Conceptual Design for the Sunniberg Bridge

Chelsea Honigmann1 and David P. Billington, Hon.M.ASCE2

Abstract: This paper explains how a major engineer, Christian Menn, conceived of his design for the 1998 Sunniberg Bridge in
Switzerland. Menn had given a brief outline of his approach in an earlier paper, and this paper works out the consequences of his ideas
by putting them into numerical form and interpreting those results. This paper brings out the close connection between aesthetic choices
and structural performance and also shows how an apparently complex form can be broken down into simple elements to justify the initial
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conceptual design. There is little published writing about the creative process of bridge engineering and nothing in detail in English on the
Sunniberg Bridge.
DOI: 10.1061/共ASCE兲1084-0702共2003兲8:3共122兲
CE Database subject headings: Bridges, cable-stayed; Conceptual design; Concrete structures; Esthetics; Switzerland.

Introduction tion of a significant achievement in prestressed concrete bridges

During the development of a bridge concept, the structural idea and the culmination of this evolution of form—a tall cable-stayed
becomes solidified through scaled diagrams of the major elements bridge with low pylons, Fig. 1.
and, parallel to this, through just enough calculations to demon- Menn developed the structural concept of the Sunniberg
strate feasibility and economy 共Menn 1998兲. This conceptual de- Bridge while serving on the jury for a design competition for the
sign stage requires the designer to develop a four-fold vision of Poya Bridge in Fribourg, Switzerland. The competition entries
the project, simultaneously considering issues of structural form, were judged in November 1989; however, not being completely
mathematical analysis, construction methods, and the relationship satisfied with the top ranked design, Menn independently devel-
of the structure to the site. oped a new bridge concept, which would later become the seed
The first of these, the form, involves determining what the for the Sunniberg Bridge design.
designer finds to be the best structural type for the bridge by The span and height of the Poya Bridge had indicated three
drawing on experience and aesthetic vision. Parallel to and depen- feasible structural types: 共1兲 an arch; 共2兲 a cantilevered girder; and
dent on the choice of form is the development of the analysis and 共3兲 a cable-stayed bridge. Given the historically and visually sig-
construction methods. The analysis method should be mathemati- nificant site outside of Fribourg, a cable-stayed bridge seemed
cally simple and sufficiently accurate. The construction method best aesthetically and also economically reasonable, although not
should be economical in its use of materials and sensitive in its having the lowest construction cost 共Menn 1991兲. The site con-
impact on the site. Finally, the overall form should be considered straints, however, required very tall piers which, when coupled
in terms of its 3D relationship to the landscape. with pylons of standard proportion, would look awkward. Menn’s
Typically, one issue becomes primary and the others are re- proposed structural concept reconciled the aesthetic, technical,
solved in relation to this one. However, the process is fluid and and economic concerns through a tall cable-stayed bridge with a
iterative, allowing feedback from all parts to inform the decisions harp cable arrangement, tall open-frame piers below the roadway,
and to refine the solution. This paper examines the process of low pylons above the roadway, and a thin deck with edge beams.
conceptual design in relation to the 1998 Sunniberg Bridge near Although this design was not chosen at Poya, the ideas devel-
Klosters, Switzerland. Specifically, the paper will show how the oped there were suitable to the site of the proposed Sunniberg
engineer, Christian Menn, developed a design by starting with an Bridge near Klosters. Here too, the importance of the landscape
initial aesthetically driven but structurally reasonable decision and the prominence of the bridge site demanded a structure that
about form. was sensitive to the surrounding environment 共Bänziger et al.
1998兲. Given similar site constraints, the Poya Bridge ideas
Development of Structural Concept worked well as a starting point for the conceptual design of the
In October 1998, construction ended on the Sunniberg Bridge Sunniberg Bridge design whose site, however, included the addi-
near the resort area of Klosters, Switzerland, marking the comple- tional demand of a curved roadway.
The first bridge proposal at Sunniberg, made over 20 years
1
Former Graduate Student, Princeton Univ. Dept. of Civil and Envi- ago, did not satisfy environmental concerns and was reevaluated a
ronmental Engineering, Princeton, NJ 08544. number of times. The final proposal incorporated more stringent
2
Gordon Y.S. Wu Professor of Engineering, Princeton Univ., Dept. of environmental standards that emphasized the importance of pre-
Civil and Environmental Engineering, Princeton, NJ 08544. serving the natural landscape 共Figi et al. 1997兲. In 1993, the Can-
Note. Discussion open until October 1, 2003. Separate discussions
ton of Graubünden gave its approval to invite three firms to com-
must be submitted for individual papers. To extend the closing date by
one month, a written request must be filed with the ASCE Managing pete for the Sunniberg Bridge design. The three designers
Editor. The manuscript for this paper was submitted for review and pos- submitted their concepts but Menn, again believing that a differ-
sible publication on April 10, 2001; approved on July 2, 2002. This paper ent design would be more appropriate for the Klosters site, pre-
is part of the Journal of Bridge Engineering, Vol. 8, No. 3, May 1, 2003. sented his Poya Bridge concept and model to the Highway De-
©ASCE, ISSN 1084-0702/2003/3-122–130/$18.00. partment architectural consultant, whose enthusiastic

122 / JOURNAL OF BRIDGE ENGINEERING © ASCE / MAY/JUNE 2003

J. Bridge Eng. 2003.8:122-130.

 Fig. 3. Transverse cross section through the road deck
Fig. 1. Overall view of Christian Menn’s Sunniberg Bridge

The horizontal curvature of the road in plan permits the deck

endorsement helped convince the Highway Department. In this to be cast as a monolithic slab with no need for expansion joints
somewhat unorthodox manner, the Highway Department decided either throughout the spans or at the abutments. Under the effects
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that one of the three firms should make the final calculations, of thermal expansion and contraction, the deck deforms as an arch
drawings, and specifications for this conceptual design. Thus, the in the horizontal plane, flexing about the vertical axis to permit
firm of Bänziger Bacchetta Partner carried out the final detailed movement without large internal forces 共Menn 1998兲.
design. The continuity of the road deck also acts to restrain the piers
For this site, a tall cable-stayed bridge with low pylons above against longitudinal and lateral deflections 共Bänziger et al. 1998兲.
a curved road deck has several important aesthetic and technical The pier form is affected by this restraint because the moment
consequences. The efficient pylon height in a typical cable-stayed down the pier decreases linearly. Menn shapes the pier form in
bridge is approximately 25% of the main span length, which, for relation to this diagram—the pier tapers to its narrowest dimen-
the Sunniberg Bridge, would lead to a pylon projecting 35 m sion at approximately 1/3 height where the moment is minimal
above the roadway. This ratio leads to efficiency in terms of cable and flares above and below this point as the moment increases,
forces but creates a visually overpowering design when coupled Fig. 4共a兲 共Bänziger et al. 1998兲.
with the tall piers and relatively short spans required for the Sun- Additionally, the road curvature produces eccentricity between
niberg. In this case, efficiency and aesthetics must be mediated. the cable forces at their point of connection to the deck and the
By reducing the ratio to 10–15% of the main span length, Fig. 2, pier-girder connection. These eccentricities cause transverse
the aesthetic issue is solved but the cable forces increase signifi- bending moments that must also be compensated for by the form
cantly. In typically flexible pylons under unbalanced live loading, of the piers and pylons. The T-shaped cross section of the pier and
the increase in cable forces due to low cable angles produces pylon, stiffened in the transverse direction, Fig. 4共b兲, resists these
correspondingly larger pylon deflections and therefore larger moments. Opposing piers are also tied at three points along their
girder deflections. However, through careful design of the pylon height, Fig. 4共c兲, preventing buckling under axial loads and pro-
form, making it stiffer against longitudinal bending, these nega- ducing additional stiffness due to the transverse pier spacing
tive effects can be reduced and the girder deflection limited pre- 共Bänziger et al. 1998兲. However, to allow rotation of the road
dominately to cable elongation. deck 共about the vertical axis兲 due to thermal effects, some trans-
An increase in the cable force due to low pylons and shallow verse flexibility is required so that the piers can deform with the
cable angles necessarily also produces an increase in the axial deck. Thus, rather than using a single solid member for the piers,
force component in the bridge girder. Thus buckling of the rela- the more flexible open-frame vierendeel design, shown in Fig.
tively slender girder, Fig. 3 becomes an issue requiring extra 4共c兲, allows this moderate flexure 共Menn 1991兲.
girder strengthening in the region near the pylon where the cu- An initial, aesthetically driven choice of structural form can
mulative axial force from the cables is largest. drive the conceptual design process, leading to a highly devel-

Fig. 2. Sunniberg Bridge elevation and plan

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J. Bridge Eng. 2003.8:122-130.

 Fig. 5. Effect of horizontal restraint on moment at pier base due to
spanwise loading: 共a兲 restrained; 共b兲 unrestrained
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Because of the very high piers 共up to 62-m tall兲, Menn 共1998兲
decided that the pylons should be as low as possible above the
roadway, about 14.8 m, which led him to choose the relatively flat
cable slope of tan ␣⫽0.2. Because flat cables could lead to unrea-
sonable girder deflections, which would be accentuated by the
multispan, cable-stay form, Menn 共1998兲 concluded that the su-
perstructure and piers must form a rigid, continuous construction.
Additionally, the top of the piers cannot move longitudinally and
thus the bending in that direction is substantially less than if the
deck could translate longitudinally, Fig. 5.
The thickness of the road deck was chosen based on a trans-
verse span of approximately 10 m between the cables. For this
span, Menn 共1998兲 determined that a road deck thickness of 40
Fig. 4. Pier and pylon P2: 共a兲 longitudinal section; 共b兲 horizontal cm with 80 cm edge stiffening would suffice. Additionally, he
cross section at 3 levels showing T-shape; 共c兲 transverse section specified the distance between the cable anchorages as 6 m in
consideration of the pylons, the cable size, and the girder stability
共buckling resulting from axial force兲.
Menn 共1998兲 made the conceptual calculations based only on
oped vision of the design and a form that could not have been the dead load and live load, as given below.
conceived by theory alone. The process is based more on experi- • Dead: g⫽190 kN/m 共dead load, including 17 cm wearing
ence and intuition about structural behavior than detailed struc- surface兲
tural analysis. • Live: q⫽4 kN/m2 •9 m 共roadway width兲⫽36 kN/m;
The following section formulates more precisely the concep- Q⫽300 kN⫹80% impact⫹80% eccentricity⫹60% reduction
tual ideas discussed above. We have made simple hand calcula- The 80% eccentricity factor accounts for the condition where the
tions based on static analysis to illustrate the basis for form and concentrated 共truck兲 loading Q is in one lane nearest the cable to
dimension. These calculations, based on conceptual formulas be designed. Although the 80% impact factor may seem high in
given in Menn’s article 共Menn 1998兲, result from using the rel- comparison to American standards, it is applied only to the con-
evant loads, material properties, and dimensions for the Sunniberg centrated load and not, as in American practice, to the uniform
Bridge. The calculations are incorporated into the text to empha- load as well.
size the relationship between the conceptual design and the final
detailed design. By giving Menn’s assumptions and formulations, Determination of the Cable Cross Section
we can show the thought process of a major bridge engineer as he
made this conceptual design. To determine the preliminary structural dimensions and critical
loads, Menn 共1998兲 examined the effects of these loads on a
single pier with cantilevered girder spans on each side. The maxi-
Calculations for Conceptual Design mum cable load is generated by the superposition of full dead and
live loading. Considering the tributary area of a single cable
共where a⫽6 m is the cable anchorage spacing兲 and allowing for a
In the case of the Sunniberg Bridge, the initial structural idea was
60% reduction of the concentrated load 共assuming 20% of the
that of a multispan, cable-stayed bridge with a narrow roadway.
load is transferred to each of two neighboring cables兲, the vertical
In reference to the landscape, topography, and the design lines of
load applied at one cable is
the road, it was clear to Menn 共1998兲 that the best solution would
have four piers 共five spans兲 with an average height of approxi-
mately 65 m to the level of the roadway. Additionally, the posi-
tioning of the piers in consideration of the river’s path resulted in
Q c⫽ 冉 冊
g⫹q
2
Q
共 a 兲 ⫹ 共 1.8兲共 1.8兲共 0.6兲
2

a cantilever projection of approximately 70 m from both middle

piers, corresponding to a main span of 140 m in the center of the
bridge. The overall length amounts to 526 m with a radius of
⫽ 冉
190 kN/m⫹36 kN/m
2 冊
共 6 m兲 ⫹
300 kN
2 冉 冊
共 1.8兲共 1.8兲共 0.6兲

curvature of 503 m. ⫽678 kN⫹291.6 kN

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J. Bridge Eng. 2003.8:122-130.

 Fig. 6. Cable geometry

Fig. 7. Moment in first span due to concentrated load

Q c ⫽969.6 kN 共 218 kips兲
From the cable geometry shown in Fig. 6, the resulting force in
the cable T is given by 0.4-m thick slab spanning between 0.8-m deep edge beams. Near
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the pier connection where the axial force is greatest, however,

Qc 969.6 kN
T⫽ ⫽ Menn 共1998兲 increased the resisting area of the girder cross sec-
sin ␣ sin 11.3° tion to about 9 m2 so that the stress would be
T⫽4,948 kN 共 1,112 kips兲
The allowable tensile stress is given by Menn 共1998兲 as ␴ c
␴ G,1⫽ 冉 96,960⫻103 N
9 m2 冊冉 1 m
1,000 mm 冊 2

⫽0.45 f sy , where the maximum stress f sy

␴ G,1⫽10.8 N/mm2 共 1,566 psi兲
⫽1,600 N/mm2 (232 ksi). Then
Additionally, Menn 共1998兲 makes the assumption that the concen-
␴ c ⫽0.45共 1,600 N/mm2 兲 ⫽720 N/mm2 共 104 ksi兲
trated load on the first 6-m cable span is taken by bending of the
The minimum cable area, A min , is then road deck between the first cables and the pylon. The resultant
stress can be estimated by considering the effect of a concentrated
共 4,948 kN兲共 1,000 N/kN兲
A min⫽ load 共here with only the impact factor兲 centered on a typical beam
720 N/mm2 fixed at one end 共the pier兲 and simply supported at the other, as
A min⫽6,872 mm2 共 10.7 sq in.兲 shown in Fig. 7. The moment at the fixed end is

Using 7-mm diameter wires, the required number of wires in the 3Q•a 3 共 540 kN兲共 6 m兲
M pier⫽ ⫽ ⫽607.5 kN-m
cables is 16 16
6,872 mm2 The stress at the fixed end due to this moment is then
number of wires⫽ ⫽179 wires
1 M pier M pier 607.5 kN-m
␲ 共 7 mm兲 2
冉 冊 冉 冊
␴ G,2⫽ ⫽ 2 ⫽
4 S bh 共 1 m strip共 0.72 m兲 2
In the final detailed design, the allowable tensile stress was taken 6 6
as 0.5f sy , thus leading to the cables with between 125 and 160
wires 共Däniker 1998兲, providing a cable area of 4,810 to 6,157 ⫽7,031 kN/m2
mm2 共7.46 to 9.54 sq in.兲. The reduction in area indicates that the ␴ G,2⫽7.0 N/mm2 tension/compression
loads and load factors considered in the conceptual design stage
are conservative. Thus, the total stress in the bottom fiber of girder at the pylon
connection would be about 18 N/mm2 共2,611 psi兲. The compres-
sive strength of the concrete in the Sunniberg Bridge is given as
Estimate of Concrete Stresses in Girder Cross Section 23 N/mm2 共3,336 psi兲 共Bänziger et al. 1998兲. These results do not
at Piers consider the potential buckling of the cross section under axial
The cable force calculated above determines the axial force in the compression. Stability issues will be considered in a separate sec-
girder. Each cable contributes a component of axial force to the tion.
girder. Therefore, the critical girder section, at the connection
between the pier and the girder where the cumulative effect is Determination of the Pylon Form
largest. The compression in the girder due to a single plane of
cables is given by In developing the conceptual design of the bridge, Menn 共1998兲

冉 冊
gave particular care to the shaping of the pylon, which must resist
Qc longitudinal and lateral bending stresses as well as the axial load-
N⫽ 共 number cables兲共 N G 兲 ⫽ 共 number cables兲
tan ␣ ing from the cables. Longitudinal bending is maximum when live

冉 冊
load is applied to just the main span. The worst-case axial force
969.6 kN and transverse bending 共due to bridge curvature兲 results from
⫽ 共 10 cables兲
tan 11.3° dead load plus uniformly distributed live load on both sides of the
pylon.
N⫽48,480 kN 共 10,899 kips兲
To resist these stresses, Menn 共1998兲 designed the cross sec-
Because axial compression is carried by the entire girder cross tion as a T-form with outer rectangular flanges and a reinforced
section, this axial force must be doubled to account for the two rectangular web, Fig. 8. To estimate the cross-sectional dimen-
parallel cable planes. Therefore, the total axial force in the girder sions, he assumed that the transverse bending is carried by the
is 96,960 kN. Along most of the span, the girder is made up of a rectangular web, the longitudinal bending is carried by the two

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J. Bridge Eng. 2003.8:122-130.

 Based on the average eccentricities, e l ⫽2.54 m and e r ⫽0.21 m
共Appendix兲, we have
M T,l ⫽ 共 2.54 m兲共 20 cables兲共 678 kN兲 ⫽34,442 kN-m
M T,r ⫽ 共 0.21 m兲共 20 cables兲共 678 kN兲 ⫽2,848 kN-m
giving a total transverse moment of 37,290 kN-m 共27,504 kip-ft兲,
which is reasonably close to the previous calculation.

Fig. 8. Cross section of a pylon 共piers have similar T-form兲 Determination of Stresses in Pier Cross Section at Pier
Top
The maximum longitudinal moment in the piers is caused by full
live load on only the main span. Due to the rigid pier-girder
rectangular flanges, and all three rectangular elements together
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connection, this moment decreases linearly down the pier shaft.

carry the axial compressive force.
Thus, the value at the pier base is only one half the value at the
The torsional moment of the cantilever system 共girder with
road level. Additionally, there is a zero point at approximately
cable stays兲 results in a transverse bending moment at the pylon
one-third height. The moments at the pier top and base are calcu-
base. Therefore, the total transverse moment can be calculated
lated below
similarly to a curved beam with both longitudinal bending and ql 2 共 36 kN/m兲共 70 m兲 2
torsional moment, as shown in Fig. 9共a兲. However, the transverse M long,top⫽ ⫽
moment at the base of each pylon can also be calculated directly 2 2
using the sum of the cable end loads, Q c 共here without the con- ⫽88,200 kN-m 共 65,053 kip-ft兲
centrated load兲, and their transverse distance from the pylon, as
1
shown in Fig. 9共b兲. M long,base⫽ M long,top⫽44,100 kN-m 共 32,526 kip-ft兲
From Fig. 9共a兲, Menn 共1998兲 defines the total transverse mo- 2
ment by the following equation, where l is the length supported Using his assumption that the flange sections of each pier resist
by cables: these longitudinal moments, Menn 共1998兲 varied the spacing be-
M T ⫽2• 冕 lM

0 R
ds⫽2•
0
冕
l 共 g⫹q 兲 s 2

2R
ds
tween the flanges to follow the bending moment. As a result, the
cross-sectional width is smallest at the one third height and flares
above and below with the increasing moment. The moments
1 共 g⫹q 兲 l 2 above are based on loading over the entire bridge width 共two pier
⫽2• •l legs兲. We have calculated the stress in one pier leg at the top to be
3 2R
M •c top 共 44,100 kN-m兲共 2.79 m兲
1 共 190 kN/m⫹36 kN/m兲共 63 m兲 2 ␴ long,top⫽ ⫽
⫽2• I long,top 13.3 m4
3 2R
⫽9,251 kN/m2 共 1,342 psi兲 tension/compression
• 共 63 m兲
The tensile stress in the flange is only partially compensated for
M T ⫽37,450 kN-m 共 27,621 kip-ft兲 by the compressive stress due to dead load, g, as we calculate
Alternatively, from Fig. 9共b兲, one can determine the transverse below
moment in each pylon using the following equations where the
subscripts l and r denote the left 共outside兲 and right 共inside兲
cables, respectively: ␴ g⫽
1
2
g•L
⫽
冉1
2 冊
•190 kN/m 共 140 m兲

A pier,top 5.4 m2
M T,l ⫽e l • 兺 Q c,l and M T,r ⫽e r • 兺 Q c,r ⫽2,463 kN/m2 共 357 psi兲 compression

Fig. 9. Two methods for determining the transverse moment, M T , in the pylon: 共a兲 M T calculated from torque, m t ; 共b兲 M T calculated from
vertical components of cable force

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 Fig. 10. Axial load in the piers created by transverse moment.
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Fig. 11. Girder buckling diagram with the equivalent load, ⌬T, from
The total tensile stress in the flanges due to longitudinal bending cable elongation
and dead load is 6,788 kN/m2 共985 psi兲, which must be resisted
by steel reinforcing and prestressing. The total compressive stress
is 11,714 kN/m2 共1,699 psi兲, which is well below the allowable Determination of the Critical Girder Axial Force
compressive stress of 23 N/mm2 共23,000 kN/m2 or 3,336 psi兲, as
given previously for the concrete. Menn 共1998兲 chose the spacing, a, of the cable anchors so that
girder buckling over a single cable span length is not critical. On
the other hand, because of the relatively flat cables, girder buck-
Effect of the Cross Beam at the Top of the Pier ling can become critical over two span lengths. The elongation of
The cross beam at the top of the pier transfers the transverse the cables due to girder bending generates only a relatively small
bending moment at the pylon base 共through bending兲 into axial resisting force in the cable. The second span is the critical buck-
force in the piers, Fig. 10. Therefore, the cross beam must be ling section due to the large axial load and the flexibility of the
sized to resist the full transverse moment. system, Fig. 11.
Based on a 12-m spacing between the pier legs at the level of The deflection, f, of the girder under live load generates two
the roadway, and using the value of M T calculated from the tor- simultaneous effects which act to maintain equilibrium within the
sional moment 共37,450 kN-m兲, the amount of axial force at the structure: 共1兲 the internal cable force caused by the elongation of
top of each pier leg is the cables; and 共2兲 the internal girder force generated by the bend-
ing of the girder. These ‘‘stabilizing’’ forces can be idealized as a
M T 37,450 kN-m
N pier⫽ ⫽ ⫽3,121 kN 共 702 kips兲 uniform load, q st , distributed over 2a, acting in the upward di-
d 12 m rection. The stabilizing contribution of cable and girder is based
The stress at the top of the piers due to transverse bending is then on the relative stiffness of each.
the relatively low value of On the other hand, the deflection, f, of the girder creates insta-
bility under the axial compressive force N. Again, this ‘‘destabi-
N pier 3,121 kN lizing’’ force acts as a uniformly distributed load q dst , however,
␴ trans,top⫽ ⫽
A pier,top 5.4 m2 in the downward direction. Equilibrium, and therefore stability, in
the girder is maintained as long as the destabilizing load does not
⫽578 kN/m2 共 84 psi兲 tension/compression exceed the stabilizing load. Therefore, the critical girder axial
Again, the dead load compressive stress of ␴ g ⫽2,463 kN/m2 cal- force, N G,crit , is determined by equating the stabilizing and desta-
culated above eliminates the tension in the outside pier leg, re- bilizing equivalent loads q. Out of the equation 兺 q st ⫽ 兺 q dst ,
sulting in a net compressive stress of 1,885 kN/m2 共273 psi兲. In one can then solve for N G,crit .
the inside pier leg, the total compressive stress is 3,041 kN/m2
共441 psi兲. These stresses are well below the capacity of the con- Stability Effects
crete. First, the girder deflection, f, corresponds to a cable elongation,
⌬L c ⫽ f •sin ␣, as shown in Fig. 12. The resulting increase in
cable force, ⌬T, is given by the following equation where E c A c is
Spacing of Cross Beams in Piers the cable stiffness and L c is the cable length
In addition to transferring forces, the cross beams serve to reduce E c A c •⌬L c E c A c • f sin ␣
the buckling length of the pier. Based on a maximum spacing of ⌬T⫽ ⫽
Lc Lc
22.17 m between cross beams, a factor of 0.6 to account for the
rigidity of the connection between pier and cross beam, and an The vertical component, ⌬T y , is then
average radius of gyration of about 0.45 m, the slenderness of the
E c A c • f •sin2 ␣
pier is ⌬T y ⫽⌬T•sin ␣⫽
Lc
kL 共 0.6兲共 22.17 m兲
␭⫽ ⫽ ⫽33 This vertical cable force can be converted to an equivalent uni-
r 0.45 m
form load, q ⌬T , on the girder of length 2a. In order to be equiva-
Thus, for the highest pier, two cross beams suffice to reduce the lent, the distributed load should produce the same girder deflec-
slenderness of the pier leg to less than 50, a value used by Menn tion at midspan, Fig. 13. The deflection of the girder of length 2a
共1998兲 to avoid the danger of buckling. under a concentrated load, ⌬T y , is

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J. Bridge Eng. 2003.8:122-130.

 Fig. 12. Girder deflection due to concentrated and uniformly distrib-
uted loads Fig. 13. Girder axial force, N, and eccentricity, f, create instability
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effect

⌬T y 共 2a兲 3
f ⌬T y ⫽
48E G I G
The deflection of a girder under a distributed load, q ⌬T , is shown in Fig. 13. This axial load can be replaced by an equivalent
distributed load, q dst , which produces the same moment at mid-
5q ⌬T 共 2a兲 4 span for a girder of length 2a. Equating the moments for the two
f q ⌬T ⫽
384E G I G load types gives
Equating these two expressions and solving for q ⌬T gives q inst共 2a兲 2
N• f ⫽
⌬T y E c A c • f •sin2 ␣ 8
q ⌬T ⫽0.8 ⫽0.8 (1)
a aL c The equivalent destabilizing load is then
The bending of the girder produces a second component of stabi- 2N• f
lizing force. The force due to girder flexure is equivalent to a q inst⫽ (4)
a2
uniform load on the deck which produces a deflection, f, at mid-
span. The equivalent load on a simply supported girder of length
2a is given by the following equation, where E G I G is the girder Critical Axial Force
bending stiffness. Equating these stability and instability loads 关Eqs. 共3兲 and 共4兲兴
results in the critical axial force in the girder
384f •E G I G 4.8f •E G I G
q G⫽ ⫽ (2) 0.8E c A c • f •sin2 ␣ 4.8f •E G I G 2N G,crit• f
5 共 2a 兲 4 a4 ⫹ ⫽
aL c a4 a2
The total stability load, q st , is the sum of the components due to (5)
cable elongation and girder deflection 关Eqs. 共1兲 and 共2兲兴. 0.4aE c A c •sin ␣ 2.4E G I G
2
N G,crit⫽ ⫹
0.8E c A c • f •sin ␣ 4.8f •E G I G
2 Lc a2
q st ⫽ ⫹ (3)
aL c a4 This critical axial force is calculated based on the entire cross-
sectional width and two opposing cables.
Instability Effects Based on standard material properties of steel and concrete, as
A destabilizing effect is generated by the axial force, N, which well as the structural dimensions determined from Menn’s paper
acts at an eccentricity, f, creating a moment around Point A as 共1998兲, we can determine the critical axial force in the girder

0.4共 6,000 mm兲共 205 kN/mm2 兲共 2⫻6,872 mm2 兲 •sin2 共 11.3° 兲 2.4共 10 kN/mm2 兲共 2.61⫻1011 mm4 兲
N G,crit⫽ ⫹
12,237 mm 共 6,000 mm兲 2

⫽21,217 kN⫹174,000 kN
共 cables兲 共 girder兲

N G,crit⫽195,217 kN 共 43,887 kips兲 tion, and are the main source of deflection in the cable-girder
system.
Comparing the total critical axial load to the actual axial load
The vertical resisting force in the cables is only 1/8 the resistance in the girder, one can determine a buckling safety factor. The axial
of the girder to flexure. Thus, the cables contribute only limited load in the second cable span due to dead and live load acting on
stability to the system, especially at such a low angle of inclina- both planes of cables is

128 / JOURNAL OF BRIDGE ENGINEERING © ASCE / MAY/JUNE 2003

J. Bridge Eng. 2003.8:122-130.

 Table 1. Calculation of Average Cable Eccentricities
Cable number 共i兲 si ␪ i,l (rad) y i,l (m) ␪ i,r (rad) y i,r (m)
1 6 0.0118 0.0354 0.0121 0.0369
2 12 0.0236 0.1417 0.0241 0.1477
3 18 0.0354 0.3187 0.0362 0.3322
4 24 0.0472 0.5666 0.0482 0.5905
5 30 0.0590 0.8852 0.0603 0.9226
6 36 0.0708 1.2746 0.0723 1.3283
7 42 0.0826 1.7345 0.0844 1.8077
8 48 0.0945 2.2651 0.0964 2.3607
9 54 0.1063 2.8663 0.1085 2.9871
10 60 0.1181 3.5378 0.1205 3.6870
Note: y l,ave⫽1.363; y r,ave⫽1.391
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36 kN/m⫽ 共 0.0108 kN/mm2 兲 • f

Therefore,
共 36 kN/m兲共 1 m/1,000 mm兲
w o⫽ f ⫽
0.0108 kN/mm2
w o ⫽3.33 mm
From Eq. 共6兲, the second order girder deflection w⫽(1.8)
⫻(3.33 mm)⫽6 mm (0.24 in.).

Fig. 14. Transverse width of bridge cables

Conclusions

N⫽ 共 number cables兲共 N G 兲 ⫽ 共 number cables兲 冉 冊

Qc
tan ␣
It becomes evident from the above discussion that simple concep-
tual calculations can lead from the initial structural concept to
detailed conclusions about form and dimension. The transparent

⫽ 共 2 planes⫻9 cables兲 冉 969.6 kN

tan 11.3° 冊 and slender form of the Sunniberg Bridge can be justified on this
basis in that it achieves Menn’s 共1998兲 aesthetic goals while keep-
ing within the discipline of structural feasibility and efficiency.
N⫽87,343 kN 共 19,635 kips兲 Specifically, the aesthetic choice of high piers, low pylons, a rela-
Therefore, the safety factor against buckling is approximately tively flat cable inclination, and a slender deck affects the behav-
2.25. ior of the structure but can be compensated for by relatively
Once this critical girder axial force is known, the second order simple structural detailing as summarized below.
girder deflection due to live load can be calculated simply using The use of a low cable angle, which causes increased axial
the following equation: force in the deck, is compensated for by increasing the cross
section of the deck near the pylon such that the compressive stress
1 1 is below the limits of the concrete and the buckling safety factor
w⫽w o ⫽w o
N 1 is above two. The continuity of the deck girder and the fixed
1⫺ 1⫺ pier-girder connections decrease the deflection arising from pier
N G,crit 2.25
(6) rotation by limiting the horizontal displacement of the pier tops.
w⫽1.8w o The shaping of the pier, with increased longitudinal stiffness near
The term w o in Eq. 共6兲 is the first-order deflection due to live load the level of the roadway, further reduces pier rotation. Addition-
only. One can calculate this deflection using the equation for the ally, the girder deflection is limited by increasing its longitudinal
stability load, q st . Under equilibrium conditions, the stabilizing bending stiffness near the pier-girder connection. Therefore, the
force generated by the cable and girder deflection must equal the deflection at midspan is essentially due to cable elongation.
applied live load. Using a live load of 36 kN/m, as previously As shown by the calculation of the axial forces in the piers, the
specified, the deflection is calculated as follows: curvature of the bridge plan, and therefore the eccentricity of the
cable loads, produce relatively small transverse moments. The
0.8E c A c • f sin2 ␣ 4.8• f •E G I G transverse stiffness of the pier cross section and the transverse
36 kN/m⫽ ⫹
aL c a4 spacing of the pier legs are sufficient to reduce the stresses caused
by the transverse moments. Additionally, the horizontal arch ac-
0.8共 205 kN/mm2 兲共 2⫻6,872 mm2 兲 • f •sin2 共 11.3° 兲 tion produced by the bridge curvature in plan allows the use of a
⫽
共 6,000 mm兲共 12,237 mm兲 monolithic deck slab, eliminating the need for expansion joints to
control deflections due to thermal effects.
4.8• f • 共 10 kN/mm2 兲共 2.6⫻1011 mm4 兲 Thoughtful conceptual design is essential to the creation of
⫹
共 6,000 mm兲 4 fine bridges. This paper has tried to provide an insight into one

JOURNAL OF BRIDGE ENGINEERING © ASCE / MAY/JUNE 2003 / 129

J. Bridge Eng. 2003.8:122-130.

 such bridge and could help stimulate more studies of the many where i represents the cable number and ␪ i is given by the arc
exemplary bridges being built worldwide. length (s i ⫽i•6 m cable spacing兲 and radius
si i•6 m
␪ i⫽ ⫽
Acknowledgments R 共 503 m⫾5.2 m兲
The calculation of the eccentricities is shown in Table 1.
The writers are deeply indebted to Christian Menn for his consid- Adjusting for the location of the cable-pylon connection 共1.18
erable help in preparing this paper. The writers also wish to ac- m, as shown in the Fig. 14兲, the average eccentricities are then
knowledge Heinrich Figi, Chief Structural Engineer 共General • e l (outer)⫽1.363 m⫹1.18 m⫽2.54 m
Planning兲, Cantonal Highway Administration, Chur, Switzerland, • e r (inner)⫽1.391 m⫺1.18 m⫽0.21 m
Dialma Jakob Bänziger, Civil Engineering 共Final Design兲, and
Aldo Bacchetta, Civil Engineer, Bänziger⫹Bacchetta⫹Partner,
Zurich, Switzerland for their role in the design and documentation References
of this bridge.
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Bänziger, D. J., Bacchetta, A., and Baumann, K. 共1998兲. ‘‘Statik und

konstruktion 共statics and construction兲.’’ Schweizer Ingenieur und Ar-
chitekt, Lausanne Switzerland, 44 共in German兲.
Appendix. Calculation of average cable eccentrici-
Däniker, J. 共1998兲. ‘‘Schrägkabel 关the stay cable兴.’’ Schweizer Ingenieur
ties, e l and e r und Architekt, Lausanne Switzerland, 44 共in German兲.
Figi, H., Menn, C., Bänziger, D. J., and Bacchetta, A. 共1997兲. ‘‘Sunniberg
The radius of curvature to the longitudinal centerline of the bridge Bridge, Klosters, Switzerland.’’ Struct. Eng. Int., Zurich, Switzerland,
is given as R⫽503 m. The transverse width between the cables is 7, 6 – 8.
10.4 m, Fig. 14. Based on trigonometry, the eccentricity of each Menn, C. 共1991兲. ‘‘An approach to bridge design.’’ Eng. Struct., 13, 106 –
individual cable with respect to the pylon connection, is given by 112.
Menn, C. 共1998兲. ‘‘Generelle berechnung 关general calculations兴.’’ Sch-
weizer Ingenieur und Architekt, Lausanne, Switzerland, 44 共in Ger-
y i ⫽R• 共 1⫺cos ␪ i 兲 man兲.

130 / JOURNAL OF BRIDGE ENGINEERING © ASCE / MAY/JUNE 2003

J. Bridge Eng. 2003.8:122-130.