Structural Engineering and Mechanics, Vol. 41, No. 1 (2012) 139-155

139

Time-dependent effects on dynamic properties ofcable-stayed bridges
Francis T.K. Au* and X.T. Si
Department of Civil Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong, China(Received May 11, 2011, Revised September 17, 2011, Accepted December 13, 2011)
Abstract. Structural health monitoring systems are often installed on bridges to provide assessments of
the need for structural maintenance and repair. Damage or deterioration may be detected by observation ofchanges in bridge characteristics evaluated from measured structural responses. However, construction
materials such as concrete and steel cables exhibit certain time-dependent behaviour, which also results inchanges in structural characteristics. If these are not accounted for properly, false alarms may arise. This
paper proposes a systematic and efficient method to study the time-dependent effects on the dynamic
properties of cable-stayed bridges. After establishing the finite element model of a cable-stayed bridge
taking into account geometric nonlinearities and time-dependent behaviour, long-term time-dependent
analysis is carried out by time integration. Then the dynamic properties of the bridge after a certain
period can be obtained. The effects of time-dependent behaviour of construction materials on the dynamic
properties of typical cable-stayed bridges are investigated in detail.
Keywords: cable-stayed bridges; concrete creep; geometric nonlinearities; structural health monitoring
systems; time-dependent behaviour

1. Introduction
Bridges are important links in almost every transportation system. As they may be damaged
during their service lives due to service loads, and environmental and accidental actions, it isdesirable to conduct regular condition assessment of important bridges to obtain information on theoccurrence, geometric location and severity of any structural damage at the earliest possible stage toprevent these structures from any potential catastrophic events (Liu et al. 2009, Kim et al. 2011).
Among various major crossings built over the past four decades, cable-stayed bridges have become
very popular not only because of their remarkable structural efficiency but also their aestheticallypleasing appearance. With advances in material technology and the increasing use of high-strength
materials together with slender structural members, the time-dependent behaviour associated withthe higher stress levels has become an increasing concern.
In parallel with the gradual but steady increase in span lengths in cable-stayed bridges in recent
decades, structural health monitoring (SHM) systems are increasingly installed on such bridges tomonitor their performance and safety by observation of any changes in bridge characteristics caused*Corresponding author, Professor, E-mail: francis.au@hku.hk

140Francis T.K. Au and X.T. Si

by damage (Doebling et al. 1996, Zhang et al. 2008, Kim et al. 2011). However there are various
components in cable-stayed bridges with time-dependent behaviour. For example, concrete structuralmembers exhibit various time-dependent phenomena, including ageing which manifests as increasein elastic modulus (Cluley and Shepherd 1996), creep in the form of gradual increase of inelasticstrain under sustained stress, and drying shrinkage in the form of shortening as a result of loss ofmoisture. Stay cables and prestressing tendons may experience stress relaxation or creep behaviour
with time (Kmet et al. 2011). These time-dependent phenomena will affect the internal force
distribution in the structure and its long-term performance, which also result in changes in bridge
characteristics. If these issues are not considered properly, false positive or false negative alarm of
SHM systems may arise. Therefore it is necessary to take such effects into account carefully inorder to build a reliable monitoring system.
The use of finite element method in conjunction with time integration serves as a reliable tool fortime-dependent analysis of concrete structure (Ghali et al. 2002, Au et al. 2009). In this method, thestructural concrete components are usually represented as beam-column elements while the steel
tendons embedded in concrete are modelled as truss elements with nodes connected to the beam
nodes by rigid arms (Aalami 1998, Elbadry and Ghali 2001, Au et al. 2009). In order to account for
the interaction among concrete creep, concrete shrinkage and cable relaxation accurately, theequivalent creep model for tendons proposed by Au and Si (2009) is used together with the creep
and shrinkage models of concrete. The long-term performance of concrete structures can then be
predicted accurately taking into account the time-dependent behaviour of concrete and steel tendons.
Although much work has been done on the time-dependent static behaviour of concrete structures,the effects of time-dependent behaviour on the dynamic performance of concrete structures havereceived relatively little attention. One of the first publications in this area was the investigation of
creep and shrinkage effects of concrete on dynamic behaviour of reinforced concrete slab-and-beam
structures (Sapountzakis and Katsikadelis 2003). Ma et al. (2011) studied the creep effects on
dynamic behaviour of concrete-filled steel tube arch bridge. Both of their results showed that thenatural frequencies would decrease gradually with time as some kind of long-term effective
modulus (Trost and Wolff 1970, Bažant 1972) was adopted. As there has been little work in thisrespect done on cable-stayed bridges, it is necessary to develop a systematic and reliable method toinvestigate the effect of time-dependent behaviour due to concrete ageing, creep and shrinkage, andcable relaxation on the dynamic properties of such structures. It not only improves ourunderstanding of the long-term performance of this kind of bridges of emerging importance, but it
also ensures that any such SHM system can be made reliable in the long run. However it should bepointed out that the present analysis does not take into consideration the effects of damage,deterioration due to corrosion, etc. as they are based on totally different mechanisms.

2. Methodology for analysis of time-dependent behaviour
2.1 Equivalent creep coefficient for steel tendons
The intrinsic stress relaxation ∆σpr in a steel cable is the loss of stress at constant strain. It
depends on both the duration of sustained tension t (hours) and the ratio of the initial prestress σpi tothe “yield” strength of steel fpy. The equation commonly adopted for the stress-relieved strands orwires (Magura et al. 1964) for σpi/fpy ≥ 0.55 is

Time-dependent effects on dynamic properties of cable-stayed bridges

141∆σpr ( t )log ( t ) σ---------------- = – -------------- ⎛ -----pi- – 0.55⎞⎠
σpi
10 ⎝ fpy(1)

In the derivation of equivalent creep coefficient for steel tendons, it is assumed that the creep
coefficient is independent of age and the modulus of elasticity Es remains constant. Given the typeof tendon and based on the intrinsic stress relaxation, the creep coefficient ϕs ( t ) of a tendon
stressed initially to σpi can be worked out at regular time intervals ∆t as (Au and Si 2009)–∆σpr ( ∆t )ϕs ( ∆t ) = --------------------------------------σpi + ∆σpr ( ∆t ) ⁄ 2(2)

σ pi ϕs ( k∆t )ϕs [ ( k + 1 )∆t ] = --------------------------------------σpi + ∆σpr ( ∆t ) ⁄ 2[ ∆σ pr [ ( k + 1 )∆t ] – ∆σ pr ( k∆t ) ] [ 1 + ϕs ( ∆t ) ⁄ 2 ] – ∆σpr ( ∆t )ϕs [ ( k – 1 )∆t ] ⁄ 2– -------------------------------------------------------------------------------------------------------------------------------------------------------------------------σ pi + ∆σpr ( ∆t ) ⁄ 2n∑ [ ∆σpr ( i∆t ) – ∆σpr [ ( i – 1 )∆t ] ] { ϕs [ ( k – i + 2 )∆t ] – ϕs[ ( n – i )∆t] } ⁄ 2
i=2– ------------------------------------------------------------------------------------------------------------------------------------------------------------------- ( k = 1,2...n )σ pi + ∆σpr ( ∆t ) ⁄ 2(3)

Therefore, the equivalent creep coefficients of tendons can be obtained by a step-by-step procedure.
Note that, although the creep coefficient ϕs ( t ) depends on σpi/fpy, this parameter has been dropped
for convenience in notation.
2.2 ACI recommendations for modelling creep and shrinkage of concrete
The prediction of creep and shrinkage effects in concrete structures as well as effects of various
relevant variables related to material properties, climate and member size are provided in detail in areport by ACI committee 209 (1997). For brevity, it is referred to as ACI 209 hereafter. A summaryof the recommendations is given below.
This report suggests that the compressive strength fc(t) (MPa) at age t (days) can be expressed interms of the corresponding value fc(28) (MPa) at age 28 days by
1 - f ( 28 )fc ( t ) = -----------c
a + bt(4)

where the parameters a and b that depend on the cement type and method of curing arerecommended in ACI 209. The modulus of elasticity of the concrete Ec(t) (MPa) can be estimatedfrom the compressive strength fc(t) and density of concrete w (kg/m3) by
3Ec ( t ) = 0.043 [ w fc ( t ) ]

1⁄2

(5)

The creep coefficient ϕc (t,t0) at time t (days) for a constant stress applied at the loading age t0(days) is
0.60( t – t0 )ϕ c ( t,t0 ) = --------------------------------ϕ
0.60 cu
10 + ( t – t0 )

(6)

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Francis T.K. Au and X.T. Si

where ϕcu is the ultimate creep coefficient that, in the absence of specific data for local aggregatesand material conditions, can be estimated as ϕcu = 2.35γc in terms of the correction factor γc related
to ambient relative humidity, average thickness of the member or its volume-surface ratio, andtemperature.
The recommended equation for prediction of shrinkage strain εsh ist – tsu( εsh )εsh = --------------------f + ( t – ts )(7)

where f (days) is a constant dependent on curing conditions (i.e., 35 for moist cured concrete, 55 forusteam cured concrete), ts is the age of concrete at initiation of drying (days), and ε sh is the ultimate-6
shrinkage strain that, in the absence of shrinkage data, can be taken as 780γsh × 10 m/m in terms ofthe correction factor γsh related to ambient relative humidity, average thickness of the member or its
volume-surface ratio, and temperature.
2.3 Time integration method for time-dependent behaviour of cable-stayed bridges
An accurate model taking into account the time-dependent behaviour of construction materials is
indispensable for the accurate prediction of dynamic responses of cable-stayed bridges under various
environmental and imposed loading many years after their completion. Based on the above equivalent
creep model of tendons and other models for concrete creep and shrinkage, a finite element method
in conjunction with time integration has been developed to predict the long-term behaviour ofcommon concrete structures such as prestressed concrete girder bridges (Au and Si 2011).
When the finite element method is used to model a bridge, concrete members are usually
represented as beam-column elements while tendons are idealized as truss elements connected tohypothetical rigid arms attached monolithically to the beam axis to ensure compatibility. FollowingeT
the traditional finite element method, the incremental nodal load vector { ∆q } s = [ ∆f1 ∆f2 ] s of atendon element for the time interval from t to (t+∆t) can be derived ase{ ∆q } s = [ k ] s { ∆u } s + { ∆f }ϕs(8)

in terms of the stiffness matrix [ k ]s , incremental displacement vector { ∆u }s and incremental load
vector due to tendon creep { ∆f } ϕs . Note that the variable t has been omitted for brevity hereafter.
The stiffness matrix [ k ] s (Au and Si 2011) is given byEs ( ∆t )As 1 –1
[ k ]s = -------------------ls
–1 1

(9)

where As is cross sectional area, ls is the length of tendon element, and Es ( ∆t ) is the mean modulus
of elasticity over the time interval ∆t taking into account stress relaxation, which can be expressedin terms of the modulus of elasticity of the steel tendon Es as
Es
Es ( ∆t ) = ----------------------1 + ϕs ( ∆t )The incremental load vector due to cable relaxation { ∆f }ϕs is given by(10)

Time-dependent effects on dynamic properties of cable-stayed bridges

143·⎧ Ns ( τ ) ⎫ ⎫⎧ Ns ( t0 ) ⎫ t
Es ( ∆t ) ⎧{ ∆f }ϕs = -------------- ⎨ [ ϕ s ( t + ∆t – t0 ) – ϕ s ( t – t0 ) ] ⎨
+
[
(
t
+
∆t
–
τ
)
–
ϕ
(
t
–
τ
)
]
ϕ
⎨ ·
⎬dτ ⎬
⎬ ∫ s
s
Es ⎩
⎩ –Ns ( τ ) ⎭ ⎭⎩ –Ns ( t0 ) ⎭ t0
(11)·
where Ns ( t0 ) is the axial force of the element at time t0 and Ns ( τ ) is the derivative of the axial
force with respect to the dummy time variable τ.
Similarly neglecting body forces, the incremental load vector of concrete beam-column elementeT{ ∆q } c = [ ∆f1 ∆f2 ... ∆f12 ] c can be obtained ase{ ∆q } c = [ k ] c { ∆δ }c + { ∆f }ϕc + { ∆f }cs(12)

in terms of beam stiffness matrix [ k ] c , incremental displacement vector { ∆δ }c , incremental load
vector due to concrete creep { ∆f } ϕc and incremental load vector due to concrete shrinkage { ∆f }cs .
Using the conventional finite element method, the stiffness matrix and the incremental load vector
due to creep and shrinkage can be formulated in terms of the mean modulus of elasticity Ec and themean modulus of rigidity of concrete Gc over the time step from t to (t + ∆t), which can be
expressed respectively as[ Ec ( t ) + Ec ( t + ∆t ) ] ⁄ 2
Ec ( t ) = -------------------------------------------------1 + ϕc ( t + ∆t,t ) ⁄ 2(13)

Ec ( t )Gc ( t ) = -------------------2 ( 1 + vc )(14)

where vc is the Poisson’s ratio of concrete. Further details are available in Au and Si (2011).
2.4 Modelling of cable-stayed bridges for free vibration analysis
The analysis of concrete cable-stayed bridges taking into account time-dependent effects of concrete
creep and shrinkage, and cable relaxation, as well as time-independent nonlinear effects including
cable sag, large displacements, etc., has been reported by Curley and Shepherd (1996). A concrete
cable-stayed bridge is first discretised using the finite element method. The bridge deck and towers
are usually modelled as Bernoulli-Euler beam-column elements (Au et al. 2001). Each stay cable ismodelled as either a single truss element with an effective modulus or a series of cable elementswith the original modulus. One common practice to model a sagging cable is to use a single truss
element with a length equal to that of its chord and an equivalent modulus of elasticity to allow for
sag and elastic strain. A widely used expression (Au et al. 2001) for this equivalent modulus Eeq isgiven byEca
-3
Eeq = --------------------------------------------------------2
1 + ( wHca ) Aca Eca ⁄ 12T(15)

where Hca is the horizontal projected length, Aca is the cross-sectional area, Eca is the effective
modulus of elasticity, w is the weight per unit length and T is the updated cable tension of the cable.
The main effect of large displacement is the “P-delta effect” in the bridge girders and towers. This

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Francis T.K. Au and X.T. Si

is due to the coupling of large lateral deflections with high compressive axial forces. This coupling
reduces both the axial and rotational stiffnesses of the corresponding members. An efficient approachto consider this effect is to adopt the geometric stiffness matrix [k]G of each beam-column element(McGuire et al. 2002) to modify its elastic stiffness matrix [k]E. The resulting tangential stiffness
matrix [k]T to be used as the element stiffness matrix in the analysis then becomes[k]T = [k]E + [k]G

(16)

After building up the local tangential stiffness matrix and consistent mass matrix of each element,
they are transformed to the global coordinates system and assembled to form the global tangential
stiffness matrix [K]T taking into account the effect of cable sag and large displacement as well as
the global mass matrix [M]. Finally the governing equation for analysis of dynamic properties of anundamped cable-stayed bridge appears as
2( [ K ]T – ω [ M ] ) { D } = { 0 }(17)where ω is the circular frequency and { D } is the eigenvector.
The modulus of elasticity to be used in free vibration analysis is a key issue to address. Based on
the principle of superposition, the total concrete strain εc ( t ) due to the initial applied stress σc ( t0 ) ,creep and shrinkage is given by Ghali et al. (2002) as
1 + ϕ c ( t,t0 )- +εc ( t ) = σ c ( t0 ) ------------------------Ec ( t0 )∆σ c ( t )1 + ϕc ( t,τ )- dσ c ( τ ) + εcs ( t,t0 )∫ -----------------------Ec ( τ )

(18)

0

where ϕ c ( t,t0 ) is the creep coefficient at time t for concrete loaded at time t0, Ec(t) is the modulus
of elasticity of concrete at time t, ∆σc ( t ) is the stress increment from time t0 to t and εcs ( t,t0 ) is
the free shrinkage from time t0 to t. To predict the long-term performance of concrete structures
efficiently, one may rewrite Eq. (18) for convenience as∆σ c ( t )1 + ϕc ( t,t0 )- + εcs ( t,t0 )ε c ( t ) = σc ( t0 ) ------------------------- + ---------------Ec ( t0 )Ec ( t,t0 )(19)where the age-adjusted elasticity modulus (AAEM) Ec ( t,t0 ) to be used in the calculation of thetotal strain increment, instantaneous plus creep, due to a stress increment of magnitude developing
gradually from zero to a value ∆σc ( t ) is given by in terms of an ageing coefficient χ c ( t,t0 ) asEc ( t0 )Ec ( t,t0 ) = ------------------------------------------1 + χc ( t,t0 )ϕc ( t,t0 )(20)

In other words, the AAEM takes into account the long-term deformation of concrete, primarily
creep, over the period from time t0 to t as a matter of years, which explains why it gradually drops
with time. Therefore it is obvious that the free vibration analysis of a concrete bridge at time t many
years after completion, which involves variations of structural responses within fractions of asecond, should be based on the instantaneous modulus of elasticity of concrete Ec(t) then, instead of
the AAEM from time t0 to t as in some previous work.

Time-dependent effects on dynamic properties of cable-stayed bridges

145

2.5 Finite element implementation
The three-dimensional free vibration analysis of concrete cable-stayed bridges considering geometricallynonlinearity and time-dependent behaviour can be carried out in accordance with the flowchart
shown in Fig. 1 as explained below:a) The finite element model of the cable-stayed bridge is built up first. The bridge girder andtowers are represented by beam-column elements taking into account geometric nonlinearities.
Each stay cable is modelled as a single truss element considering cable sag under its own weight.
The prestressing tendons in the concrete girder are modelled with truss elements attached to thegirder axis with rigid arms to approximate the tendon profiles.
b) The time-dependent analysis from completion of the cable-stayed bridge to the time of interestis carried out using the established finite element model by time integration taking into accountcreep, shrinkage and ageing of concrete, and stress relaxation in prestressing tendons and stay
cables. The prevalent bridge geometry and internal forces at the time of interest will then be
noted.
c) The finite element model at the time of interest is updated using the bridge geometry andinternal forces obtained from the time-dependent analysis. In particular, the global tangential
matrix and global mass matrix are based on the updated model and the instantaneous Young’s
modulus of each material at the time of interest.
d) Free vibration analysis of the bridge is then carried out using subspace iteration method or

Fig. 1 Flowchart for free vibration analysis of cable-stayed bridges taking into account time-dependent behaviour

146

Francis T.K. Au and X.T. Si

similar based on the updated global stiffness and mass matrices to obtain the natural frequencies andmodal shapes.

3. Case studies
3.1 Dynamic properties of a cable-stayed concrete cantilever considering time-dependent
behaviour
The dynamic properties of a hypothetical cable-stayed concrete cantilever as shown in Fig. 2
under long-term effects are studied using the proposed method. The cantilever is adapted from an
example provided by Ghali et al. (2002). It is 10 m in length with a square cross section of 1 m × 1 m.
Its unit weight is 25 kN/m3. The characteristic compressive strength of concrete is fck = 36 MPa.
Moist curing is carried out until Ts = 3 days after which shrinkage begins. The relative humidity istaken as 70% throughout. For the concrete in the cantilever, it is assumed that Type I cement isused, the water cement ratio is 0.45 and the percentage of fine aggregates is 60%. The stay cable isa stress relieved tendon with a cross sectional area As = 250 mm2, Young’s Modulus Es = 195 GPa,and unit weight ρs = 78 kN/m3. The initial tension P0 = 210 kN is applied at time t0 = 28 days withreference to the age of cantilever. The initial prestressing ratio σpi/fpy is taken as 0.8, which isunusually high to demonstrate the effect of relaxation. The parameters of ACI 209 (1997) are
adopted for analysis. In view of the relatively small dimensions of the structure, the sag effect ofcable and geometric nonlinearities of the cantilever are ignored. The cantilever is modelled by 4
identical beam-column elements while the cable is modelled by a truss element. First, the initial
natural frequencies of this structure at Day 28 are calculated without accounting for any timedependent behaviour. Then the dynamic properties of global vibrations at Day 300 are calculated.
To investigate the effects of various sources of time-dependent deformation, a few hypotheticalcases are worked out, namely (a) Case A: concrete ageing only; (b) Case B: concrete ageing andcable relaxation only; (c) Case C: concrete ageing, creep and shrinkage, and cable relaxation; and(d) Case D: cable relaxation only.
The percentage losses of cable tension with time in Cases C and D are shown in Fig. 3. The otherinternal forces also vary accordingly. The first three frequencies and their percentage differencesfrom the initial frequencies are shown in Table 1. Comparing results of various cases with the initial

Fig. 2 A cable-stayed cantilever

Time-dependent effects on dynamic properties of cable-stayed bridges

147

Table 1 Global dynamic properties of cable-stayed cantilever at Day 300
Case A

Case B

Case C

Case D

Mode

Initial
freq.
(Hz)

Freq.
(Hz)

Diff.
(%)

Freq.
(Hz)

Diff.
(%)

Freq.
(Hz)

Diff.
(%)

Freq.
(Hz)

Diff.
(%)

1

6.453

6.664

3.26

6.664

3.26

6.666

3.31

6.453

0.00

2

38.84240.2203.55

40.2203.55

40.237

3.59

38.8420.00

3109.071112.949

3.56

112.949

3.56

112.9943.60

109.071

0.00

Fig. 3 Percentage losses of cable tension with time inCases C and D

Fig. 4 Frequency increase with time due to effects oftime-dependent behaviour

results shows that the natural frequencies increase slightly under concrete ageing effect and its
combination with other time-dependent deformations, while cable relaxation has little additional
effect. It is also observed that concrete ageing has the most important influence on the long-term
dynamic properties among various time-varying factors by comparing Cases A, B and C. Comparing
results of Cases B and C demonstrates that the interaction among concrete ageing, creep and shrinkage
has more effect on the dynamic characteristics than the interaction between concrete ageing andcable relaxation. Comparison of the results of various cases also shows that the interaction among
various time-varying factors is greater than their individual effects. Therefore it is desirable to take
into account the interaction among various time-varying factors when long-term analysis of dynamic
behaviour is performed. One important observation is that all the frequencies go up primarily
because of concrete ageing, namely the gradual increase in modulus of elasticity of concrete.
Fig. 4 shows that the first three frequencies of the structure increase gradually with time under theeffects of cable relaxation, and ageing, creep and shrinkage of concrete. The combined timedependent behaviour increases the frequencies by up to 3.6% within the first year after completion.
In particular, the frequencies increase relatively rapidly in the first year but tend to become stable
after a few years.
3.2 Dynamic properties of a cable-stayed bridge considering time-dependent behaviour
A typical modified fan cable-stayed concrete bridge similar to that studied by Au et al. (2001)
with a main span of 364 m, as shown in Fig. 5, is modelled as a two-dimensional structural system

148

Francis T.K. Au and X.T. Si

Fig. 5 Schematic structural arrangement of a typical concrete cable-stayed bridgefor dynamic analysis. The bridge deck is hinge-supported on Tower 1 but roller-supported on Tower
2. The corresponding properties of the bridge deck and towers are given in Table 2. The propertiesof stay cables and the finite element discretisation are shown in Tables 3 and 4. The cable forces are
Table 2 Properties of deck and towers of cable-stayed bridge
Cross sectionalarea
A (m2)

Second moment ofarea
I (m4)

Young’s modulus ofconcrete at Day 28
Ec28 (MPa)

Densityρ (kg/m3)

Bridge deck

6.00

4.19

32 0902550

Bridge tower-above

14.2

30.0

32 0902550

Bridge tower-below deck

35.8

40.0

32 0902550

Part of structures

Table 3 Cross sectional areas of stay cables of cable-stayed bridge
Cable no.

Area (m2)Cable no.

Area (m2)

1 & 240.060

7 & 180.013

2 & 230.023

8 & 170.016

3 & 22

0.020

9 & 160.019

4 & 21

0.017

10 & 150.023

5 & 200.014

11 & 140.026

6 & 190.011

12 & 130.029

Time-dependent effects on dynamic properties of cable-stayed bridges

149

Table 4 Discretisation scheme of a cable-stayed bridge
Components

No. ofelements

SchemeDeck

26

Length from left to right ends: 6 × 26 m, 6 × 28 m, 2 × 14 m, 6 × 28 m, 6 × 26 m

Tower

11

Length from bottom to top: 3 × 11.33 m, 3 × 17.47 m, 5 × 3.6 m

Cable

1

Whole length of each cableadjusted by iterations so that the deck and tower geometry under permanent loading is that asshown in Fig. 5. The working stress ratios σpi/fpy of the stay cables are kept low enough so thatcable relaxation is negligible, like what is normally adopted in practical cable-stayed bridges.
However in view of the much larger dimensions, the sag effect of cables is always taken into
account by the use of equivalent modulus Eeq. Some simplifying assumptions are made in this study
which focuses on the effects of time-dependent behaviour on the dynamic properties of cable-stayed
bridges. It is assumed that the bridge is constructed over a short period of time and the construction
sequence can be ignored. The characteristic compressive strength of the concrete used in the deckand towers is fck = 40 MPa. Wet curing is carried out until Ts = 3 days after which shrinkage begins.
The parameters of ACI 209 (1997) are adopted. The relative humidity is taken as 80% throughout.
For the concrete in the deck and towers, it is assumed that Type I cement is used, the water cement
ratio is 0.45 and the percentage of fine aggregates is 60%.
A few hypothetical cases are worked out. Firstly, the initial natural frequencies of global vibrationsof the bridge at Day 28 are calculated without considering geometric nonlinearities of the deck andtowers and any time-dependent behaviour. Secondly, the natural frequencies at Day 28 are obtained
considering geometric nonlinearities (Case A). Then the dynamic properties at Day 300 are obtainedfor various cases considering geometric nonlinearities, namely (a) Case B: concrete ageing only; (b)Case C: concrete ageing and creep only; and (c) Case D: concrete ageing, concrete creep andshrinkage. The first ten natural frequencies and the percentage changes compared with the initialvalues are shown in Table 5 and Fig. 6 respectively.
Table 5 The first ten natural global frequencies (Hz) of cable-stayed bridge
Mode

Initial (Day 28)

Case A (Day 28)

Case B (Day 300)

Case C (Day 300) Case D (Day 300)

1

0.337

0.335

0.339

0.341

0.343

2

0.461

0.454

0.461

0.465

0.470

3

0.687

0.676

0.684

0.687

0.689

4

0.748

0.739

0.746

0.747

0.749

5

0.839

0.826

0.836

0.840

0.846

6

1.047

1.028

1.047

1.051

1.061

7

1.219

1.202

1.230

1.226

1.230

8

1.281

1.256

1.283

1.281

1.282

9

1.338

1.313

1.342

1.340

1.347

10

1.389

1.373

1.408

1.418

1.426

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Francis T.K. Au and X.T. Si

Fig. 6 Variation of the first 10 frequencies for various hypothetical cases

Table 5 and Fig. 6 show that geometric nonlinearities tend to reduce the natural frequencies by amaximum of 1.94%. However, the natural frequencies increase under the combined effects of timedependent behaviour and geometric nonlinearities by a maximum of 2.66%. It indicates that the
time-dependent behaviour more than offsets the effects of geometric nonlinearities on dynamic
properties of this bridge. Comparing Cases B, C and D shows that concrete ageing plays an importantrole in increasing the long-term natural frequencies. The effects of creep are higher than those ofshrinkage. The interaction between concrete creep and ageing effects should therefore be considered
carefully for long-term analyses of concrete structures.
Fig. 7 shows that the frequencies of the first 10 modes of this bridge increase with time withrespect to Case A under the time-dependent effects of ageing, creep and shrinkage of concrete
coupled with geometric nonlinearities. It is also observed that the time-dependent behaviour has
different effects on various frequencies of the bridge, ranging from 1.42% to 4.37% at Day 3,000. It
further confirms that time-dependent behaviour increases the natural frequencies of this concreteFig. 7 Variation of the first 10 frequencies with time under all time-dependent effects

Time-dependent effects on dynamic properties of cable-stayed bridges

151

bridge. It is therefore obvious that the long-term behaviour of concrete and its interaction withgeometric nonlinearities should be properly taken into account for reliable damage identification inany vibration-based structural health monitoring systems.
3.3 Effects of cable relaxation on dynamic properties of cable-stayed girders with multiple
cablesTo investigate particularly the effects of cable relaxation on cable-stayed girders with multiple
cables, the cable-stayed cantilever shown in Fig. 8 and the cable-stayed girder with a hinged end
shown in Fig. 9 are further considered. Each of them is supported by two stay cables such that theirinitial cable forces provide the vertical reactions to the deck when considered as a continuous beamwith the cable supports replaced by roller supports. The cable lengths have been adjusted by
iteration so that the initial vertical deflections at deck level cable anchorages are effectively zero.
The cross sectional areas of stay cables are then adjusted such that the initial prestressing ratios σpi/fpy
all have a relatively high value of 0.8. The properties of the concrete girders and other assumptions
are the same as those in the example in Section 3.1. The weight of concrete girder is supported byfalsework until the time t0 = 28 days with reference to the age of concrete when the cables are
tensioned. The cross sectional areas of cables and initial cable forces are determined accordingly
and shown in Tables 6 and 7. The girder is modelled by 8 identical beam-column elements while
each cable is modelled by a truss element.
To investigate the effects of various sources of time-dependent deformation, a few hypotheticalcases are studied, namely (a) Case A: concrete ageing only; (b) Case B: concrete ageing and cablerelaxation only; (c) Case C: concrete ageing, creep and shrinkage, and cable relaxation; and (d)Fig. 8 A cable-stayed cantilever with two cables

Fig. 9 A cable-stayed girder with two cables and a hinged end

152Francis T.K. Au and X.T. Si

Table 6 Cable-stayed cantilever with two cables
Case C at Day 300

Cable

Sectional area
(mm2)

Initial cable force at
Day 28 (kN)

Loss in cable force (%)

Deflection at anchorage (mm)

AC

488

409.8

10.5

18.1

DC

763

641.0

9.1

6.2

Table 7 Cable-stayed girder with a hinged end and two cables
Case C at Day 300

Cable

Sectional area
(mm2)

Initial cable force at
Day 28 (kN)

Loss in cable force (%)

Deflection at anchorage (mm)

AC

468

392.9

4.5

43.7

DC

833

700.1

-2.7

21.1

Case D: cable relaxation only. The variations of cable forces and deflections at anchorage in Case C
at Day 300 shown in Tables 6 and 7 will shed light on the behaviour of the girders. Because of therelatively high stiffness of the concrete cantilever, relaxation of cables gives rise to relatively highlosses in cable forces but relatively little deck deflections. However, the girder with a hinged end
has relatively little constraints against displacement, in particular the bodily rotation about thehinged end. Relaxation of cables causes relatively large deflections and substantial redistribution ofcable forces, including increase in force in the interior cable.
The dynamic properties of global vibrations at Day 300 are also calculated. The first threefrequencies and their percentage differences from the initial frequencies are shown in Tables 8 and 9
for the cantilever and girder with hinged end respectively. Comparing results of various cases with
the initial results shows that conclusions drawn in Section 3.1 for the natural frequencies are
Table 8 Global dynamic properties of cable-stayed cantilever with two cables at Day 300
Case A

Case B

Case C

Case D

Mode

Initial
freq.
(Hz)

Freq.
(Hz)

Diff.
(%)

Freq.
(Hz)

Diff.
(%)

Freq.
(Hz)

Diff.
(%)

Freq.
(Hz)

Diff.
(%)

1

1.880

1.925

2.42

1.925

2.41

1.926

2.44

1.880

-0.01

2

9.837

10.175

3.44

10.175

3.44

10.179

3.48

9.837

0.00

3

27.12228.086

3.55

28.085

3.55

28.098

3.60

27.1220.00

Table 9 Global dynamic properties of cable-stayed girder with a hinged end and two cables at Day 300
Case A

Case B

Case C

Case D

Mode

Initial
freq.
(Hz)

Freq.
(Hz)

Diff.
(%)

Freq.
(Hz)

Diff.
(%)

Freq.
(Hz)

Diff.
(%)

Freq.
(Hz)

Diff.
(%)

1

1.210

1.211

0.07

1.210

0.01

1.210

0.00

1.209

-0.07

2

6.951

7.186

3.37

7.185

3.37

7.188

3.41

6.951

0.00

3

21.97322.753

3.55

22.753

3.55

22.763

3.60

21.9730.00

Time-dependent effects on dynamic properties of cable-stayed bridges

153

generally valid even for cable-stayed girders with multiple cables acting at high initial prestressing
ratios. In particular, changes in frequencies in Case D show that the effect of cable relaxation is
negligible. The effects of time-dependent behaviour on global dynamic properties of cable-stayed
bridges are essentially through changes in instantaneous stiffness as well as geometric stiffness
resulting from changes in geometry. The time-dependent factors of concrete ageing, creep andshrinkage, and cable relaxation all contribute to minor changes in geometry and hence minor changes
in dynamic properties. A more significant effect is concrete ageing which gradually increases theinstantaneous stiffness of concrete members. An exception to this is the first mode of the cablestayed girder with hinged end, which results largely from rotation of the girder about the hinged
end. One may also note that the instantaneous material stiffness of steel cables remains unchanged
with time, which explains why the effects of cable relaxation on global dynamic properties are
relatively small.
To provide better understanding of the overall dynamic behaviour, the local cable vibrations inCase C of the cable-stayed cantilever with two cables are worked out. Assuming that bare steel
cables are used, the initial frequencies of local vibrations of cables AC and DC are 7.958 Hz and14.677 Hz respectively, which will decrease to 7.529 Hz and 13.994 Hz respectively at Day 300
because of various time-dependent effects. The local cable frequencies tend to be higher than the
global structural frequencies. Moreover, the opposing trends of global structural frequencies and local
cable frequencies can be explained by the fact that the former result from the holistic structural
behaviour while the latter are governed by cable forces.

4. Further verification
To further verify the above numerical findings, an experiment has been conducted to monitor the
long-term development of dynamic properties of a simply supported post-tensioned concrete beamof length 2100 mm, span 2000 mm, breadth 100 mm and depth 150 mm. The concrete had cylinder
strength of 54.2 MPa and Young’s modulus of 24764 MPa at Day 21. An initial prestressing forceof 100 kN was applied at Day 14 by a straight 7-wire super strand of 12.9 mm diameter with 25
mm eccentricity. The preliminary results of the frequency of the first mode in Fig. 10 clearly show
an upward trend, confirming the dominant effect of concrete ageing compared with other factors.

Fig. 10 Variation of frequency of first mode of a post-tensioned beam

154

Francis T.K. Au and X.T. Si

However, the validity of the numerical model presented is only as good as the models for creep andshrinkage of concrete and relaxation of steel tendons. Actually Neville (2004) and Brooks (2005)
have presented experimental results of more than 20 years to show the increase of concrete strengthand modulus of elasticity with time. Therefore it is expected that the presented model will also bevalid for a long time until the structure suffers from damage, corrosion, etc.

5. Conclusions
A systematic and efficient method is proposed to investigate the dynamic properties of cable-stayed
bridges considering the effects of long-term time-dependent behaviour due to concrete ageing, creep
and shrinkage together with any possible cable relaxation. The proposed time integration method
can cope with time-dependent finite element analyses of cable-stayed bridges by proper use of thetime-dependent constitutive model of concrete and the equivalent creep model for cables while
taking account of various geometric nonlinearities. Free vibration analysis for the time of interest
can be carried out by means of subspace iteration method or similar based on the instantaneousmaterial properties, and the updated internal forces and geometry of the bridge then. Numericalexamples are presented to illustrate the application of the proposed method as well as to investigate
the behaviour of typical cable-stayed concrete bridges. Results show that, although geometric
nonlinearities tend to reduce the natural frequencies, the time-dependent behaviour of concrete morethan offsets it and tends to increase the natural frequencies in the long run. Therefore, whether
accounting for the geometric nonlinearities or not, the estimated natural frequencies of such
structures increase gradually with time due to concrete ageing effect alone, its interaction with creep
and shrinkage of concrete, and cable relaxation, or their combined effects. Furthermore, it is foundthat cable relaxation has comparatively little effect on the natural frequencies. These results also
indicate that concrete ageing has the most important influence on the dynamic properties among
various time-varying factors. The interaction between concrete ageing effect and effect of concrete
creep, cable relaxation or their combined effects are generally greater than their individual effects.
Hence the interaction among various time-varying factors should be considered carefully duringlong-term dynamic analyses of concrete cable-stayed bridges. Besides, the effect of time-dependent
behaviour on dynamic properties varies from mode to mode. Therefore, the long-term variations ofdynamic characteristics due to time-dependent behaviour should be investigated in detail in order toensure reliable damage identification in any vibration-based structural health monitoring systems.

Acknowledgements
The work described in this paper has been supported by the Research Grants Council (RGC) of
the Hong Kong Special Administrative Region, China (RGC Project No. HKU 7102/08E).

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