Research Article  Open Access
Panagis G. Papadopoulos, Andreas Diamantopoulos, Haris Xenidis, Panos Lazaridis, "Simple Program to Investigate Hysteresis Damping Effect of CrossTies on Cables Vibration of CableStayed Bridges", Advances in Civil Engineering, vol. 2012, Article ID 463134, 15 pages, 2012. https://doi.org/10.1155/2012/463134
Simple Program to Investigate Hysteresis Damping Effect of CrossTies on Cables Vibration of CableStayed Bridges
Abstract
A short computer program, fully documented, is presented, for the stepbystep dynamic analysis of isolated cables or couples of parallel cables of a cablestayed bridge, connected to each other and possibly with the deck of the bridge, by very thin pretensioned wires (crossties) and subjected to variation of their axial forces due to traffic or to successive pulses of a wind drag force. A simplified SDOF model, approximating the fundamental vibration mode, is adopted for every individual cable. The geometric nonlinearity of the cables is taken into account by their geometric stiffness, whereas the material nonlinearities of the crossties include compressive loosening, tensile yielding, and hysteresis stressstrain loops. Seven numerical experiments are performed. Based on them, it is observed that if two interconnected parallel cables have different dynamic characteristics, for example different lengths, thus different masses, weights, and geometric stiffnesses, too, or if one of them has a small additional mass, then a single pretensioned very thin wire, connecting them to each other and possibly with the deck of the bridge, proves effective in suppressing, by its hysteresis damping, the vibrations of the cables.
1. Introduction
The pretensioned cables in a typical cablestayed bridge of medium size [1], as they are very long with a length of magnitude order 100 m, and a pretension axial force of magnitude order 1000 kN, exhibit, perpendicularly to their axis, a very small geometric stiffness, corresponding to their fundamental vibration mode, of a magnitude order only 50 kN/m. Also perpendicularly to their axis, they exhibit a very small intrinsic damping, due to their material internal friction. For the previous reasons, they are often subjected to large amplitude vibrations. And, if the external excitation is approximately periodic, with a period close to a natural period of the cable, for example, the fundamental one, then resonance may happen, and vibration amplitudes increase excessively and are maintained, with no significant reduction for a long time, unless special measures are taken.
Two usual reasons for the previous cable vibrations of cablestayed bridges are the following.(1)A pretensioned cable exhibits a sag under its selfweight. Because of traffic, the ends of the cable, on pylon and deck, are subject to a variation of their displacements; thus the elongation and axial force of the cable vary, which implies variation of its geometric stiffness, too, as well as variation of the sag of the cable. This vibration, due to variation of geometric stiffness, is called parametric excitation.(2)The successive pulses of a wind pressure exert a drag force, perpendicularly to the vertical plane of cables, at one side of the bridge. The variation of this drag force causes vibration of the cables.
In [2], a complete description of the problem of cable vibrations of cablestayed bridges is presented, as well as a state of the art of various types of dampers for cable vibrations (viscous dampers, crossties, and others), along with case studies of dampers on real bridges.
The viscous dampers, although widely mentioned in literature, present some problems: usually, they are installed at the ends of a cable, where they are not very helpful; it seems that their main role is a slight reduction of cable’s length, thus a slight increase of its geometric stiffness. Rarely, they are installed at intermediate points of a cable, where they are more helpful; however, this installation is difficult.
On the other hand, the crossties are preferable, for the following reasons: they are light and cheap, they are easily installed and pretensioned, and they easily replaced when damaged. And a great advantage of them is that although they are very thin, with a ratio of crosssection area of the cable to that of the crosstie of a magnitude order 1000, however, the axial elastic stiffness, of a single pretensioned crosstie, is comparable in magnitude with the geometric stiffness of a cable, that is of magnitude order 50 kN/m, along the same direction, perpendicularly to cable axis. Also, as the crossties are very thin, they are almost invisible, so they do not harm the aesthetics of the bridge.
For the previous reasons, recently many researchers recommend the use of crossties to suppress large amplitude cable variations of cablestayed bridges. In [3–6], analytical studies are performed on crossties or hybrid systems consisting of viscous dampers and crossties.
Here, a simplified analytical method is proposed [7], in order to investigate the hysteresis damping effect of crossties, where, for every individual cable, an SDOF model is adopted, approximating its fundamental vibration mode. The geometric nonlinearity of the cables is taken into account by their geometric stiffness. At same time, the proposed method is accurate, as it includes the material nonlinearities of the crossties, by their compressive loosening, tensile yielding, as well as hysteresis stressstrain loops.
A short computer program (only about 120 Fortran instructions), fully documented, is presented for the stepbystep dynamic analysis [9] of isolated cables or couples of parallel cables, connected to each other and possibly with the deck of the bridge, by very thin pretensioned wires (crossties) and subjected to variation of the axial forces of cables due to traffic [8] or to successive pulses of wind drag force.
Seven numerical experiments are performed. And, based on them, observations are made on the effectiveness of a single pretensioned very thin wire, connecting a couple of cables of a cablestayed bridge, in suppressing, by its hysteresis damping, their large amplitude vibrations.
2. Equations of the Problem
Figure 1 shows a part of a typical cablestayed bridge [1], with a pylon consisting of two vertical legs, which are connected by two transverse beams, a part of the deck with a slender rectangular plate section, two couples of pretensioned parallel inclined cables at each side of the bridge, connected by their ends to the pylon and the deck and two pretensioned very thin vertical inplane crossties, at two sides of bridge, which intend to suppress parametric vibration of cables due to traffic loads , as well as two outofplane horizontal crossties, which intend to suppress cable vibrations due to wind forces .
In the following, the equations of nonlinear dynamic analysis will be written, for a specific cable structure consisting of two parallel pretensioned cables (1 and 2 in Figure 1), connected by two very thin pretensioned crossties (3 and 4 in Figure 1) to each other and with the deck of the bridge and subjected to parametric vibration, due to traffic. In subsequent applications, by simple and obvious modifications of these equations, other cable structures (combinations of main cables and crossties) subjected to traffic or wind excitation will be treated, too.
In the following analysis, the inclination of the cables will be ignored for the sake of simplicity. So, in Figure 2(a), the couple of horizontal parallel pretensioned main cables represents the inclined cables 1 and 2 of Figure 1, whereas the very thin pretensioned vertical wires represent the inplane crossties 3 and 4 of Figure 1.
2.1. Geometric Equations
For every individual cable, a simplified SDOF model is adopted, which approximates its fundamental vibration mode. This unique DOF, for every cable, is the displacement of its center perpendicularly to its axis, that is, the vertical displacements downwards and in Figure 2(a), for the upper and lower cables, respectively. The geometric equations, relating the displacements of cables with the elongations and strains of crossties, are the following, according to Figure 2(a): where and are elongations of upper and lower crossties, respectively, and their design (nominal) lengths, and and are their initial undeformed lengths. And the strains and of upper and lower crossties, respectively, are
2.2. Constitutive Equations
Figure 2(b) is the primary stressstrain diagram, which describes the axial nonlinear stressstrain law of a crosstie, made of the same highstrength steel as the main cables. This σε law includes compressive loosening, tensile yielding, as well as hysteresis stressstrain loops, resulting from the obvious in loadingunloading rule Figure 2(b). There is only one constitutive variable, the plastic strain of the crosstie. The present stress is an obvious function of present strain ε and present value of plastic strain : whereas the variation of plastic strain can be expressed as a function of present strain ε, variation of present strain , as well as present value of plastic strain , in an obvious manner by the loadingunloadingreloading rule of Figure 2(b):
2.3. Static Equations
The axial forces of the crossties are for the upper and lower ties, respectively, where is the crosssection area of the very thin wires (crossties), whereas the vertical nodal forces applied at the centers of the cables, upper and lower one, respectively, are, according to Figure 2(a) where the downwards direction has been taken as positive, and are weights at centers of upper and lower cables, respectively, and are their geometric stiffnesses, where is the given timehistory of the axial forces of the cables.
2.4. Dynamic Equations
Damping is ignored, as the material internal friction of the cables is meaningless. The vertical accelerations at the centers of upper and lower cable are where and are lumped masses at centers of cables (Figure 2(a)), whereas the upper dots mean derivation with respect to time.
2.5. External Excitation
Within the input data of the problem, the timehistory of external excitation is given, which is here the variation of axial forces of cables due to traffic. The function is assumed to be described by a piecewise linear curve, as shown in Figure 2(c). And within each time interval, between two successive nodes, a linear interpolation is performed, in order to find, from a specific time instant , the corresponding axial force of the cables.
2.6. Initial Value Problem
A state vector is introduced: consisting of the vertical displacements (Figure 2(a)) and velocities of the centers of upper and lower cables, respectively, as well as of the constitutive variables , which are the plastic strains (Figure 2(b)) of upper and lower crossties, respectively.
By combining all the previous equations, (1) up to (8), a system of firstorder ordinary nonlinear differential equations is obtained: which, along with the initial value of the state vector for time , and with sought function the timehistory of the state vector , constitutes an initial value problem.
2.7. Proposed Algorithm
For the stepbystep dynamic analysis (direct time integration) of the previous initial value problem of (9a) and (9b), the algorithm of trapezoidal rule is proposed: where and are two successive steps of the algorithm. This coincides with the algorithm of constant average acceleration of Newmark’s group of algorithms for stepby step dynamic analysis.
The aforementioned algorithm is combined with a predictorcorrector technique, with two corrections per step, PE(CE)^{2}, where, in this symbol, means prediction and correction of the state vector y, whereas means evaluation of the function of (9a). In more detail, the proposed predictorcorrector technique can be written, within any th step of the algorithm, as follows:PredictionFirst correction Second and final correction
Thanks to the aforementioned predictorcorrector technique, no solving of algebraic system is needed, within each step of the algorithm.
The stability criterion of the proposed algorithm is [9] that is, ; otherwise a divergent solution results, whereas the accuracy criterion is at least that is, ; otherwise a significant accumulated truncation error appears, which is expressed as amplitude decay of the vibration, as well as period elongation.
3. Computer Program
Based on the proposed algorithm of previous Section 2.7, a simple and very short computer program has been developed, with only 115 Fortran instructions, consisting of the MAIN program (79 instructions) which performs the algorithm of stepbystep dynamic analysis, and of three subroutines: Subroutine EVAL (17 instructions) which evaluates the present strain and stress state of the cable structure under consideration, subroutine SE (9 instructions) which describes the nonlinear uniaxial stressstrain law of a crosstie, and subroutine NHIST (10 instructions) describing the given timehistory of the external excitation, which is, here, the variation, with respect to time, of the axial force of cables due to traffic.
A full documentation of the previous computer program is presented as Appendix, consisting of the description of program line by line in Appendix A.1, of the complete list of Fortran instructions in Appendix A.2, and the variables explanation in Appendix A.3. The documentation of the computer program is completed by the series of seven applications in Section 4. The program is particularly oriented to the specific third application of Section 4.3, as already mentioned in the equations of problem in Section 2. However, by simple and obvious modifications of the computer program, all the other numerical experiments, in Section 4, can be treated, too.
4. Applications (Numerical Experiments)
Seven applications (numerical experiments) follow, on the dynamic analysis of isolated pretensioned cables of a cablestayed bridge or couples of parallel cables connected to each other and possibly with the deck of the bridge by very thin pretensioned single wires (crossties). Three of these cable structures are subjected to variation of axial forces of cables due to traffic (parametric excitation) and four of them are subjected to successive pulses of drag force due to a strong wind.
As already mentioned, the previously presented algorithm, in Section 2, is oriented only to the specific third application of Section 4.3. However, by simple and obvious modifications of this algorithm, all the other applications present in Section 4, which are presented herein after, have been analysed, too.
4.1. First Application: Isolated Cable Subject to Traffic
As shown in Figure 3, an isolated cable is subjected to a periodic variation of its axial force with a period equal to its fundamental one. Up to the fifth cycle that the excitation lasts, the vibration amplitude of the cable increases up to about 1.6 m and then remains constant; only a slight algorithmic damping is observed.
4.2. Second Application: Couple of Interconnected Cables Subject to Traffic
The cable of first application is, in Figure 4, connected to another shorter parallel cable by a thin crosstie. Both cables are subjected to a periodic variation of their axial force with a period equal to the fundamental one of the cable system. Up to the fifth cycle that the excitation lasts, the vibration amplitude of the upper cable increases up to 1.4 m and that of the shorter lower cable up to 0.7 m. Then, both amplitudes are gradually reduced, the upper one up to 0.4 m and the lower one up to 0.3, in 10 sec. Because of the different lengths of the two cables, thus different geometric stiffness, masses, and weights, too, large differences of displacements at the ends of the crosstie are obtained, thus large stressstrain loops with a total width mm/m, which are responsible for the significant hysteresis damping which is achieved.
4.3. Third Application: Couple of Cables Connected to Each Other and to Deck, Subject to Traffic
The cable system of the second application is, in Figure 5, supplied by one more thin crosstie connecting the lower cable with the deck of the bridge. Because of the stiffness of this additional crosstie, the maximum vibration amplitudes of both cables are significantly reduced, that of the upper cable to 0.7 m and that of the lower cable to 0.3 m. However, at same time, this reduction of displacements has a consequence less wide stressstrain loops in the upper crosstie with total width mm/m and very thin stressstrain loops in the lower crosstie with only mm/m, resulting in low values of hysteretic damping.
4.4. Fourth Application: Isolated Cable Subject to Wind
As shown in Figure 6, an isolated cable is subjected to a resonant periodic wind drag force. Up to the fifth cycle that the excitation lasts, the vibration amplitude increases up to about 3.0 m and then remains constant; only a slight algorithmic damping is observed.
4.5. Fifth Application: Couple of Interconnected Cables Subject to Wind
Two identical parallel cables are, in Figure 7, interconnected by a thin crosstie. A wind drag force acts on one cable only; initially, this cable exhibits larger displacements, but gradually the movement is transferred to the other cable, too. So, the displacements are divided by two, compared with those of previous fourth application. During the initial stage of displacements transfer from one cable to the other, stressstrain loops of the crosstie with a total width of medium size mm/m appear. From this point on, as the two cables are identical and perform similar movements, no more yielding of the crosstie appears, as shown in Figure 7(g), thus no more stressstrain loops and hysteretic damping, too.
4.6. Sixth Application: Couple of Interconnected Cables, Additionally Connected by Diagonals to Pylon and Deck, Subject to Wind
The cable system of the fifth application is, in Figure 8, supplied with diagonal ties connected with the pylon and the deck of the bridge. These diagonal ties offer a small additional stiffness, perpendicularly to the cables, which slightly reduces their displacements. However, at same time, this restriction of displacements further reduces the total width of stressstrain loops of the crosstie to mm/m, thus reducing the hysteretic damping.
4.7. Seventh Application: Couple of Interconnected Cables Subject to Wind, with Small Additional Mass on One Cable
The cable system of fifth application is in Figure 9, supplied by a small additional mass to the one of the two cables. So, the two cables have now different dynamic characteristics and they perform significantly different movements. As a consequence, large differences of displacements at the ends of the crosstie result, thus wide stressstrain loops with a total width mm/m, which implies significant hysteretic damping. The displacement amplitudes of the cables are now only about onefourth of those of the fifth application.
5. Conclusions
Cable vibrations of cablestayed bridges have been examined. Either isolated cables or couples of parallel cables, connected to each other and possibly with the deck of the bridge, by a very thin pretensioned wire (crosstie), have been considered. External excitation is either traffic, which causes displacements of cable ends on deck and pylon, thus variation of axial forces, geometric stiffnesses and sags of cables, too (parametric excitation), or successive pulses of drag force due to a strong wind, perpendicularly to a vertical cables’ plane at one side of the bridge.
The proposed analytical model is on the one hand simplified, as an SDOF oscillator is adopted for every individual cable, approximating its fundamental vibration mode. However, on the other hand, the proposed analytical model is accurate, as it takes into account the geometric nonlinearity of the cables by their geometric stiffness; also it includes the material nonlinearity of the crossties by their compressive loosening, tensile yielding, and hysteretic stressstrain loops.
The equations of the problem of dynamic analysis, oriented to a specific cable structure, have been written, consisting of the geometric, constitutive, static, and dynamic ones, as well as of the given timehistory of the external excitation. By combining these equations, an initial value problem is obtained. For the stepbystep dynamic analysis of this problem, the algorithm of trapezoidal rule is proposed, combined with a predictorcorrector technique, with two corrections per step. So, no solving of algebraic system is required within each step of the algorithm.
Based on the proposed algorithm, a short computer program has been developed, with only 115 Fortran instructions, consisting of the main program and three subroutines. A full documentation is given for this program, which means transparency of computation.
Seven numerical experiments have been performed by the aforementioned program, three with variation of axial forces of cables due to traffic (parametric excitation) and four with successive pulses of drag force due to a strong wind.
On the basis of previous series of numerical experiments some observations with practical usefulness are made. (These are not strict theoretical conclusions, but simple observations based on the results of numerical experiments.)
It is confirmed by the series of numerical experiments, the great advantage of pretensioned crossties, that although they are very thin, with ratio of crosssection area of a cable to that of a crosstie of magnitude order 1000, however, they possess an axial elastic stiffness comparable in magnitude to the geometric stiffness of cables, with magnitude order 50 kN/m, along the same direction, that is perpendicularly to cables axes.
The inplane crossties (within a vertical cables plane) are intended to suppress cables’ variations from parametric excitation due to traffic, whereas the outofplane crossties (transverse ones connecting cables at two sides of bridge) are intended to suppress cables vibrations from successive pulses of drag force due to a strong wind.
General observation from all numerical experiments: in a couple of parallel cables connected to each other and possibly with the deck of bridge by crossties, even a single crosstie proves effective by its hysteresis damping (due to stressstrain loops) in suppressing large amplitude cable vibrations under the following circumstances: if the two cables have different dynamic characteristics, for example, different lengths which imply different masses, weights, and geometric stiffnesses, too, or if one of them has a small additional mass.
Appendices
A. Documentation of the Proposed Computer Program
In this appendix, documentation is given for the proposed computer program, for the stepbystep dynamic analysis of the third application, that of a couple of parallel cables connected to each other and to the deck of bridge by a thin crosstie and subject to a variation of their axial forces.
A.1. Description of Program Line by Line
The description refers to the complete numbered list of Fortran instructions of Algorithms 1, 2, 3, 4.




MAIN Program
The first seven lines include nonexecutable statements. Particularly, in the three first lines, the COMMON instructions connect the MAIN program with the three subroutines, by their common variables.
In the next 17 lines, 8 up to 24, the input data are read: geometric data and density of steel in lines 89, the parameters of σε law of steel along with the pretension stresses of cables and crossties in line 16, and the timehistory of axial forces of cables given by the coordinates of nodes of piecewise linear curve in lines 21–24. In lines 10–15 and 17–20, some simple preliminary calculations are performed to determine crosssection areas and pretension forces of cables and crossties, as well as masses and weights of cables and yield strain of steel. In lines 25–35, the initial characteristic equation of the cable structure is solved, so that to find its natural frequencies and periods. In lines 3637, the time scale of the given time history of external excitation is expanded so that to obtain a period equal to the fundamental one of the structure, in order to cause resonance, whereas in line 38, the minimum natural period of the structure dictates the timestep length of the algorithm, so that to assure accuracy of computation.
In lines 39–49, the initial conditions are established: time in line 39, determination of initial static displacements of cables in lines 4041, evaluation of undeformed lengths of crossties in lines 42–44, zero initial velocities of cables in lines 4546, zero initial plastic strains of crossties in lines 4748, and evaluation of initial strain and stress state of structure by calling subroutine EVAL in line 49.
In line 50, any step of algorithm begins by increasing time by . In line 51, by calling subroutine NHIST, the present value of axial force of cables is determined. In lines 52–55, the prediction of values of displacements and velocities is performed, and in lines 56–58 by calling subroutine EVAL the corresponding plastic strains and accelerations are found. So in lines 52–58 the prediction of state vector within a step of algorithm is performed. In lines 59–65 the first correction of value of state vector is made by use of trapezoidal rule. And in lines 66–70, the second and final correction. In lines 71–73, the output data of present step of algorithm are written (time , axial force of cables, displacements and of cables, axial forces and of crossties, strains and stresses ε_{o}σ_{ο} and ε_{u}σ_{u} of crossties).
In lines 74–79 if a maximum time has been exhausted, the algorithm is interrupted. Otherwise, we continue to the next step of the algorithm.
Subroutine EVAL
Lines 1–3 are nonexecutable statements. In lines 4–7, from the displacements of cables, the elongation, strain, stress by calling subroutine SE, and axial force of upper crosstie are determined. In lines 8–11, the corresponding quantities are found for the lower crosstie. In lines 1213, the vertical nodal forces on centers of upper and lower cables are determined and in lines 1415 the corresponding accelerations.
Subroutine SE
In lines 3–5, the new plastic strain of the crosstie is found. In lines 67, the stress of the crosstie is determined.
Subroutine NHIST
In line 5 is found the time interval where present time is included. In line 6, a linear interpolation is performed between the two endnodes of the previous time interval, in order to find the axial force of cables corresponding to present time .
A.2. List of Fortran Instructions
MAIN Program
See Algorithms 1, 2, 3, and 4.
A.3. Variables Explanation
MAIN Program
coefficient of characteristic equationAC: crosssection area of a cableAW: crosssection area of a wire (crosstie) coefficient of characteristic equation coefficient of characteristic equation coefficient of characteristic equationDC: crosssection diameter of a cableDENS: density of steelDHO: elongation of upper tieDHU: elongation of lower tieDW: crosssection diameter of wire (crosstie)DT , time steplength of algorithmELAST: initial elasticity (Young) modulusEO: strain of upper tieEOPL: plastic strain of upper tieEOPLP: prediction of EOPLEOPL1: first correction of EOPLEU: strain of lower tieEUPL: plastic strain of lower tieEUPLP: prediction of EUPLEUPL1: first correction of EUPLEVAL: subroutine for evaluation of strain and stress state of the structureEW0: pretension strain of wires (crossties)EY: yield strain of steelFO: vertical nodal force at center of upper cableFU: vertical nodal force at center of lower cableGO: vertical acceleration at center of upper cableGOP: prediction of GOGO1: first correction of GOGU: vertical acceleration at center of lower cableGUP: prediction of GUGU1: first correction of GUHO: design (nominal) length (height) of upper crosstieHO0: undeformed length (height) of upper crosstieHU: design (nominal) length (height) of lower crosstieHU0: undeformed length (height) of lower crosstieLO: length of upper cableLU: length of lower cableMO: mass of upper cableMU: mass of lower cable: axial force of a cableNHIST: subroutine for given timehistory of NK: ordinate of a node of piecewise linear curve NN: number of nodes of piecewise linear curve SO: stress of upper crosstieSTIF1: elements of stiffness matrixSTIF2: elements of stiffness matrixSTIF12: elements of stiffness matrixSU: stress of lower crosstieSW0: pretension stress of wires (crossties)SY: yield stress of steel: timeTIEIN: input fileTIEOUT: output fileTK: abscissa of a node of piecewise linear curve TMAX: , maximum time of observationTO: axial force of upper crosstieTU: axial force of lower crosstie: extreme natural periods of structure: extreme natural periods of structureUO: vertical displacement of center of upper cableUOP: prediction of UOUO1: first correction of UOUU: vertical displacement of center of lower cableUUP: prediction of UUUU1: first correction of UUVO: vertical velocity of center of upper cableVOP: prediction of VOVO1: first correction of VOVU: vertical displacement of center of lower cableVUP: prediction of VUVU1: first correction of VUWO: weight at center of upper cableWU: weight at center of lower cableW1: extreme natural frequencies of the structureW2: extreme natural frequencies of the structure.
Subroutine EVAL
Only the following variable is different from those of MAIN program:SE: subroutine for stressstrain law of a crosstie.
Subroutine SE
Only the following variables are different from those of MAIN program:: strain of a crosstieEPL: plastic strain of a crosstie: stress of a crosstie.
Subroutine NHIST
All the variables are the same as in MAIN program.
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Copyright
Copyright © 2012 Panagis G. Papadopoulos et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.