interaction whena variation of the cable’s configuration is induced by live andenvironmental loads p acting upon the supported structure S, adisplacement vector s is enforced at the end-point P and a newequilibrium configuration � is attained. It is worth observing thatwhen the stay-supported structure S has a complex spatial cabledisposition (e.g., cable-stayed bridges with A-shaped towers),displacement s and the actual cable’s configuration � could notlie in π. Accordingly, a global Cartesian frame (O, x1, x2, x3)can be usefully introduced, with {e1, e2, e3} the correspondingorthonormal basis.
As notation rules, symbols “⊗”, “·” and “×” denote, in thefollowing, tensor, inner and vector product operators, respec-tively, |w| = (w · w)1/2 for every w ∈ V , V being the vectorspace associated to the three-dimensional Euclidean point space.Moreover, double-line capital character identifies a second orthird order tensor, I represents the identity tensor, the notation“sp{v1, v2, . . .}” denotes the span generated by v1, v2, . . ., and“(A)⊥” is the vector subspace orthogonal toA. When necessary,Einstein’s summation convention is adopted.
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CABLES’ NONLINEAR RESPONSE IN CABLE-STAYED STRUCTURES 459
FIG. 2. Reference and actual stay configurations in a global Cartesian frame:notation.
Let be e2 = j and δ the angle between i and e1 (Figure 2). Ac-cordingly, the chord unit vector c in the reference configurationcan be rewritten as
c = cos α cos δ e1 + sin α e2 + cos α sin δ e3 = β(d e1+e2+ζe3)(13)
where
β = (1 + ζ2 + d2)−1/2 = sin α (14)
h = (P − O) · e2 (15)
ζ = (P − O) · e3/h (16)
d = (P − O) · e1/h (17)
and i = cx/|cx |, with cx = (I − e2 ⊗ e2)c.The actual equilibrium configuration � associated to s = si ei
lies in the vertical plane π = sp{ i, e2}, where i = cx/|cx | andcx = (I−e2 ⊗e2)c, c being the actual chord direction.This latteris defined as in Eq. (13) by the actual angles α and δ
tan α = 1 + S2√(d + S1)2 + (ζ + S3)2
, tan δ = ζ + S3
d + S1(18)
S = Si ei = si/h ei being the dimensionless displacement vec-tor.
Denoting by P the actual position of P , the cable’s chordkinematics induced by s can be described as:
(P −O) = (P −O)+s = RaU(P −O) = VRa(P −O) (19)
where Ra ∈ Orth+ is the second-order rotation tensor with axisa = c × s, and U(s), V(s) are the chord’s right and left stretchtensors, respectively:
U = I + (λ − 1)c ⊗ c, V = I + (λ − 1)c ⊗ c (20)
with c = Rac and where λ denotes the chord stretch, obtainedonce a measure of the along-the-chord strain �ε is assumed.Green-Lagrange and logarithmic strain measures are herein con-sidered:
where � is the actual chord length and �εn = �/� − 1 is thenominal along-the-chord strain measure. Accordingly, stretchλ = �/� can be evaluated as:
λGL =√
1 + 2�εGL (23)
λln = exp(�εln) (24)
It is worth observing that, when P ∈ B ={w ∈ V : |w| = �}, a chord’s rigid displacement field appears,that is �ε = 0 and U = V = I.
The actual cable-structure interaction force is described byfP = NP tP , where the actual cable’s tension in P can be thoughtof as the corresponding value in � plus a variation, i.e., NP =NP + �N , whereas fO = NO tO , NO resulting from Eq. (8)applied to �.
If it is omitted for the sake of brevity, the dependency on Si ,the actual horizontal cable’s tension T and the dimensionless pa-rameter τ can be expressed as functions of τ, that is as functionsof the along-the-chord stress σ in the reference configuration:
T (τ) = F(τ) cos α = [F + �F(τ)] cos α
= Ac [σ + �σ(τ)] cos α (25)
τ(τ) = γcAcL
T= τ
λ σ
σ + �σ(τ)(26)
where �σ = �F/Ac is the τ-depending variation of the along-the-chord apparent stress corresponding to the variation of thecable’s configuration. It is worth observing that if the cable’schord cross section is assumed to be undeformable, �σ can beregarded at the same time as both a Cauchy- and a Kirchhoff-type measure of the apparent stress variation.
From Eqs. (25–26), given a reference configuration � and adisplacement s, the actual configuration � as well as the corre-sponding constraint reactions and cable tensile stress can be ex-actly evaluated through Eqs. (2), (6) and (7) applied to �, oncethe along-the-chord stress variation �σ has been determined.
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460 G. VAIRO
For a chord-strain measure �ε different from zero, �σ can berelated to �ε by introducing an apparent constitutive equationbased on the so called secant equivalent elastic modulus.
Differentiating the compatibility Eq. (10) and taking into ac-count that dLu = 0, the exact tangent equivalent elastic modulusat the stress level σ turns out to be:
E∗t (σ) = dσ
dε= −�
[∂
∂�(L − �L)
] [∂
∂σ(L − �L)
]−1
(27)
where dε = d�/� is the infinitesimal chord-strain. Moreover,integrating the tangent relationship
dσ = E∗t (σ) dε (28)
the exact secant equivalent modulus is obtained
E∗s (σo,�σ) = �σ
[ ∫ σo+�σ
σo
1
E∗t (σ)
dσ
]−1
(29)
σo denoting the reference along-the-chord cable’s traction.Therefore, the secant elastic cable’s response can be representedas:
�σ = E∗s (σo,�σ) �ε (30)
where it has been emphasized that �σ depends on the secantmodulus which depends itself on �σ. Accordingly, iterativeprocedures are needed. For what follows it is worth observingthat
lim�σ→0
E∗s = E∗
t |σ=σo,
∂E∗s
∂�σ
∣∣∣∣�σ=0
= 1
2
∂E∗t
∂σ
∣∣∣∣σ=σo
(31)
When exact expressions (11) and (12) for L and �L areused, tangent and secant equivalent moduli (27), (29) cannotbe represented by simple closed-form expressions and numer-ical approaches have to be employed. Nevertheless, a certainsimplification is obtained considering that in usual cable-stayedstructural applications high stress levels appear in cables, i.e.,τ � 1. Accordingly, exact expressions (11) and (12) can bereplaced by their second order expansions with respect to τ.Therefore, replacing Eq. (10) by
� + γ2c�
3
24σ2cos2 α = Lu + σ�
Ec
+ γ2c�
3
12Ecσ(32)
equivalent tangent modulus turns out to be (see Eq. (27)):
E∗t = Ec
1 + τ2 cos2 α/8 − ω(1 + τ2/4)
1 + a(ζ2 + d2)(1 − ω/ cos2 α)(33)
where
a = τ2
ω
β2
12= Ecγ
2ch
2
12σ3(34)
Moreover, in steel stays usually employed in cable-stayedstructures, the cable’s material is characterized by small ratiosbetween allowable stress and Young modulus (about 3 − 4 ×10−3). Accordingly, Eq. (33) can be simplified in the limit ofω and τ2 tending to zero. Nevertheless, when long cables areconsidered (e.g., � greater than 250 m), ratio τ2/ω is usuallynot negligible. Therefore, in these cases τ2 and ω are assumedto have the same infinitesimal order:
lim(τ,ω)→(0,0)
τ2
ω= 12
a
β2 = Ecγ2c�
2
σ3(35)
Accordingly, the well-known Dischinger’s equivalent tan-gent modulus is obtained:
E∗td (σ) = Ec[1 + a(ζ2 + d2)]−1 (36)
Combining Eqs. (29) and (36), the Dischinger’s secant mod-ulus results in:
E∗sd (σo,�σ) = Ec
[1 + a(ζ2 + d2)
2 + �σ/σo
2(1 + �σ/σo)2
]−1
(37)
where the sag parameter a is evaluated at the initial stress levelσo.
As a notation rule, for a generic quantity G depending on τ
and ω, Gd denotes the limit
Gd = lim(τ,ω)→(0,0)
G(τ,ω) (38)
under the assumption (35). Accordingly, the following equationscan be verified (see Eqs. (9)):
td = c, td = c (39)
fP d = Fc, fP d = (F + �Fd ) c = Ac(σo + E∗sd�ε)c (40)
�Nd = �Fd = Ac E∗sd �ε, (41)
where �F(τ) = [F(τ) − F]. In other words, in the frame-work of the Dischinger’s theory, the fictitious along-the-chordquantities �F , �σ and �ε assume the meaning of real varia-tions and therefore Eqs. (28) and (30) can be considered as realconstitutive relationships.
As it clearly appears from Eqs. (30) and (37), the secantrelationship between the stress increment and the strain one isimplicit. As a consequence, in order to obtain an accurate eval-uation of the cable-structure interaction, iterative procedureshave to be employed, so that closed-form relations cannot beproduced. On the other hand, if the variation of the apparent
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CABLES’ NONLINEAR RESPONSE IN CABLE-STAYED STRUCTURES 461
stress �σ is very small, then the cable’s behavior can be ex-plicitly represented through a first order approximation in �σ
of the apparent secant constitutive relationship, that is by thefictitious tangent modulus (see Eqs. (28) and (36)). As previ-ously discussed, this approach leads to a softer cable responsewhen significant stress variations occur in the stay, and doesnot provide a a suitable approximation of the stay’s nonlinearresponse. Therefore, a reasonable second order approach in �σ
should be established.
3. QUASI-SECANT EQUIVALENT APPROACHLet the residual function R corresponding to a reference
stress level σo to be defined as
R(�σ,�ε) = �σ − E∗s (�σ)�ε (42)
vanishing when �σ and �ε satisfy Eq. (30). Series expansionof R around �σ = 0 and �ε = 0 gives
R(�σ,�ε) = �σ − E∗t |σ=σo
�ε
− 1
2
∂E∗t
∂σ
∣∣∣∣σ=σo
�σ �ε + o(�σ2,�ε2) (43)
where (see Eq. (31))
R(0, 0) = 0,∂R∂�σ
∣∣∣∣(0,0)
= 1,∂R∂�ε
∣∣∣∣(0,0)
= −E∗t |σ=σo
(44)
∂2R∂�ε2
∣∣∣∣∣(0,0)
= ∂2R∂�σ2
∣∣∣∣∣(0,0)
= 0,
∂2R∂�σ ∂�ε
∣∣∣∣∣(0,0)
= −1
2
∂E∗t
∂σ
∣∣∣∣σ=σo
(45)
Accordingly, from the bilinear part in �σ�ε of Eq. (43),disregarding terms of order o(�σ2,�ε2) and introducing the“quasi-secant” modulus E∗
qs as
E∗qs = E∗
t |σ=σo+ �Es�σ (46)
with �Es = (1/2) (∂E∗t /∂σ)|σ=σo
, the following apparentconstitutive relationship arises:
�σ = E∗qs(�σ) �ε (47)
It is worth noting that the quasi-secant modulus represents afirst order approximation in �σ for the secant one E∗
s and it pro-duces a cable constitutive description up to terms of order two inalong-the-chord stress and strain variations. Following this ap-proach, the relationship between �σ and �ε can be represented
through the following explicit expression:
�σ = E∗t |σ=σo
�ε
1 − �Es �ε(48)
If Eq. (33) is employed, �Es can be written as:
�Es = 1
2
∂E∗t
∂σ
∣∣∣∣σ=σo
= 3
2
E∗t (1 − E
∗t )
ω
×{
1 + ω
[2
β2(ζ2 + d2)
(2
3− E
∗t
)
+ 5
E∗t (1 − E
∗t )
(E
∗t
2 − 2
3E
∗t + 1
10
)]}
= 3
2
E∗t (1 − E
∗t )
ω[1 + O(ω)] (49)
where both the dimensionless tangent modulus E∗t = E∗
t /Ec
and ω are evaluated for σ = σo. Due to the smallness of ω, inEq. (49) it has been emphasized that the first term is dominantwith respect to the other ones. Accordingly, in the frameworkof the Dischinger’s approach, Eq. (49) reduces to:
�Esd = 1
2
∂E∗td
∂σ
∣∣∣∣σ=σo
= 3
2
E∗td (1 − E
∗td )
ω(50)
It is worth observing that �Es allows to take into accountthe nonlinear variation of the cable’s stiffness during the defor-mation process.
4. QUASI-SECANT SECOND-ORDER DISPLACEMENTMODEL
As previously discussed, stays of cable-stayed structures gen-erally work in a Dischinger-compatible way. Moreover, the be-havior of cable structures is usually analyzed considering theadditional hypothesis of small structural displacements. Underthese assumptions, it would seem reasonable to consider alsofor the stays a first-order displacement chord-strain measure.But, following this approach, a complete agreement between thecatenary’s results and the Dischinger-based ones appears onlywhen the cable’s traction variation is produced by structuraldisplacements acting along the cable-chord direction. Never-theless, even if components of s could be considered as small inthe analysis of the supported structure S, the effect of the finitechord rotation in very long and quasi-horizontal cables shouldbe described by means of a large displacement approach, able toinclude such significant geometrical non-linear effects in cable-structure interaction.
4.1. Cable StressA second order expansion around S = 0 of the dimensionless
quantity �σ/Ec is considered. Since �σ = 0 when S = 0, the
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462 G. VAIRO
following result holds:
�σ
Ec
= G · S + 1
2H · (S ⊗ S) + o(|S|2) (51)
where G ∈ V and H ∈ Sym+
G = ∇(
�σ
Ec
)∣∣∣∣S=0
, H = ∇[∇
(�σ
Ec
)]∣∣∣∣S=0
(52)
and where the symbol ∇ has the meaning of gradient operatorwith respect to the dimensionless displacement components. Inthe framework of the quasi-secant approach (see Eq. (48)), thefollowing expressions can be obtained:
being �F |S=0 = 0. In Eq. (64) the operator “sym∗” applies onthird order tensors and it is defined as:
sym∗[A] = 1
2(A + A
T ∗) ∀ A ∈ Lin (65)
where AT ∗ is the unique third order tensor satisfying the prop-
erty: (AS) · v = (AT ∗S
T ) · v, for every second order tensor S
and for every v ∈ V. Accordingly, (AT ∗)ijk = (A)ikj and then
∇(v ⊗ w) = (∇v ⊗ w)T ∗ + (v ⊗ ∇w) ∀ v, w ∈ V (66)
If the quasi-secant constitutive law (48) is used and in theframework of the Dischinger’s assumptions, i.e., employing Eqs.(39–41), the second order expansion around S = 0 for �Fd
results in (see Eqs. (63–64)):
�Fd = (c ⊗ Gd )S
+{
1
2c ⊗ Hd + sym∗[∇ c|S=0 ⊗ Gd ]
}(S ⊗ S)
+ o(|S|2) (67)
For every second order tensor S and for every v, w, u ∈ V ,the following properties can be proved:
(S ⊗ v)(w ⊗ u) = (v · u)Sw = (S ⊗ v)T ∗(u ⊗ w) (68)
(v ⊗ S)(w ⊗ u) = (v ⊗ w)Su = [S · (w ⊗ u)]v (69)
v‖ · v = |v‖|2, v⊥ · v = |v⊥|2 (70)
Moreover, after some algebra, the following result is obtained(see Eqs. (13) and (18)):
∇ c|S=0 = β P⊥ (71)
Accordingly, from Eq. (51) and the properties (68–71), theexpansion (67) become:
�Fd =(
�σ
Ec
)d
c + E∗tdβ
2 (S · c) S⊥ (72)
As a conclusion, from Eqs. (55–57) and from the analysisof the Eqs. (60–61) and (72) the following remarks can besummarized:
i) the quasi-secant second-order-displacement cable stressvariation can be thought as the sum of the tangent
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CABLES’ NONLINEAR RESPONSE IN CABLE-STAYED STRUCTURES 463
second-order one, i.e., that one corresponding to the tan-gent constitutive law (28), and of a constitutive refinement,depending on �Es and |S‖|2. Accordingly, it clearly ap-pears that the quasi-secant constitutive refinement with re-spect to the tangent behavior is possible only if at leasta second-order displacement theory is considered. Hence,the first-order quasi-secant displacement theory coincideswith the first-order tangent one.
ii) If the tangent constitutive model—employed in the case ofa second order displacement approach—is considered, thestress variation and the cable-structure interaction forcesare refined in an explicit way only through the contributiondue to the finite rotation of the cable’s chord. On the otherhand, a significant constitutive refinement is obtained onlyif the quasi-secant model is considered.
iii) The tangent terms depend on the choice of the strain mea-sures, whereas this does not occur for the constitutive re-finement.
iv) The last contribution in Eq. (72) represents a tangentsecond-order term which couples the stretch-componentsrelated to |S‖| and |S⊥|, and produces a deviation of theinteraction force from the reference chord direction c.
5. VALIDATION AND COMPARISONSIn order to show effectiveness and accuracy of the proposed
refined approach, several applications based on stays typicallyemployed in long-span cable-stayed bridges are herein dis-cussed.
In Figure 3 several models based on the Dischinger’s as-sumptions are put in comparison with the exact catenary results(see Eqs. (27) and (29)) by the plots of the dimensionless equiv-alent modulus E∗/Ec versus the angle α. Different values ofthe dimensionless ratios τ and ω, and different values of thedimensionless cable stress variation �σ/σo are investigated.It is worth observing that, for a steel cable with � = 600 m,γc = 7860 kg/m3, Ec = 206 GPa, the four herein-employed
values for τ and ω correspond to reference cable stress σo vary-ing from 100 to 400 MPa. These values can be considered astypical for long steel cables in long-span cable-stayed bridges[1, 3, 29, 30].
Proposed results confirm that Dischinger’s secant and tangentmoduli are very close to those obtained by the exact catenarytheory. Moreover, tangent modulus is not properly able to repro-duce the elastic behavior of the cable when high stress variationsoccur. On the other hand, the quasi-secant model clearly appearsto be in a good agreement with the secant catenary approach,showing a certain precision lack only when high stress variationsand reference cable stress are considered. Nevertheless, quasi-secant results are always better than tangent ones. For example,in the case of �σ/σo = 0.6, τ = 0.12 and ω = 1.94 × 10−3
(σo = 400 MPa for a steel cable with � = 600 m), quasi-secantmaximum percentage error with respect to the exact secant ap-proach is about 8%, whereas for the tangent model is about24%.
With reference to the notation introduced in Figure 2, let astay undergoing to the displacement vector s = s eη at the endpoint P to be considered, such that the actual configuration � liesin the same plane of �. The unit vector eη = (cos η i + sin η j)is introduced, η being the angle between s and i. The stay is as-sumed to be characterized by τ = 0.128 and ω = 1.75 × 10−3
(i.e., σo = 360 MPa when a steel cable with � = 600 m isconsidered). Different cable reference configurations, i.e., dif-ferent values of the chord angle α, and different displacementdirections, i.e., different values of η, are investigated in or-der to evaluate the dimensionless cable stress variation �σ/Ec
as well as horizontal (along i) and vertical (along j) compo-nents of the dimensionless variation �F of the cable-structureinteraction.
For given values of α and η, let the relative quadratic errorEg corresponding to the model g be defined as:
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464 G. VAIRO
TABLE 1Relative quadratic error E (%) between the dimensionless cablestress variation �σ/Ec computed for different models, and theexact catenary value. Dimensionless displacements Smin andSmax (×103) correspond to �σ/σo = −0.5 and �σ/σo = 1,
respectively. td: Dischingher’s tangent; sd: Dischinger’ssecant; qs: quasi-secant; (1): first order displacement
approximation; (2) second order displacement approximation
where Qe denotes the quantity under investigation evaluatedthrough the exact catenary approach, Qg is the same quan-tity evaluated by the model g, and (Smin, Smax) is the compar-ison interval. In detail, Smin is the dimensionless displacement(s/h when α �= 0, s/� when α = 0) producing in the cable�σ/σo = −0.5 and Smax is the dimensionless displacementproducing �σ/σo = 1. In the case of a steel cable for long-span cable-stayed bridges, these values of the ratio �σ/σo canbe considered as limit conditions compatible with bridge func-tional behavior as well as with the structural safety of the stay[1, 3, 29, 30].
Table 1 summarizes the values of E for the quantity Q =�σ/Ec computed by the quasi-secant approach, when a secondorder displacement formulation and different along-the-chordstrain measures are considered, as well as by tangent and secant
TABLE 2Relative quadratic error E (%) between the dimensionless
variation of the horizontal (�Fx) and vertical (�Fy)cable-structure interaction computed through the second order
quasi-secant approach, and the exact catenary values. Thesymbols are the same as Table 1
Dishinger’s theory, applied considering a first-order strain mea-sure. It should be observed that in the case α = 0◦, η = 90◦,since �σ/σo is an even function with respect to dimensionlessdisplacement S, Smin has been assumed to be null. Values of Efor components �Fx and �Fy of the cable-structure interactionand corresponding to the quasi-secant model are summarized inTable 2.
Moreover, referring to the previously introduced comparisonintervals, Figures 4 to 6 show the dimensionless cable stress vari-ation, the horizontal and vertical projections of �F, evaluatedthrough different models, versus the dimensionless displace-ment. For the sake of brevity, only some representative casescorresponding to α = 30◦ are herein presented.
Proposed results clearly highlight that the quasi-secant ap-proach provides a suitable description of the cable non-linear
FIG. 4. Dimensionless cable stress variation �σ/Ec versus the dimensionless displacement in the case α = 30◦ (τ = 0.128, ω = 1.75 × 10−3). (•): catenaryresults; (�): quasi-secant second order model (logarithmic strain measure); (∗): quasi-secant second order model (Green-Lagrange strain measure); (− − −):Dischinger’s tangent model (first order strain measure); (−−−−−): Dischinger’s secant model (first order strain measure).
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CABLES’ NONLINEAR RESPONSE IN CABLE-STAYED STRUCTURES 465
s / h x 103 s / h x 103 s / h x 103
Fx x 103 Fx x 103 Fx x 103
-6 -4 -2 0 2 4 6-2
-1
0
1
2
-6 -4 -2 0 2 4-2
-1
0
1
2
-9 -6 -3 0 3 6 9-2
-1
0
1
2
FIG. 5. Dimensionless horizontal variation �Fx of the cable-structure interaction force versus the dimensionless displacement in the case α = 30◦ (τ = 0.128,ω = 1.75 × 10−3). The symbols are the same as Figure 4.
behavior, with slightly more accurate results when logarithmicstrain measure is employed than Green-Lagrange. Quasi-secantresults, obtained considering different displacement directions,are in good agreement with those obtained by the catenary for-mulation both for quasi-horizontal stays and quasi-vertical ones.Moreover, present analyses confirm that the proposed approachexhibits a good accuracy not only for positive displacements,i.e., acting to increase the stress level in the cable, but also fornegative values of s. Finally, results concerning displacementvector in agreement with the comparison interval and with asignificant component orthogonal to the cable chord, show thatthe quasi-secant approach properly takes into account the ge-ometrical nonlinear effects induced by the finite cable’s chordrotation.
From a quantitative point of view, estimates of relativequadratic error in quasi-secant applications are always less than1.1%, except when horizontal cables are considered (α = 0◦).Nevertheless, the great values of E when α = 0◦ and η = 90◦
can be justified considering the large value of Smax , that is anon realistic comparison interval. Indeed, when values of Smax
comparable with the other cases (of about 10−2) are analyzed,error estimates highly reduce (to about 1% with a logarithmicstrain measure) for both �σ/Ec and �Fx . Moreover, high val-ues of E corresponding to �Fy when α = 0◦ can be considered
not significant because of the negligible values of the verticalinteraction force (smaller than 10−6).
On the other hand, a perfect matching into the compari-son intervals appears when the secant approach is considered,whereas the application of the Dischinger’s tangent formula-tion combined with a first-order strain measure leads to greaterror estimates. In detail, present results confirm that tangentapproach does not allow to describe properly the elastic stay’sbehavior and the corresponding cable-structure interaction, es-pecially in quasi-horizontal cables and also in case of smalldisplacement values (E greater than 11%).
6. CONCLUDING REMARKSIn this paper an analytical refined model for the analysis
of elastic stays in cable structures has been proposed. It rep-resents a second order approximation both with respect to thecable stress variation and the apparent chord-strain one, allow-ing to describe geometrical nonlinear effects in cable’s behavior.Moreover, through a second order displacement approach, thepresent model leads to a closed-form description of the cable-structure interaction, depending only on the reference config-uration of the cable. Accordingly, considering different finitestrain measures, explicit formulas useful for technical designapplications are proposed.
s / h x 103 s / h x 103 s / h x 103
Fy x 103 Fy x 103 Fy x 103
-6 -4 -2 0 2 4 6-1.0
-0.5
0.0
0.5
1.0
1.5
-6 -4 -2 0 2 4-1.0
-0.5
0.0
0.5
1.0
1.5
-9 -6 -3 0 3 6 9-1.0
-0.5
0.0
0.5
1.0
1.5
FIG. 6. Dimensionless vertical variation �Fy of the cable-structure interaction force versus the dimensionless displacement in the case α = 30◦ (τ = 0.128,ω = 1.75 × 10−3). The symbols are the same as Figure 4.
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466 G. VAIRO
Several test cases, based on stays usually employed in long-span cable-stayed bridges, have confirmed the good agreementof the present formulation with the exact catenary model and theclassical secant one, which both need iterative procedures to beapplied. On the other hand, the classical tangent model, gener-ally adopted to build up closed-form solutions for cable-stayedstructures, does not furnish a suitable quantitative approxima-tion of the cable’s behavior and cable-structure interaction. Ac-cordingly, the proposed approach opens the possibility to refineexisting closed-form solutions for the analysis of cable-stayedstructures, allowing a better quantitative agreement with the realstructural response.
ACKNOWLEDGEMENTSThe author would like to thank professor Franco Maceri for
valuable suggestions and fruitful discussions on this paper.This work was developed within the framework of La-
grange Laboratory, a European research group comprisingCNRS, CNR, the Universities of Rome “Tor Vergata,” Calabria,Cassino, Pavia, and Salerno, Ecole Polytechnique, Universityof Montpellier II, ENPC, LCPC, and ENTPE.
REFERENCES1. M.S. Troitsky, Cable Stayed Bridges: An Approach to Modern Bridge
Design, Van Nostrand Reinhold Co., New York, 1988.2. H.M. Irvine, Cable Structures, MA: MIT Press, Cambridge, 1992.3. N.J. Gimsing, Cable Supported Bridges: Concept and Design, 2nd ed., J.
Wiley & Sons, New York, 1997.4. Z. Agocs and J. Brodniansky, Cable Structures, J. Construct. Steel Res.,
vol. 46(1), pp. 488–488, 1998.5. J.W. Leonard, Tension Structures, McGraw-Hill, New York, 1988.6. H.A. Buchholdt, An Introduction to Cable Roof Structures, Cambridge
University Press, Cambridge, 1985.7. J. Kadlcak, Statics of Suspension Cable Roofs, A.A. Balkema, Rotterdam,
1995.8. W. O’Brian, General Solution of Suspended Cable Problem, J. Struct. Div.-
ASCE, vol. 93, pp. 1–126, 1967.9. B. McDonald and A. Peyrot, Sag-tension Calculation Valid for Any Line
Geometry, J. Struct. Eng.-ASCE, vol. 116(9), pp. 2374–2387, 1990.10. A.H. Peyrot and A.M. Goulois, Analysis of Cable Structures, Comput.
Struct., vol. 10, pp. 805–813, 1979.11. H.B. Jajaraman and W.C. Knudson, A Curved Element for the Analysis of
Cable Structures, Comput. Struct., vol. 14, pp. 325–333, 1981.12. R. Karoumi, Some Modeling Aspects in the Nonlinear Finite Element
Analysis of Cable Supported Bridges, Comput. Struct., vol. 71, pp. 397–412, 1999.
13. K.S. Kim and H.S. Lee, Analysis of Target Configurations Under DeadLoads for Cable-Supported Bridges, Comput. Struct., vol. 79, pp. 2681–2692, 2001.
14. N.B. Kahla, Response of a Guyed Tower to a Guy Rupture Under no WindPressure, Eng. Struct., vol. 22, pp. 699–706, 2000.
15. D. Bruno and A. Leonardi, Nonlinear Structural Models in Cableway Trans-port Systems, Simulat. Pract. Theory, vol. 7, pp. 207–218, 1999.
16. F. Dischinger, Hangebrucken fur schwerste Verkehrslasten (I and II) (inGerman), Der Bauingenieur, vol. 24(3), pp. 65–75, 107–113, 1949.
17. J.H. Ernst, Der E-modul Von Seilen Unter Berucksichtigung Des Durch-hanges (in German), Der Bauingenieur, vol. 40(2), pp. 52–55, 1965.
18. D. Bruno, F. Maceri, and R.S. Olivito, Analysis of the Elastic Response ofStays and Stayed Systems, IABSE Proceedings P-143/90, pp. 29–44, 1990.
19. H.M. Ali and A.M. Abdel-Ghaffar, Modelling the Nonlinear Seismic Be-havior of Cable-stayed Bridges with Passive Control Bearings, Comput.Struct., vol. 54, pp. 461–492, 1995.
20. A.S. Nazmy and A.M. Abdel-Ghaffar, Three-dimensional Non-linear StaticAnalysis of Cable-stayed Bridges, Comput. Struct., vol. 34, pp. 257–271,1990.
21. J. Cheng, J. J. Jiang, R. C. Xiao, and H. F. Xiang, Nonlinear AerostaticStability Analysis of Jiang Yin Suspension Bridge, Eng. Struct., vol. 24,pp. 773–781, 2002.
22. W.K. Song and S.E. Kim, Analysis of the Overall Collapse Mechanism ofCable-Stayed Bridges with Different Cable Layouts, Eng. Struct., vol. 29,pp. 2133–2142, 2007.
23. J.D. Yau and Y.B. Yang, Vibration Reduction for Cable-Stayed BridgesTraveled by High-Speed Trains, Finite Elem. Anal. Des., vol. 40, pp. 341–359, 2004.
24. G. Tibert, Numerical Analyses of Cable Roof Structures, Trita-Bkn, Bulletin49, 1999.
25. Y.M. Desai and S. Punde, Simple Model for Dynamic Analysis of CableSupported Structures, Eng. Struct., vol. 23, pp. 271–279, 2001.
26. A.M.S. Freire, J.H.O. Negrao, and A.V. Lopes, Geometrical nonlineari-ties on the Static Analysis of Highly Flexible Steel Cable-Stayed Bridges,Comput. Struct., vol. 84, pp. 2128–2140, 2006.
27. F.T.K. Au, Y.S. Cheng, Y.K. Cheung, and D.Y. Zheng, On the Determinationof Natural Frequencies and Mode Shapes of Cable-Stayed Bridges, Appl.Math. Model., vol. 25, pp. 1099–1115, 2001.
28. P. Warnitchai, Y. Fujino, and T. Susumpow, A Non-Linear Dynamic Modelfor Cables and its Application to a Cable-Structure System, J. Sound Vib.,vol. 187, pp. 685–712, 1995.
29. F. De Miranda, A. Grimaldi, F. Maceri, and M. Como, Basic Problems inLong-Span Cable-Stayed Bridges, Dep. of Structures, Univ. of Calabria,Report n. 25, 1979.
30. M. Como, A. Grimaldi, and F. Maceri, Statical Behaviour of Long-SpanCable Stayed Bridges, Int. J. Solids Struct., vol. 21(8), pp. 831–850, 1985.
31. D. Bruno, On the Dynamic Behaviour of Long-Span Cable-Stayed bridgesunder moving loads, Costruzioni Metalliche, vol. 2, pp. 3–15, 1985.
32. D. Bruno, A. Leonardi, and F. Maceri, On the Nonlinear Dynamics of Cable-Stayed Bridges, Cable-Stayed Bridges Experiences & Practice 1987: Proc.Int. Conf. on Cable-Stayed Bridges, Bangkok, Thailand, pp. 529–544, 1987.
33. F. Maceri, D. Bruno, and A. Leonardi, Aerodynamic Behavior of Long-Span Cable-Stayed Bridges, Microcomputers in Civil Engineering, vol. 5,pp. 85–94, 1990.
34. D. Bruno and V. Colotti, Vibration Analysis of Cable-Stayed Bridges, Struc-tural Engineering International, vol. 4(1), pp. 23–28, 1994.
35. D. Bruno and A. Leonardi, Natural Periods of Long-Span Cable-StayedBridges, J. Bridge Eng., vol. 2(3), pp. 105–115, 1997.
36. G.T. Michaltsos, A simplified Model for the Dynamic Analysis of Cable-stayed Bridges, In Mechanics, Automatic Control and Robotics, vol. 3(11),pp. 185–204, Facta Universitatis, Nis, Serbia, 2001.
37. G.T. Michaltsos, J.C. Ermopoulos, and T.G. Konstantakopoulos, Prelimi-nary Design of Cable-Stayed Bridges for Vertical Static Loads, Struct. Eng.Mech., vol. 16(1), pp. 1–15, 2003.
38. M. Como, Stabilita Aerodinamica dei Ponti di Grande Luce (in Italian),in Elio Giangreco—Ingegneria delle Strutture, vol. II, pp. 159–330, Utet,Torino, 2002.
39. F. Maceri and G. Vairo, Modelling and Simulation of Long-span BridgesUnder Aerodynamic Loads, In M. Fremond and F. Maceri (eds.), NovelApproaches in Civil Engineering, Lect. Notes Appl. Comput. Mech., vol.14, pp. 359–376, Springer-Verlag, Berlin, 2004.
40. D. Bruno, F. Greco, and P. Lonetti, Dynamic Impact Analysis of LongSpan Cable-stayed Bridges Under Moving Loads, Eng. Struct., vol. 30, pp.1160–1177, 2008.
41. M. Virlogeux, Recent Evolution of Cable-Stayed Bridges, Eng. Struct., vol.21, pp. 737–755, 1999.
42. G. Vairo, A Quasi-secant Continuous Model for the Analysis of Long-spanCable-stayed Bridges, Meccanica, vol. 43, pp. 237–250, 2008.
Downloaded By: [Vairo, Giuseppe] At: 08:37 22 July 2009
