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Full scale dynamic testing of the Annacis cable stayed bridge Vincent, Douglas Hillyard Charles 1986

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FULL SCALE DYNAMIC TESTING OF T H E ANNACIS C A B L E STAYED BRIDGE  By  DOUGLAS HILLYARD CHARLES VINCENT B.A.Sc, University of British Columbia, 1984  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in  THE FACULTY OF GRADUATE STUDIES (Department of Civil Engineering)  We accept this thesis as conforming to the required standard  The University of British Columbia September 198C ©  Douglas Hillyard Charles Vincent, 1086  In presenting this thesis in partial fulfilment of the requirements for an advanced degree at The University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.  Department of Civil Engineering The University of British Columbia 2324 Main Mall Vancouver, British Columbia Canada  V6T 1W5  September 1986  ii  Abstract  Cantilever construction techniques and reduced deck bending moments have made cable stayed bridges popular for long spans. Recently; however, large amplitude wind induced cable vibration problems have occurred on several of these long span bridges. One of these bridges had several of its longest cables fitted with discrete dampers near the lower end cable socket. Although effective, these discrete dampers, which resemble large shock absorbers, are obvious and very unappealing. Some of these large cable oscillations are caused by eddy shedding but others appear to be created by some other means of excitation. Past research suggests that wind induced support motion combined with low damping in the cables may produce local vibration problems. Many authors have proposed closed form and analytical solutions to the dynamic response of a cable with sag. Due to a lack of measured values, estimates of cable damping have been used in these analyses. Although seme scale model tests of a cable stayed bridge have been done, no actual full scale tests have been performed. With the Annacis cable stayed bridge being constructed in the Vancouver area a unique opportunity arose to perform full scale dynamic tests of the cables as well as the entire bridge. Through a series of tests of cables of varying length, the natural frequency and damping value of each cable was determined. A parameter study was undertaken in order to determine the nature of the damping present. Each of the cables supporting the Annacis bridge is fitted with a split neoprene ring in the cable anchorage assembly at each end. Hidden from view, this neoprene ring was designed to seal the top socket and provide a discrete damper at each end of the cable. Tests done before and after these dampers were in place determined their effectiveness. The possibility of large cable motions being caused by wind induced support motion as suggested by the previous literature was investigated. Tests were carried out to determine if cable motions were harmonically related to the motion of the towers or deck. Finally a simple pendulum was used to harmonically excite the completed bridge in several of its lower modes. This test provided a check on the natural frequencies of these modes as well as providing full bridge damping values from the free vibration response.  iii  Table of Contents  Abstract List of Figures List of Tables Notation Acknowledgements  .  ii iv vii viii x  1  Introduction  1  2  Background  6  2.1 General Cable Problem  6  2.2 Taut Cables - Static Analysis  7  2.3 Taut Cables - Dynamic Analysis  9  3  Objectives  28  4  Selection of Test Procedure  28  5  Experimental Test Series  33  5.1 Parameter Study  33  5.2 Damper Effectiveness  34  5.3 Component Interaction  35  5.4 Full Bridge Tests  35  6  Analysis of Recorded Data  39  7  Test Results  48  7.1 Parameter Study  48  7.2 Damper Effectiveness  50  7.3 Component Interaction  51  7.4 Full Bridge Tests  53  8  Conclusions  95  9  Future Work and Recommendations  98  References  100  JV  List of Figures  1.1  Acnacis Cable Stayed Bridge  4  1.2  Cable end socket detail showing split neoprene ring  5  2.1  Catenary cable  17  2.2  Catenary cable with change in end thrust  18  2.3  Behaviour of a suspended cable under changes in load  19  2.4  Component interaction  20  2.5  Mode shapes of system  20  2.6  Combined mode shapes of system  20  2.7  Variation of cable amplitude with frequency ratio f /f,  2.8  Cable with a damper of stiffness w  21  2.9  A plot of Dynamic Amplification Factor vs. Frequency  22  2.10  Cable under a tension T with an applied harmonic load p{x,t)  2.11  Optimum value of damper stiffness for e = 1/10  2.12  Deflected shape of a cable with the optimum damper  2.13  Motion of a cable with its end supports moving parallel to its axis at 2/  2.14  Region of large cable motion and its dependence on  c  and mass ratio p  .  .  .  .  21  22 23  .  .  .  . c  .  23  .  24  support motion and frequency  24  2.15  Cable amplitude resulting from strainless state  25  4.1  Shop drawing of three wheeled cart  31  4.2  Cart mounted on cable at lower socket prior to test  32  4.3  Instrumentation used for damping test  32  5.1  Typical test set-up showing accelerometers on the cable, deck and tower  6.1  Sine wave with exponential decay  44  6.2  Damping program C A B D A M P  45  6.3  Flow chart showing collection and analysis of data  46  6.4  Free body diagram of an elemental length of string  47  7.1  Cable 12 (1-damper). Response spectrum and acceleration - time record  .  57  7.2  Cable 22 (1-damper). Response spectrum and acceleration - time record  .  58  7.3  Cable 23 (1-damper). Response spectrum and acceleration - time record  .  59  .  38  List of Fitpires  7.4 7.5  Cable 24 (1-damper). Response spectrum and acceleration - time record Cable 26 (1-damper). Response spectrum and acceleration - time record  60 61  7.6  Cable 27 (1-damper). Response spectrum and acceleration - time record  62  7.7  Cable 28 (1-damper). Response spectrum and acceleration - time record  63  7.8  Cabie 29 (1-damper). Response spectrum and acceleration - time record  64  7.9  Cable 30 (1-damper). Response spectrum and acceleration - time record  65  7.10  Cable 39 (1-damper). Response spectrum and acceleration - time record  66  7.11  Cable 40 (1-damper). Response spectrum and acceleration - time record  G7  7.12  Cable 42 (1-damper). Response spectrum and acceleration - time record  68  7.13  Cable 44 (1-damper). Response spectrum and acceleration - time record  69  7.14  Cable 45 (1-damper). Response spectrum and acceleration - time record  70  7.15  Cable 46 (1-damper). Response spectrum and acceleration - time record  71 72  7.16 7.17  Free body diagram of a taut string with lateral deflection of unity  73  7.18 7.19  72  Cable 27 (2-dampers) . Response spectrum and acceleration - time record . . . .  74  7.20  Cable 28 (2-dampers). Response spectrum and acceleration - time record . . . .  75  7.21  Cable 29 (2-dampers). Response spectrum and acceleration - time record . . . .  76  7.22  Cable 30 (2-dampers). Response spectrum and acceleration - time record . . . .  77  7.23  Cable 39 (2-dampers). Response spectrum and acceleration - time record . . .  78  7.24  Cable 40 (2-dampers). Response spectrum and acceleration - time record . . . .  79  7.25  Bar graph showing the increase in damping due to the addition of a second damper  80  7.26  Deck motion at socket 26. Response spectrum and acceleration - time record  81  7.27  Deck motion at socket 27. Response spectrum and acceleration - time record  82  7.28  Deck motion at socket 28. Response spectrum and acceleration - time record  83  7.29  Deck motion at socket 29. Response spectrum and acceleration - time record  84  List of Figures  7.30  vi  Deck motion at socket 30. Response spectrum and acceleration - time record  .  .  .  85  7.31  North half of bridge showing cables 4C and 45C and towers Nl and N2  .  86  7.32  Motion of tower? Nl and N2 necessary to create the "parameter" effect  .  85  7.33  Response spectra for lateral mode pendulum test  37  7.34  Response spectra for first lateral mode test  87  7.35  First lateral mode pendulum test. Response spectra of the east and west girders  7.36  Free response plots from two tests of thefirstlateral mode  7.37  Response spectra for the east and west girders  88 .  .  .  .  resulting from oscillation in thefirsttorsional mode 7.38 7.40  90  Free response plots from two tests of the first torsional mode  7.39 Vertical response spectrum resulting from cyclic drop test Mode shapes of completed Annacis bridge  89  . . . .  .  .  .  91 92 93  VIJ  List of Tables  Important parameters for the cables tested with one damper in place  94  Effectiveness of a second neoprene damper on the six cables tested  94  viii  Notation  A  area  A  c  amplitude of cable  t  amplitude of support (tower or deck)  A c  arclength  DAF  dynamic amplification factor  E  Young's modulus  e  distance from end support to location of damper  f  natural frequency of cable  c  f„  natural frequency of support  g  acceleration of gravity  H  cable thrust (horizontal)  h  difference in elevation between cable supports  k  lateral stiffness of cable  kg  lateral stiffness of support  L  horizontal distance between cable supports  /  length of string or cable  c  m  mass of cable  m  mass of support  q  restoring force  5  tangent length between cable supports  T  cable tension  xv  weight per unit length of cable  W  stiffness of discrete damper  y  lateral deflection of cable or string  6  logarithmic decrement  A  deflection vector from computer r u n  e  e/L for a horizontal cable  K  dimensionless parameter used to describe cable thrust  fi  mass ratio  c  g  (m /m ) c  t  Notation  p  mass per unit length  u  c  circular natural frequency of cable  u  t  circular natural frequency of support  LOA  damped circular natural frequency  c  damping ratio  ix  X  Acknowledgements  I would like to thank Professor S.F._ Stiemer for his support throughout this research. The involvement in this project of C B A - Buckland and Taylor and P C L Paschen - Pike was greatly appreciated. Special thanks to the Ministry of Transportation and Highways of British Columbia for supporting this investigation. I would like to thank Dr. Peter Taylor, Dr. Stanley Hutton, Dave Boyd and Don Bergman for their interest and assistance in the bridge damping tests. I am indebted to the many graduate students and other friends who assisted in this research. Very special thanks must go to Bernie Merkli for his workmanship, suggestions and assistance, and to Bruce Jeffery3 for his valuable help on numerous site visits. Finally, I would like to thank my parents and my friend Barb Harris for their continuing encouragement throughout this endeavor.  CHAPTER 1  INTRODUCTION  Throughout history bridges have been some of the most challenging structures for engineers. As the need for longer span bridges increased, suspension, and more recently cable stayed bridges came to the forefront. These bridges consist of a light superstructure suspended by cables, which when closely spaced, reduce deck bending moments allowing longer spans. In the past, the light superstructure associated with these bridges created aerodynamic problems culminating in the collapse of the Tacoma Narrows bridge in 1940. Since that time a great deal of work has been done in the area of deck aerodynamics using models and wind tunnek to examine the effects of eddy shedding and such phenomenon! as buffeting and galloping. As well as wind induced deck motion there have also been cable oscillations due to wind effects. In fact two recently constructed cable stayed bridges have had severe cable motion problems. At the Brotonne bridge in France these cable motions became so large that two of the longest cables began to oscillate out of phase and came into contact with one another [11]. These long cables are almost parallel to one another and are 1957 mm apart at the center. Thus the amplitude of motion had to have been at least 1000 mm. The solution to this problem was the installation of discrete dampers near the lower socket 1  2  on some of the longer cables.  Although effective, these dampers, which resemble large  shock absorbers, are very unappealing. Most of the work done on cable vibration has been in the areas of suspension cables, transmission lines and mooring lines. All of these areas involve cables with large sag and dynamic analysis is complicated due to the geometric nonlinearities which arise. Cable stays are taut compared to suspension cables and little research has been done on the dynamic response of inclined cables with low sag. For small oscillations at least, these cables are assumed to remain linear with respect to both material and geometry. A simple closed form solution exists for an inextensible taut string with fixed ends. Closed form and analytical solutions for inclined cables with large sag have also been proposed. However, the cable stay lies somewhere between these two extremes. The half bandwidth method was used to determine bridge damping values for some of the lower modes at the Pasco-Kennewick bridge; however, no cable damping values were given [12]. Although some analytical and scale model work has been done on cable stays, no full scale tests have been carried out to verify these values. With the Annacis cable stayed bridge under construction in the Vancouver area a unique opportunity existed to undertake full scale cable tests. The bridge spans the south arm of the Fraser River and when completed its main span will be the longest of its type in the world. The Annacis bridge consists of cables in two planes in a modified fan arrangement supporting a 465 m main span, two 182.75 m side spans and two 50 m transition spans (Figure 1.1). There are a total of 192 main cables and 8 tie down cables consisting of long lay galvanized bridge strand sheathed with black polethylene. The strands are 7 mm diameter galvanized steel with an ultimate tensile strength of 1520 MPa. Every cable has a zinc filled cast steel socket at both ends. The cables terminate at tie beams in the towers where provision is made for jacking and adjustment.  Cable  lengths range from 49.5 m to 237.5 m and outer diameters vary between 99 mm and 149 mm. In order to increase damping each cable is fitted with a split ring neoprene damper  3 contained in the cable anchorage assembly at both ends (Figure 1.2). These, along with the polethylene cover were used to achieve a more desirable result than steel strand grouted in a steel casing such as that used at Brotonne.  ^28.0  £2L  35.0 m  FIGURE 1.1  Annacis Cable Stayed Bridge, (a). Elevation (b). Tower section. Typical cross section, (from reference [28]) .  5  8mm thick  FIGURE 1.2  Cable  end  (b). Bottom  socket  detail  socket,  showing  (from  split  reference  neoprene [29}).  ring.  (a). Top  socket,  6  CHAPTER 2  BACKGROUND  2.1  G E N E R A L  C A B L E  P R O B L E M  Closed form and analytical solutions for both the static and dynamic analysis of cables with large sag are contained in previous literature.  A n investigation into the  static and dynamic analysis of mooring lines was done by Chang and Pilkey [1]. The work involved a two dimensional static analysis considering forces normal to the line only and a dynamic solution using incremental integration. All of the solutions were for lines with large sag and thus low tension. Research into suspension cables provided several closed form solutions. Irvine and Griffin [2] used a linear mathematical approach to solve for the response of a cable with its end points being excited sinusoidally. The solution assumes no damping and ignores second order effects and out of plane motion. Henghold, Russell and Morgan [3] studied the free vibrations of a cable in three dimensions. It was discovered that for a cable with large sag, crossovers occur between in-plane and sway modes. However, for low sag cables the in-plane and sway mode frequencies are the same and the cable behaves like a taut string. Wilson and Wheen [4] assumed a parabolic shape and derived design expressions for an inclined taut cable under self weight, uniform dead loads and point loads.  7  Analytical solutions were proposed by West, Geshwindner and Suhoski [5] who modelled the cable using a linkage of straight pin ended bars; and Henghold and Russell [6] who developed a three node cable element and used a non-linear finite element approach to the problem.  2.2  TAUT CABLES - STATIC ANALYSIS An analysis of the proposed A . L . R . T . cable stayed bridge crossing from A n -  nacis Island to Surrey by Hooley [7] contained a cable element defined by dimensionless parameters. For the catenary cable shown in Figure (2.1) the following expressions hold. c  =\l  -^-  k 2 +  smb  (2  -  1)  or  wL r = ——. 2H  Elastic elongation = A . wh?  1  wL  H  . , wL  - — - smh H 2HL coth —— 2H -f 2- "+ 2wL  AE  ^'^  r  2  3  The above expressions are exact but very inconvenient to work with. For tight cables (r < 1/2) it is convenient to expand the hyperbolic functions as power series. The series below are used and terms up to and inluding r r r sinhr^r+^-r — + 3  r  are retained.  4  5  r  2  ••• r  4  6  cosh r = l + — + — + —— -I 2  24  720  1  r  r  2r  r  3  45  coth r = - H  3  1  5  945  1  From these expressions we get: \/c  „\  2  - h? =  fco& e\ 2  V+ 2  - + — + 6 120  /cos 0 2  ••] /  cos ^\ 4  r  4  +  (2.4)  8  and, A = A  We define USL  •4-^-4)  HL sec 9 2  AE  (2.5)  as the unstressed length of the cable when the thrust is H  0  USLQ = c — A  0  where c = arclength.  Arclength is invarient and does not change unless temperature changes. USLQ = S ( 1 + £ cos ij  -  2  S  ec  Neglecting the r / 3 term and recalling that r = wL/2H  we get:  2  U  S  L  t , S I  -  q  »  =  s  S  ^4HJ\S^)-'AE{L^)  +  +  2 « ^ - A k  (-» 2  As the load on the structure changes (i.e. thrust changes from H  to H  Q  1 +  2  as shown in Figure  6  the deck load) the cable end  (2.2).  The elongation of the cable  (A = USLQ — USL) is given as the relative displacement of points A and B in direction S. This A is the deflection vector from the static computer run. w L*  HQS  f  2ASHI  AEL  [  2  +  S  2  2  w L*  2ASH  +  w L AE\  (  2  ~ AEL V  S  B  24S # ) 3  HS ]  2  2  AEL\  2  (  w L AE\ 2  AEL \  2  2  0  B  245 /J ) 3  2  If we introduce two new parameters, _ a  S  2  ~ AEL  3  P  _  ~  w L AE 2  h  245  3  Then we have;  Equation (2.7) can be put into dimensionless form by substituting H = K@ and H  Q  = /c /?, 0  where K and /c are dimensionless variables which describe the thrust for different loadings. 0  9  The final equation relating cable deflection to cable thrust using dimensionless paremeters is:  If we call the left hand side of Equation (2.8) F then for any load on the structure we know F since: are constant  or,/? K  is known from initial conditions  A  is known from output  Q  These parameters were used to study cable behaviour under increasing load. An ultimate load analysis was carried out using the load case D . L . + A L . L . and a plot of F vs.  K was made for the cable most affected by changes in live load (Figure  2.3).  From this plot we see that cables with K greater than 1.5 are considered stiff. In other words, for these cables there is a linear relationship between deflection and load, whereas for a slack cable this relationship becomes non-linear. This geometric nonlinearity arises because large changes in deflection result in changes in the equivalent axial stiffness of the cable. The cables at the Annacis Island bridge are similar to those chosen for the proposed A . L . R . T . bridge. By calculating the value of K (for the D . L . condition) for the 0  cables at the Annacis Island site we see that KQ lies between 3.38 and 5.59, which puts all of the cables well into the so called stiff range.  If the ends of the cables essentially  remain fixed, they will act as taut strings (i.e. linear deflection theory holds), thus all cable tensions in this thesis have been calculated using taut string theory.  2 . 3  T  A  U  T  C  A  B  L  E  S  -  D  Y  N  A  M  I  C  A  N  A  L  Y  S  I  S  A recent paper by Kovacs [8] discusses the effects of component interaction, and introduces two possible causes of large cable vibrations.  10  The first of these is caused by excitation of the supports perpendicular to the cable axis. For interaction to occur the natural frequency of the support and the cable must be almost equal (ie f : / , w 1 : 1). c  This phenomenon can be modelled as a two degree of freedom system as shown in Figure (2.4). The equations of motion for this system are; m u + k u + (v — ipu)(—k ip) = 0  (2.9)  + (v — Tpu)k = 0  (2.10)  8  g  c  mv c  Divide Equation (2.9) by m  c  and Equation (2.10) by m  8  —  — u.,  and substitute;  c  m.  m.  to obtain the following equations of motion. ti + u u + (a — rpu)(—fnl)u ) 2  = 0  2  t; + (v — i>u)u = 0 2  In order to find the natural frequencies of the combined system we use det |fc/m — u \ 2  det  u] + \ity u 2  2  — u  -V-ty^l  2  W2  0  U  2  C  or, w  - [u + w ( l  4  2  2  + nit )] u 2  2  + uu 2  2  = 0  This equation has solutions;  " i = \j\ { K  2  "  +  2  =  vH  "'  [  +  + ^ )\ ~ \JW+  ^ c ( l + / / V ) ] - 4w2«a J 2  2  +  ^  2  )  1 +  V ( /  w 2+  w  2  2  c (l + ^ )] -4o; a; 2  2  2  2  2 c  |  (2.11)  (2.12)  There are two distinct mode shapes for the two degree of freedom system as shown in Figure (2.5). If we let; u — U  smut  v = V sin ut  11  Then for mode 1 we get, (2.13) and for mode 2 we get, rpu U  2  (2.14)  u - uj 2  2  However, the final motion of the system when u  » u  g  is a combination of these two  c  modes as shown in Figure (2.6). The following expressions describe the resultant motion. (2.15)  U + U = A.l  2  (2.16) By combining Equations (2.11), (2.12), (2.13), (2.14), (2.15) and (2.16) we can solve for Vi and V . 2  1  1 +  (2.18)  [u; +u, (l+/«V )] 2  V = -i>A, { 2  2  2  1-  (2.19) 2uvw»  '1  L[w2+ 2(iW)]J w  A plot of A /A rp Cmax  t  is shown in Figure (2.7). By combining Equations (2.17) and (2.18)  we find; A  1  max  c  T 2  1-  (2.19)  0"c  1 + ^ ( 1 W )  F r o m the plot we see that as u> approaches u we get the m a x i m u m amplitude s  c  of cable vibrations to be; A  ' 4  A  (2.20)  T h u s line amplitude of the cable oscillations depends on the amplitude of the tower motion and the ratio of the cable mass over the tower mass.  12  Although these cable motions may become large as we can see from Equation (2.20) they can be reduced with the installation of a discrete damper. Kovacs also considered the effectiveness of a discrete damper on a cable with zero damping in the system. Both the location and stiffness of this discrete damper were examined to achieve the optimum (lowest) dynamic amplification factor  (DAF).  Obviously, the closer the damper is to the center of the cable, the more effective it will be in reducing first mode vibrations. Consider a cable with a discrete damper located at a distance e from the end of the cable (Figure 2.8). There are two extreme cases for the stiffness (w) of the damper. If w = 0 there is no effect on damping and oscillations will approach infinity at resonance. If w = oo the only effect that the damper will have is to reduce the cable length from / to (/ — e) and thus change the natural frequency. However the damper will not reduce the amplitude of oscillations. From an analytical model the author [8] showed that for a damper located a fraction e, where s = e/l, from the end of the cable we get the following natural frequencies for the two extreme cases. For w = 0 the damped natural frequency (f ) is; d  fi = fc and, DAFjg^  (2.21)  = 1  f?  and for w — oo we have, fd =  fc (1-S)  and, DAF  1 rc=O0  l-(l-*) g  (2.22)  2  As shown by the plot (Figure 2.9) curves for all values of damping must intersect at the optimum value for DAF. If we equate Equations (2.21) and (2.22) we will  13  find the optimum value of natural frequency and the optimum value of the DAF. 1 1  1  V  This gives the solution; opt  DAF  (2.23) *  opt  -  For a cable with tension T oscillating under a harmonic load p(x, t) as in 0  Figure (2.10) we wish to find the optimum D A F for the cable and the corresponding optimum stiffness (ty) of an applied discrete damper, assuming that its location is already fixed at some distance (e) from the support. The overall effectiveness of a discrete damper increases as it moves toward the center of a cable; however, bridge dampers are only practical up to an e value of approximately 1/10. If e is greater than 1/10 the damper becomes very large and difficult to install. An analytical study with the parameter e asssuming values of 1/50, 1/20 and 1/10 was performed to determine these optimum values. The results of this work showed that; DAF  opt  «  1  ~  (2.24)  4sf l  (2.25)  c  and the corresponding logarithmic decrement is;  8  opl  « e • ir  (2.26)  The result for s = 1/10 is shown in Figure (2.11) as the dashed line. For any value of / the DAF always lies below the optimum value of 1/e which shows that we have the correct stiffness for the damper. With this optimum value of damping we see that when displacement is at a maximum the cable acts as if its support where located at a distance of e/2 from the actual support (Figure 2.12).  14  The other important interaction is caused by support motion parallel to the cable axis causing harmonic changes in cable tension. Structural interaction will occur if u : u « 2 : 1 and a description of this interaction is shown in Figure (2.13). For a a  c  continuous system with fixed ends, such as a cable, motion perpendicular to the axis is described by; y = Y sin — sin (2wf t) 0  (2.27)  c  The harmonic support force is given by; T = T - AT sm(4icf t) 0  max  (2.28)  c  where TQ is the original cable tension at rest. This harmonic end force creates a restoring force Aq along the cable.  Ag = -Ar sin(4T/ *)0 mai  Substituting for  c  from Equation (2.27) we get;  (j)  7r \ 2  7T!E  r sin sin(27r/ 0 0  T  c  and through manipulation;  A<?=(y)  Y AT Q  COS(2TT/ £) c  N  2  cos(67r/ t)l c  2  . irx  sin —  (2.29)  This restoring force is shown in Figure (2.13) as the solid line and the first term (underlined) is shown as the dashed line in the graph. From the first graph in the Figure we see that the displacement function is a sine wave and thus the velocity function is a cosine wave. The first term of the force Aq and the velocity of the cable are in phase and if we divide the first term of the restoring force by the velocity of the cable y we get a constant term.  This constant is equivalent to a negative logarithmic damping function 6. &= " ^ ^ ^  (2-31)  15  Where the damping is negative because it is in phase with the velocity. Thus, for a cable with its ends moving parallel to its axis at twice the first natural frequency of the cable, we have the following expression to describe its motion. 7TX  y = Y sin y Q  , v-TT*™)  S  in(27rf t) + --c  When the supports are not moving at exactly 2f we have the following. c  r = r -Ar 0  m a x  sin47r/*  /#/  c  and the maximum cable amplitude will be;  1  Ag /  2  (2.32)  max  by combining this with the first term in Equation (2.29) we find;  ^ WHS))" •©*(«) The relationship between A T  m o x  / 7 o and n is shown for various values of 8  in Figure (2.14). From this we see that as 6 increases the range of / over which increasing dynamic response occurs broadens. Thus the region of instability increases. For the case when / = f  c  we find that the maximum amplitude of cable  vibration A depends only on the amplitude of the support motion A c  g  and the cable  length (Figure 2.15).  ^.u«.«fv^U  (2-34)  This maximum amplitude is attained when the so called strainless situation occurs. At this point the motion of the supports no longer forces the cable past its equilibrium deformed shape (ie when the cable is at maximum amplitude the motion of the end supports do not effect the tension in the cable and the entire system is in equilibrium throughout the motion). Although this motion will eventually stabilize it will continue at this stable amplitude until the support motion ceases or changes frequency. Because the energy which  16 creates the support motion can come from any part of the structure and the driving force is parallel to the cable axis, this cable motion can not be controlled with the installation of a discrete damper. A cable damper could not be expected to dissipate the amount of energy necessary to create the support motion. A paper by Maeda, Maeda and Fujiwara [9] discusses cable deck interaction. Each of the cables has a different natural frequency and the authors claim that the oscillation of the deck is interrupted by the cables when f  c  « / „ . When this interaction  occurs there is a transfer of energy from the deck to the cables. The deck thus sees an apparent increase in damping and the cable sees a decrease in damping. This transfer of energy causes an increase in the DAF of the cable and a decrease in the DAF of the deck. This phenomenon is referred to as "system damping". The authors claim that by properly tuning the cables to the various natural frequencies of the bridge, these oscillations can be damped by the cables. However, this tuning, although it decreases bridge motion, may lead to large local cable vibrations. Godden [10] undertook a seismic model study of the proposed Ruck-a-Chucky bridge, which is a curved box girder bridge supported by 48 cable stays. The research indicates that local cable vibrations do not have a significant influence on the overall response of the bridge and that cable forces are primarily related to the vertical oscillation of the bridge as a whole. The cables also exhibited an overall linear response. The study concluded that cable vibration, though visible under all conditions of ground motion, has little effect on the gross behaviour of the bridge.  17  FIGURE 2.1  Catenary cable.  FIGURE 2.2  Catenary  cable with change in end  thrust.  FIGURE 2.3  Behaviour of a suspended  cable under changes in load.  20  FIGURE 2.4  Component interaction, (a). Actual system, (b). Idealized system, (c). Relative displacements of the two masses, (from reference [8]). fx  FIGURE 2.5  Mode shapes of system, (a). No interaction. (/ ^ /„) (b). Combined mode shapes. (f = / , ) . (from reference [8]). c  c  FIGURE 2.6  Combined  mode shapes of system. (f tts /,). (from reference c  [8]).  21  FIGURE 2.8  with a damper of stiffness w. (a), location of damper, (b). deflected shape with w = 0. (c). deflected shape with w = co. (from reference [8j). Cable  22  FIGURE 2.0  A plot of Dynamic  plx.t) 'p(x).  FIGURE 2.10  Amplification  Factor vs.  Frequency.  sfo(2irnt]  Cable under a tension T with an applied harmonic load p{x,t). reference [8j).  (from  23  Deflected shape of a cable with  the optimum damper,  (from  reference  24  FIGURE 2.13  Motion of a cable with its end supports moving parallel to its axis at 2/ . (from reference [8]). c  To sin Anft  -A  0.5  \~&T  mtuc  am Anft  0.i 0.3 0.2 0.1  negative stable  logarithmic decrement  0  FIGURE 2.14  Region of large cable motion and its dependence and frequency, (from reference [8j).  on support  motion  FIGURE 2.15  Cable amplitude resulting from strainless state, (from reference [8]).  26  CHAPTER 3 OBJECTIVES  Through tests performed both during the construction phase and after completion of the Annacis cable stayed bridge the following objectives were undertaken. (1) Determine the first natural frequency of cables of various lengths. Using the results from these tests and the well known taut string theory calculate the tension in each cable at the time of the test. (2) By examining free vibration decay plots determine the critical first mode damping values for cables between 50 m and 220 m in length and compare these to estimated cable damping values for similar bridges. (3) Undertake a parameter study in order to determine what type of damping is present.Investigate how specific parameters can be altered to increase damping values if necessary and how these changes affect other variables such as cost. (4) Investigate possible interaction between cable, deck and tower and determine what effect interaction may have on the natural frequency and damping of the individual components or the combined system. (5) Through tests of cables with none, one and both split neoprene rings in place in the end sockets determine the effectiveness of these as discrete dampers.  27 (6) B y isolating the first vertical, first torsional and first lateral modes of the completed bridge determine the frequencies and damping values associated with these modes.  28  CHAPTER 4  S E L E C T I O N OF T E S T PROCEDURE  Since most serious cable vibration problems involve first mode oscillation, a test which induces the first mode was required. The most readily available instrumentation devices were strain gauge type accelerometers and to measure first mode oscillations one had to be placed at the mid point of the cable. The accelerometer could have been placed at some other point but the accelerations expected were small ( 0.02 - 0.10g at the mid point ) and the best results would be obtained if measurements were taken at mid length. The simplest possible test was an impact test but such a test may have damaged the polyethylene sheathing. A pull back test, also relatively simple, was considered; however, in order to get measurable accelerations the pull back force would have been quite large and may have caused local sheathing damage at the connection point. Also problems with an adequate release mechanism were encountered. The next suggestion was a motorized cart with rotating masses which could travel up the cable, excite it at its natural frequency until a desired response is obtained, and then be switched off so that the free vibration response could be recorded. There were several problems with the motorized cart solution. As well as being costly, weight was a problem due to a possible altering of the cable natural frequency and damping from the effect of added mass. Also since erection loads changed daily, exact cable tensions and thus frequencies were not known prior to testing, making  29  proper tuning of the exciting force difficult. The final solution to the problem was a hand cart equipped with an accelerometer. The cart could be pulled into position allowing hand tuned manual excitation of the cable. In order to keep weight to a minimum and avoid any added mas3 effect the cart was constructed of aluminum.  With cable diameters varying between 99 mm and  149 mm the cart had to be constructed in two parts with spacers lo accommodate for the diameter changes. The final design used three nylon rollers to track the cable. These rollers were tapered on the latjie to ensure a larger contact surface. In case of any small cable irregularities the bolts connecting the two halves of the cart were fitted with springs to allow a slight expansion of the system. Figure (4.1) shows a drawing of the cart and the completed cart is shown in Figure (4.2). The cart had a rope connected to the upper end by which it was pulled into position. A clownrigger was connected to the lower end to provide a bottom end tie back and also to measure the actual location of the cart on the cable, through a built in counter. The accelerometer was mounted on the top cross plate and a pull rope was connected below the bottom wheel. Three different linear strain gauge type accelerometers were used to measure the response of the various components. A ± 2 . 5 g Stratham type A5 accelerometer with an overload capacity of ± 7 . 5 g was used on the cart to measure cable motion. Deck motions were monitered using a ± 5 g C E C accelerometer and perpendicular tower motion was recorded using a ± 0 . 2 5 g Stratham accelerometer.  The reason the ± 0 . 2 5 g accelerometer  was not used to measure cable and deck motions was that it became saturated in a l g environment (ie its overload capacity was ± 0 . 7 5 g ) . The raw signals from all three of these accelerometers were amplified to a desirable level using variable gain Daytronic amplifiers. This field data was then recorded on one of two portable analogue/analogue tape recorders. The first was a Precision Instruments PI-6200 portable instrumentation recorder capable of recording up to 4 channels on standard 1/4 in audio tape. The second was a Phillips  30 portable recorder capable of recording up to 7 channels on a 1/2 in cartridge tape. In order to continuously monitor the output signal throughout the test, a two channel oscilloscope was used.  T h e instrumentation used in a typical test is shown in Figure  (4.2).  Due  to the amount of ambient vibration in the deck during all phases of construction the instrumentation equipment was calibrated in the laboratory before and after each series of tests.  900  FIGURE 4.1  Shop drawing of three wheeled cart.  32  33  CHAPTER 5 EXPERIMENTAL TEST SERIES  5.1  P A R A M E T E R STUDY The first series of tests was done on cables of varying lengths in order to  determine natural frequency and damping values. The natural frequency was then used to calculate cable tension using taut string theory. The data was gathered so that a parameter study could be undertaken to determine what parameters directly affect damping. The construction sequence is such that a cable is lifted into place in the upper socket, jacked to its desired working tension and then the upper neoprene damper is put into place. Since the damper is placed immediately after jacking, so that the staging may be jumped up the tower to the next cable socket, all of the tests in this series were done on cables with one damper already in pln.ce. In order to perform the test, the cart, mounted with the ± 2 . 5 g accelerometer, was placed on the cable at the lower socket, and then hoisted into position at the mid point of the cable. This hoist rope was then tied off to the staging near the upper socket of the cable. The downrigger cable connected to the lower end, to ensure the cart is at the proper location, was then tightened to prevent motion of the cart parallel to the cable axis. When  the cart was properly tied off, the exciting rope was held taut and a zero voltage signal was attained using the balance control on the amplifier. The entire set-up is shown in Figure  (5.1). At this point the recording equipment was turned on and the cable was  excited by hand at its natural frequency. As the cable was excited the output response was monitored and when a predetermined level was reached the excitation stopped. From this point on the free vibration response of the cable was recorded.  Approximately 30  seconds of free response are recorded. After the test has been completed it is viewed on the oscilloscope to determine if damping can readily be obtained from the free vibration decay. If not, perhaps because higher modes are also evident, the test would be repeated until a clean first mode response was obtained. After a satisfactory result is achieved, the cart is lowered down to the bottom socket, removed and placed on the next cable to be tested. Appropriate spacers will be installed if the next cable is of different diameter.  5.2  DAMPER EFFECTIVENESS The second series of tests was designed to determine the effectiveness of  the discrete neoprene dampers in the sockets at each end of the cable.  Some increase  in damping was expected as first one and then two dampers were added to the system. However, as mentioned earlier, the cable is jacked to its working tension and then the top damper is immediately put in place. Thus the cable is only partially jacked when the top damper is not in place and fully jacked when it is. These large differences in tension between the zero and one damper cases makes it difficult to compare the damping values obtained from these two sets of tests. The cables do; however, have approximately the same tension when the second damper is placed as when the first damper was installed. From tests of a series of cables with one and then two dampers in place the effectiveness of the single bottom damper can be determined. Although the main span of the bridge was closed before this research was completed none of the bottom dampers had been put  35  in place. In order to test their effectiveness six of these bottom dampers were installed so that tests could be carried out. The test procedure for this series of tests was exactly the same as that used in the parameter study.  5.3  COMPONENT INTERACTION The final series of cable tests performed investigated the possible interaction  of the cable, deck and tower. Several tests were done on some of the shorter cables with the ± 0 . 2 5 g accelerometer mounted in the tower at the upper socket of the cable being tested. The response of the tower was recorded simultaneously along with the response of the cable to see if there was any relationship between their predominant natural frequencies. The set-up and procedure were the same as for the parameter study except that there was an extra accelerometer mounted in the tower. Due to the high stiffness of the tower the motion of the cable had no effect on it and only a low ambient signal was obtained throughout the experiments. Interaction between the cable and the bridge deck seemed more probable since the deck is not nearly as stiff as the tower. Also its mass per unit length is much lower. In order to investigate this possibility a series of tests was carried out with the ± 5 . 0 g accelerometer placed on the deck beside the lower socket of the cable being tested. The test procedure was again the same except that the extra accelerometer was placed on the deck rather than in the tower.  5.4  FULL BRIDGE TESTS As well as cable frequencies and damping these same values were of interest  for some of the lower modes of the entire bridge. The predicted lowest mode was the first vertical with the next two modes being the first lateral and first torsional. Two tests were performed in order to excite and investigate the characteristics of the first vertical mode. The first of these tests was performed before the main span  36  was closed and thus intended to measure the frequency of the half bridge in its vertical mode. The travelling derrick was moved to the free end of the half bridge, as the land side had already been landed on tower S2, and a reel of bridge cable was connected to it. The reel was then lifted above the deck, the boom was lowered quickly and the brakes were suddenly applied. It was hoped that this braking force would supply enough energy to excite the half bridge in its first vertical mode. The ± 2 . 5 g and ± 5 . 0 g accelerometers were mounted on the girder on opposite sides of the deck to measure vertical accelerations and determine if a torsional mode was present. When the main span of the bridge was closed a second test was carried out to determine the first vertical mode of the bridge. Rather than using a braking force to supply the input an impact test was used. A larger force could then be applied and the exciting mass would remain in contact with the bridge. For this test a large dumptruck filled with concrete blocks was driven to the center of the main span. The rear axle of the truck was then rolled onto 6 inch high wooden pads. The exciting force was produced by driving the truck off the blocks and braking immediately after the rear wheels made contact with the deck. The same instrumentation was used in this test as was used in the first bridge test. A third test was carried out to determine the natural frequency and damping value for the first lateral and first torsional modes. This test, like the cable tests, used a harmonic input force to excite the bridge. A variable length 2000 kg pendulum was used as the exciting force. A rough terrain crane mounted at the center of the main span provided the variable length pendulum which oscillated perpendicular to the bridge in order to excite the lateral and torsional modes. The instrumentation for this test consisted of two ± 0 . 2 5 g strain gauge type accelerometers for measuring lateral motion and two ± 2.5g accelerometers for measuring vertical motion. The signals from all four accelerometers were amplified with Daytronic variable gain amplifiers. The signals from any two of these could be analysed simultaneously using a Nicolet spectral analyser.  The Nicolet spectral analyser was used to  37 accurately determine the frequency of the lateral and torsional modes so that the pendulum could be tuned to the correct exciting frequency. The Nicolet analyser also displayed displacement-time curves and hard copies were made on site using an H P plotter. The four accelerometers were placed at different locations throughout the tests in order to determine if a mode was purely lateral, torsional, or a combination of the two. A n impact test was also performed to check the natural frequency of the first vertical mode.  38  Accelerometers (I indicates direction of sensitivity) 1 = ±2.5g on cable 2 = ±0.25g in tower 3 = ±5.0g on deck  FIGURE 5.1  Typical tower.  test set-up showing  accelerometers  on the cable, deck and  39  CHAPTER 6  ANALYSIS OF R E C O R D E D DATA  When a test series is completed it is desirable to reduce the recorded data to graphical form for analysis. The analogue signal from the portable tape recorder was digitized using a Digital Equipment PDP 11/04 computer. The system has a built in low pass filter which can vary between 2.5 V and 100 V. A special program (ACQUIRE) digitizes a test record and stores it in a buffer for further manipulation. The software also allows for a variable sampling frequency and length of record. All of the data analysed in this thesis were sampled at a frequency of 100 Hz. Once stored in the buffer the digitized data can be manipulated using the program OMFORM. A constant can be added to the buffer in order to achieve a zero mean value for the digitized record. This is a very useful feature in that when a test is performed it is impossible to get a perfect zero start point using the balance control on the amplifier. The entire record, or any part of it, can then be produced on a plotter (CALCOMP 565). The modified contents of the buffer are then formatted and stored on a floppy disk for later use. A third program (EDSPEC) receives data from thefloppydisk and computes a Fast Fourier Transform (FFT). The F F T is calculated using the first 1024 points in a given record. Once the F F T has been calculated either the Power Spectral Density or the  40  Fourier Spectrum for that record can be plotted. Given the free vibration decay plot and the Fourier Spectrum, the displacement record can easily be calculated by integrating the acceleration record twice with respect to time. This simply involves division by w , since for all tests the first natural 2  frequency was dominant. Also, by reading off the values of consecutive peaks, first mode damping can be obtained using the logarithmic decrement method. Consider a decaying sine wave such as that shown in Figure  (6.1). The  equation for this curve is;  x{t) = X e~  iut  Q  son u t d  (6.1)  Then the ratio of any two successive positive peaks is; ^ -  =  (6.2)  Taking the natural logarithm of both sides we get the logarithmic decrement 5.  6 = l n - ^ - = 27T£— x  (6.3)  n+l  and, w = a/\/l d  c  (6.4)  2  If we combine Equations (6.3) and (6.4) we get;  vi-<:  2  For low damping we can approximate Equation (6.5) by;  6 w 2TCC  (6.6)  Now using this approximation we can expand Equation (6.2) as a series.  = e « e ^ = 1 + 2 ^ + -2££)1 + ... 6  «n  +l  2  2!  (6.7)  41  For low values of ( sufficient accuracy can be obtained by retaining only the first two terms in the series, then;  For very low values of damping, such as the ones evident in this thesis, a simplified expression of damping over m cycles can be obtained. (6.9) A F O R T R A N program ( C A B D A M P ) was written to obtain displacement and damping values for each test. The program is shown in Figure (6.2) The complete flow of data from the actual test, to the lab where it is digitized, to the first mode damping values is shown in the flow chart of Figure (6.3). Once the frequency, tension and damping values have been obtained for a given cable the information is stored in a spreadsheet. With the information from the plots recorded in a spreadsheet, the frequency, mass per unit length and length of each cable were used to calculate an approximate tension at the time of the test. As discussed earlier, Hooley [7] developed a set of dimensionless parameters and produced a plot which displays the behaviour of any cable under increases in load (Figure 2.3). As seen from the plot, cables which have the dimensionless parameter KQ greater than 1.5 are considered stiff, and have linear load deflection characteristics. Recall that the Annacis cables under dead load are well into the stiff range and can be analysed using inextensible string theory. Consider a flexible string of mass p per unit length stretched under tension T . By assuming the lateral deflection of the string y to be small, changes in tension due to deflection are negligible and can be ignored. A free body diagram of an elemental length dx of the string is shown in Figure (6.4). Assuming all deflections and slopes are small, the governing equation for motion in the y direction is; (6.10)  42  or, 89 _ p d y 2  dx~  T dt  2  (  Since the slope of the string (9) can be expressed as 9 = dy/dx,  6  '  U  )  Equation (6.11) reduces  to;  (en) dx where c = \jTjp  ~ c dt  2  2  2  1  J  and is the velocity of wave propagation along the string. One method  of solving the partial differential equation is separation of variables, where we look for a solution of the form, y{x,t) = Y{x)G{t)  (6.13)  By substituting this expression into Equation (6.12) we obtain, 1 d Y _ 1 1 d?G Y dx ~ c G di* 2  2  2  ( 6 - 1 4 )  where the left hand side of the equation is independent of t and the right hand side is independent of x. Thus each side must be a constant. If we call this constant —(w/c) we 2  obtain two ordinary differential equations. d?Y  a?  /w\2 +  (T)  dG  ¥  =  0  <- > 6  15  2  dt  2  + uG = 0 2  (6.16)  with the general solutions,  Y = As'm—x + Bcos—x c c  (6-17)  G = C8'mujt + Dcosut  (6.18)  The arbritrary constants A , B, C , D depend on the boundary conditions. For our case we have a string of length / between two fixed supports, so the boundary conditions are,  1,(0,0  =  0  y(i,t) = o  43  With these boundary conditions the solution becomes; ujl sin—= 0  (6.19)  c  or, —— = n.7T c  n=  1,2,3,...  (6.20)  Each n represents a normal mode vibration with natural frequency determined from the equation; "  f  =  » = U,3,..  Il =Tl\p C  (6.21)  The mode shapes are sinusoidal with the distribution, . nirx Y = sin —  , . (6.22)  Thus if the frequency is known from testing, the corresponding tension can be calculated using; T =  Equation (6.23) has been used throughout the analysis to calculate cable tensions.  (6-23)  FIGURE 6.1  Sine wave with exponential  decay, (from reference [25]).  /COMPILE C PROGRAM TO A N A L Y S E THE DECAY P L O T S OF C A B L E S . T H I S PROGRAM C C A L C U L A T E S THE MEAN DAMPING V A L U E AS WELL AS THE E F F E C T OF C D I S P L A C E M E N T ON DAMPING. C R E A L DEC,SUM,AVDEC,TOT,DAMP,W1,SCALE DIMENSION A ( 6 0 ) , X ( 6 0 ) , D ( 6 0 ) C R E A D { 5 , * ) N,W1,SCALE,NC,ND W R I T E ( 6 , 6 0 ) NC 60 F O R M A T ( ' C A B L E # ' , I 2 ) W R I T E ( 6 , 7 0 ) ND 70 F O R M A T C N O . OF DAMPERS = ' , 1 1 ) WRITE(6,80) N 80 F O R M A T C N O . OF C Y C L E S = ' , 1 2 ) W R I T E ( 6 , 9 0 ) W1 90 FORMAT('NATURAL F R E Q . = ' , F 7 . 4 ) WRITE(6,100) SCALE 100 F O R M A T C C O N S T = \ F 7 . 4 ) C READ(5,*) ( A ( I ) , I = 1 , N ) C 1=1 SUM=0 PI=3.1415926 C C C A L C U L A T I O N OF MEAN V A L U E OF DAMPING OVER N C Y C L E S . C II=N-1 DO 20 1=1,11 DEC=(ALOG(A(I)/A(I+1)))/(2*PI) SUM=SUM+DEC 20 CONTINUE AVDEC=SUM/(N-1) AVDEC=AVDEC*100 W R I T E ( 6 , 1 0 1 ) AVDEC 101 F O R M A T ( ' 0 ' , ' A V G . DAMPING V A L U E ( % C R I T ) =',F10.7 ) K=N-2 1=3 C C V A R I A T I O N OF DAMPING WITH D I S P L A C E M E N T I N MM.. C WRITE(6,200) 200 FORMAT('0',T9,'CYCLE #',T22,'DISPLACEMENT(MM)',T44, +'DAMPING(%CRIT)') WRITE(6,*) ' ' C DO 30 1=3,K,2 X(I)=A(I)*9810*SCALE/(W1**2) J=1 TOT=0 DO 40 J=1 ,4 JJ=J-1 DAMP=(ALOG(A(l-2+JJ)/A(I-2+J)))/(2*PI) TOT=TOT+DAMP 40 CONTINUE D(I)=TOT/4*100 WRITE(6,*) I , X ( I ) , D ( I ) 30 CONTINUE STOP END /EXECUTE  FIGURE 6.2  Damping  program  CABDAMP.  SITE Accelerometer  Accelerometer  on cable  on deck  1  Daytronic  Daytronic  amplifier  amp ifier  ± 1 Portable tape  —————  recorder  p  Oscilloscope  LAB Portable tape recorder  ACQUIRE  •  OMFORM Record stored on disk  Plotter  EDSPEC  i  3rrrr_ C ^ o v e r n cycles.  Response  Acceleration — time  spectrum  record  *  CABDAMP f vs x every three cycles.  FIGURE 6.3  Flow chart showing collection  and analysis of data.  47  FIGURE 6.4  Free body diagram 126}).  of an elemental length of string,  (from  reference  CHAPTER 7  T E S T RESULTS  7.1  P A R A M E T E R STUDY Since the problem of detrimental cable motion at Brotonne involved first  mode oscillations of the longest cables, the characteristics of this mode were investigated. Cables varying in length from 50 m to 220 m were tested in order to determine if the longer cables had inherently lower damping values than the shorter ones. For each of the cables tested a free vibration decay and a Fourier Spectrum were plotted and these are shown in Figures (7.1 - 7.15). The results of these tests, along with important cable parameters are included in Table (7.1). This table was prepared in a spreadsheet environment using L O T U S 1-2-3. Since the top socket neoprene damper is installed immediately after jacking all of these tests were performed with the top damper in place. Cable tensions are generally lower than final dead load tensions since all of the tests were performed before the 50 mm concrete deck overlay was in place. However, some of the cables had tensions greater than their final dead load tensions due to the location of live loads (cranes, trucks, etc.) at the time of the test.  49 The damping values obtained for all of the cables tested, with one damper in place, range between 0.27 and 0.52 percent of critical damping. Critical damping is defined as the lowest value of damping for which no oscillation about the zero deflection position occurs in the free response of the system. Since the DAF « 1/2$ large oscillations are in fact a possibility. A parameter study was undertaken to determine what actually causes the damping evident in these cables. All of the tests performed were for small amplitude oscillations, and it was first expected that damping would increase with amplitude. However, this relationship was not evident in these low amplitude linear tests. For very large amplitudes geometric relationships may become non-linear and an increase in damping may be expected. Damping in most structures is idealized as viscous damping, where the damping force is proportional to the velocity of motion (Fj = cy). In this expression y is the velocity of the system and c is a damping constant. In order for the equations of motion to uncouple, the damping constant c must be of the form; c  = o[m] + ai[k]  (7.1)  0  This is known as Rayleigh damping . Thus if the damping constant c is proportional to structural mass and/or stiffness, then it satisfies the orthogonality condition and the following relationship holds;  = ir ° T a  +  a i  n  -  (7  2)  where c is the damping ratio in any mode n and u> is the natural frequency of that n  n  mode. The first term, l / 2 w „ a , corresponds to mass proportional damping and the second 0  term, oj /2a , n  l  corresponds to stiffness proportional damping. From a plot of Damping vs  Frequency (Figure 7.16)  we see that damping is proportional to frequency and not the  inverse of frequency. Therefore, the damping in these cables can be said to be stiffness proportional.  50  Consider a cable with no sag as a single degree of freedom system with its mass concentrated at the center. From the free body diagram of Figure (7.17) we get the following relationship for lateral stiffness.  * = f  Ignoring the constant a plot of Damping vs Stiffness {T/S)  ('3)  (Figure  7.18)  shows that damping is directly related to stiffness. For large cable oscillations, tension in the cable will increase due to non-linear effects, and damping will increase slightly. However, even if the tension doubles, which is highly unlikely, the damping value will only double. This leaves a damping value below one percent critical and the DAF will still be very large.  7.2  DAMPER EFFECTIVENESS In order to determine the overall effectiveness of the neoprene dampers, tests  with none, one, and two dampers in place were performed. Due to the large difference in tension between the none and one damper cases only the effectiveness of the second damper was examined. With the damper placed approximately 1.5 m from the end socket of the cable e values varied between 1/35 for the short cables to 1/140 for the long cables. Recalling that the effectiveness of the damper depends on its distance of application from the end point {DAF  «  1/e:), large increases in damping were not expected with the  installation of a second damper.  It was also expected that the benefits of the second  damper would be more evident in the shorter cables where the e value is larger. The free vibration and spectral plots for six cables with the second damper in place are shown in Figures (7.19 - 7.24). The natural frequencies, and thus tensions sometimes varied slightly between the one and two damper tests for a given cable. However, these slight differences in tension should not have any effect on damping. Thus the damping  51  values for both cases are compared in Table (7.2). They are also displayed graphically in Figure (7.25). As can be seen, the addition of a second damper brings only a slight increase in overall cable damping (approximately a 10 percent increase over the 1-damper value on average). This increase in damping is consistent in all cables tested; thus the shorter cables do not achieve a greater increase in damping due to their larger e value. These tests involve low amplitude oscillations and for larger displacements these neoprene dampers may be more effective. Due to their location, the dampers, even during large first mode vibrations, see relatively low amplitude motions compared to the amplitude of motion at the center of the cable. However, for higher mode excitation, the relative amplitudes at the centerline and the line of action of the damper could be similar, and thus these dampers may be more effective in damping out these higher modes. The wind speeds measured at the site will tend to excite the higher modes of these longer cables and thus the dampers should be quite effective in reducing these wind induced oscillations.  7.3  COMPONENT INTERACTION Tests to investigate component interaction were carried out with accelerom-  eters on the deck and in the tower. The available instrumentation allowed either the cable and tower, or the cable and deck to be analysed simultaneously. Cable tests performed with a ± 0 . 2 5 g accelerometer in the tower showed that there was no increase in tower motion over the ambient level before excitation of the cable. Since the deck is much more flexible than the tower it was thought that cable motions may excite certain deck modes. A series of tests with an accelerometer mounted on the deck at the lower cable socket was done and the results are shown in Figures (7.26 - 7.30). These spectra and response curves can be compared with those for cables 26 through 30 v/hich are shown in Figures(7.5 - 7.9). The level of local deck motion increased due to the harmonic input of the cable; however, this induced deck motion did not  52  appear to be harmonic itself. No evidence of either a 1:1 or a 2:1 ratio of deck frequency to cable frequency was noted. This lack of interaction was not surprising since cable motions were small and the energy input into the cable is not nearly enough to induce the lower modes of the bridge. In order to investigate this phenomenon in depth large amounts of energy would have to be input to the deck or tower. However, evidence of component interaction occurred twice during the erection. The first occurrence was just before the main span was completed in a moderate (18 knt) SSW wind. Cable 4C, on the north end of the bridge, began to oscillate in the direction of the wind in its first mode. A SSW wind acts almost parallel to the cable axis, and a check of the frequency of eddy shedding proved that the wind was not directly exciting the cable. Cable 4C is one of the longer cables on the north side and its sockets are near the top of tower N l and directly above the bent N2 as shown in Figure (7.31). Cable 4C oscillated in its first mode, parallel to the bridge, with a steady state amplitude of approximately 300 mm. This steady state condition lasted for several hours, during which time smaller motions were, also evident in main span cable 45C which shares a tie beam with cable 4C. The approximate first mode period of cable 4C was 3.1 seconds as measured with a stopwatch. This motion is obviously created by support excitation and its steady state nature would indicate that it is the parameter effect discussed by Kovacs [8]. In order to excite the cable in its first mode the two towers N l and N2 would have to be moving 180 degrees out of phase with a period of approximately 1.55 seconds, as shown in Figure (7.32). However, since N2 is far stiffer than N l it is more likely that the cable socket at N2 remains stationary and tower N l oscillates at 2/ . Since the energy used to excite the cable is not c  input directly to the cable by the wind, but rather comes from support excitation due to wind energy being input at some other location in the bridge, this motion cannot be reduced by discrete dampers acting on the cable. A discrete damper could never dissipate the amount of energy necessary to excite the tower or deck; however, this parameter  excitation will eventually reach a steady state level above which amplitudes will no longer increase. This steady state level depends on the amplitude of the support motion (in this case the towers) according to Equation (2.33).  A  c. u tab  «  (2.33)  This phenomenon occurred a second time after the main span had been completed. Under a similar but slightly stronger wind cables 3C and 4C began to oscillate laterally in a direction parallel to the bridge. However, this time amplitudes were considerably smaller (approximately 75 mm) and the resulting motion seemed to contain higher modes, perhaps the second or third. Although much more difficult to verify (eg mode shape, period, etc) these motions were also evident for several hours at what appeared to be a steady amplitude. Both of these visual records indicate that a steady amplitude which lasted for several hours existed. This steady amplitude corresponds to the so called "strainless" state which a cable reaches due to parameter excitation.  7.4  FULL BRIDGE TESTS The natural frequency and damping value for the first vertical mode of the  bridge were of high interest. In order to determine these parameters a test was performed using the braking force from a crane as the vertical input. The resulting response and spectral plots showed the predominant frequency to be 1.37 Hz. This high frequency (much higher than the predicted first vertical » 0.25 Hz) was that of the mass swinging as a pendulum. The high local accelerations caused by this swinging overpowered any first vertical mode oscillations that may have been present due to the braking force. In order to eliminate this pendulum effect, another test was planned in which the exciting mass would remain in contact with the deck after the initial impulse.  54 This second test was the truck drop test described earlier.  A dumptruck  rolled off a set of blocks onto the bridge deck and then remained in contact with the bridge. The results of this test showed the predominant response frequency to be 2.35 Hz. Although the truck remained in contact with the bridge, it acted as a spring-mass-dashpot system and the large local accelerations at 2.35 Hz are due to the truck oscillating on its springs. These high local accelerations again predominated over any first vertical mode response which may have occurred. After this test it was concluded that the best way to excite the bridge in a given mode is to harmonically excite it at the estimated frequency of the desired mode. This way energy could be put into one particular mode and once a desirable level of response was produced, the exciting mass could be stopped, placed on the deck and the free vibration response of the bridge could be measured. Since it was difficult to produce a low frequency vertical exciting force the decision was made to excite the first lateral mode of the bridge instead of the first vertical mode. A pendulum with a relatively large mass swinging perpendicular to the bridge had successfully been used to excite the Lions Gate bridge, so a similar set-up was chosen. The pendulum used at the Annacis site consisted of a rough terrain crane, positioned at center span, with a 2000 kg section of curb at the end of its line. Since the exact frequency of the first lateral mode was not known, the crane allowed for variability in the length of the pendulum to permit tuning. Ropes on either side of the mass allowed two men, positioned on opposite sides of the bridge, to maintain a relatively constant input force. Analytical models indicated that the first lateral and first torsional modes were close together and perhaps coupled. In order to determine this, four accelerometers (2 vertical and 2 lateral), were used throughout the experiment.  The instrumentation  consisted of the four accelerometers mentioned, two Daytronic variable gain amplifiers, a Nicolet spectral analyser and an H P plotter. The pendulum was first adjusted to the estimated frequency of the first lateral mode and the bridge was excited at this frequency for several minutes. The mass  55  was then placed on the deck and a spectral analysis of the response was done. The spectrum revealed two main frequencies, one at 0.35 Hz, and the other at 0.475 Hz. The pendulum was then adjusted to a frequency of 0.35 Hz and the test was repeated.  For this test  two vertical accelerometers were placed cn the girder on opposite sides of the bridge, and one lateral accelerometer was located on the center line of the bridge. Response spectra for one of the vertical accelerometers and the lateral one at the center line are shown in Figure (7.33). There is a large peak at 0.35 Hz in the lateral response spectra but a very small peak at this frequency for the vertical response spectra. This indicates an uncoupled first lateral mode at 0.35 Hz. By taking more sample points and using a smaller band for the spectra the exact first lateral mode frequency was found to be 0.325 Hz. A smaller peak at 0.225 Hz was evident in some of the response spectra; however, lateral response spectra for the crane and deck (Figure 7.34) revealed that this was a fundamental frequency of the crane. After exciting the bridge at 0.325 Hz for approximately five minutes an amplitude of between 50 mm and 100 mm was achieved and the mass was brought to rest so that the free vibration response could be obtained. A lateral response spectrum at each girder (Figure 7.35) showed two strong peaks at 0.325 Hz. and the displacement-time plot showed that the two responses were in phase. This again indicated a lateral mode, providing the shear center is at deck level. The free response plots (Figure 7.36) showed an average damping value of 0.46 percent of critical for the first lateral mode. The next mode to be isolated was the first torsional mode with a frequency of 0.475 Hz. After several minutes of excitation, amplitudes of approximately 100 mm were obtained and slight cable motions due to this excitation were observed. At this point the mass was brought to rest snd the free vibration response was again recorded. The resulting spectrum (Figure 7.37) shows peaks for both vertical accelerometers at 0.475 Hz. The displacement-time trace (Figure 7.38) showed that the two responses were 180 degrees out of phase, indicating the first torsional mode. The average damping value obtained from these plots was 0.30 percent of critical.  56 The resulting damping values for the first lateral and first torsional modes are taken from relatively low amplitude tests, and for large amplitude oscillations the damping values should be higher. A vertical impact test was carried out with the 2000 kg mass being dropped onto the deck every 30 seconds. These cyclic drops allowed for continual updating of the response spectrum. The resulting spectrum is shown in Figure (7.39). The large peak at 0.275 Hz indicates the first vertical mode of the bridge.  A damping value for the  first vertical mode could not be obtained from this impact test because the free vibration response contained many higher modes. The first vertical, first lateral and first torsional modes shapes for the completed four lane bridge are shown in Figure (7.40).  57  CABLE » 1 2 ( 1 DRAPER. OCT 1 . 1 9 8 5 . FOURIER SPECTRUM.  NOT F U L L Y  JRCKED)  o  r~  -1—  0.0  3.D  4.5  1  1  fi.Cl  7.5  1—  • FREQUENCY iHZ)  CRBLE " 1 2 ( 1 D R M E R , NOT F U L L Y OCT 1 , 1 9 8 5 . FREE V I B R A T I O N DECRY P L O T . P  9.0  ill. 5  12.0  JRCKED)  32.0  FIGURE 7.1  Cable 12 (1-damper). record.  i3.  Response  spectrum  and acceleration  3R.0  - time  58  cr  FIGURE 7.2  Cable record.  22 (1-damper).  Response  spectrum  and acceleration  - time  FOURIER SFECTRUM CfiBLE "23 FREE VIBRfiIC?N TEST T  FIGURE 7.3  Cable 23 (1-damper).  record.  JULY  16.1965.  Response  spectrum  and acceleration  -  60  FOURIER SPECTRUM A N A L Y S I S ' CABLE 2 4 J U L Y 1 5 1 9 8 5 K  >;  [? !";  CABLE * 2 4 J U L Y 1 5 1 9 3 5 FREE V I B R A T I O N TEST M I D - S P A N A C C E L E R A T I O N DECAY  ie.c  FIGURE 7.4  Cable 24 (1-damper). record.  Response  spectrum  and acceleration  - time  61  FIGURE  7.5  Cable 26 (1-damper). record.  Response  spectrum  and acceleration  - time  FIGURE 7.6  Cable 27 (1-damper). record.  Response  spectrum  and acceleration  -  time  63  • CfiQLF * 2 8 I 1 - D H M P E R ) FILTFREDfi 2.5 HZ. FOUR IFR SPECTRUM. T  ~~  N0V  13.1385.  '  x  FIGURE 7.7  Cable 28 (1-damper). record.  Response  spectrum  and acceleration  - time  64  CRBLE <=23 l l-DfiMPERJ NGV 12.1965" FILTERED M 2.5 HZ. • "• COURIER SPECTRUM. — t  FIGURE 7.8  Cable 29 (1-damper). record.  Response  spectrum  and acceleration  - time  65  CR5LE "30 (1-DRMER) NGV 13.1985. FILTERED RT 2.5 HZ. FOURIER SPECTRUM. r~. P  FIGURE 7.9  Cable 30 (1-damper). record.  Response  spectrum  and acceleration  - time  66  i i-DRMPERf"EB 6.1986/:. FOURIER SPECTRUM 12. D HZ F I L T E F O r  CR3LE: "*39  :  LU  1  0. L'  =*  1  1  r—  FREQUENCY .'HZ1  10.5  12.0  13.5  CRBLE *39 t i - D R M P E R ) FED 6,1966. FREE V I G R R T I I 3 N DECRY P L O T . FILTFRED R 2 . 5 H Z . : — T  27.0  FIGURE 7.10  Cable 39 (1-damper). record.  Response  spectrum  and acceleration  - time  67  13.5  35.0  FIGURE 7.11  Cable 40 (1-damper). record.  Response  spectrum  and acceleration  - time  68  FOURIER  LU  SPECTRUM  A. U.G  FIGURE 7.12  J..U  Cabie record.  (2.5  HZ  4. r  FILTER)  r  1  i  FREQUENCY ' it  42 (1-damper).  Response  spectrum  and acceleration  - time  69  LHBLF. M 4 1 1 - D R M E R ) F F R I L 8 . 1 9 8 B . FOURIER SPECTRUM ( 2 . 5 HZ F I L T F R J P  FIGURE 7.13  Cable record.  44  (1-damper).  Response  spectrum  and  acceleration  - time  70  FIGURE 7.14  Cable 45 (1-damper). record.  Response  spectrum  and acceleration  - time  71  CABLE" "46 tT-DHMPER) APRIL 6.1986. FOURIER SPECTRUM (2.5 HZ FILTERJ :  FIGURE 7.15  Cable 46 record.  (1-damper). Response spectrum  and acceleration  - time  FREQUENCY vs CABLE  DAMPING  0.94  Damping (X critical)  STIFFNESS vs CABLE DAMPING 60 - i  FIGURE 7.18  —  A plot of stiffness {T/S)  vs.  —  damping.  T4  CRQLE "27 RPRIL 1.4. 1985 ; R E E . v i B f i n n o N DECRY  12-BRMPERS  FIGURE 7.19  I T  2 . 5 KZ F I L T E R !  Cable 27 (2-dampers). record.  Response spectrum  and acceleration  - time  75  in  C.J L  "|  CRBLF "2$ RPRIL 14.iStiS FOURIER SPECTRUM • 12 -0DHPER5 2.5 HZ FILTER)  !  1  ! •  :  i.  ;  -  ' •  !  1  •j ':  i  i  •  i  i  "  i . I 1 , T. ~  n  i •  i!  C-3  i  i Ct'.'V "  i  .  !  • •1 j  ! -  :  II. 5 .'•  :i  '  1 ;1  ;  j ! .i  i 2'. 25 i  ! |  .  1  !  !...• ! ' | '  _. I  ;  :  1  ; , y ;  1  FREQUENCY iHZ) ! ""."".'] " T :  " 'j  |  ;  1  •  -1  I . I •5.25 i  i  • !  •  ~r '  , '  ' i  !  8  •  i  I Fi.L!  r ' {i-BLE *2b RPRIL Ii4,l965 i :FREETVIBRRTIQN DECRY PLOT! —l-2=DftMPEf?S--2:5 KZ. FILTER) J  FIGURE 7.20  Cable 28 (2-dampers). record.  Response  spectrum  and acceleration  - time  FIGURE 7.21  Cable 29 (2-dampers). record.  Response  spectrum  and acceleration  - time  77  : i  COBLE «3d) APRIL 14,1086 T FOURIER SPECTRUM : • ! "T2--DinMFERS--2.5 HZ FILTER]  fi.0  FIGURE 7.22  Cable 30 (2-dampers). record.  Response  spectrum  fi.  and acceleration  75  - time  78  :mu «39 npRiL u. lass  0URIFR SPECTRUM i j V2 -DHMPERS" 'LVJ HZ V1LTE RJ r  FIGURE 7.23  Cable 39 (2-dampers). record.  Response spectrum  and acceleration  - time  79  FIGURE 7.24  Cable 40 (2-dampers). Response spectrum and acceleration - time record.  80  Effectiveness of Neoprene Damper 0.45 - i  c "5. E  o a  27  28 V/\  FIGURE 7.25  1-Dompw  29 Cabl« Number ^///A  30  39  40  2-Dampera  Bar graph showing the increase in damping second damper.  due to the addition  of a  FIGURE 7.28  Deck motion at socket 26. Response spectrum and acceleration record.  - time  82  FIGURE 7.27  Deck motion record.  at socket  27. Response  spectrum  and acceleration  - time  83  DECK R SOCKET *28 N0V 13,1385.. FOURIER SPECTRUM. T  i/|  —  — \ " ~ " '  f-.i-  "7.5  i  1 — - — i  It. 5  9.U  FREOUENC'' ihZi 1  DECK HOT IONfi SOCKET *28; NOV 13,1385. .r:i T  ' I ME ::SEC(3NDS'... .2.0  FIGURE 7.28  Deck motion at record.  ;•-  ,5.-:<  .  socket 28.  2U.0 j , . .  2».&  .;.  ..-AA*  ,j  32.0  Response spectrum and acceleration  3ft U  - time  84  GECK MOT ION RT SOCKET 29R: D U R l N f i i : T f S : T d r ; : — T I ~ - : OF CRESLE =29 - NOV 13. 1985. ^——tzzziz:.::-hiry;" ;  i  FIGURE 7.29  Deck motion at socket 29. Response spectrum and acceleration record.  - time  85  DECK RT SOCKET =30 NOV 13. :985. ' FOURIER SPECTRUM. v.:";  FIGURE 7.30  Deck motion record.  at socket 30. Response spectrum and acceleration - time  86  FIGURE 7.31  North half of bridge showing cables AC and 45C and towers Nl and N2.  FIGURE 7.32  Motion effect.  of towers Nl and N2 necessary to create the  "parameter"  87  0.35000 HZ A M  7  —H  1  f-  87.5-06 £  -I  h  r-  - t  C  h i  SU 99  1  W  RS B -M -  _J  1  FIGURE 7.33  1  1  1  4  1  8 12.-06 E -I  A M SU 99  —U-iif—!Lii  1  HZ  Response spectra for lateral mode pendulum test. Upper trace horizontal response spectra. Lower trace - vertical response spectra.  0.22500 HZ  RS  J  1  C  1—  I—  r  1  •\  MUA/W\  B M -I  -I FIGURE 7.34  +  1  HZ  Response spectra for first lateral mode test. Upper trace - lateral response of deck. Lower trace - lateral response of crane.  88  FIGURE 7.35  First lateral mode pendulum west girders.  test. Response spectra of the east and  89  90  FIGURE 7.37  Response spectra for the east and west girders resulting from tion in the first torsional mode.  oscilla-  91  FIGURE 7.38  Free response plots from two tests of the first torsional  mode.  92  1 2 #AVG G . 2 7 5 0 0 H2 I A  1-  h  62 . 4-Q6 74.G-06 f — -  h  (-  t: F  VLG C  h  I-  t-  1  4-I  M : -L  i  J  i  4  v  t  4-  A  vt  f  1  I  -1  /  1  0 . 1A  FIGURE 7 . 3 9  1  0. 1A  1  1_  |__:  A/1 6  Vertical response  spectrum  1  HZ  1  A  _,  1  10  resulting from cyclic drop test.  03  FIGURE 7.40  Mode shapes of completed Annacis bridge, (a). First vertical (b). First lateral, (c). First torsional. (Courtesy of Buckland and Taylor).  94  Cable no. 45 42 39 44 12 30 46 23 40 29 23 27 22 26 24  Damp i ng ( ocr i t )  Dens I t y !kg/m)  Length (m)  0 . 2794 C . 3067 0 . 3119 0.3148 0.3154 0 .3223 C .3303 0.3374 0.3621 0 .3676 0.3703 0.3754 0.4172 0.4238 0.5238  5 4.330 54.330 50.255 54.330 46 . 535 36.595 54.330 36 .595 50.255 36 .595 3S.595 40 . 469 36.595 40.469 46.535  212 .345 135 .960 159 .141 20 3 .791 141 .729 83 .346 221 .393 63 . 165 168 . 510 75 .906 56 . 104 61 . 558 61 .751 55 . 267 50 .433  a  TABLE 7.1  Cable number  27 28 29 30 39 40  TABLE 7.2  Important parameters  Damping (% c r i t i c a l ) (1-damper) 0.3754 0.3374 0.3676 0.3223 0.3119 0.3621  Effectiveness  Freq (Hz)  Tens i o n (kN)  0 . 496 0 . 579 0.699 0 . 585 0.650 1.162 0 . 496 1. 470 0 .701 1. 320 1.610 1. 490 1. 560 1.740 2 .34  2422 2517 2488 3085 1557 1374 2633 1470 2806 1470 1198 1353 1358 1497 2594  Mass (kg) 11564 10103 7993 11072 6602 3050 12055 2494 8468 2778 2053 2491 2260 2237 2349  Stiffness (kN/m) 11.379 13 .535 15.634 15.138 10.986 16 . 485 11.866 21.565 16.652 19 . 366 21.353 21.979 21.992 27 .087 51.435  for the cables tested with one damper in place.  Damping (% c r i t i c a l ) (2-dampers) 0.4131 0.3611 0.4015 0.3476 0.3453 0.4152  Increase in damping (% c r i t . ) 0.0377 0.0237 0.0339 0.0253 0.0334 0.0531  of a second neoprene damper on the six cables tested.  95  CHAPTER 8  CONCLUSIONS  In this thesis an investigation into the dynamic behaviour of a cable stayed bridge and its various components was presented.  It included a discussion of previous  work in the area of bridge dynamics and more specifically cable vibrations. Full scale tests were aimed at determining natural frequencies and damping values for the lower modes of the cables and the bridge, as a whole. Cable tests also investigated the type of damping present and the effectiveness of a discrete damper when added to the system. Findings and recommendations from the experimental results and analysis can be summarized as follows. (1) Dynamic cable testing provides a fast and simple method of determining cable tensions during the erection of a bridge. Since erection loads are changing constantly, this approach can be used to monitor the change in tension of certain specified cables. When the bridge is completed the tensions calculated from dynamic tests can be used in conjunction with a computer to "smooth" the profile of the bridge by adjusting cable lengths and thus tensions to obtain a perfect parabolic curve. (2) With the top neoprene damper in place, damping values for all of the cables tested were found to be between 0.27 and 0.52 percent of critical damping.  96 These values are similar to the damping values obtained for the lower modes of the entire bridge and those obtained from the data collected at the PascoKennewick bridge [12].  As discussed earlier DAF  « l/2f, thus these low  damping values help to explain why large local cable vibrations have occurred in the past. (3) Damping appears to be frequency and thus stiffness dependent. In order to increase the damping in a cable its stiffness must be increased. This can be achieved by using fewer cables with increased area; however, this increase in damping due to an increase in cable spacing and area must be balanced against other constraints. For instance, if the cable spacing increases, the bending moments in the deck also increase, which would require deeper girders. Also, increased cable loads may introduce anchorage problems in the deck and tower. For large cable oscillations non-linear effects may cause increases in tension and thus stiffness. However, even if stiffness doubles due to these dynamic effects, damping will only double, leaving cable damping values below 1.0 percent of critical. (4) The addition of a discrete neoprene damper near the cable end socket causes a slight increase in damping (an average of about 10 percent above the damping value with one damper in place) for the six cables tested. These dampers do not appear to be more effective on the shorter cables even though the value e (s — e/S)  is greater for these cables than the long ones. The tests  conducted were for small amplitude vibrations and since the dampers are near the end sockets they resisted only very small relative cable motions. For larger oscillations of all modes the amount of cable displacement at the location of the damper would be larger and it would be more effective. (5) From the cable tests performed during the course of thi3 thesis no evidence of component interaction was evident. However, the energy was being put  97 into the system through the cable and the small amount of energy required to produce cable motion was not enough to induce motion in the stiffer and more massive deck or tower.  In order to properly investigate component  interaction a test which inputs energy into either the deck or tower would be required. Although not evident in direct tests, wind induced cable motions did occur during the erection. The so called parameter effect was evident on two occasions during a moderate steady wind. The resulting cable motions of approximately 150-300 mm could not have been produced by eddy shedding. The wind speeds necessary to induce first mode oscillations are very low and could not provide the required energy. Instead, these steady state motions were produced by harmonic support motion produced by wind energy being transferred to the bridge at some other location. As discussed earlier this motion is dependent only on the amplitude of support motion at a 2:1 frequency ratio and it cannot be reduced by the addition of discrete dampers in the system. The energy being input to the system through the supports far exceeds that which could be dissipated by a discrete damper. (6) Overall damping values for the first lateral and first torsional modes of the bridge were 0.46 and 0.30 percent of critical damping respectively. These values are very low, even compared to a suspension bridge, making aerodynamic considerations extremely important. The damping tests were performed with amplitudes of approximately 50 mm at center span and the damping should increase as the amplitude of oscillation increases. The pendulum test used in this work was very successful.  The modal frequencies can be isolated  and a properly tuned harmonic input force (even if the force i3 quite small) produces measurable motions in a short period of time.  98  CHAPTER 9  FUTURE WORK A N D RECOMMENDATIONS  Several areas of cable and bridge dynamics should be investigated to provide more insight into the cause of large cable vibrations. Cable tests similar to those done in this thesis, but producing larger amplitudes, should be carried out to determine the effectiveness of various types of discrete dampers in damping out first and other higher mode vibrations. These tests could also provide insight into whether cable damping increases as the amplitude of vibration increases. Interaction between cable, deck and tower should be studied in detail with emphasis on support induced cable motions. One possible means of examining component interaction would be to use the pendulum test described in this thesis with a slightly larger mass. By simultaneously monitoring the deck, cable and tower the existence of interaction in the first torsional and perhaps other lower modes could be investigated. The work done in this thesis shows that a relatively small harmonic exciting force is capable of producing quite large oscillations if applied for a long enough period of time. This method can be used to accurately determine the damping values for the lower modes of a bridge. By performing damping tests throughout the construction phase of a bridge the increase in damping associated with the addition of certain elements (such as precast panels or a deck overlay) can be monitered.  99  In order to perform tests such as these quickly and accurately a more efficient portable data acquisition would be desirable. With the advances in microcomputers, hardware has become available which can convert a personal computer to a data acquisition system at a relatively low cost. Graphics software is also available so that displacementtime plots and response spectra can be produced on site.  100  REFERENCES  [1] Chang, P.Y., Pilkey, W.D., "Static and Dynamic Analysis of Mooring Lines", Journal of Hydronautics,  Vol 7,  1973.  [2] Irvine, H . M . , Griffin, J . H . , " O n the Dynamic Response of a Suspended Cable", Internationa] Journal of Earthquakes  and Structural  Dynamics,  Issue  4, April-June 1976. [3] Henghold, W . M . , Russell, J.J., Morgan, J . D . , "Free Vibrations of a Cable in Three Dimensions", A.S.C.E.  Structural  Division Journal, June 1976.  [4] Wilson, A.J., Wheen, R.J., "Inclined Cables Under Load - Design Expressions", A.S.C.E.  Structural  Division Journal, May 1977.  [5] West, H . H . , Geshwindner, L . F . , Suhoski, J.E., "Natural vibrations of Suspension Cables", A.S.C.E.  Structural  Division  Journal, Nov 1975.  [6] Henghold, W . M . , Russell, J.J., "Equilibrium and Natural Frequency of Cable Structures (A Non-linear Finite Element Approach)", Computers Structures,  Vol 6,  and  1976.  [7] Hooley, R.F., "Catenary Cable", Civil Engineering455,  Class Notes 1985/86.  [8] Kovacs, I., "Zur Frage der Seilschwingungen und der Seildampfung", Die Bautechnik,  Heft 10,  1982.  [9] Maeda, Y., Maeda, K., Fujiwara, K., "System Damping Effect and its Application to Design of Cable Stayed Girder Bridge", Technology Reports the Osaka University,  of  Vol 33, No. 1699, March 1983.  [10] Godden, W . G . , "Seismic Model Studies of the Ruck-a-Chucky Bridge", Solicitation  No. DS 7159 United States Department  [11] Taylor, P.R., Personal  of the Interior,  1977.  Consultation.  [12] "Pasco-Kennewick Cable-Stayed Bridge Wind and Motion Data", U.S. Department of Transportation,  Report No. FHWA/RD-82/067 Feb, 1983.  [13] Irvine, H . M . , "Free Vibrations of Inclined Cables", A.S.C.E.  Structural  Di-  vision Journal, Feb 1978. [14] Irvine, H . M . , "Natural Vibrations of Suspension Cables", A.S.C.E. tural Division Journal, June 1976.  Struc-  REFERENCES  101  [15] Russell, J.J., Morgan, J.D., Henghold, W . M . , "Cable Equilibrium and Stability in a Steady Wind", A.S.C.E.  Structural  Division  [16] Fox, G . 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