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A short computer program, fully documented, is presented, for the step-by-step dynamic analysis of isolated cables or couples of parallel cables of a cable-stayed bridge, connected to each other and possibly with the deck of the bridge, by very thin pretensioned wires (cross-ties) 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 cross-ties include compressive loosening, tensile yielding, and hysteresis stress-strain 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.

The pretensioned cables in a typical cable-stayed bridge of medium size [

Two usual reasons for the previous cable vibrations of cable-stayed bridges are the following.

A pretensioned cable exhibits a sag under its self-weight. 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.

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 [

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 cross-ties 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 cross-section area of the cable to that of the cross-tie of a magnitude order 1000, however, the axial elastic stiffness, of a single pretensioned cross-tie, 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 cross-ties 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 cross-ties to suppress large amplitude cable variations of cable-stayed bridges. In [

Here, a simplified analytical method is proposed [

A short computer program (only about 120 Fortran instructions), fully documented, is presented for the step-by-step dynamic analysis [

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 cable-stayed bridge, in suppressing, by its hysteresis damping, their large amplitude vibrations.

Figure

Part of a typical cable-stayed bridge [

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

In the following analysis, the inclination of the cables will be ignored for the sake of simplicity. So, in Figure

(a) Geometric, static, and dynamic parameters of a cable structure. (b) Primary

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

Figure

The axial forces of the cross-ties are

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

Within the input data of the problem, the time-history of external excitation is given, which is here the variation of axial forces of cables due to traffic. The function

A state vector is introduced:

By combining all the previous equations, (

For the step-by-step dynamic analysis (direct time integration) of the previous initial value problem of (

The aforementioned algorithm is combined with a predictor-corrector technique, with two corrections per step, PE(CE)^{2}, where, in this symbol,

Prediction

Second and final correction

Thanks to the aforementioned predictor-corrector technique, no solving of algebraic system is needed, within each step of the algorithm.

The stability criterion of the proposed algorithm is [

Based on the proposed algorithm of previous Section

A full documentation of the previous computer program is presented as Appendix, consisting of the description of program line by line in Appendix

Seven applications (numerical experiments) follow, on the dynamic analysis of isolated pretensioned cables of a cable-stayed bridge or couples of parallel cables connected to each other and possibly with the deck of the bridge by very thin pretensioned single wires (cross-ties). 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

As shown in Figure

First application: isolated cable subject to traffic. (a) Given geometry and loading. (b) Axial stress-strain diagram of high-strength steel. (c) Initial static analysis. (d) Parameters of dynamic analysis. (e) Given time-history

The cable of first application is, in Figure

Second application: couple of interconnected cables, subject to traffic. (a) Given geometry and loading. (b) Crosssections of main cables and cross-tie. (c) Initial static analysis. (d) Parameters of dynamic analysis. (e), (f) Resulting time-histories of vertical displacements of centers of upper and lower cables. (g) Resulting time-history of axial force of cross-tie. (h) Resulting hysteresis stress-strain loops of the cross-tie.

The cable system of the second application is, in Figure

Third application: couple of cables, connected to each other and to deck, subject to traffic. (a) Given geometry and loading. (b) Cross sections of main cables and cross-ties. (c) Initial static analysis. (d) Parameters of dynamic analysis. (e), (f) Resulting time-histories of displacements of upper and lower cables. (g), (h) Resulting time-histories of axial forces of upper and lower cross-tie. (i), (j) Resulting hysteresis stress-strain loops of upper and lower cross-ties.

As shown in Figure

Fourth application: isolated cable subject to wind. (a) Given geometry and loading. (b) Parameters of dynamic analysis. (c) Given time-history of wind drag force. (d) Resulting time-history of displacement of center of cable.

Two identical parallel cables are, in Figure

Fifth application: couple of interconnected cables subject to wind. (a) Given geometry and loading. (b) Cross sections of main cables and cross-tie. (c) Initial static analysis. (d) Parameters of dynamic analysis. (e), (f) Resulting time-histories of displacements of two main cables. (g) Resulting time-history of axial force of cross-tie. (h) Resulting hysteresis stress-strain loops of the cross-tie.

The cable system of the fifth application is, in Figure

Sixth application: couple of cables interconnected by cross-tie, additionally connected by diagonals to pylon and deck, subject to wind. (a) Given geometry and loading. (b) Cross sections of main cables, cross-tie and diagonals. (c) Initial static analysis. (d) Parameters of dynamic analysis. (e), (f) Resulting time-histories of displacements of two main cables. (g), (h) Resulting time-histories of axial forces of diagonal bars. (i) Resulting time-history of axial force of cross-tie. (j) Resulting stress-strain loops of the cross-tie.

The cable system of fifth application is in Figure

(a) Seventh application: Couple of inter-connected cables subject to wind with additional small mass on one cable. (a) Given geometry, loading and additional small mass. (b) Comparison of cross-sections of cables and cross-tie. (c) Initial static analysis. (d) Parameters of dynamic analysis. (e), (f) Resulting time-histories of displacements of two cables. (g) Resulting time-history of axial force of cross-tie. (h) Resulting hysteresis stress-strain loops of the cross-tie.

Cable vibrations of cable-stayed 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 (cross-tie), 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 cross-ties by their compressive loosening, tensile yielding, and hysteretic stress-strain 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 time-history of the external excitation. By combining these equations, an initial value problem is obtained. For the step-by-step dynamic analysis of this problem, the algorithm of trapezoidal rule is proposed, combined with a predictor-corrector 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 cross-ties, that although they are very thin, with ratio of cross-section area of a cable to that of a cross-tie 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 in-plane cross-ties (within a vertical cables plane) are intended to suppress cables’ variations from parametric excitation due to traffic, whereas the out-of-plane cross-ties (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 cross-ties, even a single cross-tie proves effective by its hysteresis damping (due to stress-strain 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.

In this appendix, documentation is given for the proposed computer program, for the step-by-step 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 cross-tie and subject to a variation of their axial forces.

The description refers to the complete numbered list of Fortran instructions of Algorithms

1 COMMON/A/AW, HO, HO0, DHO, EO, SO, TO, HU, HU0, DHU, *EU, SU, TU, WO, WU, N,

LO, LU, FO, FU, MO, MU

2 COMMON/B/EY, ELAST

3 COMMON/C/NN, TK, NK

4 REAL LO, LU, MO, MU, N, NK

5 DIMENSION TK(20), NK(20)

6 OPEN(5,FILE = ^{“}TIEIN.TXT”)

7 OPEN(6,FILE=^{“}TIEOUT.TXT”)

8 READ(5,1)LO, LU, HO, HU, DC, DW, DENS

9 1 FORMAT(7F10.0)

10 AC = 3.14159*DC**2/4.

11 MO = DENS*AC*LO/2./10000.

12 WO = −MO*10.

13 MU = DENS*AC*LU/2./10000.

14 WU =

15 AW = 3.14159*DW**2/4./10000.

16 READ(5,1)SY, ELAST, SC0, SW0

17 EY = SY/ELAST

18

19 TO = SW0*AW

20 TU = SW0*AW

21 READ(5, 2)NN, TMAX

22 2 FORMAT(I5, F10.0)

23 READ(5,3)(TK

24 3 FORMAT(40F5.2)

25 STIF1 = 2. *

26 STIF2 = 2. *

27 STIF12= −ELAST*AW/HO

28

29

30

31

32

33

34

35

36 DO 4

37 4 TK

38 DT=

39

40 UO = (WO−TO)/(2. *N/(LO/2.))

41 UU = (WU + TO − TU)/(2. *N/(LU/2.))

42 VO = 0.

43 VU = 0.

44 EOPL = 0.

45 EUPL = 0.

46 EW0 = SW0/ELAST

47 HO0 = (HO + UO − UU)/(1. + EW0)

48 HU0 = (HU + UU)/(1. + EW0)

49 CALL EVAL(UO, UU, EOPL, EUPL, GO, GU)

50 5

51 CALL NHIST

52 UOP = UO + VO*DT

53 UUP = UU + VU*DT

54 VOP = VO + GO*DT

55 VUP = VU + GU*DT

56 EOPLP = EOPL

57 EUPLP = EUPL

58 CALL EVAL (UOP, UUP, EOPLP, EUPLP, GOP, GUP)

59 UO1 = UO + (VO + VOP)/2. *DT

60 UU1= UU + (VU + VUP)/2. *DT

61 VO1 = VO + (GO + GOP)/2. * DT

62 VU1 = VU + (GU + GUP)/2. *DT

63 EOPL1 = EOPL

64 EUPL1 = EUPL

65 CALL EVAL (UO1, UU1, EOPL1, EUPL1, GO1, GU1)

66 UO =UO + (VO+VO1)/2. *DT

67 UU = UU + (VU + VU1)/2. *DT

68 VO = VO + (GO+GO1)/2. *DT

69 VU = VU + (GU + GU1)/2. *DT

70 CALL EVAL (UO, UU, EOPL, EUPL, GO, GU)

71 WRITE (6,6)T, N, UO, UU, TO, TU

72 WRITE (6,6)EO, SO, EU, SU

73 6 FORMAT(1X,5(E10.4,1X))

74 IF(T.GT.TMAX) GO TO 7

75 GO TO 5

76 7 CLOSE

77 CLOSE

78 STOP

79 END

1 SUBROUTINE EVAL(UO, UU, EOPL, EUPL, GO, GU)

2 COMMON/A/AW, HO, HO0, DHO, EO, SO, TO, HU, HU0, DHU, EU, *SU, TU, WO, WU,

N, LO, LU, FO, FU, MO, MU

3 REAL LO, LU, MO, MU,

4 DHO = HO + UO − UU − HO0

5 EO = DHO/HO0

6 CALL SE(EO, EOPL, SO)

7 TO = SO*AW

8 DHU = HU + UU − HU0

9 EU = DHU/HU0

10 CALL SE(EU, EUPL, SU)

11 TU = SU*AW

12 FO = WO − 2. *

13 FU = WU − 2. *

14 GO = FO/MO

15 GU = FU/MU

16 RETURN

17 END

1 SUBROUTINE SE(E, EPL,

2 COMMON/B/EY, ELAST

3 IF(E,GT.EPL + EY)EPL=E-EY

4 IF(E.LT.EPL)EPL=E

5 IF(EPL.LT.0.)EPL=0.

6

7 IF(S.LT.0.)

8 RETURN

9 END

1 SUBROUTINE NHIST(T,N)

2 COMMON/C/NN, TK, NK

3 DIMENSION TK(20), NK(20)

4 REAL

5 DO 1

6 IF((

7

8 RETURN

9 1 CONTINUE

10 END

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 8-9, the parameters of

In lines 39–49, the initial conditions are established: time

In line 50, any step of algorithm begins by increasing time _{o}-_{ο} and _{u}-_{u} of cross-ties).

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.

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 cross-tie are determined. In lines 8–11, the corresponding quantities are found for the lower cross-tie. In lines 12-13, the vertical nodal forces on centers of upper and lower cables are determined and in lines 14-15 the corresponding accelerations.

In lines 3–5, the new plastic strain of the cross-tie is found. In lines 6-7, the stress of the cross-tie is determined.

In line 5 is found the time interval where present time

See Algorithms

AC: cross-section area of a cable

AW: cross-section area of a wire (cross-tie)

DC: cross-section diameter of a cable

DENS: density of steel

DHO: elongation of upper tie

DHU: elongation of lower tie

DW: cross-section diameter of wire (cross-tie)

DT

ELAST: initial elasticity (Young) modulus

EO: strain of upper tie

EOPL: plastic strain of upper tie

EOPLP: prediction of EOPL

EOPL1: first correction of EOPL

EU: strain of lower tie

EUPL: plastic strain of lower tie

EUPLP: prediction of EUPL

EUPL1: first correction of EUPL

EVAL: subroutine for evaluation of strain and stress state of the structure

EW0: pretension strain of wires (cross-ties)

EY: yield strain of steel

FO: vertical nodal force at center of upper cable

FU: vertical nodal force at center of lower cable

GO: vertical acceleration at center of upper cable

GOP: prediction of GO

GO1: first correction of GO

GU: vertical acceleration at center of lower cable

GUP: prediction of GU

GU1: first correction of GU

HO: design (nominal) length (height) of upper cross-tie

HO0: undeformed length (height) of upper cross-tie

HU: design (nominal) length (height) of lower cross-tie

HU0: undeformed length (height) of lower cross-tie

LO: length of upper cable

LU: length of lower cable

MO: mass of upper cable

MU: mass of lower cable

NHIST: subroutine for given time-history of

NK: ordinate of a node of piecewise linear curve

NN: number of nodes of piecewise linear curve

SO: stress of upper cross-tie

STIF1:

STIF2:

STIF12:

SU: stress of lower cross-tie

SW0: pretension stress of wires (cross-ties)

SY: yield stress of steel

TIEIN: input file

TIEOUT: output file

TK: abscissa of a node of piecewise linear curve

TMAX:

TO: axial force of upper cross-tie

TU: axial force of lower cross-tie

UO: vertical displacement of center of upper cable

UOP: prediction of UO

UO1: first correction of UO

UU: vertical displacement of center of lower cable

UUP: prediction of UU

UU1: first correction of UU

VO: vertical velocity of center of upper cable

VOP: prediction of VO

VO1: first correction of VO

VU: vertical displacement of center of lower cable

VUP: prediction of VU

VU1: first correction of VU

WO: weight at center of upper cable

WU: weight at center of lower cable

W1:

W2:

Only the following variable is different from those of MAIN program:

SE: subroutine for stress-strain law of a cross-tie.

Only the following variables are different from those of MAIN program:

EPL: plastic strain of a cross-tie

All the variables are the same as in MAIN program.