See discussions, stats, and author profiles for this publication at: https://www.researchgate.
net/publication/220291887
Cable optimization of a long span cable stayed bridge in La Coura (Spain)
Article in Advances in Engineering Software · July 2010
DOI: 10.1016/j.advengsoft.2010.05.001 · Source: DBLP
CITATIONS READS
66 1,158
4 authors:
Aitor Baldomir Santiago Hernandez
University of A Coruña University of A Coruña
82 PUBLICATIONS 570 CITATIONS 284 PUBLICATIONS 1,764 CITATIONS
SEE PROFILE SEE PROFILE
Felix Nieto Jose Angel Jurado
University of A Coruña University of A Coruña
86 PUBLICATIONS 561 CITATIONS 93 PUBLICATIONS 605 CITATIONS
SEE PROFILE SEE PROFILE
Some of the authors of this publication are also working on these related projects:
Optimum crossing cable system in multi-span cable-stayed bridges View project
Reliability Based Design Optimizaiton of suspension bridges considering the correlation among the points that define flutter derivatives View project
All content following this page was uploaded by Aitor Baldomir on 07 October 2019.
The user has requested enhancement of the downloaded file.
Cable optimization of a long span cable stayed bridge in
La Coruña (Spain)
A. Baldomir*, S. Hernandez, F. Nieto & J.A. Jurado
School of Civil Engineering, University of Coruña
Campus de Elviña s/n – 15071 – La Coruña, Spain
abaldomir@udc.es | Tlf: (+34) 981 167 000 – 5476 | Fax: (+34) 981 167 170
hernandez@udc.es | Tlf: (+34) 981 167 000 – 1406 | Fax: (+34) 981 167 170
fnieto@udc.es | Tlf: (+34) 981 167 000 – 1426 | Fax: (+34) 981 167 170
jjurado@udc.es | Tlf: (+34) 981 167 000 – 1410 | Fax: (+34) 981 167 170
Abstract
This document describes an optimization problem of cable cross section of a cable
stayed bridge considering constraints of cable stress and deck displacement. Since the
bridge is still in the design phase, the geometry and the mechanical characteristics are
subjected to changes. In order to avoid creating different structural models, a computer
code was written to produce a model from geometrical and mechanical data and solve
the optimization problem. At the end of the document, two examples are included to
show the capabilities of the methodology presented.
Keywords: Optimization, Cable Stayed bridges, Civil Engineering
1. Introduction
Optimization techniques are not commonly employed in professional life in civil
engineering, specifically in the area of bridge design, and the majority of consulting
firms do not apply these techniques in their works. Nevertheless, in design of large
structures, these techniques are gaining more importance due to their impact on
reduction of material cost.
The presented work comes from a study of road system around A Coruña, in particular,
the connection of the surrounding areas with the city. After analyzing traffic in different
roads, their intensity, and future projects of hub connections, it was concluded that it
would be necessary to construct a new road able to efficiently relieve the current heavy
traffic.
* Corresponding author 1
The new connection requires the erection of a bridge over the Coruña estuary, which
represents the most important part of the road. Therefore a study was conducted on the
bridge typology trying to find the best from the technical and aesthetic points of view
while taking also in account the environmental conditions of the area. After considering
different proposals, a cable stayed bridge was found to be the most adequate.
This class of bridges are used more and more to cross over long span and cases as the
Stonecutters or the Sutong bridges have span length longer than one kilometre [1]. The
reasons to use such typology are its good structural functionality, visual lightness, great
aesthetic component, and its low impact on the environment, which was a key issue at
the location of the bridge over.
Work presented in this document describes the conceptual design of the bridge and the
optimization procedure carried out for minimizing the amount of steel in the cables.
2. Bridge description
The solution adopted considers the construction of a cable stayed bridge of 1198 m of
total length which is divided into two lateral spans of 270 m and a main span of 658 m.
An elevation of the bridge is presented in Figure 1.
Figure 1. Side view of the bridge
The main elements that compose the bridge are shown in Figure 2 and can be described
as follows:
1. Deck: made of steel with an aerodynamic cross section 3 m depth and 34 m.
wide, which allows a configuration of 3 lanes in each direction. Difference of
altitude between the two end points is 15 m.
2. Towers: the bridge has two towers: each one fixed in the foundation with two
twin concrete piers separated by the deck width. From the deck two steel masts
of lambda shape come out. The cables are anchored to the vertical upper part of
the tower.
2
3. Cables: the bridge deck is sustained by 80 pair of cables that connect to the
towers. The cable configuration is a hybrid class of harp and fan.
a) Cross section of the deck
b) Tower details
Figure 2. Details of the bridge deck and towers
Conceptual design of the bridge was made taking special care in aesthetic
considerations. For doing that a very precise CAD model was created using Maya
software [2]. This model contained all visual characteristics of the construction, even
road signals, safety barriers, lampposts and so on. A few digital pictures of the bridge
can be shown in Figure 3.
Figure 3. Visualization of the new bridge over the Coruña estuary
3
Figure 3. Visualization of the new bridge over the Coruña estuary (cont’ed)
3. Structural model and analysis
The optimization process of the cables will require successive structural analysis of the
bridge. Therefore a finite element model with beam elements was generated by the
Abaqus/CAE v6.8 code [3]. As will be seen later, the model properties were not fixed,
instead a parameterization procedure was carried out, making easier to modify model
characteristics necessaries for eventual changes in the bridge design.
Three node bar elements B31 [3] were used to create a 3D structural model of the
bridge. The rotations at the beam ends for the cable elements were released in order to
cancel the transmission of bending moment between cables and deck. Material is steel
for the deck and the masts of the towers and concrete for the lower part of the towers.
Both materials were considered isotropic linear elastic defined by the Young’s module
and Poisson’s ratio.
The bridge will be built by balanced cantilever construction starting from each tower.
Therefore during the deck erection, the prestressing forces in the cables need to be
modified to satisfy the requirement that the deflection of the deck at cable ends cancels
out.
The definite values of these forces will be those corresponding to the end of the
balanced cantilever construction when only the central segment of the deck left. The
weight of that segment can be simulated by two equivalent forces at both tips of the
cantilever. The structural model for that analysis appears at Figure 4.
4
Figure 4. Structural model for evaluation of initial prestressing forces
The value of the prestressing forces is obtained as follows:
1. Deck displacements produced by self-weight without initial prestressing forces
in the cables are calculated at points where cables are attached to the deck.
2. A unit force is applied to the first pair of cables and displacement is obtained at
nodes of the deck. This operation is repeated for the rest of the cable pairs.
3. Prestressing forces are obtained by solving the following system of linear
equations:
n
w j pp Pi w j i 0 j 1,..., n (1)
i 1
where
w j pp : deck displacement j due to the self-weight
w j i : deck displacement at position j due to unit force of the cable pair i
Pi : initial prestressing force in the cable pair i
Once the bridge completed, a structural model undergoing both self weight and live
load needs to be analyzed in order to evaluate the bridge deflections and stresses and
also the elastic stability of the towers that did not have any transverse bracing in the first
conceptual design as presented in Figure 5.
5
Figure 5. Structural model of the full bridge
Elastic stability of the bridge can be found out by solving the following eigenvalue
problem
det[ K E KG ] 0 (2)
where KE and KG are the elastic and geometric stiffness matrix respectively. The lowest
value of λ gives the classical buckling load for the structural model.
The first eigenvalue of the expression (2) was λ=0.334, lower than unity and therefore
indicating that the buckling load was lower than the actual forces acting in the bridge.
Consequently to stiffen the structural model a transverse bracing at top towers vertex
was defined and the eigenvalue problem solved again. The lowest value of λ was
λ=2.013, showing that after the stiffening the bridge model was already safe from that
point of view. Figure 6 shows a part of the structural model and a detail of the tower
after incorporating the transverse beam.
Figure 6. Detail of the tower model showing the transversal bracing
6
4. Formulation of the optimization problem
The objective of this task was to create a computer program able to generate a generic
finite element model of a cable stayed bridge (according to the structural scheme
proposed in the previous section), to be calculated in Abaqus, and from the data
obtained in this calculation, to perform optimization process on the cable cross sectional
area.
The optimization problem is defined by the following elements [4-5]:
Design variables
The design variables are each of the cable cross sectional areas. The number of design
variables is reduced to half of the number of cables due to symmetry about the
longitudinal plane; however, it is not symmetrical about the central plane of the bridge
because the bridge deck is not horizontal as was aforementioned. For better
performance of the optimization process, new design variables, being the inverse of the
areas were used to linearize the stress constraints of the cables.
xi 1/ Ai i 1,..., n (3)
Being n the total number of design variables.
Design constraints
Three types of design constraints were taken into account:
1. Stress constraints of the cables
k ( xi ) M i 1,..., n k 1,..., LC (4)
where:
k ( xi ) : tensile stress in the cable i for the load case k
M : maximum allowable tensile stress in the cables
LC: total number of load cases
2. Displacement constraints of the deck
wk ( xi ) wmax i 2,..., n 1 k 1,..., LC (5)
where:
7
wk ( xi ) : deck displacement at the cable position i for the load case k
wmax : maximum allowable displacement (positive value)
3. Displacement constraints at towers top in the longitudinal direction
u k Towerj umax j 1, 2 k 1,..., LC (6)
where:
u k Towerj : displacement in x direction at the top of the tower j for the load case k
umax : maximum displacement in x direction at the top of the tower (positive
value)
Objective function
The aim of the optimization was to reduce the quantity of steel for the cables. Thus the
objective function is expressed as:
n
1
min F 2 Li (7)
i 1 xi
where:
Li: length i-esime cable
As it can be seen, this is an optimization problem with n design variables, 2n x LC
inequality constraints, and an objective function.
5. Program description
The problem formulated previously was solved by a main program created in MATLAB
[6] using also Abaqus software for the structural analysis.
The program basically consists of three parts that are described later:
1. Generation of finite element models
2. Calculation of prestressing forces in the cables for the self weight
3. Optimization of the cable sections according to the established load
combinations
At the beginning the user introduces the initial geometric and mechanical properties
data of the bridge and the program generates the files that contain the finite element
8
model to be analyzed in the Abaqus code. Then, the next step starts for obtaining the
prestressing forces in the cables corresponding to zero vertical displacement in the deck
and zero longitudinal displacement of the upper part of towers for bridge self weight.
Afterwards, the optimization phase starts and the algorithm modifies the cross section
area of cables. With that values the procedure goes back to find out a new set of
prestressing forces in the cables and moves again to the optimization phase. This
procedure is repeated until achieving the convergence criteria of the algorithm. Figure 7
shows the flowchart of this program.
Figure 7. Flow chart of optimization process
9
Generation of structural model
At this step discretization of each of the elements that compose the structural model is
made and geometrical and mechanical characteristics are assigned.
This part of program consists of three subroutines, “inputtablero”, “inputtorres” and
“inputcables” which are in charge of writing Abaqus code according to the data
provided initially. The number of parameters is quite large, about 30, which allow to
change almost every aspects of the model as well as the span between towers,
longitudinal elevation of the deck, height of towers, mechanical properties of each
element, number of bar elements in deck and towers, number of cables and its
positioning on deck, amongst others.
It is important to note that the initial values of design variables are assigned to each
cable section in this part of the program, specifically in “inputcables” subroutine.
Prestressing forces in the cable
At each iteration of the process prestressing forces at the cables that are consistent with
no vertical displacements of the deck and no horizontal displacements at the towers top
for the bridge self weight are calculated for the current values of the areas of the cables
cross section as mentioned in a previous paragraph.
Optimization phase
The optimization process is carried out using an optimization module implemented in
MATLAB. The function to perform this process is called fmincon that minimizes
functions with various variables subject to inequality constraints using a sequence of
quadratic problems.
This optimization subroutine requires certain input values such as the design variables,
the constraints, and the objective function. For obtaining better results, it also provides
gradients of the objective function and the design constraints. Those gradients are
calculated by finite difference, the most costly part of the program.
6. Application examples
The first example presented is a fictitious case of a bridge with the same longitudinal
profile as the initial bridge described in section 2, but with noticeably reduced number
of design variables (n=8).
10
The second example shows the bridge configuration described at the beginning of the
document. The number of variables in this case is much greater than the previous
example (n=80).
In both examples the optimization of the cable is performed for three simple load cases
as:
• Load case 1: live load of 4 kN/m2 on the left lateral span
• Load case 2: live load of 4 kN/m2 on the main span
• Load case 3: live load of 4 kN/m2 on the right lateral span
Those load cases are not the only ones required by the code of practice they serve to
verify the program performance and the methodology proposed as any other load
combination may be included easily. The addition of new load cases in the program
does not present any difficulty since these cases simply need to be entered in the
subroutine, “casos_carga”.
6.1 Bridge with 16 cables
The simplicity of this example allows studying easily the methodology described.
Figure 8 shows the structural model and the node labels where displacements
constraints are considered as well as the label of the cables.
Figure 8. Node and cable labels in the structural model
11
Design variables
The design variables are the inverse of the areas of eight cables in the model. Initial
value of 0.1 m2 was used.
1 1
xi 0 10 i 1,...,8 (8)
A cable(i ) 0.1
0
Design constraints
The design constraints considered for this example are summarized in the following
equations:
kN
k ( xi ) 800000 i 1,...,8 k 1, 2,3 (9)
m2
L1
wk ( xi ) i2 k 1, 2,3 (10)
550
L2
wk ( xi ) i 3,..., 6 k 1, 2,3 (11)
550
L3
wk ( xi ) i7 k 1, 2,3 (12)
550
Hj
u k Towerj j 1, 2 k 1, 2,3 (13)
550
where L1, L2 and L3 are the length of the left span, main span, and right span
respectively and H1 and H2, the tower height. All dimensions are in metres.
Objective function
The objective function is the total volume of steel cables that is to be minimized.
8
1
F 2 Li (14)
i 1 xi
Numerical Results
The optimization process converged after 28 iterations and the evolution of the cable
areas is shown in Figure 9. The optimum values of the design variables are represented
in Figure 10 and give an idea of the cross sectional area size of cables along the bridge.
12
Figure 9. Evolution of the cable areas
Figure 10. Distribution of cable area
The optimization results lead to larger area for cables attached to both ends of the deck
just as occurs usually in the design of cable stayed bridges. Those cables are the ones
that brace the towers. Likewise, the cables close to the centre of the main span have a
cross section larger than the rest, which can be observed in the dimensioning of this
class of bridges due to limiting the deck deflection. Nevertheless the proper value of the
cross section of each cable could not be found out without optimization procedures.
With regards to the objective function, it can be observed in Figure 11 that the initial
model had insufficient volume so values moved from 400 m3 to an amount close to 900
m3. Evidently, these results do not correspond to any real case since bridges are never
designed with such small number of cables and this example was intended simply to see
the program performance.
13
Figure 11. Evolution of the objective function
After the optimization, it was thorough fully observed how the design constraints were
satisfied. It turns out that the displacement constraints in the main span were active for
the overload case on this part of the bridge. Table 1 shows the vertical displacements of
the deck nodes, the longitudinal displacements of upper part of towers indicated in
Figure 8 for each load case and the maximum values allowed.
LOAD CASE 1 LOAD CASE 2 LOAD CASE 3
Deck node uz (m) |uz|max (m) uz (m) |uz|max (m) uz (m) |uz|max (m)
3 -0.2407 0.49 0.2573 0.49 0.1365 0.49
9 -0.0461 1.19 -0.5627 1.19 -0.0972 1.19
11 -0.6409 1.19 -1.1998 1.19 -0.7240 1.19
15 -0.7108 1.19 -1.1943 1.19 -0.5497 1.19
17 -0.1029 1.19 -0.5938 1.19 -0.0030 1.19
23 0.1869 0.49 0.3617 0.49 -0.2458 0.49
Tower node ux (m) |ux|max (m) ux (m) |ux|max (m) ux (m) |ux|max (m)
20025 0.1046 0.2809 0.2548 0.2809 0.1129 0.2809
20525 0.1046 0.2809 0.2548 0.2809 0.1129 0.2809
21025 -0.1396 0.3013 -0.2848 0.3013 -0.0947 0.3013
21525 -0.1396 0.3013 -0.2849 0.3013 -0.0946 0.3013
Table 1. Vertical displacements (uz) and longitudinal displacements (ux) at the optimum
Tensile stress in each cable for the optimum solution is listed in Table 2 for the three
load cases. It can be observed that with live load acting on a lateral span, only one cable
reaches the maximum stress allowable, while with live load is acting on the main span
two stress constraints, corresponding to two pairs of cables are active.
14
LOAD CASE 1 LOAD CASE 2 LOAD CASE 2
Cable A (m2) N (kN) σ (kN/m2) N (kN) σ (kN/m2) N (kN) σ (kN/m2)
1 0.3620 42724.9 118014.8 77863.4 215074.4 54860.6 151536.1
2 0.0796 63661.7 800001.3 50185.6 630654.6 49528.9 622402.2
3 0.0716 44391.2 620370.1 57244.5 799995.8 44268.5 618655.3
4 0.2432 61030.7 250938.3 76236.0 313457.5 60986.5 250756.5
5 0.2708 62732.2 231680.8 78145.9 288606.2 62780.8 231860.3
6 0.0727 44969.2 618550.0 58160.0 799989.0 45091.9 620237.7
7 0.0744 44742.6 601193.2 43757.5 587956.7 59538.0 799994.6
8 0.3924 59084.5 150556.8 84139.2 214400.2 46755.6 119140.8
Table 2. Tensile stress (σ) and axial force (N) in cables at the optimum
6.2 Bridge with 160 cables
This example corresponds to the structural model described at the beginning of the
document to be built in Coruna including the transversal beam as shown in Figure 6.
Geometrical and mechanical characteristics of towers and deck are identical to those in
example 1 and it differs only on the number of cables and their locations.
Figure 12 shows the evolution of the cable areas during the optimization process along
69 iterations starting from an initial area of 0.1 m2. Numerical results differs from the
other example in terms that the cables at the extreme ends of the bridge have areas much
larger than the rest, and the two cables close to the centre of the main span have
approximately half of that value.
The optimum values of cross section of the complete set of cables are shown in Figure
13 to give an idea of material distribution along the bridge. In addition to that, Figure 14
allows observing the evolution of the objective function along the optimization process.
More than thirty iterations are needed to start the convergence phase, what is an
indication of the problem complexity.
15
Figure 12. Evolution of the cross sections
Figure 13. Distribution of areas
16
Figure 14. Evolution of the objective function
Since the constraints are not violated at any moment, the design requirements are
satisfied. The most limiting constraint is the displacement at the centre of the main span
for the distributed load on that span. Although the stress constraints are not active, some
cables have values vary close to the imposed limit.
The optimization proves to be useful to give more efficient design than that obtained by
heuristic rules.
6. Conclusions
In this work, the results obtained from a study on the optimization of cable cross section
of a cable stayed bridge have been presented.
Some conclusions can be drawn from the results:
1. Due that this type of structure are very statically indeterminate, the optimization
process is not monotonous presenting discontinuities in the design variables as
well as in the objective function. A small modification in the design variables
can affect the fulfilment of constraints far away from the location where the
changes are produced.
2. This design procedure gives an idea to a design engineer about how much cross
sectional area is necessary for each cable under certain stress and displacement
17
constraints. Since these values are not intuitive a priori at all, they serve as a
very good design tool.
3. The distribution of optimized cable cross section along the bridge agrees with
that of actual cable stayed bridges [7].
4. After the optimization process, it can be observed that the areas of cables
attached to the extreme ends of the deck are much larger than the rest of the
cables. Those cables serve to improve the longitudinal bracing of the towers and
at the same time they have an important influence for vertical deck
displacements.
5. The cross section of the cables located close to the centre of the main span have
larger area than the rest , except those at the extreme ends, due to need fulfilling
the deck displacement constraints.
6. The optimization process provides a result where the objective function has a
minimum (local minimum), however, it does not guarantee that it is the best
solution among all the possible ones (global maximum). Nevertheless, as long as
a reduction of steel volume in comparison to other conventional techniques is
achieved, the optimization proves to be very useful to reduce bridge cost.
7. The computer code created allows changes on the finite element model
(geometry, mechanical properties, boundary condition, etc) in a simple way.
Besides it permits to include new load combinations within the optimization
process.
The conclusions drawn here on dimensioning of the cables comply exclusively with the
terms in which the optimization process is defined. This study can be enhanced from
many points of view such as:
1. Including geometry optimization to obtain optimum cable location on the deck
2. Including other structural elements such as the deck, towers, etc as design
variables.
3. Considering other types of structural analysis such as buckling, vibration or
dynamic analysis
7. References
[1] GIMSING N. J., Cable Supported Bridges. J. Wiley and Sons, Second Edition,
1997.
18
[2] Audodesk Maya 2008. Getting Started with Maya, Autodesk, Inc. USA, 2007.
[3] Abaqus/CAE v6.8, Inc. Abaqus/CAE User´s Manual, 2008.
[4] HERNÁNDEZ S., Métodos de Diseño Óptimo de Estructuras, Colegio de
Ingenieros de Caminos, Canales y Puertos, 1990.
[5] ARORA J. S., Introduction to Optimum Design, Second Edition, 2004.
[6] MATLAB v6.5-13. MATLAB Documentation.
[7] Honshu-Shikoku Bridge Authority. The Tatara Bridge. Design and Construction
Technology for the World’s Longest Cable-Stayed Bridge, Japan, 1999.
19
View publication stats
