The Cable-Stayed Alamillo Bridge, Spain

A Combined Method for Measuring Cable Forces:
The Cable-Stayed Alamillo Bridge, Spain
Summar
The paper presents a combined method for measuring the forces in the cables of
cable-stayed bridges based on measuring the natural frequencies of the cables in
a free-damped vibration and subsequent calculation of the forces using simplified vibrating chord theory. The theoretical background and the practical and
economic advantages of this method are discussed, as well as its most relevant
limitations. The method is applied to an existing bridge in order to demonstrate
that it can obtain reliable values for the forces in the cables.
Juan R. Casas
Assoc. Prof.
Technical Univ. of Catalunya
Barcelona. Spain
Juan R. Casas. born in 1960. received his
civil engineering degree from the Technical University of Catalunva (UPC),
Barcelona. Spain. in 1984. where he completed his doctorate in 1988. Since then,
he has been professor of bridge engineering at the School of Civil Engineering
at UPC. His research interests include the
design. dynamic analysis and field testing
of bridges, as well as bridge safety and
reliability.
Introduction
If the vibration method is to obtain accurate results in these circumstances, it
During the construction and service
life of a cable-stayed bridge, it is ex-
must be combined with other mea-
the forces in the cable stays. This
knowledge is critical to the correct
surement techniques, each. however,
with its own inherent errors or deficiencies. A methodology for combining measurement techniques which
alignment and distribution of internal
forces in the finished bridge. At pre-
compensates for the errors of one
method with the results of another.
sent. there are several techniques to
and vice versa, is proposed in this pa-
assess these forces. including measurement of the force in a tensioning jack,
per.
tremelv important to accurately define
application of a ring load-cell, topographic measurements, elongation of
the cables during tensioning and installation of strain gauges in the
strands. In spite of their simple theoretical bases, each of these methods is
highly complex in its practical applica-
tion. Moreover, some of these techniques are extremely expensive to apply and, in any case, produce results of
insufficient accuracy [1, 2].
A relatively simple, quick and inexpensive method of measuring cable
forces in cable-stayed bridges is to use
the vibrating chord theory, obtaining
the cable force from the cable's natural
frequency of vibration, the mass and
the real length. This method has been
successfully applied [1, 2] to relatively
short cable-stayed bridge cables —
mainly locked and with very simple anchoring devices — as well as to external
prestressing cables [3].
However, as cable length increases
+r
Peer-reviewed by international
experts and accepted by the IABSE
Publications Committee
Structural Engineering International 4/94
and/or the anchorage devices become
more complicated — as with cables
formed by a great number of strands,
where the deviators, dampers, etc.
make it more difficult to obtain accurate knowledge of the cables real vibrating length — the actual vibration
behaviour deviates substantially from
the vibrating wire theory.
The proposed methodology uses a
least-square minimisation procedure
to minimise global measurement error
and enable a usefully accurate definition of forces in stay cables. The
method is applied to the cable-stayed
Alamillo Bridge in Sevilla, Spain. in
order to check its feasibility and reliability.
The Alamillo Bridge
The Alamillo Bridge. completed in
1992. is a 200 m long cable-stayed
bridge with a 134 m tall pylon inclined
32° and 13 pairs of parallel cable stays
forming a harp shape (Fig. 1). The
deck is an hexagonal steel box girder
with a depth of 4.40 m. The width of
the steel box is variable. Every 4 m
there are two lateral cantilevers
13.20 m wide which are formed of steel
ribs. These cantilevers support a reinforced concrete slab carriageway
23 cm thick, with walkway above the
upper flange of the steel box (Fig. 2).
Each cable connecting the deck to the
pylon consists of 60 strands of 0.6" di-
ameter, except the last pair (cables
no. 13), which have 45 strands 0.6' in
diameter. The pylon is a composite
(steel + concrete) structure. Its variable cross section contains a circular
Science and Technology
235
void (Fig. 3). Both deck and pylon are
embedded in a base which, in turn, is
supported by 54 piles, 2 m in 0
and
47.5 m long.
The Bridge and the Method
In
a cable-stayed bridge with this
structural configuration. where the
load in the deck is balanced by the
weight and inclination of the pylon by
means of the cable-stays. it is fundamental to know the actual cable forces
or. at least, to know if they are within a
definite tolerance bandwidth. Any
small modification of cable forces re-
sults in a large variation in internal
forces both in the pylon and in the
deck.
Fig. 1: The cable-stayed Alamillo Bridge
Fig. 4 shows the effects of a 10% increase or reduction in cable forces on
the bending moments of the pylon and
the deck. A reduction of only 10%
causes a variation in the bending moment of the bottom part of the pylon
from 600 MNm to 1228 MNrn (more
than 100%) and from —224 MNrn to
16000
i2Oor0
250
10500
It—
325 608
—I
Ii000I
I—
I
1875
—434 MNm (almost 100%) in the maximum moment of the deck. Even more
-t
'4,
'0
-
dramatic is the variation in maximum
negative moment in pylon (from —167
to —647 MNm. i.e., 287%) when cable
forces are 10% higher than the design
forces for the permanent state (dead
load plus permanent load).
1623
Fig.
2: Cross section of the deck (mm)
Limitations of the Vibration
Method
Before discussing the application of
the method to the Alamillo Bridge,
Notation
some discussion of the method's limitations is useful. If the ideal vibrating
A
chord theory is assumed for a com-
E
=
=
=
=
f
F
H=
J
k
L
=
=
=
L* =
I
rn
s
a
=
=
=
=
=
a0 =
a'
236
=
=
=
=
section area of structural material of the cable
modulus of elasticity of cable material
natural frequency of vibration
force in the cable
horizontal component of force in the cable
sum of quadratic errors
relation between weighing and resistant area of the cable (k 1)
horizontal distance between vibration nodes in the cable
sum of horizontal distances between dampers and vibration nodes
cable length between vibration nodes
mass of the cable per unit length
sag ratio
weighting factor
angle between horizontal and cable tangent at anchorage point
angle between horizontal and cable chord connecting endpoints
relative error
relative error in cable force =
tension in the steel of the cable in the permanent state
angular frequency of vibration (= 2rJ)
cross
Science and Technology
plete horizontal cable, the cable force
Fcan be deduced using [4]:
F=
41fm
(1)
In an inclined cable (Fig. 5), the cable
force can be evaluated using [5]:
H = 4Lf2m
H
4Lf2m
F=
cosa
cosa
(2)
Moreover, if the vibrating wire theory
is applicable, there is a linear correspondence between vibration modes
and natural frequencies (Fig. 6) resulting in
a so-called non-dispersive"
model. In both cases the main assumptions of the theory are:
Structural Engineering International 4/94
_____—
— Dead load + Permanent load + Design cable forces
cable has a negligible flexural
stiffness (i.e., is perfectly flexible).
As a consequence a perfect hinge
can be assumed as the bearing condition at the cable ends.
— The
—
- - - - Dead load + Permanent load + 0.9
Dead load + Permanent load +
*
Design cable forces
1.1 *
Design cable forces
There is no relative displacement of
the points where the cable is an-
1228
chored. i.e., the possibility of coupling between vibration modes of
pylon or deck (in a cable-stayed
bridge) and the cables themselves
does not exist.
—
The transverse in-plane deflections
of symmetrical modes do not gener-
ate additional tension in the cable
(the cable is inextensible).
A detailed study dealing with the effects any deviation from the theoretical straight line in Fig. 6 is presented in
[2]. For instance, if a relative displacement between anchoring points exists
(because of bending of the deck or py-
lon). the natural frequencies of the
lowest modes are slightly greater than
when the anchoring points are fixed.
Values in MN
m
Fig. 4: Variation in bending moments due to ±10% variation in cable forces
fact. an error of only 0.5% for a cable
with a natural frequency of 0.5 Hz requires a total signal length of
Also if the cable has a more than a
1
negligible flexural stiffness, the natural
Ltf
frequencies of the higher modes are
0.005x0.5
mated cable force [4] using Eq. (1). the
factor mgL = pL must have a maximum value as expressed in Table 1, de-
pending on sag ratio (s) and cable inclination in respect to the horizontal
(a0). Table I has been derived with a
=400s
stress of 650 MPa in the steel cables.
greater than predicted by the vibrating
(or 6 minutes), which is not realistical-
chord theory. A study [3] concludes
that Eq. (1) can only be applied if the
seven lowest frequencies, measured
ly available for a free-damped vibra-
If a relative error instead of absolute
tion of a real bridge cable.
error is considered, an equation can be
with a 0.5% accuracy, lie in a straight
line as in Fig. 6. However, for long cables (with low natural frequencies) it
can be difficult, if not impossible, to
achieve such a level of accuracy. In
933.6
-J
Concerning the influence of cable inextensibility, which affects the symmetrical modes, the natural frequencies of symmetric modes of in-plane
deflections are embedded in the roots
of the equation [5]:
derived relating the relative error in
the cable force (because of neglecting
4(3
2 22
tan— = — — —I —
2
—
(.o=a)
mL2
•1
-L
EA
H)
H(1+852)
(3)
Fig. 5: Definition of terms in an inclined
cable
where
s = sag ratio of the cable = eIL (Fig. 5)
while the antisymmetric modes have
frequencies:
w,1
=2nir1—-- n=1,2,3...
mL
(4)
according to vibrating chord theory.
34
Fig. 3: Typical pylon cross section (cm)
Structural Engineering International 4194
The influence of cable extensibility is
5
2
negligible for short cables, because A
Mode number (n)
tends to 0, and therefore i = ir but
could lead to important errors for longFig. 6: Relationship between ,nodes and frecables. In fact, in order to achieve val- quencies of vibration in the vibrating chord
ues with ± 30 kN accuracy in the esti- theory (non-dispersive model)
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237
sag rati o (s)
aO
0.004
36.7
33.9
27.7
0.00 1
200
146.0
134.6
109.9
77.7
40.3
0.06
30°
45
60°
75
90°
0.007
0.009
21.2
16.8
19.6
15.5
16.1
11.5
12.8
9.2
5.0
0.57
19.7
10.3
6.2
0.45
0.26
due to the error involved in the hy-
different values for the cable forces de-
pothesis of cable inextensibility. it is
unrealistic to look for very high accuracies in the experimental measurement of the fundamental frequency of
pending on the method, it was finally
decided to perform a vibrating test of
the cables in order to determine the fi-
long cables, which in turn is difficult to
achieve because of their low funda-
the tolerance limits stated in the de-
mental frequency.
sign.
Correct application of the vibrating
An accelerometer was attached to the
lower end of the cable-sheathing pipe.
The excitation of the cable was
achieved by releasing a hanging weight
in a rhythm similar to the natural frequency of the cable. The recorded signal was checked in situ by means of a
spectrum analyser and the time signal
was recorded and digitalised for later
chord theory requires accurate knowlTable 1: Values of pL (k.\YmXni) as afunclion of a0 and s
tension increment) and cable characteristics (with E = 190000 NIPa):
tanx=x—
=
4acosa0
1.216x
.(2 + EF)
(5)
Consequently. extensibility is important on such long cables. According to
Eq. (5) — assuming an average value
for long cables of cr = 650 MPa. a0 =
order to achieve an accuracy of
0.5% in frequency (1% in tension) it is
necessary to establish a value for s:
30° — in
x= Jr/4(2+0.01)= 1.57865
4x56292
—127.32=1.57865—
1.216 x 109s2
x3.934 =s=2.3x 10-s
pL
8H
7.85kAL
8acosa0A
edge of the real length of the cable.
With the diverse elements that are
found in normal anchoring devices, determining this important factor can itself be a challenge.
Experimental Estimation
Methods
-
10's
132
=L=—
k
The epoxy-coated strands used in the
cables of Alamillo Bridge do not permit the attachment of strain gauges directly over the strand wires, as can often be done on other bridges. It was
decided to place strain gauges at the
anchorage blocks (Fig. 7). The intention was to correlate strain in this zone
with the cable forces by means of a finite element model. However, the extremely complex strain and stress field
of the anchorage zone, and the high
sensitivity of the results to the boundary conditions assumed in the mesh, as
derived by the finite element study,
rendered results of limited reliability.
Even in the most favourable case of
During the tensioning of the cables.
the pressure in the hydraulic jacking
k = 1 (normally k> 1). the cables with
pump and the elongation of the
L> 132 m have an error greater than
0.5% in frequency because of extensibility. This effect is less important for
strands were also measured. Because
of the problems detected with these
two measuring techniques during construction of the bridge, which led to
higher frequencies. As a consequence.
nal cable forces after completion of the
bridge and to check if they were within
analysis.
Application of the Combined
Method
Starting from the acceleration records
and via FF1' (Fast Fourier Transformation). the frequencies of the different
vibration modes were deduced, as appear in Table 2. Due to the excitation
procedure and damping characteristics
of the cables, the total recorded length
was not long enough to achieve a high
resolution, as also presented in Table 2
(values of M). The shorter cables vibrated for a shorter period of time and
thus yielded shorter recorded lengths.
Table 2 further shows how in the
longest cables the fundamental frequency is a slightly higher than the i-th
natural frequency divided into i, show-
ing that the non-dispersive model of
the vibrating wire theory (Fig. 6 was
not achieved. This dispersion pattern
was hardly detected because of the inadequate resolution in frequency
achieved. In fact. the cable most prone
to this phenomena is no. 12.
To detect the effect of cable extensibility would require a measured frequency error of less than 3%, as shown be-
low. In fact, this is the difference between the values for the fundamental
frequency derived using Eq. (2) or (3).
For this cable, the characteristic parameters are
a0 = 24.518°. m = 77 Kg/ui.
L = 236.1 rn and H = 4028 kN.
deriving the values s =
0.006064 and
A = 0.75954
and, therefore.
fi =
0.484
from Eq. (2) and f
= 0.499
in Eq. (3).
Fig. 7: View of the anchorage block in the deck and the position of the strain gauges
238
Science and Technology
To detect this error of 3% would require a frequency resolution of .\f =
Structural Engineering International 4/94
Cable
1L
2L
3L
4L
5L
6L
7L
8L
9L
IOL
11 L
12 L
13 L
1R
2R
3R
4R
5R
6R
7R
8R
9R
10 R
11 R
12 R
13 R
f
\f
fi
f:
0.050
0.026
0.015
0.020
0.015
0.020
0.015
0.015
0.015
0.015
0.015
0.015
0.015
2.20
1.69
4.34
3.38
2.83
2.26
1.97
1.7
1.54
1.38
0.050
0.030
0.020
0.020
0.020
0.015
0.015
0.015
0.015
0.015
0.015
0.015
0.015
2.29
1.42
1.13
0.99
0.84
0.77
0.69
0.63
0.56
0.52
0.46
0.50
1.25
1.11
1.02
0.91
0.99
—
5.04
4.23
3.34
2.95
2.54
2.32
2.07
1.86
1.66
1.52
1.36
1.49
Li
f
7.99
6.75
5.64
4.47
3.93
3.37
3.09
2.77
2.49
2.21
2.02
1.82
1.99
9.68
8.39
7.03
5.58
4.92
4.23
3.85
3.45
3.12
2.76
2.54
2.26
2.48
8.83
—
f
f6
11.48
10.05
8.47
6.68
5.89
5.07
4.62
4.15
3.74
3.32
3.04
2.73
2.98
—
—
Li
—
13.43
—
Li
—
—
—
tance between cable dampers. the
forces in column 6 of Table 3 were deduced. These values are always lower
when compared with he values derived
from the other two experimental techniques used: jack pressure during tensioning and strain gauges placed on anchorage (Table 3). Nevertheless, there
4.84
4.37
3.87
3.54
3.18
3.46
8.92
7.83
6.77
6.16
5.52
5.0
4.43
4.04
3.62
3.96
—
—
—
—
—
—
—
—
—
they are almost equal for right and left
cables. Therefore, the combined
methodology is challenged to identify
7.76
8.84
7.67
6.74
6.06
5.48
4.93
4.43
3.99
3.62
3.98
—
an inaccuracy that is common to the
7.81
—
5.93
5.41
—
—
7.63
6.94
6.20
5.61
4.98
4.56
4.07
4.45
is a clear relationship between the
forces deduced with the three methods. In fact. the greater the force value
deduced from the frequency. the
greater the force deduced with the other two methods.
Moreover, the experimental values of
frequency are very reliable because
4.55
3.42
2.86
2.25
1.93
1.69
1.52
1.38
1.23
1.71
1.43
1.12
0.96
0.86
0.77
0.69
0.64
0.55
0.52
0.47
0.50
1.11
0.98
0.91
0.99
6.73
5.16
4.29
3.35
2.87
2.54
2.27
2.07
1.87
1.66
1.50
1.35
1.49
—
8.56
7.12
5.57
—
5.69
4.45
3.84
3.38
3.06
2.76
2.49
2.22
1.99
1.79
1.99
10.3
8.52
6.68
5.74
5.06
4.56
4.12
3.72
3.32
3.0
—
4.25
3.79
3.45
3.10
2.79
2.48
2.26
2.48
—
5.91
5.3
4.81
4.34
3.88
3.5
3.17
3.48
2.7
2.98
—
7.58
6.82
6.19
5.55
4.99
4.48
4.06
4.47
Table 2: Frequencies of vibration of the cables of the A larnillo Bridge as deduced from the
vibration test. L = cable on left side; R = cable on the right
0.03 x 0.455 = 0.013 Hz, which is roughly the resolution achieved. For this rea-
Assuming an exact evaluation of cable
length and mass
son, the influence of cable extensibility
=
is clear in this cable. For this cable.
with an experimental measured value
f1 = 0.47 Hz. the average measured
natural frequencies of modes 1—7 is
f = 0.455 Hz, resulting in = (0.47—
0.455)/0.455 = 3%, as previously evaluated. For this reason, jointly with the
error in the first natural frequency due
to its small value in the long cables,
and the short record length in the short
ones, and the fact that at least 6 fre-
L
in
— 0).
Table 3 also presents the error in the
evaluation of cable force. Using Eq.
(2). with the experimental measured
frequency and L = the horizontal dis-
Cable
f1
=
e,
estimation of the distance between the
cable nodes (points without vibration)
for all the cables when considering the
distance between the dampers
(=
0)
or the estimation of cable mass. This
last possibility is very unlikely because
there is no injection in the cable.
Therefore the differences in the cable
force must come from inaccuracies in
the values off and L. This last hypothesis is supported if the location of the
damper is not identical to the vibration
node. If the nodes are located between
the damper and the anchorage plate.
the additional length of vibration
should be the same in all the cables.
because all the anchoring devices are
similar.
LtF(%) L(m)
F2 (kN)
(jack)
4091
5817
5955
6112
5189
5297
5072
5709
5494
5327
5111
4709
5376
5189
4836
4503
4326
4905
quencies are available in the whole cables with no significant difference be-
tween right and left cables. the value
for the fundamental frequency was
evaluated by means of:
1
2
3
4
(6)
5
for both right and left cables, with f,
corresponding to the left cables, and
7
obtaining the values of Table 3 jointly
with the error in column 3.
The relative error in cable force evaluated with Eq. (2) is:
EF =
iF = 2 if +2 iL + im
F
f
L
rn
(7)
Structural Engineering International 4/94
6
8
9
10
11
12
13
2.170
2.5
5
1.683
1.411
1.120
1.5
3
2
0.984
0.845
0.770
0.690
0.624
0.554
0.509
0.455
0.496
1.5
3
2.5
2.0
2.0
2.5
2.5
3.0
3.0
3.0
5
1.0
2.0
4
4
4
5
5
6
6
6
50.7
68.6
85.7
102.4
119.1
135.8
152.4
169.2
186.0
202.6
219.5
236.1
253.0
F3(kN)
(strain
gauges)
F1 (kN)
(vibration)
4503
4944
4434
4630
4434
4630
4581
4532
4228
4189
3875
4061
5072
495-I
4650
4473
4356
4689
—
4993
—
Table 3: Cable forces from vibration test (F1) (L = distance between dampers) and from
other measurement techniques (F2, Fj
Science and Technology
239
To confirm this hypothesis, the error in
unique variable in the least-square
frequency and vibrating length must
be obtained, thus to derive the correct
minimising problem.
force in the cables. The proposed
problem: the value L* and the rela-
methodology calls for using all avail-
tive error Sf in the estimation of
the frequency of vibration, being
able experimental data (from jack
pressure. strain-gauges and cable vibration) recorded with some amount
of error. Consequently, the procedure
to derive the real vibrating length of
the same in the whole cables:
J'=fj(1+f).
Bearing in mind that the force val-
—
the cables was as follows:
two variables in the
— Considering
ues coming from the strain-gauges
placed in the anchoring pipes are
A nonlinear least-square interpolation
problem was resolved according to the
expression:
less reliable than those coming from
the pressure in the jack during tensioning.
From all the possibilities, the best glob-
(8)
with
= weight factor of error assumed in
cable i
the sum of the horizontal distances between dampers and vibration nodes in
the upper and lower anchorages of the
cable, or alternatively L* and f, which
minimises J.
The resolution of this least-square
problem has been done with the following possibilities taken into account:
Different weight factors in the cables: For each cable, the factor is related with the error in the measurement of its natural frequency.
1
Efj/tflifl(Efi)
I
horizontal
distance
between
fabrication of the cable used.
obtaining the value of L*, defined as
=
the
damper and anchorage plate in the
Freai, = real force in cable i
—
al result (minimum value of J) is obtained using the values of pressure in
the jacks, with different weighing factors depending on the cable, and the
equivalent length as the unique variable, being L* = 9.6 m. This is a quite
reasonable value, taking into account
(9)
The real forces in the cables derived
from applying the proposed methodol-
ogy (cable vibration + least-square
minimisation) are presented in Table 4
and compared with those derived from
alternative techniques. These forces
are very close to those defined as optimal in the design stage for the service
operation of the bridge, which correspond to the forces that were attempted to be introduced in the cables with
the jack (F2). They are, in any case,
within the acceptable tolerances as
stated in the design requirements for
the construction procedure in order to
achieve the appropriate internal forces
in the pylon and the deck.
Conclusions
In the paper the problems arising in
the application of vibration measure-
ments in the cables of cable-stayed
bridges to obtain actual forces are described. In particular, the influence of
extensibility of very long cables in obtaining accurate values of frequency of
vibration is shown in one example.
Other problems are related to the difficulty of obtaining experimental vibration records in short cables with a
sufficient time length to derive the natural frequencies of vibration with required accuracy.
The vibration method itself is not accurate enough to derive the real forces
in the cables when the cables are extremelv short, or long, or have complex anchoring devices. The methodol-
ogy presented, however, combining
the force values derived from vibration measurements with other experimental techniques (strain gauges, pressure in tensioning jacks) can define the
real forces in the cables with sufficient
accuracy and reliability to determine if
the cable forces are within permissible
boundaries defined during the design
process.
The simplicity of the installation required for the vibration method makes
it very attractive from an cost point of
view. It is an efficient, cheap and rela-
tively easy way to monitor possible
changes in the cable forces during the
service life of a bridge, once the real vibrating lengths of the cables have been
obtained.
References
[1] KYSKA. R.; KOUTNY. V.; ROSKO, P.
where
= the relative error in the evaluation of the frequency in cable i.
The same weight factor for all cables: 2)• This assumption is equiva-
—
lent to the assumption of equal experimental error in the evaluation
of real force (value coming from
jack pressure or strain gauges) and
F1
F
F
(kN)
(kN)
(kN)
5778
5847
5817
5709
5494
2
5955
5
6
6112
5307
5405
5082
6112
5189
5297
7
5239
8
3
4
5072
Assuming the force deduced from
9
the jack pressure transducers as the
real force in the cable.
5121
5013
10
4640
4650
11
4473
Assuming the force deduced from
12
strain gauges as the real force in the
13
4562
4199
4365
—
—
cable.
Taking the value of the equivalent
vibrating length, L1 + L*. as the
240
Science and Technology
.\FI
(1F1—F,J)
Tension Measurement in Cables of CableStayed Bridges and in Free Cables. Proc. of
the Second Conf. on Traffic Effects on
Structures and Environment, Zilina, Slova-
1
5111
5072
49S4
the natural frequency for all cables.
—
Cable
4356
4689
39
kia, April 1991. pp. 190—194.
108
[2] DE MARS: P.: HARDY. D. Mesuredes
—
0
118
efforts dans les structures a cables. Annales
TP Belgique 6. Bruxelles. 1985, pp. 515—
4993
108
5327
—
4709
5376
5189
4836
4503
4326
4905
10
128
49
59
531.
[3] ROBERT. J.L.; BRUHAT, D.; GERVAIS, J. P. 1esure de la tension des cables
par méthode vibratoire. Bulletin de Liaison
des Laboratoires des Ponts et Chaussées,
173. Paris, 1991, pp. 109—114.
10
[4] JAVOR, T Damage Classification of
89
157
324
Concrete Structures. Report: RILEM Technical Committee 104—DCC Materials
and Structures. 24. 142. Paris, 1991, pp.
253—259.
Table 4: The actual forces in the cables (F1)
after completion according to the proposed
methodology compared with design forces
[5] LEONARD. J.W. Tension Structures:
Behavior and Analysis. McGraw Hill. New
York, 1988.
Structural Engineering International 4/94