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On the dynamic properties of cable-stayed bridges
On the Dynamic Stayed Bridges T. A. Wyatt, READER, IMPERIAL
Properties
PhD, CEng COLLEGE OF SCIENCE AND TECHNOLOGY
SYNOPSIS In the design of cable-stayed bridges, it is generally necessary to take account of the dynamic properties of the structure. The lowest natural frequency for vertical oscillation is shown to be closely constrained by the normal design criteria. so that the designer can make signijicant change only by altering the overall configuration. By introducing the concept of a design stress level under dead load to provide a standardised design basis, it is possible to compare various configurations; particular attention is paid to the radiating-stay conjguration having a- large number of stays, in comparison with layouts having a small number of stays. Generalised results are presented for these configurations for the stlgness contribution from the stay system. The combination of the stay system with the deck girder iz discussed, and charts are presented for selected geometric proportions, giving values of a convenient general non-dimensional frequency parameter, as a function of the stin;less of the beam relative IOthe stQj%essof the stay system. In all practical cases any deviation from the uni/brm dead load design stress basis is effectively averaged-out over the length of the bridge, either by the deck beam stlflness or by coupling caused by interconnection at the tower. The charts presented thus provide a generally applicable design tool. A brief discussion is included of practical applications.
later
in this
statical
INTRODUCTION
One
of
structional
the
notable
computer
this
conventional principles
steelwork
been the development
in
the
field
over the last twenty-five of the cable-stayed
of
The
majority of bridges in the span range 250 m to 450 m built in recent years have been of this type, and the upper economic limit seems likely to increase. There is a trend to increase the number of separate stay cables, whether in the harp layout or in the radiating configuration which is discussed extensively
leads
With
does
not
design
been published.
to
a high
the
aid
constitute
analysis;
an
The complexity
a
of
digital
obstacle
an extensive
in existing
order
of
to
survey
of
designs”)
has also
of interaction
between
the deck beam and the stay system is presumably. however. the reason why generalised formulae have not hitherto
been
estimates
of overall
dynamic
properties.
presented
to
give
behaviour,
The structural form and the metaphorical
first-approximation
especially
in relation
to
is flexible, in both the rigorous sense. In the deflection/load
sense, the cable-stayed layout is generally less flexible than an orthodox suspension bridge for the same function, forms
but is more flexible
of bridge;
important, of vortex slender
than the other competing
the dynamic
properties
and may be critical
wind-induced
oscillation
shedding examples
be of concern.(J’
must be considered,
or of flutter.‘**” wind
results
for
deformation),
the lowest
in dynamic
associated
with this structural
motion
may also
of this paper is to present
vertical
problems
as a result
With exceptionally
gust-induced
The object
(symmetric
are not sensitive
are therefore
to the design. The risk of
generalised
natural
frequency
and to discuss some
properties
that
are specifically
form. The results shown
to the assumptions
made and will give
good results in application to practical designs. The cable-stayed structural form is also flexible in the sense that a wide range of choice is open to the
con-
years has bridge.
which
and practice
It is helpful to list some major
(i) stays successes
paper,
indeterminacy.
designer.
1
of Cable-
grouped
in
a few
options:
cables/many
separate
stays, (ii) stays in one plane/stays
in two planes,
(iii)
deck beam of high torsional stiffness,
stiffness/low
(iv)
main towers at both ends of span/at
torsional
one end only.
Some examples are presented in Table 1, in association with Figure 1. Whether the stays lie in a single plane along the bridge centre line, or lie in two planes, is immaterial
to
On the dynamic properties of cable-stayed bridges Table
11
!
Figure (a)
(b) (c) (d) (e) (f) (g)
Example
L
Wye Onomichi Luangwa Leverkusen Sara Knie St Nazaire
235 215 223 280 254 320 404
L'
2h
n.L
0.37 0.40 0.18 0.38 0.20 n.a. 0.39
0.26 0.36 0.30 0.33 0.37 0.30" 0.33
0.48 0.58 0.50 0.40 0.64 0.38 0.31
1.50 1.66 0.82 0.65 1.08 0.54 1.34
65 64 62 53 75 56 49
L = main span, m; L' = side span, m; h = tower height, m n~ = calculated lowest vertical natural frequency, Hz n e = calculated lowest torsional frequency, Hz * h/L (single tower layout)
the case of vertical oscillations. Examples of both layouts appear in the table. The simplest configuration may be termed 'singlestay', Figure l(a), in that from each tower the deck is supported at a single point along the main span. Each stay-support can be treated as an independent spring. The logical development of the single-stay layout is the two-stay configuration; layout l(b) is attractive with relatively long side spans in counterpoise with the main span but is then significantly more flexible. The interconnection at the tower top of the stays supporting various points along the beam considerably complicates the analysis; this system is treated briefly in this paper. Layout 1(c) is a stiffer alternative. In the so-called 'harp' configuration, intermediate stays are laid parallel to the longest stays on each side of the tower; they thus intersect the tower at intermediate points, as in Figure l(d). This layout is not advantageous in respect to dynamic behaviour because
,ol
"--
Ib)
I------A
(c)
L_----J-
a-..i
(gl Figure 1
downward 'inertia loading' on the main span will be associated with upward loading on the side spans, bending the tower and giving high overall flexibility. This difficulty is clearly eliminated if the side span can be subdivided with fixed anchor points for the intermediate stays as in Figure l(f). The latter also illustrates a 'single-sided' main span, using only one tower. While very useful on certain sites, the singletower configuration is clearly more flexible than a two-tower configuration for the same span. Owing to the interaction with tower bending, results for the harp layouts are less readily generalised than for the radiating configuration, and they are not treated in this paper. Another layout with improved stiffness is illustrated by Figure l(e). The final example, Figure l(g), typifies the recent development of the radiating configuration, which permits reduction of the stiffness of the deck structure to a very low value. Since the stiffness is now almost
i.f
I
_
A
Cable-stayed bridge configurations
(e)
(f)
L
~--.._I1
i
~1
II
_~.=.~-
I
11_
_
_
I _
12
Journal oJConstructional Steel Research: Vol. 1, ?~b. 1." September 1980
wholly dependent on the stays, the single-tower layout is uncommon. Hybrid configurations, in which the stays converge towards the tower but do not meet at a single point, approach the dynamic properties of the true radiating configuration. Given a consistent basis for proportioning the stay system, it has proved possible to present in chart form a generalised natural frequency parameter for the radiating configuration.
2
C1 :
~zm U ~'EI
3
11
,J" /
M b-~-L
2
09
1
O7
THE SINGLE-STAY SYSTEM
2.1 A simple treatment of beam-stay interaction The single-stay system, as illustrated by Figure 2(a), provides the simplest introduction to the flexural natural modes and frequencies of cable-stayed bridges. The backstays are generally connected to the deck beam in such a manner that the eccentricity from the support pier can be neglected, and the common practical case in which the bridge is symmetric about mid-span can be modelled simply as a beam, generally continuous as shown in Figure 2(b), with elastic supports replacing the stays. Because the deck beam is unsupported over lengths of about one-third of the main span, this type must have a beam of considerable stiffness: the non-dimensional relative stillness parameter S, which is defined by equation 8 below, is typically in the range 0.4 to 0.7. The value of this parameter corresponds very roughly to the ratio of the deflection of the stay system when supporting a load uniformly distributed over the main span (with a deck system distributing the load to the stays as required but not providing overall support) to the deflection of the beam unassisted if subjected to the same load. The combined system is thus typically three times as stiff as the beam acting alone. The limiting cases shown in Figures 2(c) and 2(d) can readily be solved exactly. It is found that although the overall stiffness is dominated by the stay system (as indicated by the value of S), the fundamental mode
(O)~L
Id
?
'
(b) a
L" _l
"a"
a L
,
~
A
.[
:~
"A"
(c)
a
i
0 0
~5 Side sp~n roho CIL 0
Figure3 Frequency and gt'neralised mass parameters for a continuous beam
shape remains remarkably close to the shape of the beam mode. The system generalised mass (related to unit mid-span deflection) is thus almost unchanged, and the strain energy in the stay system can be added to an estimate of the strain energy of the beam based on the beam mode and equated to the kinetic energy similarly based. This gives an equation for the natural frequency of the combined system =
+ _ _
M
= circular natural frequency (rad/s) of the system (.% = circular natural frequency of deck beam without stays M = system generalised mass based on the beam mode k = spring stiffness (vertical deflection per unit load) of stay & = ordinate at the stay attachment of the beam mode shape function.
The term 'beam mode' is applied to the fundamental mode of the beam, taking account of continuity but disconnected from the stays. This equation can be used for a wide range of cases, giving frequency within 1 per cent of the exact value over the likely practical range; the error increases if the beam is very flexible, reaching about 2 per cent at S = 0.2. The beam frequency can be obtained from Figure 3, which gives the factor CI for the equation n~ E l mL ~
i Z ! !i
Figure 2
'L
The single-stay configuration
'-
-4
(1)
in which ~
~o~ = C , (dl
O
(2)
as a function of the side span ratio L ' / L . In this equation, m = mass per unit length of deck L = main span length L' = side span length.
On the dynamic properties of cable-stayed bridges It is not u n c o m m o n for the side spans ( L ' ) to be further continuous with approach spans. If the side span ratio is less than 0.4, this can be treated without significant loss of accuracy by considering the effective length L', (say) that would offer the same static rotational restraint over the main pier; for example, assuming similar values of E1 in the approach spans and the typical case o f approach span length two-thirds of L', Figure 3 should be entered with an effective length of L', = 0.85L'. For the most c o m m o n configurations, 0.25L < L ' < 0.4L, the E1 values at the tower and at the middle of the main span should be given approximately equal weighting if it is necessary to estimate an effective value for a varying-section beam. The cable attachment point ( L " ) is likely to be close to L/3. At L " = L/3, the mode shape function varies little over the practical range of side span ratio, from 0.818 to 0.828 for L ' = 0.25L and L' = 0.4L respectively. A careful sketch based on these values will generally give a sufficiently accurate value of p~ when L" is not exactly equal to L/3. The value C,, = 2M/mL is also shown on Figure 3.
13
Cz t..O
i
4
35
30
\
\
Side spon
25
L
0 hO ~
\
I
t
0 10 0 15 0 20 025 Tower height rotio h/L Figtgre 4 Singh'.stay configuration: thefrequency parameter C, The frequency equation (equation I) thus becomes
2.2 T h e c o n c e p t of dead load design stress The effective vertical stiffness of the single stay can, of course, readily be evaluated given the geometry of the system, and the relevant modulae of elasticity and cross-sectional areas. If the axial strains of the deck and the tower are neglected, A"
+
~t';~
(3)
in which E~ = effective Young's modulus of the stays ~t,A = stay length and cross-sectional area and primes identify backstay and forestay, as applied to L', L". However, the cross-sectional areas should not be regarded as independent variables. The live load on major bridges is not large compared to the dead load, and stays are generally proportioned to support the dead load from their 'contributing length' at a stress (fo, say) which is uniform within quite close limits. In the single-stay system, the 'contributing length' is ½(L - L"), giving a vertical load to each forestay of ½mg(L - L"). Choosing A", A' such that this load causes stress fr,, on substitution into equation 3 and with some rearrangement,
k = C 2 Es mgh
(4)
3./'9 L with
C2=
3LL'(L-L") 2 ( L ' + L " ) ( L ' L" + h 2)
(5)
l+--
L'/
C
2 -
-
3M
J~, L 2
E! / E~ gh S=--/ tnL 4 f l, L2
?5
+
which is shown graphically on Figure 4.
(6)
(7)
(8)
and 2,!zLp~ C3 - - 3M
(9)
the frequency equation reduces to
E~ gh
¢o2 = ( c I S + C 2 C 3 ) fl, L2
(10)
The parameter Cj can be taken as unity if L" = L/3, within the normal range 0.25 < L ' / L < 0.4. To allow for strain in the deck girder due to axial loads, a correction equal to f ^ E/Es should be added to the stay stress to arrive at fo, where fA is the stress in the deck girder due to the axial load. This is typically about 5 per cent on fo, and thus - 2 per cent on frequency, for single-stay systems. To illustrate the application of the above formula, consider E s = 150 k N / m m 2, fo = 450 N / m m 2, L' = L" = L/3, h = L/8; C I = 2.9 (Figure 3) and C 2 = 4.0 (Figure 4). Converting equation 10 from ~o(rad/s) to n(Hz) gives n = 6.4(0.73S + I)t/2/Ll'2
For L " = L/3, this reduces to C2=I
2mLp~ E~gh +
Introducing the notations
[ ,3 L,,Z)
( d ,,3 k = h 2E,
n"t E1 ~o2 = C 1- mL 4
(1 I)
in which L is measured in metres. For S = 0.6 and L = 235 m, which correspond approximately to the Wye Bridge, n is thus 0.50 Hz. The deck beam of the Wye Bridge is in fact continuous into approach spans, which
14 Journal of Constructional Steel Research: Vol. 1, No. 1: September 1980 would increase C~ to 3.2, and increase the frequency by 1.3 per cent. The observed fundamental vertical natural frequency of the Wye Bridge is 0.46 Hz. ~5~The relative insensitivity of the frequency to deck beam stiffness (i.e. the effect of S) will be noted; the deck beam effect raises the frequency by only 20 per cent in this example. Correspondingly about 70 per cent of the energy of motion is taken up by the stays, so that stay axial force-extension hysteresis can provide a very favourable overall value of natural damping.
2.3
E s gh co2= 2 . 3 - -
Higher modes of oscillation
In contrast to the parabolic-cable suspension bridge, the lowest frequency for motion antisymmetric about mid-span will always be higher than the frequency of the symmetric mode discussed above. The antisymmetric modes are thus of very limited practical significance. The second symmetric mode of a single-stay system may be of some interest, however, .as nodes of this mode may fie close to the stay attachment points. The stays thus make tittle contribution to the stiffness, and perhaps more important, can make little contribution to the natural damping. For the case of L' = L" = L/3, without deck beam continuity with approach spans,. this mode is clearly exactly sinusoidal with three half-waves in the main span and no cable participation at all. Thus C~ = 8 I, giving a natural frequency of 9n~(3 + C J S ) v k i.e. ty.pically about three times the frequency of the fundamental symmetric mode. 3
The solution for the stay system, corresponding to the C 2 term in equation 10, is nevertheless of interest. The specific case L' = 2L/5 and h = L/6 is shown on Figure 5. If mass mL/5 is associated with each stay and the beam is otherwise ignored, the common connection to the tower provides a coupling between the motions of the masses. The lowest natural frequency and the corresponding shape can be found by the usual methods. If the stays are proportioned on the dead load stress principle discussed above, the lowest frequency for this case is given by
THE TWO-STAYS CONFIGURATION
The direct extension of the single-stay system is illustrated by Figure l(b). The weight of the side span is now mobilised as a counterpoise to the main span. However, it is clear that in the fundamental vertical mode of oscillation, when the main span is deflected downwards, the side span deflection will be upwards. The 'inertia loading' follows the deflection, and the main span and side span loadings are cumulative in their effects, not self-cancelling. The side span stay only assists the back stay in restraining movement of the tower to the extent that the deck beam in the side span is sufficiently stiff to resist the sum of the side span inertia loading and the dynamic stay tension. It is apparent that the two-stay configuration will give a lower natural frequency than a single-stay configuration designed on the same basis. This will be particularly significant where the side spans are long; the configuration illustrated by Figure I(b) is clearly likely to suit side spans about two-fifths of the main span length. It is also apparent that interaction between beam and stay behaviour is much more complex than in the single-stay system. It has not proved possible to produce a simple general approach on the lines of equation 1.
(12)
foL:
This can be refined on the assumption that the continuous deflected form of the deck will correspond to the static deformation shape fitted to the required stay displacements, and that the forces applied to the
&
A
unst,lfened ~ staYhapde I"igure 5
A
-.~//-
beammodeshape
"l'ht" t w o xla_v COtlJigur~ltio/t
stays are statically equivalent to the corresponding distributed inertia loading. This leads to
E s gh off = 2.5 - -
./~, L 2
(13)
and the mode shape shown on Figure 5. The difference from the beam mode shape is clear. The final results including the elastic beam interaction for this specific case have been added to Figure 10. 4 THE R A D I A T I N G - S T A Y S CONFIGURATION 4.1 Stiffness of the stay s y s t e m In the radiating-stay configuration, the deck beam is supported at close intervals, and it is accordingly possible greatly to reduce the beam stiffness. Values of the relative stiffness parameter S as low as 0.02 have been reached in practical designs. At the same time, there is now no difficulty in visualisation of the natural mode of the stay system independent of beam stiffening; by treating the stays as a continuously distributed support system, the assumptions which were necessary to produce equation 13 now become immaterial. The dead load design stress principle can be applied as before to proportion the stay system. The stays radiating to the side span will clearly be equal to, and in
On the dynamic properties of cable-stayed bridges dead load balance with, the stays in the main span up to a distance L' from the tower. The direct restraint of the tower is provided by the backstay, which is aligned to intersect the beam at the abutment. The backstay is assumed to be proportioned to resist the remaining part of the main span weight, that is the weight of length (½L - L ' ) adjoining the mid-span point. The stiffness of the system is thus notably sensitive to the side span ratio, especially if L' approaches L/2. An analytic solution for the natural frequency and mode shape can now be obtained. Consider an element dx of the deck, supported by a stay element of cross-sectional area dA, as shown in Figure 6. For oscillation at circular frequency co,
x/h
u=
.0z ~Z/h5 in which .0z = hf D co5/Esg. i
u
15
20 ~ j z f o Lz
Egh
Slde span ratio
L
__
~ :o ~o-~
!
tO
o 4.
,
0 15
i
,
,
I
,
0 20
,
,
,
25
Tower height ratio h/L
Figure 7 Unstiffened parameter C,
radiating-stay configuration." frequency
(14)
-
i
The resulting horizontal force acting at the tower can readily be evaluated, and integrated to give the net
C~ =
20 (u z fo Lz
Egh Tower top dlspt u
7|
h |
~
stoy te.gthC
t
~area
10
hf o
it ~
displacement v 0'.']
Figure 6 A typical element of the radiating.stay system
dynamic load on the backstay. The compatibility of the backstay extension with the tower-top movement u implies a condition on .0 5 which can be shown to be equivalent to solving the following equation for A :
--X)(A-L in w h i c h X = - 2h'
- - x ( X + Y+ Z)=O L'
(15)
y z ( x z _ yz)
Y=--, h
Z-
2(1 + yz)3,z
and
so that continuation of the existing notation
E s gh
~5 = C4 ~
fD L5
(16)
requires 4X 5
C4 = - I+A 5
0/.
05
Side ,,pan r a h o t~l L
(17)
Charts giving C, as a function of the side span ratio L'/L and of the tower height h/L are presented as Figures 7 and 8.
Figure 8 UnstiJfened radiating-stay cor~iguration: frequency parameter C~
4.2 I n t e r a c t i o n b e t w e e n the d e c k b e a m and t h e stay s y s t e m An example of the mode shape for the unstiffened stay system is given on Figure 9. The disparity from the beam mode is even more pronounced than in the two-stays configuration. The action of the actual, probably rather flexible, beam is basically to eliminate the discontinuities at the abutment and at mid-span; that at the abutment proves to be considerably more important than the other, associating some of the side-span stays in the stiffening role of the backstay. An approximation which is valid for very flexible decks can be obtained by considering the deck as a beam on an elastic foundation (i.e. neglecting the coupling effect of tower movement on stay forces). The 'foundation stiffness', i.e. the vertical force per unit length per unit displacement, is k' = E s mgh/fo/z, so that from the definition of S (equation 8), the length parameter 2 = (4El/k') '/4 = (I/~z)V/(2LI)S TM. This approach may therefore be useful if S < (L'/2L) 2. The additional strain energy arising from the beam end shear and mid-span moments that will eliminate the discontinuities can then be added to the strain energy of
16
Journal of Constructional Steel Research: Vol. 1. ,Vo. 1: September 1980 actuo( mode s : 0 0 2 ~ (to reduced scale)
mode • :o
o~,~ot
beom mode
i
removal of the vertical support of the deck on the tower pier, would have very little effect on the dynamic properties.
oz@r"--~_-'"
_ _ /
5
GENERAL
DISCUSSION
Figure 9 ,*,lodeshapefor the radiating-stay configuration
the stay system in the unstiffened deflected form. Writing the frequency equation as w ' - = (1 + C ~ S V * ) C ~
E s gh
(18)
foL z
the correction parameter can be approximated in 0.25L < L' < 0.4L by C5 = 15(L'/L) 2
(19)
For larger values of S, the beam-stay interaction modifies the mode shape to a degree that requires specific solution for each combination. The frequency solutions as a function of S for two values of the geometric parameters are given on Figure 10, These curves should, of course, be used in preference to the approximation given by equations 18 and 19. Judicious interpretation of the trends of variation of frequency with the geometric paramctcrs (Figures 7 and 8), taking account of this approximation, will give a good cstimate of the lowest natural frequency over the range of practical design cases. The discontinuity of the unstiffened mode shape at the abutment indicates that the stresses resulting from dynamic response may be most severe at that point, at least in cases with long side spans. This is caused by the direction of the inertia loading such that main span and side span loads have cumulative effect in distinction to the counterpoised action of gravity loads. The tower is not a critical point in such cases, and the mode shape indicates that "floating' the deck from the stays, i.e.
(,J "fo L" Egh
6
T SI"~GLE"STAY
I
'r.2L
h-
I
T
"~,~..,I"~'~_.../-j~,, ~'~"
L
I T,.o-s,.¥
ol
0
__.-
",
I X
~
,'
Ol
;
i
;
I
RAO,ArJNG STAY
"
"k'
,I
L
L
,i
iI
;•
02 03 OL 05 Relahve stiffness pGrqmet~'r S
Figure I0 I~e.Y(I)'t!ehart f ) r selected geometric ratios
1
06
The foregoing sections have presented methods of estimating the fundamental natural frequency for vertical oscillation. It is apparent that the solution is closely fixed by the basic geometry and the normal and economic values adopted for the working stress in the stays. Within normal economic constraints, little freedom of manoeuvre is left to the designer. The frequencies are low, although not as low as a parabolic-cable suspension bridge. Consider the example of 450 m span. The first antisymmetric mode of an unstiffened parabolic-cable bridge having dip equal to one-tenth of the span would have a natural frequency of 0.165 Hz. The deck of a suspension bridge is, of course, effectively continuously supported, and for very long spans a modcst stiffening girder suffices which does not greatly augment the fundamental vertical natural frequency, tlowcvcr, for this span we may consider a stiffening girder having E1 equivalent to about S = 0.1 on the cable-stayed alternatives. This would raise the antisymmetriomode frequency to 0.28 tlz; the lowest symmetric mode frequency would bc similar, tile actual value dcpcnding on the side span ratio and beam end conditions. Consider now cable-stayed designs based on E~ = 150 k N / m m 2 and f~) = 400 N / m m 2. The single-stay configuration (admittedly unlikely to be used for such a span) with S = 0.4, L' = L/3 and h = L/6, would have a natural frequency of 0.38 flz. A radiating-stay design, making full use of that form to reduce the stiffening beam to (say) S = 0.05 would have frequency 0,30 Hz, reducing to 0.25 Hz if the side spans were extended to L' = 2L/5. The frequency formulae indicate that for given values of j~), S and h/L the natural frequency would be inversely proportional to the square root of the span. In practice, the required value of S is likely to decrease with span, and f o may increase, due mainly to the decreasing relative importance of live load. The reduction of frequency as the span increases would thereby bc increased. For the same reason, bridges with concrete or composite decks will also tend to have smaller S values and possibly higher f o , and thus slightly lower frequencies. The low frequencies, combined with shallow deck beams, give low values of critical speed for vortex shedding resonance, typically less than 10 m/s and possibly as low as 4 m/s. At these speeds, the possibility of low-turbulence wind structure must be considered, especially at smooth-terrain sites, and response may approach the predictions made from smooth-flow wind tunnel tests.
On the dynamic properties of cable-stayed bridges On the other hand, when strands of twisted construction are used (spiral or locked-coil strand or rope) the damping can be expected to be much larger than for otherwise similar non-stayed construction, provided that the amplitudes are sufficient to overcome the limiting inter-wire friction in the stays. Although understanding of this phenomenon is increasing, it remains difficult to predict. The opinion is strongly held in some quarters ~6~that radiating-stay bridges cannot show significant resonant magnification in response to excitation such as vortex shedding, by virtue of'system damping'. It is postulated that the radiating-stay system is subject to important non-linearities, including the effects of transverse motion and associated change of sag of the individual stays. Individual stay resonances may therefore change their effective stiffness and thus 'de-tune' the system; the number of such possibilities is held to be sufficiently large that a sustained resonance would be impossible. While this may be true in some cases, tower-top movement has been shown above to exert a strong coupling on stay motions, especially in the case of long side spans where the backstay has relatively low stiffness. The individual stay resonances are also certain to be at substantially higher frequency than the predicted fundamental bridge frequency in the case of major bridges where the dead load tension fD is high; for example, with fD = 400 N / m m 2 and the longest stay 250 m (of. the cases discussed above), the lowest individual stay frequency would be 0.45 Hz. The author recommends that system damping cannot be relied upon for structures such as the examples discussed in this paper. Other non-linearities may certainly be worthy of investigation, notably in the stress-strain behaviour of the stays. 'Second order', or 'PA', effects, including the effect of the axial compressive load in the deck beam, are however generally negligible in vertical oscillations. Torsional oscillation has not been discussed in this paper, although of comparable practical importance. Torsional excitation by vortex shedding is commonly reported in smooth-flow wind-tunnel tests, but the author is not aware of any report of significant torsional response of a cable-stayed bridge in practice. Where the torsional frequency is much higher than the vertical frequency, the reduced possibility of lowturbulence conditions at the critical wind speed may be significant. The torsional natural frequency is dominant in respect to the risk of potentially catastrophic windinduced oscillation, which is commonly regarded as related to airfoil flutter. German design practice has commonly included a check relating the site design wind speed to the calculated airfoil flutter speed modified by a cross-section aerodynamic efficiency factor, as first suggested by Selberg. The United Kingdom may follow this example. A high flutter speed is ensured by a high torsional frequency in conjunction
17
with a substantial difference between torsional and vertical frequencies having a similar mode shape. This is readily ensured by a box-form deck beam: the relative stiffness parameter for a box beam of substantial enclosed area, defined in analogy with the flexural parameter S, typically substantially exceeds unity. In such cases the flutter criterion can generally readily be met. This remains true even when the stays are placed in a single plane along the longitudinal centre line and thus provide no resistance to torsion. When the deck does not comprise a torsion box, the stays must be in two planes, generally coincident with the deck stiffening members. If the stay planes are vertical, and the tower does not provide much resistance to racking between the two planes, the torsional frequency can be estimated from co2
4r z B2
~
W v
(20)
in which r = mass radius of gyration of the deck about the longitudinal centre line B = spacing between stay planes w, = fundamental vertical natural frequency. The deck beam resistance is here treated as differential bending only. Due to the axial forces in the deck induced by the dynamic changes in the stay tensions, some horizontal motion will be coupled with the torsional mode. The torsional frequency in the absence of a torsion box can be significantly increased by inclining the stay phmes to meet at the top of an A-frame tower. Symmetry clearly imposes the condition that spanwise displacement of the tower top is eliminated from the torsional mode, substantially increasing the stiffness. It is hoped to examine this arrangement in detail, together with an examination of the coupling and non-linear effects that have been noted, in a second paper.
REFERENCES 1
2
TROITSKY, M. S. Cable-Slayed Bridges: Theory and Design. Crosby Lockwood, London, 1977. WYATI', T. A. "The effect of wind on long-span bridges.' Symposium on Wind Effects on Buildings and Structures, Loughborough, 1968.
3 Interim Design arrd Workmanship Rules for Steel Box Girder Bridges. Department of the Environment, Section 4. I 1, London, 4
5
6
1973. MELBOURNE, W. H. 'Model and full-scale response to wind action of the cable-stayed box girder West Gate Bridge.' I A H R / I U T A M Symposium on Practical Experiences with Flow Induced Vibrations, Karlsruhe, 1979. EYRE, R. and sMrrH, t. J. 'Wind- and traffic-induced dynamic behaviour of some steel box girder bridges." Symposium on Dynamic Behaviour of Bridges, T R R L , 1977. ( T R R L Supplementary Report 275.) LEONEIARDT, F. 'Latest developments of cable-stayed bridges for long spans.' Bygningsstatiske Meddelelser. 45, 4, Copenhagen, 1974.
Contributions discussing this paper should be reeelved by the Editor before I January 1981.
Properties
PhD, CEng COLLEGE OF SCIENCE AND TECHNOLOGY
SYNOPSIS In the design of cable-stayed bridges, it is generally necessary to take account of the dynamic properties of the structure. The lowest natural frequency for vertical oscillation is shown to be closely constrained by the normal design criteria. so that the designer can make signijicant change only by altering the overall configuration. By introducing the concept of a design stress level under dead load to provide a standardised design basis, it is possible to compare various configurations; particular attention is paid to the radiating-stay conjguration having a- large number of stays, in comparison with layouts having a small number of stays. Generalised results are presented for these configurations for the stlgness contribution from the stay system. The combination of the stay system with the deck girder iz discussed, and charts are presented for selected geometric proportions, giving values of a convenient general non-dimensional frequency parameter, as a function of the stin;less of the beam relative IOthe stQj%essof the stay system. In all practical cases any deviation from the uni/brm dead load design stress basis is effectively averaged-out over the length of the bridge, either by the deck beam stlflness or by coupling caused by interconnection at the tower. The charts presented thus provide a generally applicable design tool. A brief discussion is included of practical applications.
later
in this
statical
INTRODUCTION
One
of
structional
the
notable
computer
this
conventional principles
steelwork
been the development
in
the
field
over the last twenty-five of the cable-stayed
of
The
majority of bridges in the span range 250 m to 450 m built in recent years have been of this type, and the upper economic limit seems likely to increase. There is a trend to increase the number of separate stay cables, whether in the harp layout or in the radiating configuration which is discussed extensively
leads
With
does
not
design
been published.
to
a high
the
aid
constitute
analysis;
an
The complexity
a
of
digital
obstacle
an extensive
in existing
order
of
to
survey
of
designs”)
has also
of interaction
between
the deck beam and the stay system is presumably. however. the reason why generalised formulae have not hitherto
been
estimates
of overall
dynamic
properties.
presented
to
give
behaviour,
The structural form and the metaphorical
first-approximation
especially
in relation
to
is flexible, in both the rigorous sense. In the deflection/load
sense, the cable-stayed layout is generally less flexible than an orthodox suspension bridge for the same function, forms
but is more flexible
of bridge;
important, of vortex slender
than the other competing
the dynamic
properties
and may be critical
wind-induced
oscillation
shedding examples
be of concern.(J’
must be considered,
or of flutter.‘**” wind
results
for
deformation),
the lowest
in dynamic
associated
with this structural
motion
may also
of this paper is to present
vertical
problems
as a result
With exceptionally
gust-induced
The object
(symmetric
are not sensitive
are therefore
to the design. The risk of
generalised
natural
frequency
and to discuss some
properties
that
are specifically
form. The results shown
to the assumptions
made and will give
good results in application to practical designs. The cable-stayed structural form is also flexible in the sense that a wide range of choice is open to the
con-
years has bridge.
which
and practice
It is helpful to list some major
(i) stays successes
paper,
indeterminacy.
designer.
1
of Cable-
grouped
in
a few
options:
cables/many
separate
stays, (ii) stays in one plane/stays
in two planes,
(iii)
deck beam of high torsional stiffness,
stiffness/low
(iv)
main towers at both ends of span/at
torsional
one end only.
Some examples are presented in Table 1, in association with Figure 1. Whether the stays lie in a single plane along the bridge centre line, or lie in two planes, is immaterial
to
On the dynamic properties of cable-stayed bridges Table
11
!
Figure (a)
(b) (c) (d) (e) (f) (g)
Example
L
Wye Onomichi Luangwa Leverkusen Sara Knie St Nazaire
235 215 223 280 254 320 404
L'
2h
n.L
0.37 0.40 0.18 0.38 0.20 n.a. 0.39
0.26 0.36 0.30 0.33 0.37 0.30" 0.33
0.48 0.58 0.50 0.40 0.64 0.38 0.31
1.50 1.66 0.82 0.65 1.08 0.54 1.34
65 64 62 53 75 56 49
L = main span, m; L' = side span, m; h = tower height, m n~ = calculated lowest vertical natural frequency, Hz n e = calculated lowest torsional frequency, Hz * h/L (single tower layout)
the case of vertical oscillations. Examples of both layouts appear in the table. The simplest configuration may be termed 'singlestay', Figure l(a), in that from each tower the deck is supported at a single point along the main span. Each stay-support can be treated as an independent spring. The logical development of the single-stay layout is the two-stay configuration; layout l(b) is attractive with relatively long side spans in counterpoise with the main span but is then significantly more flexible. The interconnection at the tower top of the stays supporting various points along the beam considerably complicates the analysis; this system is treated briefly in this paper. Layout 1(c) is a stiffer alternative. In the so-called 'harp' configuration, intermediate stays are laid parallel to the longest stays on each side of the tower; they thus intersect the tower at intermediate points, as in Figure l(d). This layout is not advantageous in respect to dynamic behaviour because
,ol
"--
Ib)
I------A
(c)
L_----J-
a-..i
(gl Figure 1
downward 'inertia loading' on the main span will be associated with upward loading on the side spans, bending the tower and giving high overall flexibility. This difficulty is clearly eliminated if the side span can be subdivided with fixed anchor points for the intermediate stays as in Figure l(f). The latter also illustrates a 'single-sided' main span, using only one tower. While very useful on certain sites, the singletower configuration is clearly more flexible than a two-tower configuration for the same span. Owing to the interaction with tower bending, results for the harp layouts are less readily generalised than for the radiating configuration, and they are not treated in this paper. Another layout with improved stiffness is illustrated by Figure l(e). The final example, Figure l(g), typifies the recent development of the radiating configuration, which permits reduction of the stiffness of the deck structure to a very low value. Since the stiffness is now almost
i.f
I
_
A
Cable-stayed bridge configurations
(e)
(f)
L
~--.._I1
i
~1
II
_~.=.~-
I
11_
_
_
I _
12
Journal oJConstructional Steel Research: Vol. 1, ?~b. 1." September 1980
wholly dependent on the stays, the single-tower layout is uncommon. Hybrid configurations, in which the stays converge towards the tower but do not meet at a single point, approach the dynamic properties of the true radiating configuration. Given a consistent basis for proportioning the stay system, it has proved possible to present in chart form a generalised natural frequency parameter for the radiating configuration.
2
C1 :
~zm U ~'EI
3
11
,J" /
M b-~-L
2
09
1
O7
THE SINGLE-STAY SYSTEM
2.1 A simple treatment of beam-stay interaction The single-stay system, as illustrated by Figure 2(a), provides the simplest introduction to the flexural natural modes and frequencies of cable-stayed bridges. The backstays are generally connected to the deck beam in such a manner that the eccentricity from the support pier can be neglected, and the common practical case in which the bridge is symmetric about mid-span can be modelled simply as a beam, generally continuous as shown in Figure 2(b), with elastic supports replacing the stays. Because the deck beam is unsupported over lengths of about one-third of the main span, this type must have a beam of considerable stiffness: the non-dimensional relative stillness parameter S, which is defined by equation 8 below, is typically in the range 0.4 to 0.7. The value of this parameter corresponds very roughly to the ratio of the deflection of the stay system when supporting a load uniformly distributed over the main span (with a deck system distributing the load to the stays as required but not providing overall support) to the deflection of the beam unassisted if subjected to the same load. The combined system is thus typically three times as stiff as the beam acting alone. The limiting cases shown in Figures 2(c) and 2(d) can readily be solved exactly. It is found that although the overall stiffness is dominated by the stay system (as indicated by the value of S), the fundamental mode
(O)~L
Id
?
'
(b) a
L" _l
"a"
a L
,
~
A
.[
:~
"A"
(c)
a
i
0 0
~5 Side sp~n roho CIL 0
Figure3 Frequency and gt'neralised mass parameters for a continuous beam
shape remains remarkably close to the shape of the beam mode. The system generalised mass (related to unit mid-span deflection) is thus almost unchanged, and the strain energy in the stay system can be added to an estimate of the strain energy of the beam based on the beam mode and equated to the kinetic energy similarly based. This gives an equation for the natural frequency of the combined system =
+ _ _
M
= circular natural frequency (rad/s) of the system (.% = circular natural frequency of deck beam without stays M = system generalised mass based on the beam mode k = spring stiffness (vertical deflection per unit load) of stay & = ordinate at the stay attachment of the beam mode shape function.
The term 'beam mode' is applied to the fundamental mode of the beam, taking account of continuity but disconnected from the stays. This equation can be used for a wide range of cases, giving frequency within 1 per cent of the exact value over the likely practical range; the error increases if the beam is very flexible, reaching about 2 per cent at S = 0.2. The beam frequency can be obtained from Figure 3, which gives the factor CI for the equation n~ E l mL ~
i Z ! !i
Figure 2
'L
The single-stay configuration
'-
-4
(1)
in which ~
~o~ = C , (dl
O
(2)
as a function of the side span ratio L ' / L . In this equation, m = mass per unit length of deck L = main span length L' = side span length.
On the dynamic properties of cable-stayed bridges It is not u n c o m m o n for the side spans ( L ' ) to be further continuous with approach spans. If the side span ratio is less than 0.4, this can be treated without significant loss of accuracy by considering the effective length L', (say) that would offer the same static rotational restraint over the main pier; for example, assuming similar values of E1 in the approach spans and the typical case o f approach span length two-thirds of L', Figure 3 should be entered with an effective length of L', = 0.85L'. For the most c o m m o n configurations, 0.25L < L ' < 0.4L, the E1 values at the tower and at the middle of the main span should be given approximately equal weighting if it is necessary to estimate an effective value for a varying-section beam. The cable attachment point ( L " ) is likely to be close to L/3. At L " = L/3, the mode shape function varies little over the practical range of side span ratio, from 0.818 to 0.828 for L ' = 0.25L and L' = 0.4L respectively. A careful sketch based on these values will generally give a sufficiently accurate value of p~ when L" is not exactly equal to L/3. The value C,, = 2M/mL is also shown on Figure 3.
13
Cz t..O
i
4
35
30
\
\
Side spon
25
L
0 hO ~
\
I
t
0 10 0 15 0 20 025 Tower height rotio h/L Figtgre 4 Singh'.stay configuration: thefrequency parameter C, The frequency equation (equation I) thus becomes
2.2 T h e c o n c e p t of dead load design stress The effective vertical stiffness of the single stay can, of course, readily be evaluated given the geometry of the system, and the relevant modulae of elasticity and cross-sectional areas. If the axial strains of the deck and the tower are neglected, A"
+
~t';~
(3)
in which E~ = effective Young's modulus of the stays ~t,A = stay length and cross-sectional area and primes identify backstay and forestay, as applied to L', L". However, the cross-sectional areas should not be regarded as independent variables. The live load on major bridges is not large compared to the dead load, and stays are generally proportioned to support the dead load from their 'contributing length' at a stress (fo, say) which is uniform within quite close limits. In the single-stay system, the 'contributing length' is ½(L - L"), giving a vertical load to each forestay of ½mg(L - L"). Choosing A", A' such that this load causes stress fr,, on substitution into equation 3 and with some rearrangement,
k = C 2 Es mgh
(4)
3./'9 L with
C2=
3LL'(L-L") 2 ( L ' + L " ) ( L ' L" + h 2)
(5)
l+--
L'/
C
2 -
-
3M
J~, L 2
E! / E~ gh S=--/ tnL 4 f l, L2
?5
+
which is shown graphically on Figure 4.
(6)
(7)
(8)
and 2,!zLp~ C3 - - 3M
(9)
the frequency equation reduces to
E~ gh
¢o2 = ( c I S + C 2 C 3 ) fl, L2
(10)
The parameter Cj can be taken as unity if L" = L/3, within the normal range 0.25 < L ' / L < 0.4. To allow for strain in the deck girder due to axial loads, a correction equal to f ^ E/Es should be added to the stay stress to arrive at fo, where fA is the stress in the deck girder due to the axial load. This is typically about 5 per cent on fo, and thus - 2 per cent on frequency, for single-stay systems. To illustrate the application of the above formula, consider E s = 150 k N / m m 2, fo = 450 N / m m 2, L' = L" = L/3, h = L/8; C I = 2.9 (Figure 3) and C 2 = 4.0 (Figure 4). Converting equation 10 from ~o(rad/s) to n(Hz) gives n = 6.4(0.73S + I)t/2/Ll'2
For L " = L/3, this reduces to C2=I
2mLp~ E~gh +
Introducing the notations
[ ,3 L,,Z)
( d ,,3 k = h 2E,
n"t E1 ~o2 = C 1- mL 4
(1 I)
in which L is measured in metres. For S = 0.6 and L = 235 m, which correspond approximately to the Wye Bridge, n is thus 0.50 Hz. The deck beam of the Wye Bridge is in fact continuous into approach spans, which
14 Journal of Constructional Steel Research: Vol. 1, No. 1: September 1980 would increase C~ to 3.2, and increase the frequency by 1.3 per cent. The observed fundamental vertical natural frequency of the Wye Bridge is 0.46 Hz. ~5~The relative insensitivity of the frequency to deck beam stiffness (i.e. the effect of S) will be noted; the deck beam effect raises the frequency by only 20 per cent in this example. Correspondingly about 70 per cent of the energy of motion is taken up by the stays, so that stay axial force-extension hysteresis can provide a very favourable overall value of natural damping.
2.3
E s gh co2= 2 . 3 - -
Higher modes of oscillation
In contrast to the parabolic-cable suspension bridge, the lowest frequency for motion antisymmetric about mid-span will always be higher than the frequency of the symmetric mode discussed above. The antisymmetric modes are thus of very limited practical significance. The second symmetric mode of a single-stay system may be of some interest, however, .as nodes of this mode may fie close to the stay attachment points. The stays thus make tittle contribution to the stiffness, and perhaps more important, can make little contribution to the natural damping. For the case of L' = L" = L/3, without deck beam continuity with approach spans,. this mode is clearly exactly sinusoidal with three half-waves in the main span and no cable participation at all. Thus C~ = 8 I, giving a natural frequency of 9n~(3 + C J S ) v k i.e. ty.pically about three times the frequency of the fundamental symmetric mode. 3
The solution for the stay system, corresponding to the C 2 term in equation 10, is nevertheless of interest. The specific case L' = 2L/5 and h = L/6 is shown on Figure 5. If mass mL/5 is associated with each stay and the beam is otherwise ignored, the common connection to the tower provides a coupling between the motions of the masses. The lowest natural frequency and the corresponding shape can be found by the usual methods. If the stays are proportioned on the dead load stress principle discussed above, the lowest frequency for this case is given by
THE TWO-STAYS CONFIGURATION
The direct extension of the single-stay system is illustrated by Figure l(b). The weight of the side span is now mobilised as a counterpoise to the main span. However, it is clear that in the fundamental vertical mode of oscillation, when the main span is deflected downwards, the side span deflection will be upwards. The 'inertia loading' follows the deflection, and the main span and side span loadings are cumulative in their effects, not self-cancelling. The side span stay only assists the back stay in restraining movement of the tower to the extent that the deck beam in the side span is sufficiently stiff to resist the sum of the side span inertia loading and the dynamic stay tension. It is apparent that the two-stay configuration will give a lower natural frequency than a single-stay configuration designed on the same basis. This will be particularly significant where the side spans are long; the configuration illustrated by Figure I(b) is clearly likely to suit side spans about two-fifths of the main span length. It is also apparent that interaction between beam and stay behaviour is much more complex than in the single-stay system. It has not proved possible to produce a simple general approach on the lines of equation 1.
(12)
foL:
This can be refined on the assumption that the continuous deflected form of the deck will correspond to the static deformation shape fitted to the required stay displacements, and that the forces applied to the
&
A
unst,lfened ~ staYhapde I"igure 5
A
-.~//-
beammodeshape
"l'ht" t w o xla_v COtlJigur~ltio/t
stays are statically equivalent to the corresponding distributed inertia loading. This leads to
E s gh off = 2.5 - -
./~, L 2
(13)
and the mode shape shown on Figure 5. The difference from the beam mode shape is clear. The final results including the elastic beam interaction for this specific case have been added to Figure 10. 4 THE R A D I A T I N G - S T A Y S CONFIGURATION 4.1 Stiffness of the stay s y s t e m In the radiating-stay configuration, the deck beam is supported at close intervals, and it is accordingly possible greatly to reduce the beam stiffness. Values of the relative stiffness parameter S as low as 0.02 have been reached in practical designs. At the same time, there is now no difficulty in visualisation of the natural mode of the stay system independent of beam stiffening; by treating the stays as a continuously distributed support system, the assumptions which were necessary to produce equation 13 now become immaterial. The dead load design stress principle can be applied as before to proportion the stay system. The stays radiating to the side span will clearly be equal to, and in
On the dynamic properties of cable-stayed bridges dead load balance with, the stays in the main span up to a distance L' from the tower. The direct restraint of the tower is provided by the backstay, which is aligned to intersect the beam at the abutment. The backstay is assumed to be proportioned to resist the remaining part of the main span weight, that is the weight of length (½L - L ' ) adjoining the mid-span point. The stiffness of the system is thus notably sensitive to the side span ratio, especially if L' approaches L/2. An analytic solution for the natural frequency and mode shape can now be obtained. Consider an element dx of the deck, supported by a stay element of cross-sectional area dA, as shown in Figure 6. For oscillation at circular frequency co,
x/h
u=
.0z ~Z/h5 in which .0z = hf D co5/Esg. i
u
15
20 ~ j z f o Lz
Egh
Slde span ratio
L
__
~ :o ~o-~
!
tO
o 4.
,
0 15
i
,
,
I
,
0 20
,
,
,
25
Tower height ratio h/L
Figure 7 Unstiffened parameter C,
radiating-stay configuration." frequency
(14)
-
i
The resulting horizontal force acting at the tower can readily be evaluated, and integrated to give the net
C~ =
20 (u z fo Lz
Egh Tower top dlspt u
7|
h |
~
stoy te.gthC
t
~area
10
hf o
it ~
displacement v 0'.']
Figure 6 A typical element of the radiating.stay system
dynamic load on the backstay. The compatibility of the backstay extension with the tower-top movement u implies a condition on .0 5 which can be shown to be equivalent to solving the following equation for A :
--X)(A-L in w h i c h X = - 2h'
- - x ( X + Y+ Z)=O L'
(15)
y z ( x z _ yz)
Y=--, h
Z-
2(1 + yz)3,z
and
so that continuation of the existing notation
E s gh
~5 = C4 ~
fD L5
(16)
requires 4X 5
C4 = - I+A 5
0/.
05
Side ,,pan r a h o t~l L
(17)
Charts giving C, as a function of the side span ratio L'/L and of the tower height h/L are presented as Figures 7 and 8.
Figure 8 UnstiJfened radiating-stay cor~iguration: frequency parameter C~
4.2 I n t e r a c t i o n b e t w e e n the d e c k b e a m and t h e stay s y s t e m An example of the mode shape for the unstiffened stay system is given on Figure 9. The disparity from the beam mode is even more pronounced than in the two-stays configuration. The action of the actual, probably rather flexible, beam is basically to eliminate the discontinuities at the abutment and at mid-span; that at the abutment proves to be considerably more important than the other, associating some of the side-span stays in the stiffening role of the backstay. An approximation which is valid for very flexible decks can be obtained by considering the deck as a beam on an elastic foundation (i.e. neglecting the coupling effect of tower movement on stay forces). The 'foundation stiffness', i.e. the vertical force per unit length per unit displacement, is k' = E s mgh/fo/z, so that from the definition of S (equation 8), the length parameter 2 = (4El/k') '/4 = (I/~z)V/(2LI)S TM. This approach may therefore be useful if S < (L'/2L) 2. The additional strain energy arising from the beam end shear and mid-span moments that will eliminate the discontinuities can then be added to the strain energy of
16
Journal of Constructional Steel Research: Vol. 1. ,Vo. 1: September 1980 actuo( mode s : 0 0 2 ~ (to reduced scale)
mode • :o
o~,~ot
beom mode
i
removal of the vertical support of the deck on the tower pier, would have very little effect on the dynamic properties.
oz@r"--~_-'"
_ _ /
5
GENERAL
DISCUSSION
Figure 9 ,*,lodeshapefor the radiating-stay configuration
the stay system in the unstiffened deflected form. Writing the frequency equation as w ' - = (1 + C ~ S V * ) C ~
E s gh
(18)
foL z
the correction parameter can be approximated in 0.25L < L' < 0.4L by C5 = 15(L'/L) 2
(19)
For larger values of S, the beam-stay interaction modifies the mode shape to a degree that requires specific solution for each combination. The frequency solutions as a function of S for two values of the geometric parameters are given on Figure 10, These curves should, of course, be used in preference to the approximation given by equations 18 and 19. Judicious interpretation of the trends of variation of frequency with the geometric paramctcrs (Figures 7 and 8), taking account of this approximation, will give a good cstimate of the lowest natural frequency over the range of practical design cases. The discontinuity of the unstiffened mode shape at the abutment indicates that the stresses resulting from dynamic response may be most severe at that point, at least in cases with long side spans. This is caused by the direction of the inertia loading such that main span and side span loads have cumulative effect in distinction to the counterpoised action of gravity loads. The tower is not a critical point in such cases, and the mode shape indicates that "floating' the deck from the stays, i.e.
(,J "fo L" Egh
6
T SI"~GLE"STAY
I
'r.2L
h-
I
T
"~,~..,I"~'~_.../-j~,, ~'~"
L
I T,.o-s,.¥
ol
0
__.-
",
I X
~
,'
Ol
;
i
;
I
RAO,ArJNG STAY
"
"k'
,I
L
L
,i
iI
;•
02 03 OL 05 Relahve stiffness pGrqmet~'r S
Figure I0 I~e.Y(I)'t!ehart f ) r selected geometric ratios
1
06
The foregoing sections have presented methods of estimating the fundamental natural frequency for vertical oscillation. It is apparent that the solution is closely fixed by the basic geometry and the normal and economic values adopted for the working stress in the stays. Within normal economic constraints, little freedom of manoeuvre is left to the designer. The frequencies are low, although not as low as a parabolic-cable suspension bridge. Consider the example of 450 m span. The first antisymmetric mode of an unstiffened parabolic-cable bridge having dip equal to one-tenth of the span would have a natural frequency of 0.165 Hz. The deck of a suspension bridge is, of course, effectively continuously supported, and for very long spans a modcst stiffening girder suffices which does not greatly augment the fundamental vertical natural frequency, tlowcvcr, for this span we may consider a stiffening girder having E1 equivalent to about S = 0.1 on the cable-stayed alternatives. This would raise the antisymmetriomode frequency to 0.28 tlz; the lowest symmetric mode frequency would bc similar, tile actual value dcpcnding on the side span ratio and beam end conditions. Consider now cable-stayed designs based on E~ = 150 k N / m m 2 and f~) = 400 N / m m 2. The single-stay configuration (admittedly unlikely to be used for such a span) with S = 0.4, L' = L/3 and h = L/6, would have a natural frequency of 0.38 flz. A radiating-stay design, making full use of that form to reduce the stiffening beam to (say) S = 0.05 would have frequency 0,30 Hz, reducing to 0.25 Hz if the side spans were extended to L' = 2L/5. The frequency formulae indicate that for given values of j~), S and h/L the natural frequency would be inversely proportional to the square root of the span. In practice, the required value of S is likely to decrease with span, and f o may increase, due mainly to the decreasing relative importance of live load. The reduction of frequency as the span increases would thereby bc increased. For the same reason, bridges with concrete or composite decks will also tend to have smaller S values and possibly higher f o , and thus slightly lower frequencies. The low frequencies, combined with shallow deck beams, give low values of critical speed for vortex shedding resonance, typically less than 10 m/s and possibly as low as 4 m/s. At these speeds, the possibility of low-turbulence wind structure must be considered, especially at smooth-terrain sites, and response may approach the predictions made from smooth-flow wind tunnel tests.
On the dynamic properties of cable-stayed bridges On the other hand, when strands of twisted construction are used (spiral or locked-coil strand or rope) the damping can be expected to be much larger than for otherwise similar non-stayed construction, provided that the amplitudes are sufficient to overcome the limiting inter-wire friction in the stays. Although understanding of this phenomenon is increasing, it remains difficult to predict. The opinion is strongly held in some quarters ~6~that radiating-stay bridges cannot show significant resonant magnification in response to excitation such as vortex shedding, by virtue of'system damping'. It is postulated that the radiating-stay system is subject to important non-linearities, including the effects of transverse motion and associated change of sag of the individual stays. Individual stay resonances may therefore change their effective stiffness and thus 'de-tune' the system; the number of such possibilities is held to be sufficiently large that a sustained resonance would be impossible. While this may be true in some cases, tower-top movement has been shown above to exert a strong coupling on stay motions, especially in the case of long side spans where the backstay has relatively low stiffness. The individual stay resonances are also certain to be at substantially higher frequency than the predicted fundamental bridge frequency in the case of major bridges where the dead load tension fD is high; for example, with fD = 400 N / m m 2 and the longest stay 250 m (of. the cases discussed above), the lowest individual stay frequency would be 0.45 Hz. The author recommends that system damping cannot be relied upon for structures such as the examples discussed in this paper. Other non-linearities may certainly be worthy of investigation, notably in the stress-strain behaviour of the stays. 'Second order', or 'PA', effects, including the effect of the axial compressive load in the deck beam, are however generally negligible in vertical oscillations. Torsional oscillation has not been discussed in this paper, although of comparable practical importance. Torsional excitation by vortex shedding is commonly reported in smooth-flow wind-tunnel tests, but the author is not aware of any report of significant torsional response of a cable-stayed bridge in practice. Where the torsional frequency is much higher than the vertical frequency, the reduced possibility of lowturbulence conditions at the critical wind speed may be significant. The torsional natural frequency is dominant in respect to the risk of potentially catastrophic windinduced oscillation, which is commonly regarded as related to airfoil flutter. German design practice has commonly included a check relating the site design wind speed to the calculated airfoil flutter speed modified by a cross-section aerodynamic efficiency factor, as first suggested by Selberg. The United Kingdom may follow this example. A high flutter speed is ensured by a high torsional frequency in conjunction
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with a substantial difference between torsional and vertical frequencies having a similar mode shape. This is readily ensured by a box-form deck beam: the relative stiffness parameter for a box beam of substantial enclosed area, defined in analogy with the flexural parameter S, typically substantially exceeds unity. In such cases the flutter criterion can generally readily be met. This remains true even when the stays are placed in a single plane along the longitudinal centre line and thus provide no resistance to torsion. When the deck does not comprise a torsion box, the stays must be in two planes, generally coincident with the deck stiffening members. If the stay planes are vertical, and the tower does not provide much resistance to racking between the two planes, the torsional frequency can be estimated from co2
4r z B2
~
W v
(20)
in which r = mass radius of gyration of the deck about the longitudinal centre line B = spacing between stay planes w, = fundamental vertical natural frequency. The deck beam resistance is here treated as differential bending only. Due to the axial forces in the deck induced by the dynamic changes in the stay tensions, some horizontal motion will be coupled with the torsional mode. The torsional frequency in the absence of a torsion box can be significantly increased by inclining the stay phmes to meet at the top of an A-frame tower. Symmetry clearly imposes the condition that spanwise displacement of the tower top is eliminated from the torsional mode, substantially increasing the stiffness. It is hoped to examine this arrangement in detail, together with an examination of the coupling and non-linear effects that have been noted, in a second paper.
REFERENCES 1
2
TROITSKY, M. S. Cable-Slayed Bridges: Theory and Design. Crosby Lockwood, London, 1977. WYATI', T. A. "The effect of wind on long-span bridges.' Symposium on Wind Effects on Buildings and Structures, Loughborough, 1968.
3 Interim Design arrd Workmanship Rules for Steel Box Girder Bridges. Department of the Environment, Section 4. I 1, London, 4
5
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1973. MELBOURNE, W. H. 'Model and full-scale response to wind action of the cable-stayed box girder West Gate Bridge.' I A H R / I U T A M Symposium on Practical Experiences with Flow Induced Vibrations, Karlsruhe, 1979. EYRE, R. and sMrrH, t. J. 'Wind- and traffic-induced dynamic behaviour of some steel box girder bridges." Symposium on Dynamic Behaviour of Bridges, T R R L , 1977. ( T R R L Supplementary Report 275.) LEONEIARDT, F. 'Latest developments of cable-stayed bridges for long spans.' Bygningsstatiske Meddelelser. 45, 4, Copenhagen, 1974.
Contributions discussing this paper should be reeelved by the Editor before I January 1981.