A Combined Method for Measuring Cable Forces: 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. +r Peer-reviewed by international experts and accepted by the IABSE Publications Committee Introduction During the construction and service life of a cable-stayed bridge, it is extremelv important to accurately define the forces in the cable stays. This knowledge is critical to the correct alignment and distribution of internal forces in the finished bridge. At present. there are several techniques to assess these forces. including measurement of the force in a tensioning jack, 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 application. 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 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. If the vibration method is to obtain accurate results in these circumstances, it must be combined with other measurement techniques, each. however, with its own inherent errors or deficiencies. A methodology for combining measurement techniques which compensates for the errors of one method with the results of another. and vice versa, is proposed in this paper. 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" diameter, 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 Structural Engineering International 4/94 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 Fig. 1: The cable-stayed Alamillo Bridge i2oor0 10500 -t 16000 '4, '0 250 325 608 It I I I Ii000I 1875 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 results in a large variation in internal forces both in the pylon and in the deck. 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 434 MNm (almost 100%) in the maximum moment of the deck. Even more 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). Fig. 2: Cross section of the deck (mm) 1623 Limitations of the Vibration Method Notation A E = cross section area of structural material of the cable = modulus of elasticity of cable material = natural frequency of vibration = force in the cable f F H = horizontal component of force in the cable J = sum of quadratic errors k = relation between weighing and resistant area of the cable (k 1) L horizontal distance between vibration nodes in the cable L* = sum of horizontal distances between dampers and vibration nodes I rn s = 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 a a0 = angle between horizontal and cable chord connecting endpoints = relative error a' = relative error in cable force = = tension in the steel of the cable in the permanent state = angular frequency of vibration (= 2rJ) Before discussing the application of the method to the Alamillo Bridge, some discussion of the method's limitations is useful. If the ideal vibrating chord theory is assumed for a complete 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 F= H 4Lf2m (2) cosa cosa 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: 236 Science and Technology Structural Engineering International 4/94

The 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. There is no relative displacement of the points where the cable is anchored. 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 generate 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 pylon). the natural frequencies of the lowest modes are slightly greater than when the anchoring points are fixed. Also if the cable has a more than a negligible flexural stiffness, the natural frequencies of the higher modes are greater than predicted by the vibrating chord theory. A study [3] concludes that Eq. (1) can only be applied if the seven lowest frequencies, measured 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 Ltf Dead load + Permanent load + Design cable forces - - - - Dead load + Permanent load + 0.9 * Design cable forces 1 =400s 0.005x0.5 Dead load + Permanent load + 1.1 * Design cable forces 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 (or 6 minutes), which is not realistically available for a free-damped vibration of a real bridge cable. 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]: -J 4(3 tan = I 2 2 22 ml2 (.o=a) where EA H ) H(1+852) s = sag ratio of the cable = eil (Fig. 5) while the antisymmetric modes have frequencies: w,1 =2nir1 -- n=1,2,3... (4) ml 1228 Values in MN mated cable force [4] using Eq. (1). the factor mgl = pl must have a maximum value as expressed in Table 1, depending on sag ratio (s) and cable inclination in respect to the horizontal (a0). Table I has been derived with a stress of 650 MPa in the steel cables. If a relative error instead of absolute error is considered, an equation can be derived relating the relative error in the cable force (because of neglecting (3) Fig. 5: Definition of terms in an inclined cable -L 1 m Fig. 3: Typical pylon cross section (cm) according to vibrating chord theory. The influence of cable extensibility is negligible for short cables, because A tends to 0, and therefore i = ir but could lead to important errors for long cables. In fact, in order to achieve values with ± 30 kn accuracy in the esti- 2 34 Mode number (n) Fig. 6: Relationship between,nodes and frequencies of vibration in the vibrating chord theory (non-dispersive model) 5 Structural Engineering International 4194 Science and Technology 237

ao sag rati o (s) 0.00 1 0.004 0.007 0.009 200 146.0 36.7 21.2 16.8 30 134.6 33.9 19.6 15.5 45 109.9 27.7 16.1 12.8 60 77.7 19.7 11.5 9.2 75 40.3 10.3 6.2 5.0 90 0.06 0.26 0.45 0.57 Table 1: Values of pl (k.\ymxni) as afunclion of a0 and s tension increment) and cable characteristics (with E = 190000 NIPa): 4acosa0 tanx=x - 1.216x 10's =.(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 = 30 in order to achieve an accuracy of 0.5% in frequency (1% in tension) it is necessary to establish a value for s: x= Jr/4(2+0.01)= 1.57865 127.32=1.57865 4x56292 1.216 x 109s2 x3.934 =s=2.3x 10-s pl 7.85kAL 132 =L= 8H 8acosa0A k Even in the most favourable case of k = 1 (normally k> 1). the cables with L> 132 m have an error greater than 0.5% in frequency because of extensibility. This effect is less important for higher frequencies. As a consequence. due to the error involved in the hypothesis of cable inextensibility. it is unrealistic to look for very high accuracies in the experimental measurement of the fundamental frequency of long cables, which in turn is difficult to achieve because of their low fundamental frequency. Correct application of the vibrating chord theory requires accurate knowledge 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 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. During the tensioning of the cables. the pressure in the hydraulic jacking pump and the elongation of the strands were also measured. Because of the problems detected with these two measuring techniques during construction of the bridge, which led to Fig. 7: View of the anchorage block in the deck and the position of the strain gauges different values for the cable forces depending on the method, it was finally decided to perform a vibrating test of the cables in order to determine the final cable forces after completion of the bridge and to check if they were within the tolerance limits stated in the design. 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 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, showing 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 below. 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). To detect this error of 3% would require a frequency resolution of.\f = 238 Science and Technology Structural Engineering International 4/94

Cable \f fi f: f Li f f6 f Li Li 1 L 0.050 2.20 4.34 7.99 9.68 11.48 2 L 0.026 1.69 3.38 5.04 6.75 8.39 10.05 13.43 3 L 0.015 1.42 2.83 4.23 5.64 7.03 8.47 4 L 0.020 1.13 2.26 3.34 4.47 5.58 6.68 7.81 8.92 5 L 0.015 0.99 1.97 2.95 3.93 4.92 5.89 7.83 6 L 0.020 0.84 1.7 2.54 3.37 4.23 5.07 5.93 6.77 7.63 7 L 0.015 0.77 1.54 2.32 3.09 3.85 4.62 5.41 6.16 6.94 8 L 0.015 0.69 1.38 2.07 2.77 3.45 4.15 4.84 5.52 6.20 9 L 0.015 0.63 1.25 1.86 2.49 3.12 3.74 4.37 5.0 5.61 IOL 0.015 0.56 1.11 1.66 2.21 2.76 3.32 3.87 4.43 4.98 11 L 0.015 0.52 1.02 1.52 2.02 2.54 3.04 3.54 4.04 4.56 12 L 0.015 0.46 0.91 1.36 1.82 2.26 2.73 3.18 3.62 4.07 13 L 0.015 0.50 0.99 1.49 1.99 2.48 2.98 3.46 3.96 4.45 1 R 0.050 2.29 4.55 6.73 8.83 2 R 0.030 1.71 3.42 5.16 8.56 10.3 3 R 0.020 1.43 2.86 4.29 5.69 7.12 8.52 4 R 0.020 1.12 2.25 3.35 4.45 5.57 6.68 7.76 8.84 5 R 0.020 0.96 1.93 2.87 3.84 5.74 7.67 6 R 0.015 0.86 1.69 2.54 3.38 4.25 5.06 5.91 6.74 7.58 7 R 0.015 0.77 1.52 2.27 3.06 3.79 4.56 5.3 6.06 6.82 8 R 0.015 0.69 1.38 2.07 2.76 3.45 4.12 4.81 5.48 6.19 9 R 0.015 0.64 1.23 1.87 2.49 3.10 3.72 4.34 4.93 5.55 10 R 0.015 0.55 1.11 1.66 2.22 2.79 3.32 3.88 4.43 4.99 11 R 0.015 0.52 0.98 1.50 1.99 2.48 3.0 3.5 3.99 4.48 12 R 0.015 0.47 0.91 1.35 1.79 2.26 2.7 3.17 3.62 4.06 13 R 0.015 0.50 0.99 1.49 1.99 2.48 2.98 3.48 3.98 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 reason, 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 frequencies are available in the whole cables with no significant difference between right and left cables. the value for the fundamental frequency was evaluated by means of: for both right and left cables, with f, corresponding to the left cables, and obtaining the values of Table 3 jointly with the error in column 3. The relative error in cable force evaluated with Eq. (2) is: if if il im EF = = 2 +2 + F f L rn (6) Assuming an exact evaluation of cable length and mass L = in 0). Cable f1 e, Table 3 also presents the error in the evaluation of cable force. Using Eq. (2). with the experimental measured frequency and L = the horizontal distance 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 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 they are almost equal for right and left cables. Therefore, the combined methodology is challenged to identify an inaccuracy that is common to the 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) F1 (kn) (vibration) F2 (kn) (jack) F3(kN) (strain gauges) 1 2.170 2.5 5 50.7 4091 5817 5709 2 1.683 1.5 3 68.6 4503 5955 5494 3 1.411 1.0 2 85.7 4944 6112 5327 4 1.120 2.0 4 102.4 4434 5189 5 0.984 1.5 3 119.1 4630 5297 4993 6 0.845 2.5 5 135.8 4434 5072 7 0.770 2.0 4 152.4 4630 5111 4709 8 0.690 2.0 4 169.2 4581 5072 5376 9 0.624 2.5 5 186.0 4532 495-I 5189 10 0.554 2.5 5 202.6 4228 4650 4836 11 0.509 3.0 6 219.5 4189 4473 4503 12 0.455 3.0 6 236.1 3875 4356 4326 13 0.496 3.0 6 253.0 4061 4689 4905 (7) Table 3: Cable forces from vibration test (F1) (L = distance between dampers) and from other measurement techniques (F2, Fj Structural Engineering International 4/94 Science and Technology 239

To confirm this hypothesis, the error in frequency and vibrating length must be obtained, thus to derive the correct force in the cables. The proposed methodology calls for using all available 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 cables was as follows: A nonlinear least-square interpolation problem was resolved according to the expression: with (8) = weight factor of error assumed in cable i Freai, = real force in cable i obtaining the value of L*, defined as 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. I = where 1 Efj/tflifl(Efi) = the relative error in the evaluation of the frequency in cable i. The same weight factor for all cables: 2) This assumption is equivalent to the assumption of equal experimental error in the evaluation of real force (value coming from jack pressure or strain gauges) and the natural frequency for all cables. Assuming the force deduced from the jack pressure transducers as the real force in the cable. Assuming the force deduced from strain gauges as the real force in the cable. Taking the value of the equivalent vibrating length, L1 + L*. as the unique variable in the least-square minimising problem. Considering two variables in the problem: the value L* and the relative error Sf in the estimation of the frequency of vibration, being the same in the whole cables: J'=fj(1+f). Bearing in mind that the force values coming from the strain-gauges placed in the anchoring pipes are less reliable than those coming from the pressure in the jack during tensioning. From all the possibilities, the best global 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 the horizontal distance between damper and anchorage plate in the fabrication of the cable used. The real forces in the cables derived from applying the proposed methodology (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 (9) achieve the appropriate internal forces in the pylon and the deck. Cable F1 (kn) F (kn) F (kn).\fi (1F1 F,J) 1 5778 5817 5709 39 2 5847 5955 5494 108 3 6112 6112 5327 0 4 5307 5189 118 5 5405 5297 4993 108 6 5082 5072 10 7 5239 5111 4709 128 8 5121 5072 5376 49 9 5013 49S4 5189 59 10 4640 4650 4836 10 11 4562 4473 4503 89 12 4199 4356 4326 157 13 4365 4689 4905 324 Table 4: The actual forces in the cables (F1) after completion according to the proposed methodology compared with design forces Conclusions In the paper the problems arising in the application of vibration measurements 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 methodology 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 relatively 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. Tension Measurement in Cables of Cable- Stayed Bridges and in Free Cables. Proc. of the Second Conf. on Traffic Effects on Structures and Environment, Zilina, Slovakia, April 1991. pp. 190 194. [2] DE MARS: P.: HARDY. D. Mesuredes efforts dans les structures a cables. Annales TP Belgique 6. Bruxelles. 1985, pp. 515 531. [3] ROBERT. J.L.; BRUHAT, D.; GER- VAIS, 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. [4] JAVOR, T Damage Classification of Concrete Structures. Report: RILEM Technical Committee 104 DCC Materials and Structures. 24. 142. Paris, 1991, pp. 253 259. [5] LEONARD. J.W. Tension Structures: Behavior and Analysis. McGraw Hill. New York, 1988. 240 Science and Technology Structural Engineering International 4/94