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Steelbridge 2004 Steel bridges extend structural limits

Millau, June 23-25, 2004

095/1

Tectonic Shift Effects on Suspension Bridges

Paolo CLEMENTE Structural Engineering PhD ENEA, Casaccia Research Centre Rome, ITALY [email protected]

Keywords: suspension bridges, tectonic shifts, distortions in suspension bridges.

Abstract The effects of relative displacements between the foundations of a suspension bridge are analysed. A general model is defined, which allows to analyse the structural behaviour under any kind of relative displacement system. The general relation-ships are deduced. Then, in the numerical investigation, the most interesting and likely cases are analysed. The results obtained showed that slow movements cannot create serious problems to the structure in its life time. Anyway, they can became dangerous if their effects will add to those of a very strong earthquake. Therefore, the monitoring of the foundations is suitable as well as the analysis of the effects on the design details.

1. Introduction The shifts between the foundations of a long span bridge could cause serious problems to the structure. The effects could be traumatic when large shifts are caused by very strong earthquakes, as in the case of the Akashi Kaikyo Bridge in Kobe, where the January 1995 earthquake caused a permanent displacement between the towers of about 1 m. At that moment the bridge was in construction: only the two pylons had been erected and the main ropes had been positioned.

Slow shifts can be controlled, but they certainly act on the completed structure, interesting pylons, cables and deck. The problem could be very delicate for railway bridges, as in the case of the Messina Strait Bridge.

As a matter of fact, recent studies carried out by ENEA pointed out the existence of relative displacements between the costs of Sicily and Calabria, at the zones where the foundations of the pylons of the Messina Strait Bridge should be constructed. Both relative vertical and horizontal displacements have been pointed out. For the cost in Calabria an uplift rate of about 2 mm/year has been estimated, while Sicilian costs are rising up with the rate of 0.5 mm/year. The difference, equal to 1.5 mm/year, represents the present relative uplift rate [1]. Besides, geodesic space studies by means of GPS systems in South Italy, pointed out the presence of relative horizontal displacements with a rate of about 10.0 mm/year. Of course, in these cases detailed studied are needed to control the displacement of the costs by means a monitoring system [1, 2].



095/2

In this paper the results of a study on the effects of distortions between the foundations, i.e., tower and anchorages, of a suspension bridge are shown. The analysis is carried out by referring to a general model, considering any kind of displacement. In the numerical investigation the most interesting cases are analysed.

2. Hypotheses and structural model This study is relative to very long-span suspension bridges, therefore the following hypotheses are made about the model. The deck is suspended to one or more parallel steel rows. These are anchored to the soil at their ends, as shown in Figure 1, and can move with reference to the towers in the directions orthogonal to B-B' and C-C', respectively. The deck can be suspended to the main cables along all the three spans; the beam is fixed in the transversal direction at the pylons, but it can move, with reference to them in vertical and longitudinal directions. Spherical hinges are between the main span and the side spans, so the relative rotations with reference to both the vertical and the horizontal axes are allowed. Simple supports at the end of the girder allow the longitudinal displacements at these sections.

Each cable is subject in the main span to a uniform load w, which represent essentially the dead load and is much higher than the travelling load in long-span bridges. The side spans are subjected to the loads wa and wb respectively. If the deck is suspended in the main span only, then loads wa and wb are just related to the self weight of the cable. If this is negligible in comparison to load w, then the cable, subject to high axial forces at its ends, assumes an almost straight line configuration. It is worth nothing that in very long-span bridges the cable self weight is a high ratio of the total load.

The displacements of joints A, B, C and D are independent one of the others and so are the displacements of the corresponding joints A', B', C' and D'. The pylons are able to react, at joints B, B', C and C', with a vertical force and with an horizontal force acting in the directions B-B' and C-C', respectively.

It is worth reminding that, with these hypotheses, the structural behaviour of a long-span suspension bridge is very similar to the behaviour of a couple of simple cables, the stiffening contribution of the girder being negligible [3, 4]. Therefore, the study is carried out by referring to the model of a three span simple cable and the effects on the girder are evaluated later.

α

Fig. 1 Suspension bridge model



095/3

3. Fundamental equations and preliminary design The configuration of a single cable is identified by the coordinates of the joints A, B, C and D (Figure 1). Suppose the cable to be on a plane in its natural configuration, then the geometrical characteristics can be derived from the coordinates of the joints:

ABa xxL −= BC xxL −= CDb xxL −= (1)

ABa zzh −= BC zzh −= DCb zzh −= (2)

The equilibrium equations, which state the uniqueness of the horizontal force H, can be written in the non-dimensional form:

( )LhLfwLH

281−

= (3)

( )aaaa

aa

LhLfwLLw

wLH

281−

⋅= (4)

( )bbbb

bb

LhLfwLLw

wLH

281−

⋅= (5)

This is a system of three equations in the four unknown H, f/L, fa/La, fb/Lb. Usually we assume a value for f/L and deduce H from the first equation and then fa/La and fb/Lb from the second and the third one, respectively, which can be rewritten:

−+=Lh

Lf

wLLw

Lh

Lf aa

a

a

a

a

22 (6)

−+=Lh

Lf

wLLw

Lh

Lf bb

b

b

b

b

22 (7)

If we assume the cable shape to be parabolic, then the deformed cable lengths in the three spans, under the loads wa, w and wb respectively, are given with good approximation by:

+

+= 2

22

2381

a

a

a

aaa

Lh

Lf

LL

L� (8)

+

+= 2

22

2381

Lh

Lf

L� (9)

+

+= 2

22

2381

b

b

b

bbb

Lh

Lf

LL

L� (10)

From H we deduce the maximum axial force in the cable (at B and C) and then the cross sectional



095/4

area A, when the stress σ under the load w has been fixed. From the relation-ship that gives the cross-sectional area A we can extract the non-dimensional parameter:

( ) 1618 2 +

σ==ρLf

EEAwL (11)

which is a function of the cable material properties (tension σ and Young’s modulus E) and of the sag-ratio f/L. The initial lengths of the three portions of the cable in the three spans are given by the approximate non-dimensional relations:

( )

+

+

−ρ−

+

+= 2

22

2

22

3161

22381

a

a

a

a

a

a

a

aaa

Lh

Lf

LhLfLh

Lf

LL

L� (12)

( )

+

+−ρ−

+

+= 2

22

2

22

3161

22381

Lh

Lf

LhLfLh

Lf

L� (13)

( )

+

+

−ρ−

+

+= 2

22

2

22

3161

22381

b

b

b

b

b

b

b

bbb

Lh

Lf

LhLfLh

Lf

LL

L� (14)

In all the previous equations, the contributions of the deformed lengths and those of the length variations due to the axial forces are apparent. The total length of the undeformed cable is given by the summation:

ba �� ++= (15)

4. Effects of shifts: general procedure If the restraints of the cable are subjected to a displacement system, the geometrical characteristics of the cable change as well as the axial force. Denoting the new geometrical coordinates by the subscript “1”, the new span lengths are:

( ) ( )2112

111 ABABa yyxxL −+−=

( ) ( )2112

111 BCBC yyxxL −+−= (16)

( ) ( )2112

111 CDCDb yyxxL −+−=

and the heights became:

( )111 ABa zzh −= 111 BC zzh −= ( )111 DCb zzh −= (17)

As already said, the cable is supposed to be on a plane in its natural configuration, but it is not, in general, when a displacement system has occurred. In fact, each span may develop on a different plane. The angles between the new planes and the old ones are given by (positive if counter-



095/5

clockwise, Figure 2):

−−=β

11

111

AB

ABa xx

yyarctan

−−=β

11

111

BC

BC

xxyyarctan

−−=β

11

111

CD

CDb xx

yyarctan (18)

while the directions of the horizontal reactions of the pylons are individualised by the angles of these (B-B’ and C-C’, respectively) with y axis. For the equilibrium of the horizontal forces it is important to define the directions orthogonal to the pylons, which can be individualised by their angles (positive if counter-clockwise) with x axis:

−−=β

′

′

11

11

BB

BBB xx

yyarctan

−−

=β′

′

11

11

CC

CCC xx

yyarctan (19)

β

β

β

β

β

β

β β

βββ β

Fig. 2 New configuration. Plan view

Being the cable in different planes in the three spans, the horizontal forces are not equal for all the spans. So the new equilibrium equations must be written as follows:

( )1111

11

281

LhLfLL

wLH

−⋅= (20)

( )1111

11

281

aaaa

aaa

LhLfwLLw

wLH

−⋅= (21)

( )1111

11

281

bbbb

bbb

LhLfwLLw

wLH

−⋅= (22)

Towers are supposed to have infinity stiffness in their planes, individualised by lines B-B’ and C-C’, but no stiffness in the direction perpendicular to these. So the equilibrium imposes the components of the horizontal force in the direction orthogonal to B-B’ and C-C’, for pylon B and C respectively, to have the same absolute value (Figure 2):

( ) ( )BaBa cosHcosH β−β⋅=β−β⋅ 1111 (23)



095/6

( ) ( )1111 β−β⋅=β−β⋅ CCbb cosHcosH (24)

If f1/L1 is fixed, then H1, fa1/La1 and fb1/Lb1 can be found as in the previous case, from the first equilibrium equation and from the second and third ones, which can be rewritten:

( )( )1

1

22 aB

Baa

a

a

a

a

coscos

Lh

Lf

wLLw

Lh

Lf

β−ββ−β

−+= (25)

( )( )Cb

Cbb

b

b

b

b

coscos

Lh

Lf

wLLw

Lh

Lf

β−ββ−β

−+=1

1

22 (26)

The solution can be found with an iteration procedure, which can be started by assigning a tentative value of f1/L1, calculating the new values of H1, fa1/La1 and fb1/Lb1 and verifying that the total length of the cable keeps constant. The new cable lenghts in the three spans are:

( )

+

+

−ρ−

+

+=

2

1

12

1

1

1111

2

1

12

1

111

3161

221

381

a

a

a

a

aaaaa

a

a

aaa

Lh

Lf

LhLfLh

Lf

LL

L� (27)

( )

+

+

−ρ−

+

+=

22

1

1

1111

2

1

12

1

111

21

3161

221

381

Lh

Lf

LhLfLh

Lf

LL

L� (28)

( )

+

+

−ρ−

+

+=

2

1

12

1

1

1111

2

1

12

1

111

3161

221

381

b

b

b

b

bbbbb

b

b

bbb

Lh

Lf

LhLfLh

Lf

LL

L� (29)

and the compatibility condition can be written:

baba �� ++=++ 111 (30)

The force acting at the top of the pylons in the direction of their planes are:

( ) ( )BaBaB sinHsinHH β−β⋅+β−β⋅= 1111 (31)

( ) ( )CbbbCC sinHsinHH β−β⋅+β−β⋅= 1111 (32)

HB > 0 when the force on the pylon is directed from B to B'; the two terms of HB are concord if min(βa1, β1) < βB < max(βa1, β1), otherwise they have opposite sign. Analogously, HC > 0 when the force on the pylon is from C to C'; the two terms of HC are concord if min(β1, βb1) < βC < max(β1, βb1), otherwise they have opposite sign.

5. Shift effects on the cable The previous relations, relative to the general case, can be simplified in practice. In fact, it is usually (Fig. 3):

ba hh = 0=h ba LL = (33)

AA xx ′= BB xx ′= CC xx ′= DD xx ′= (34)



095/7

In the following numerical investigation the typical geometrical values f/L = 1/11, La/L = Lb/L = 1/3 and ha/La = hb/Lb = 0.3 have been assumed. Besides, it is W/L = 1/60. The material is characterised by σ/E = 0.005. For the load ratios wa/w and wb/w the two limit values have been considered:

- wa/w = wb/w = 0, relative to short-span bridges having only the main span suspended to the rows, in which the cable self weight is negligible;

- wa/w = wb/w = 1, relative to bridges in which all the three spans are suspended.

In the case, very usual in practice, of very long-span bridges having only the main span suspended, the load ratios are lower than 1, and one can assume wa/w = wb/w = 0.5.

In the following some basic cases of relative displacements between the foundations of a suspension bridge are studied. These are the longitudinal, vertical and transversal displacements and rotation of one tower around its vertical axis.

α

Fig. 3 Investigated model

5.1 Longitudinal shifts

The previous relations semplify very much if only longitudinal displacements are acting on the structure. Two elementary cases are considered in the following. In the first one the shift is due to a distorsion in the right side span, in the second case it is supposed to occur in the main span.

5.1.1 dAx = dBx = dCx = 0, dDx ≠ 0

This is the case in which the anchor D is subjected to an horizontal displacement in x direction. Only the rigth side span changes (Fig. 4):

aa LL =1 LL =1 Dxbb dLL +=1

aa hh =1 hh =1 bb hh =1 (35)

0111 =β=β=β=β=β bCBa

In figure 5 the diagrams of the f1/L1 versus the displacement dDx/L are plotted for the two limit values of the load ratios wa/w=wb/w, while figure 6 shows the variation of the horizontal force H. The behaviour, almost linear for low displacements, become non-linear for high values of dDx/L.



095/8

Fig. 4 Effects of dDx ≠ 0

0,06

0,08

0,10

0,12

-0,015 0 0,015dDx/L

f/L

wa=wb=0wa=wb=w

1,0

1,5

2,0

-0,015 0 0,015

dDx/L

H/w

L

wa=wb=0wa=wb=w

Fig. 5 Variation of f/L due to dDx ≠ 0 Fig. 6 Variation of H/wL due to dDx ≠ 0

5.1.2 dAx = dBx = 0, dCx = dDx ≠ 0

The case of longitudinal shifts between the towers (Fig. 7) has already been analised in a previour paper [5], by means of a simplified model. It is:

aa LL =1 CxdLL +=1 bb LL =1

aa hh =1 hh =1 bb hh =1 (36)


Fig. 7 Effects of dCx = dDx ≠ 0



095/9

Figures 8 and 9 show the diagrams of the sag-ratio and the horizontal force H/wL versus the longitudinal shift dCx/L (=dDx/L). The non linear behaviour is apparent only for high values of the shifts. It is worth noting that, for a fixed value of the shift, the first case (dDx ≠ 0) causes greater effects than the second one (dCx = dDx ≠ 0).

0,02

0,06

0,10

0,14

0,18

-0,05 0 0,05dCx/L=dDx/L

f/L

wa=wb=0wa=wb=w

0,0

2,0

4,0

6,0

-0,05 0 0,05dCx/L=dDx/L

H/w

L

wa=wb=0wa=wb=w

Fig. 8 Variation of f/L due to dCx=dDx≠ 0 Fig. 9 Variation of H/wL due to dCx=dDx≠ 0

5.2 Vertical shifts

Two elementary cases of vertical shifts are considered in the following. In the first one the shift is due to a vertical displacement of joint D, in the second case the distorsion is supposed to occur in the main span, which translates in a vertical shift of both joints C and D.

5.2.1 dAz = dBz = dCz = 0, dDz ≠ 0

Suppose the anchor D to be subject to a vertical displacement dDz (Fig. 10). All the geometrical characteristics of the cable keep constant, except for the rigth side height:

aa LL =1 LL =1 bb LL =1

aa hh =1 hh =1 Dzbb dhh −=1 (37)


Fig. 10 Effects of dDz ≠ 0

In figure 11 the diagrams of the f1/L1 versus the displacement dDz/L are plotted for the two limit values of the load ratios wa/w=wb/w, while figure 12 shows the variation of the horizontal force H/wL.



095/10

0,06

0,08

0,10

0,12

-0,015 0 0,015dDz/L

f/L

wa=wb=0wa=wb=w

1,0

1,2

1,4

1,6

1,8

-0,015 0 0,015

dDz/L

H/w

L

wa=wb=0wa=wb=w

Fig. 11 Variation of f/L due to dDz≠ 0 Fig. 12 Variation of H/wL due to dDz≠ 0

5.2.2 dAz = dBz = 0, dCz = dDz ≠ 0

It is (Fig. 13):

aa LL =1 LL =1 bb LL =1

aa hh =1 Czdhh +=1 bb hh =1 (38)


Figures 14 and 15 show the diagrams of the sag-ratio and the horizontal force H/wL versus the vertical shift dCx/L (=dDz/L). As in the case of longitudinal shifts, an uplift of D only is more dangerous that the same contemporary uplift of joints C and D.

Fig. 13 Effects of dCz = dDz ≠ 0

0,08

0,09

0,10

-0,015 0 0,015dCz/L=dDz/L

f/L

wa=wb=0wa=wb=w

1,30

1,40

1,50

-0,015 0 0,015

dCz/L=dDz/L

H/w

L

wa=wb=0wa=wb=w

Fig. 14 Variation of f/L due to dCz=dDz≠ 0 Fig. 15 Variation of H/wL due to dCz=dDz≠ 0



095/11

5.3 Transversal shifts

Also for transversal shifts two cases are considered, the first relative to a displacement of D along y and the second relative to a displacement of both C and D in y direction.

5.3.1 dAy = dBy = dCy = 0, dDy ≠ 0

If anchor D is subjected to a transversal displacement (Fig. 16), then only the rigth side height and the angle βb1 change:

aa LL =1 LL =1 221 Dybb dLL +=

aa hh =1 hh =1 bb hh =1 (39)

011 =β=β=β=β CBa ( )bDyb Ldarctan=β 1

The span variations are negligible and so are the sag-ratio variations. In figure 17 the diagrams of the horizontal force H versus the displacement dDy/L are plotted for the two limit values of the load ratios wa/w=wb/w. Figure 18 shows the value of HC/wL.

β

β

β

Fig. 16 Effects of dDy ≠ 0

1,37

1,38

1,39

-0,015 0 0,015

dDy/L

H/w

L

wa=wb=0wa=wb=w

-0,10

-0,05

0,00

0,05

0,10

-0,015 0 0,015

dDy/L

HC

/wL

wa=wb=0wa=wb=w

Fig. 17 Variation of H/wL due to dDy≠ 0 Fig. 18 Variation of HC/wL due to dDy≠ 0

5.3.2 dAy = dBy = 0, dCy = dDy ≠ 0

Suppose C and D to be subject to the same transversal displacement (Fig. 19). It is:



095/12

aa LL =1 221 CydLL += bb LL =1

aa hh =1 hh =1 bb hh =1 (40)

011 =β=β ba ( )Ldarctan Cy=β1 1β=βB 1β−=βC

Span and sag-ratio variations are very low. In figure 20 the diagrams of the horizontal force H/wL versus the displacement dCy=dDy≠ 0 are plotted for the two limit values of the load ratios wa/w=wb/w, while figure 21 shows the values of HC/wL.

β

β

β β

β

Fig. 19 Effects of dCy = dDy ≠ 0

1,37

1,38

1,39

1,40

1,41

-0,05 0 0,05

dCy/L=dDy/L

H/w

L

wa=wb=0wa=wb=w

-0,10

-0,05

0,00

0,05

0,10

-0,05 0 0,05

dCy/L=dDy/L

HC

/wL

wa=wb=0wa=wb=w

Fig. 20 Variation of H/wL due to dCy=dDy≠ 0 Fig. 21 Variation of HC/wL due to dCy=dDy≠ 0

5.4 Rotation of a tower

Consider the tower B subject to a vertical counter-clockwise rotation βB around its centre (Fig. 22). All the joints keep their coordinates, except fot B and B’, whose displacements are:

BxBBx sinWdd β⋅=−= ′ 2 ( )ByBBy cosWdd β−⋅=−= ′ 12 (41)

W is the distance between the two parallel cables, as already defined. In the hypotesis of very small rotation we can assume for cable A-B-C-D:



095/13

( ) 221 ByBxaa ddLL ++= ( ) 22

1 ByBx ddLL +−= bb LL =1

aa hh =1 hh =1 bb hh =1 (42)

0≠βB 01 ≠βa 01 ≠β 01 =β=β Cb

In cable A’-B’-C’-D’ just the opposite happen: the side span redices while the main span increases, the total span being constant. The effects in terms of sag-ratio and horizontal force variations are shown in figures 23 and 24. Changes in βa1 and β1 are very low and so are the effects in terms of HB and HC, at least for real values of βB. For this reason these haven’t been drawn in figure 22.

Fig. 22 Effects of tower rotation βB

0,090

0,091

0,092

-0,5 0,0 0,5

βB

f/L

wa=wb=0wa=wb=w

1,35

1,36

1,37

1,38

1,39

1,40

-0,5 0,0 0,5

βB

H/w

L

wa=wb=0wa=wb=w

Fig. 23 f/L variation due to tower B rotation βB Fig. 24 H/wL variation due to tower B rotation βB

6. Shift effects on the girder As already said the the deck is suposed to be simply supported at its ends. The contact with the pylons allow it both longitudinal and vertical displacements, but do not allow transversal movements.

Longitudinal shifts determine variations in the sag-ratio. The beam, suspended to the cables, follows them showing a maximum displacement at the midspan, which is equal to the sag variation. This effect, which could cause excessive slope in the road profile especially at the ends, can be balanced



095/14

by means of regulation of the vertical cable length. Joints at the ends should be able to support these shifts. Forces in the vertical cables do not change and so their lentghs. Actually they assume a not vertical configuration, due to the different behaviour of cables and deck. As a result the beam is subject to tension [5].

Transversal and vertical shifts do not have any effect on the girder due to the kind of constrains between the main span and the side spans. In both cases the resulting slopes should be controlled.

7. Conclusions Relative large shifts between the foundations in suspension bridge can be very dangerous. Actually the usual values of tectonic displacements are very low, so the structural elements will not suffered dangerous increases of forces.

The results of this analysis confirm that the for shifts lower than 0.001, which had already been suggested as the limit value for safe of very long-span bridges [5], the variations of f/L and H/wL are very low and therefore not dangerous for the bridge. For higher values the variations both in terms of sag-ratio and horizontal force become very high. It is worth observing that for all the cases the structure behaves linearly for distortion values up to 0.001. Obviously, the allowable displacement values relative to the tower rotation are much smaller.

We conclude reminding that the monitoring of suspension bridges and their foundations is very important. Besides, one should accounting for the effects of such distortions on the structural details, such as joints and constraints. Maintenance interventions on these details can be carried out during the life of the bridge and this possibility should be take into account in the design phase.

8. References [1] ANTONIOLI F., SYLOS LABINI S., FERRANTI L., “Il Ponte sullo Stretto: problematiche

geologiche”, Energia, Ambiente e Innovazione, ENEA, No. 1, 2002, pp. 63-67 (in Italian).

[2] ANZIDEI et al., “Cinematica del Mediterraneo centrale da reti GPS permanenti e non permanenti”, Volume of abstracts, Workshop "Il contributo delle antiche linee di riva alla comprensione della dinamica recente" (Messina, May 5-8), 2003, pp. 50-51.

[3] CLEMENTE P., NICOLOSI G., RAITHEL A., “Preliminary design of very long-span suspension bridges”, Int. J. Engineering Structures, Elsevier Science Ltd, Vol. 22, No. 12, 2000, pp. 1699-1706.

[4] NICOLOSI G., RAITHEL A., CLEMENTE P., “Static Issues in Very Long-Span Suspension Bridge Design”, IABSE Report Long-Span and High-Rise Structures, IABSE, Zurich, No. 79, 1998, pp. 545-546.

[5] CLEMENTE P., “Ponti sospesi di grande luce: effetti di distorsioni tra le torri”, Costruzioni Metalliche, ACAI, Milano, No. 4, 2003, pp. 45-50 (in Italian).

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