Computer Based Optimising of the Tensioning of

Cable-Stayed Bridges

Arne Bruer Heinz Pircher Heinz Bokan

Civil Engineer Civil Engineer, Senior Partner Civil Engineer, Partner
Teknisk Data AS TDV GesmbH TDV GesmbH
Oslo, Norway Graz, Austria Graz, Austria

Summary
A numerical approach to reduce the calculation effort when attempting to minimise the number
of stressing operations during the erection of cable-stayed bridges is shown. The proposed
method is illustrated with sample calculations from a small example and from the Uddevalla
bridge which is currently under construction.

The Problem

The solution for the optimum tensioning strategy for long span cable-stayed bridges can be an
extremely tedious and time-consuming process for the following reasons:

Practical reasons

• Tensioning one cable affects the forces in all the other cables
• Cables can not, in reality, withstand compressive forces but stressing an adjacent cable may
apparently cause this condition.
• Stressing of the stay cables is an expensive procedure due to the difficulty in the cable
stressing procedure.

Analytical reasons

• A minimal cable tensioning strategy whilst saving a considerable amount of time and money
during the construction phase greatly complicates the analytical phase of the design process.
• Definition of the tensioning strategy is interrelated with the chosen erection method and the
simulation of the erection procedure using the structural model can be very complicated.
• The deck girder and pylon system must behave reasonably during all phases of construction.
i.e. the deflections should be neither excessive nor incompatible with type of construction.
• Creep and Shrinkage (applicable where some or all the bridge elements are concrete or
partially concrete members) greatly complicates the analytical process.

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• “Uplift conditions” could exist at the temporary supports which further complicates the
analysis. Whilst special “Non-tension” members could be used, this greatly increases the
degree of indeterminacy and hence the speed of analysis.

All of the above practical and analytical reasons obviate the need for a consistent, standard, non
trial and error type approach to the solution for these complex structures. It is possible to use a
unit load system of analysis tied to the bridge construction method, relate this to an estimate of
the max/min final live load moment envelope and through this, where possible, minimise the
cable stressing operations.

The proposed method will always achieve a solution, which must then be checked for structural
consistency by the user. The results can be structurally unacceptable as the solution is directly
achieved from a set of simultaneous equations. The structural unacceptability may arise from
such things as “compression in the cables” or “unacceptably high stresses” etc.

• Structurally acceptable results clearly demonstrate that the parameters chosen to define the
structure and its construction are correct and also define the required tensioning and
construction strategy
• Structurally unacceptable results will point the way for modification of the parameters to be
used in the next analysis. (The modification would typically be to the “Ideal Bending
Moment Diagram” – Refer below)

Choosing the System and manipulating the Moment Diagram.

The basic bridging system must be chosen and optimised before the stressing strategy design can
be found. The system is chosen through a series of considerations such as required bridge
functionality, availability and cost of materials, Clients requirements etc. The bridging system is
considered, from this analytical viewpoint, as basic given information.
Integral with the bridging system choice is the concept that almost any moment diagram can be
achieved in the deck and in the pylon by adjusting the following Degrees of Freedom:
• The tensioning forces in the stay cables and their stressing procedure
• The support movements (translation – longitudinal and vertical)
• Prefabrication shape of the deck girder and the pylon
• The erection procedure of the deck and the pylon

Finding the “Ideal Moment Diagram” for dead load.

Once the basic information has been defined in principle, the effects of the traffic / pedestrian
loads and any additional loading – balustrade / guard-railing / surfacing / etc. on this
“fundamentally defined” structure can be estimated.
The load capacities of the deck sections and the pylon sections can then be compared with the
live plus additional dead load envelopes and the “ideal” dead load force diagrams may then be
defined. The sign of the “ideal” dead load force diagrams may well be of opposite sign to the
diagrams resulting from normal load directions. This demonstrates the distinct advantage of
cable supported structures where the initial dead load moment diagram can be easily manipulated
to suit the design needs.

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Establishing the “Unit Force Equations” Principles

General

When the “ideal” dead load force diagrams have been defined then the system of unit forces can
be mathematically equated to these “ideal” dead load force diagrams.
The process for defining the tensioning sequence and amount, the deck and pylon construction
sequence as well as any required deck/pylon prefabrication effects then begins.
Process Principle:
• The unit loading system is first defined for the final stage structure in order to establish
reasonable member sizes. This process usually involves certain re-definitions of the member
sizes and program re-runs to prove structural integrity.
• Once reasonable values have been achieved the “unit force method” can be extended to the
construction stage analysis.
• Each construction stage can be checked and proven for design compliance.

Degrees of Freedom

The most commonly selected unit forces or Degrees of Freedom (N.B these “DOF´s” are not the
same as the structural system DOF´s) in the structural system include:
• A unit shortening of the cable (causing an axial cable tension) – or a unit tensioning causing
an axial cable shortening.
• A unit translation of a rigid support (transverse or longitudinal movement at a pier or
abutment support). – A longitudinal force applied at the end of the deck changes the
moments by changing the cable forces which act on the deck.

Setting up the “Unit Force Equations”

• Define the unit loading cases and the “ideal moment diagram”. The same number of unit
loading cases must be defined as the number of “Fixed Moment” points chosen on the
structural model to represent the “ideal moment diagram” (or vice-versa!).
Principles in Example below:
• The “ideal dead load bending moment diagram” is defined for the deck girder by bending
moments at 9 points along the girder (positions A, B, C ...... I).
• Nine unit loading cases are selected for setting up the simultaneous equations.
• The 8 unknown “required” stay cable forces – chosen in this case to be 1000 kN.
• One unit translation at the end support – chosen in this case to be 50 cm settlement.
The solution to the equations (the unknowns) will be the factor by which the unit loads should be
factored to achieve the “ideal dead load bending moment diagram”.

Note: There is no fixed prescription for the selection of the unit loading cases. The designer is
free to choose whatever he wishes. This flexibility is demonstrated in the example by the non-
selection of the two stay cables adjacent to the pylon (positions E and F).
The 1000 kN unit cable force was selected to be of the same order of magnitude as the final
cable force because of 2nd Order considerations. (Refer below for further description of 2nd Order
effects.)
The “ideal moment diagram” for dead load is given below and is very different from the dead
load bending moment diagram (MP) which would result if the loading was applied to the
structure with un-stressed cables.

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N.B. Care must be taken in the selection of sensible and unrelated “ideal moments” as if one is
related to another (i.e. dependent on it) then a singularity in the equations will result and
there will be no solution. Provided there is no singularity, a solution will be reached.

The following system of linear equations is set up:

MA = MP + MT1=1 . X1. + MT 2 =1 . X 2+ ........ MT 8=1 . X 8+ MT J . X 9
. .
. .
MI = MP + MT1=1 . X1. + MT 2 =1 . X 2+ ........ MT 8=1 . X 8+ MT J . X 9

MA....... MI Final stage moment at the current position (including tensioning + jacking).
MP Permanent load moment at the current position (without tensioning or jacking).
MT1=1... MT 8 =1 Bending Moment due to each unit tensioning at the current position.
MJ Bending Moment due to unit jacking of the end support at the current position.

The X 1 ...... X 9 factors set up in the unit loading cases are the unknowns in the set of linear
equations which are found by the solution of these equations.
Note that the system of equations is not symmetrical and the diagonal coefficient may be zero.
This is to be considered when solving the equations. This basic solution defines the cable forces
and the jacking force for the final stage and, at the moment, does not include the effects of:
the sequence of construction stages, the creep, 2nd Order Theory or the non linearity of the cables
due to the sagging effects.
The basic principles must therefore be extended to accommodate these effects.

Construction Stage analysis.

A similar system of unit loading cases can be defined for the construction stage analysis. The
unit loading cases are, in this case, applied to the different structural systems which exist at the
individual construction stages. The sketches below, which show a few of the construction stages,
demonstrate the principle of the method of analysis. The loading cases for each construction
stage are combined to form the set of simultaneous equations which must be solved to find the
required multiplication factors for the unit loadings.

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Creep – a rational linear approach

There is a general belief that creep design is a non-linear problem and therefore it is often
approached in an empirical manner using some “rules of thumb” or “past experience” to assess
its affects. This approach is particularly prevalent where the structural concrete is subjected to
the many and varying loads which occur during the multiple construction stages of large cable-
stayed bridges.
Taking account of creep effects using the CEB-FIP model code is even more complex than was
the case using more traditional methods. Inspite of the above statements it can be shown through
a series of mathematical equations that the effects of creep can be treated in a linear manner.
The derivation for the effects of creep is founded on the known fact:
{εe} . φ = {εc} (Elastic Strain * Creep Factor = Creep Strain)
Decomposing the structure down to element level, the above equation is applied to each
individual element by applying the generalised displacement method rules for calculating initial
strain type loads:
Define {εe} over the element.
Define {εc} = {εe} . φ over the element.
The member end displacements {δc} are found by weighted integration of the strain vector over
the element length in the usual way.
The member end forces are calculated and the system of equations are assembled and solved for
nodal displacement {δ} in the usual way.
({δ} - {δc}) . [k] = {FI} gives the internal forces due to creep.
The system of analysis is completely linear up to this point.
In the specific case of creep, cognisance must be taken of the age differences in the concrete as
well as the various ages of different parts of the structure at the time of each increment of load
application (εe is no longer constant but varies with time). A finite difference approach in time is
applied here and using a linear variation over a time interval, we can say:
{εt} = {ε0} . t 1 – t +{ε1} . t– t 0
∆t ∆t
This equation can then be put into the basic displacement equation: at time t0 , {εe} is known
and then by solving the equations for {δ} at time t = t1 a recursive formula can be derived which
results in a linear relationship at time t1 such that the equation including the effects of creep is
the same as the original equation with the exception that the modulus E is replaced by
E A detailed description of the whole procedure is given in Ref. 2 and the
1+φ*0.5 theoretical background for the finite difference approach to solve “initial value
problems” is described in Ref. 3.
The essence of the above statement is, that all the creep influences on the final distribution of
internal forces and displacements are related in a linear manner to the elastic strain which itself
initially caused the creep.
The principles of linear superposition may, in consequence, be applied and the total creep
occurring during a single time step may be decomposed into single contributions:
Considering one of the prescribed ideal moment positions:
M creep = M p + M c t =1 . X 1 + M c t =2 . X 2 + M c t =3 . X 3 .............etc.
M creep therefore consists of one part which is related to the permanent load and the other parts
are related to the unit loads described above which are linearly coupled to the same unknown
factors X 1 ...... X 9.
As before the basic concept can now be applied; the effects of creep for permanent loads and for
unit loads are decomposed into separate contributions from each time interval and then summed.
The system of equations for defining X 1 ...... X 9. therefore remains linear. The only

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approximation made is the assumption that the behaviour within any single time step is linear
which is consistent with the usual application of the “finite differences in time” approach.

Second Order Theory and cable non-linearity (due to cable sag)

Since the element stiffness depends on the axial force (in the case of 2nd Order Theory as well as
for cable sag), the basic displacement method equations become non-linear. The equations
defining the solutions for X 1 ...... X 9, which were proved to be linear for the creep case above,
also become non-linear. An iterative approach must therefore be applied:

The Simple Approach

• Estimate X 1 ...... X 9, and use this estimate of the unknowns to find the variable stiffness
which on substitution into the equations hopefully gives a solution which is close to the final
behaviour.
• Correct the estimate of X 1 ...... X 9 and calculate again.

A Better Approach

Use the tangent stiffness for calculating the influence of the application of a small increment to
each unit loading case. The equations can then be transformed to define the iterative correction
for X 1 ...... X 9 and a procedure such as the Newton Raphson method can be set up.

The tangent matrix for the 2nd Order Theory or even large deflections (with respect to suspension
bridges) can be similar to that usually applied in the “Large Deflection Theory”. E.g the
corrective term N/L is added into the appropriate position in the element stiffness matrix. The
cable sagging effects can be accommodated by deriving dS/d ∆x from the well known “Peterson
Formulae” (Ref. 4). Where S means the Cable force and ∆x is the cable extension. Convergency
is accelerated and guaranteed, when using the tangent matrix with the Newton Raphson approach
as long as a real solution exists.

The Results from the sample analysis.

This particular example was chosen not only to demonstrate the principles of analysis but also to
demonstrate the effects of 2nd Order Theory and of creep on the structure.

The results from a few selected points have been chosen for demonstrating these principles:

Final Stage cable forces (kN) resulting from the different analyses

Cables 1st order theory 2nd order theory 2nd order theory
& creep – no creep & creep
Posn E 1073.9 1079.8 718.6
Posn F 1000.8 1003.5 663.9

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Initial Stage cable forces (kN) resulting from the different analyses

Cables 1st Order Theory 2nd Order Theory 2nd Order Theory
& creep – no creep & creep
Posn B 1775.52 1484.44 1833.65
Posn I 1788.02 1468.22 1840.82

Pylon Moments (kNm) resulting from the different stage analyses (design system)

Construction 1st Order Theory 2nd Order Theory 2nd Order Theory
stage & creep – no creep & creep
1 -500.00 -41 -503.0
2 -471.0 -32 -472.0
3 -466.2 -33 -464.0
4 -463.1 1493 -538.0
5 -2151.6 -617 -2413.0
Final -2019.0 -617 -1977.0

Pylon Moments (kNm) resulting from the different stage analyses (1st system)

Construction 1st Order Theory 2nd Order Theory 2nd Order Theory
stage & creep – no creep & creep
1 0 - 0
2 0 - 0
3 0 - 0
4 10492.4 - 15657.0
5 10651.0 - 15567.9
Final 244.1 - 1126.0

Minimum Deck Girder Moment Envelope (kNm) (design system)

Construction 1st Order Theory 2nd Order Theory 2nd Order Theory
stage & creep – no creep & creep
1 -4338 -4851 -4347
2 -4129 -4877 -4223
3 -4194 -4835 -4228
4 -3951 -5018 -3982
5 -3792 -5089 -3819

Maximum Deck Girder Moment Envelope (kNm) (design system)

Construction 1st Order Theory 2nd Order Theory 2nd Order Theory
stage & creep – no creep & creep
1 1876 1238 1935
2 3232 2420 3401
3 2906 2756 3098
4 3675 2860 3775
5 3906 3146 4018

These above results highlight:

• The importance of accurate creep action assessment and shows that creep effects are critical
to the structural integrity and must be accurately calculated and can not simply be assessed

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 from some “arbitrary rules”. It can be seen that the creep in the deck affects the cable forces
which in turn affect the deck and pylon moments significantly. The pylon moments are
modified to such a degree that they are even reversed in construction stage 4.
• The importance of consistent construction stage checks as the moments in the pylon, whilst
being quite acceptable in the final stage are excessive in the construction stage under the 1st
system of analysis.
• The significant changes to the pylon moments caused by 2nd Order effects. (The pylon is
highly compressed and therefore sensitive to additional moments from the deflected shape).
• The easy parameter design check:
• Whilst a solution to the 1st system of analysis was found, the pylon failure in construction
stage 4 & 5 was easily identified.
• Inspection of the 1st system showed that the translational fixity at the pylon was the cause
of the excessive moments. Removal of this fixity proved to be an adequate modification
to the design system.

Construction Stage Analysis – forwards or backwards?

The traditional method of carrying out the construction stage analysis is to start at the “Final
stage structure” and gradually reduce the structure (going backwards) stage by stage until the
first construction stage is reached.
It is argued that this method is the most likely to achieve the fastest result as it starts from a
structurally correct solution - the final stage – which may possibly have been defined using the
unit load method described above. Whilst this argument does have much merit, it falls down
when problems are subsequently found at a particular construction stage. The check then reduces
to a trial and error method. Using the proposed unit load method, the forwards or backwards
solution are equally possible and equally simple as even the creep principles described above can
be adapted for the backwards solution by solving “the equations” for ε0 instead ε1.

The Uddevalla Cable-Stayed Bridge

A bridge which was designed using the principles described above is the Uddevalla Cable-
Stayed Bridge.
This cable-stayed bridge is the central part of a continuous 1712m crossing over the Sunninge-
sund waterway between Udevallamotet north and Udevallamotet south in Sweden.
The approach viaducts, comprising twin steel box girders with a concrete slab, are rigidly
connected to the main bridge on either side and provide overall longitudinal structural stability.
The cable-stayed bridge portion comprises a 414 metre main span, symmetrical side spans of 179
metres and two 85 metre high (above the deck girder) diamond shaped concrete pylons which
anchor the fan shaped stay cable arrangement. The stay cables, which support the bridge deck on
either side, are anchored at 13.32 metre centres in the longitudinal direction.
The bridge deck structure carries 6 lanes of traffic and comprises a composite, open steel grid
structure with a 240 mm thick concrete top slab which spans longitudinally over the diaphragms.
The deck edge beams (I type beams) also have a thin walled shell structure connected to the side
which in addition to acting as a wind spoiler provides some torsional stiffness to the edge beams.
More comprehensive descriptions and details of the Uddevalla Cable-Stayed Bridge can be
found in Ref. 4.
Given below is a summary of the principles used in the analysis of the Uddevalla Cable-Stayed
Bridge using the unit load method:

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The “Degrees of Freedom” (or unknowns) chosen for the unit load analysis were:
• All the stay cables – a unit tensioning
• Translation at one cable-stayed bridge pylon support ( “X” and “Y” directions)

The Uddevalla Cable-stayed Bridge construction requires a 3 stage stressing procedure:

Stage I stressing provides support for the new steel portion of the deck during assembly. The
cables are initially stressed to provide support and to counteract excessive deflection before
making the welded connection to the existing deck.
Stage II stressing provides support for the whole structural self weight comprising the steel plus
pre-cast concrete top slab elements. The procedure is simple as the stressing jacks from the
previous stressing operation are still connected. The first Unit load analysis to find the cable
forces is carried out at this stage.
Stage III stressing is required for counteracting the superimposed dead load and creep effects on
the pylon deflection. The procedure is required because the stringent minimal pylon moment
criteria precludes a sufficient pylon pre-camber. The second Unit load analysis to find the re-
tensioning cable forces is carried out at this stage.

The “Ideal Moment diagram” chosen for the initial dead load is shown below together with a
general bridge arrangement. Note the unusual shape in this “Ideal Moment diagram” in the deck
girder was dictated by a strict limitation prescribed for the pylon moments which takes
cognisance of the “very severe” environmental conditions for reinforced concrete
weathering/corrosion. In order to comply with this stipulation, the “Ideal Moment diagram” had
to include a minimal moment condition in the pylon.

Ideal Moment Diagram

The diagrams below show the time dependency of a few characteristic results from the analysis.
The time axis is not to scale but shows a sequence of the different actions. Stages 1-16 are the
cable tensioning and deck cantilevering stages. The deck construction is complete at the end of

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stage 16, the additional dead load is applied in stage 17 and stage 18 is for creep and shrinkage
up to “time infinity”.

Cable force variation in side span cables 5, 6, 7 and 8 (numbering from the pylon)

Main span moment variation at cable 2 north and cable 2 south (numbering from the pylon)

Moment variation in pylon at the top of the footing

References:
[1] C. Hansvold, Sunnungesund Cable-Stayed Bridge, IABSE Symposium Kobe 1998
[2] Heinz Pircher, Finite Differences to simulate creep and shrinkage in pre-stressed concrete
and composite structures, Proceedings of the Int. Conf. on computation Modelling of
Concrete Structures, edited by H. Mang, N. Bicanic, R. De Borst, Pineridge Press 1994
[3] O. C. Zienkiewicz, R.L. Taylor, The finite Element Method, Fourth Edition Volume 2
[4] C. Peterson, Abgespannte Maste und Schornsteine Statik und Dynamik, Berlin: Ernst +
Sohn 1970

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