Nonlinear Dynamic Response of Guyed Tower to Sudden Guy Rupture

ELSEVIER PII: S0141-0296(97)00173-3

Engineering Structures, Vol. 19, No. 11, pp. 879-890, 1997 1997 Elsevier Science Ltd

All rights reserved. Printed in Great Britain 0141-0296/97 $17.00 + 0.00

Nonl inear dynamic response of a

guyed tower to a sudden guy

rupture Nabi l Ben Kah la

Department of Civil Engineering, Ecole Nationale D'lng~nieurs de Gab~s, Route de Medenine, 6029 Gabbs, Tunisia (Received July 1996; revised version accepted November 1996)

The geometrically nonlinear dynamic response of a three-dimensional model of a guyed tower, when a selected guy suddenly ruptures, sending an impulsive shock to the entire structure is studied. The sudden rupture of a guy is simulated by the instantaneous removal of its end tension, applied as an external force at its mast attachment point. A 500 ft tall tower with three stay levels was analysed. The analysis was performed for a nominal wind speed acting in the plane of the ruptured guy, assumed to be a windward one. The mast was modelled by a three-dimensional space truss and the guys by three-dimensional elastic catenaries. 1997 Elsevier Science Ltd.

Keywords: guyed towers, sudden rupture, dynamic response, nonlinear analysis

1. Introduction

The sudden rupture of a guy in a guyed tower acts as a triggering mechanism, subjecting the mast to a load imbalance. The load imbalance causes an impulsive shock to the structure, giving the system an initial acceleration and set- ting it into a free vibration motion. The resulting time dependent forces and displacements can be large enough to provoke the collapse of the system.

If the guyed tower is a part of a communication network, its collapse causes the loss of serviceability and if built in an urban area, human casualties and considerable economic losses could be expected. For these reasons, the Inter- national Association For Shell and Spatial Structures (lASS) ~ recommend that guyed towers of class I be designed to withstand the rupture of one arbitrarily selected guy. The analysis is performed for a nominal wind speed acting in the plane of tile ruptured guy, assumed to be a windward one. The static wind pressure is maintained on the structure after rupturing of the guy.

The purpose of this ,audy was to investigate the geometrically nonlinear dynamic response of a guyed tower to the sudden rupture of a guy, using a detailed finite element model. This model incorporates catenary elements to represent guy cables, accounts for mechanical, structural and aerodynamic damping and includes geometrical non- linearities. The dynamic analysis is illustrated with a three- dimensional example tower having three stay levels.

2. Guyed tower model

Figure la illustrates the profile of the 500 ft tall guyed tower that was analysed. The elevation of the tower is shown in Figure lb. The triangular cross-section mast (see Figure lc) is made up of 50 identical panels, each having a width and length of 10 ft. A typical panel consisting of nine pinned end members is shown in Figure ld. The struts are 2 in diameter steel pipes, the diagonals are 3 in diameter steel pipes and the chords are 3.5 in solid rounds. The compression and tension capacities of each member are given

J

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Figure 1 Example tower: (a) profile; (b) elevation; (c) cross- section; (d) typical mast panel

879

880

Table I Compression/tension capacities of mast members

Compression capacity, Ib Tension capacity, Ib

Struts 13162 Diagonals 29798 Chords 146298

Nonlinear dynamic response of guyed tower to sudden guy rupture: N. Ben Kahla

Table 3 Windward guys' end tensions: preliminary design

Guy cable X-component, Ib Z-component, Ib

38520 G 1 22393 -28898 80280 G2 27014 -22614 346356 G3 23075 -9763

in Table 1. The mast is laterally supported at three levels by a set of three 1 in diameter guys. The guys have a break- ing tension of 122 000 lb. The guys attached to the mast at the level of 480 ft have a pretension of 11% of their break- ing tension, those at level 320 ft: 12% and finally those at 160 ft: 13%. The structure is subjected to a static wind pressure, computed according to the EIA standard 2, with a velocity of 75 mph at a reference height of 33 ft acting at 180 from the X-axis. The ground exposure is of type C.

The guys are assumed to be perfectly flexible, i.e. without any bending stiffness. They are considered to have a uniform section between their attachment points and are modelled by elastic catenaries 3. To account for the energy dissipation in the material and the friction due to inner strand rubbing, fictitious linear viscous dampers are inserted in parallel with each guy 4,5. Seven equally spaced nodes are defined on each guy attached to the mast at levels 480 ft and 320 ft and only five on those attached at level 160ft. The total number of cable elements, nodes and degrees of freedom used for the analysis is given in Table 2.

The mast is composed of individual members welded together. The welded connections provide some bending rigidities which are neglected, thus assuming the joints to act as smooth hinges. The mast is then modelled by a three- dimensional space truss made up of 450 members (labelled in descending numbers). This model does not account for mast structural damping.

3. Solution method

First, a geometrically nonlinear static analysis of the guyed tower under the wind loading was performed to determine the windward guys' end tensions. These forces are summar- ized in Table 3.

A lumped mass idealization is used at the element level and 20 frequencies and mode shapes of the intact guyed tower were obtained. The frequencies were determined for vibration of the structure about its deflected position under the static wind pressure and are listed in Table 4. This table also contains the frequencies and periods of the damaged structure (for three different cases of guy ruptures). The sudden rupture of a single windward guy is simulated as follows: the guy is replaced by its end tension which is applied as an external force to the point where the guy was

attached to the mast. At time t = 0 s, this force is removed instantaneously from the structure, while maintaining the static wind pressure on it, causing a force imbalance at the location of the rupture. This imbalance initiates the motion of the system resulting in an initial acceleration.

The highly nonlinear dynamic equations of motion are solved using a step-by-step numerical integration method. The linear acceleration method, together with an iterative scheme to ensure dynamic equilibrium within each time step, is used 5-7. The fraction of critical damping used for the fictitious linear viscous dampers was 5% and the maximum imbalance force allowed was 1 lb. The time domain was 60 s with a time step for the numerical integration of 0.0005 s.

4. Results and discussion

Table 5 summarizes the peak lateral mast displacements at guy attachment points (parallel to the direction of the wind, computed as the average of the displacements of nodes A, B and C shown in Figure lc) and the dynamic amplification factors of guys' end tensions at mast attachment points. The results are presented for three different cases of guy ruptures. The dynamic amplification factor is defined as the ratio of the peak transient guy end tension to the initial guy end tension. Table 5 gives also the initial displacements and guys' end tensions, calculated under static wind pressure, before the sudden rupture of a selected guy. The lateral mast displacements are measured with respect to the mast static unloaded equilibrium position (i.e. no wind is acting on the structure).

When windward guy G1 suddenly breaks, violent motion of the mast is observed at this guy attachment point with a peak transient lateral displacement of-13.68 ft. This can be seen in Figure 2, by inspection of the plot of X-480 (X refers to the direction of the motion, 480 the level of the mast considered). This lateral motion of the mast becomes less important for elevations lower than 320ft (see Figure 2, plots of X-240, X160 and X-80). At an elevation of 320 ft, the peak transient lateral displacement of the mast is -3.81 ft. At the time when this peak is reached, guy G2 becomes very taut, resulting in a large end tension. Figure 3 shows the time variations of the guys' end tensions at mast attachment points. The variables T7, T8 andT9 are the end tensions of the guys attached to the mast at level 480 ft, at

Table 2 Total number of cable elements, nodes and degrees of freedom

Number of cable elements Number of nodes Number of degrees of freedom

Entire guyed tower Guyed tower without G1 Guyed tower without G2 Guyed tower without G3

48 195 567 42 190 552 42 190 552 44 192 558


Table4 Guyed tower frequencies (rad/s) and period(s)

881

Intact structure Damaged structure

Rupture of G1 Rupture of G2 Rupture of G3

No Freq. Period Freq. Period Freq. Period Freq. Period

1 2.403 2.614 1.927 3.260 1.907 3.295 2.258 2.782 2 2.861 ;!. 196 2.318 2.711 2.557 2.457 2.511 2.502 3 3.108 ;!.022 3.262 1.926 2.622 2.396 2.657 2.365 4 3.508 1.791 3.316 1.895 3.162 1.987 3.357 1.872 5 3.719 1.689 3.423 1.836 3.628 1.732 3.753 1.674 6 3.869 1.624 3.615 1.738 3.784 1.661 3.832 1.640 7 4.012 1.566 3.713 1.692 4.089 1.536 3.918 1.604 8 4.688 1.340 3.822 1.644 4.546 1.382 3.992 1.574 9 4.777 1.315 4.200 1.496 4.654 1.350 4.464 1.408

10 4.864 1.292 4.542 1.383 4.859 1.293 4.537 1.385 11 4.947 1.270 4.747 1.323 4.930 1.274 4.838 1.299 12 5.038 1.247 5.397 1.164 5.036 1.248 4.952 1.269 13 5.107 1.230 5.484 1.146 5.277 1.191 5.046 1.245 14 6.095 1.031 5.547 1.133 5.485 1.145 5.263 1.194 15 6.147 1.022 5.985 1.050 6.029 1.042 5.374 1.169 16 6.389 0.983 6.380 0.985 6.259 1.004 6.029 1.042 17 6.617 0.950 6.545 0.960 6.348 0.990 6.185 1.016 18 6.918 0.908 6.662 0.943 6.475 0.970 6.500 0.967 19 7.105 0.884 6.770 0.928 6.667 0.942 6.685 0.940 20 7.205 0.872 6.837 0.919 6.801 0.924 6.727 0.934

Table5 Summary of response

Initial guy tension (Ib) at Dynamic amplification factor

Level Initial Rupture Peak mast (ft) displ. (ft) A B C of guy displ. (ft) A B C

480 -1.52 14613 14467 36559 G1 -13.68 1.51 1.53 G2 -3.37 1.44 1.48 1.76 G3 -1.86 1.38 1.41 1.23

320 -0.99 11281 11281 35231 G1 -3.81 1.47 1.46 2.84 G2 -3.43 1.79 1.79 G3 -1.64 1.47 1.48 1.43

160 -0.33 12541 12690 25056 G1 0.35 1.54 1.45 1.16 G2 -1.12 1.22 1.22 2.05 G3 -1.27 1.38 1.38

points A, B and C, respectively. The plot of T9 is left blank to remind the reader that guy G1 has ruptured. T55, T56 and T57 are the end tension of the guys attached to the mast at level 320 ft at A, B and C, respectively. Finally T103, T104 and T105 are the end tensions of the guys attached to the mast at level 160 ft.

The effect of the impulsive shock is especially felt in guy G2. The plot of guy G2 end tension (T57) gives a peak transient tension of 100 000 lb for an initial tension of 35 231 lb, therefore a dynamic amplification factor of 2.84. The time variations of the end tensions of the other guys show small peak tensions.

Table 6 summarizes the initial axial forces (I.F.), peak compression (P.C.) and ?eak tension (P.T.) forces in each member of selected mast panels. To study the effect of the sudden rupture of guy G1 on individual members of the mast, the time variations of the axial forces were plotted for each member of selected mast panels. Those plots are given in Figures 4-10 for the levels 480 ft, 400 ft, 320 ft, 240 ft, 160 ft, 80 ft and 10 ft, respectively. The first three plots of each figure represent the time variations of the axial forces in the struts, from left to right in members 1-2, 2-3 and 1-3, respectively, (with reference to Figure lc). The

second set of plots is that of the forces in the diagonals, from left to right in members 1-5, 2-6 and 3-4, respectively. The bottom three plots are of the forces in the chords, in members 1-4, 2-5 and 3-6, respectively.

The peak axial forces in the members of the panel located at level 480ft are below their compression/tension capacities as seen in Figure 4 and do not fail. Figure 5 shows that only strut 2-3 fails by exceeding its compression capacity. The peak transient compression force is -23 7491b (see Table6), while the struts compression capacity is -13 162 lb (see Table 1). The impact of the sudden rupture of guy G I is strongly observed in the panels located between guys attachment points at levels 320 ft and 160 ft. This is best appreciated by looking at Figures 6-8. The peak transient compressive forces in chords 1-4 and 2- 5 exceed their compressive capacities which were calculated to be -146 298 lb. These peak forces are, respectively, -305 370 lb and -303 908 lb at level 320 ft, -217 426 lb and -216011 lb at level 240 ft, and -166 728 lb and -167 191 lb at level 160 ft. This is true for all the chords of type 1-4 and 2-5 of all the panels between levels 320 ft and 160 ft. In addition, the peak transient tension force in chord 2-6 of the panel at level 320 ft was found to be

882

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Figure3 Time variations of guys' end tensions at mast attachment points after sudden rupture of G1


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Time variations of mast member forces in panel at elevation 480 ft after sudden rupture of G1

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Nonlinear dynamic response of guyed tower to sudden guy rupture: N. Ben Kahla 885

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Table6 Initial and peak transient forces in mast members (sudden rupture of G1)

Member

Level (ft) Type 1-2 2-3 1-3 1-5 2-6 3-4 1-4 2-5 3-6

480 I.F. 3981 11840 8634 ~,25 -6667 5306 -13742 -9537 -30920 P.C. -11221 -11378 -3077 -12837 -6872 -23376 -23005 -34553 -30920 P.T. 13620 11840 17187 13908 24378 5306 10426 254 26395

400 I.F. 111 -524 1690 -562 1440 -2653 -4182 -4719 -56585 P.C. -9535 -23749 -676 -12718 -3686 -29176 -139941 -160629 -56585 P.T. 9899 2146 21378 10315 32935 3407 268026

320 I.F. 4538 10205 12702 -965 -5693 3362 -44487 -41085 -27935 P.C. -3661 -11538 -28464 -216 -305370 -303908 -27935 P.T. 9129 23542 31670 10435 164 24560 498371

240 I.F. 533 -20 1839 -1056 556 -2792 -38804 -38935 -47050 P.C. -8522 -2643 -14522 -11911 -21674 -6034 -217426 -216011 -50649 P.T. 9243 15504 3901 9527 2956 21229 324705

160 I.F. 6964 9816 12563 -1412 -3459 679 -70141 -67666 -18275 P.C. -2535 -9686 -22383 -2821 -166728 -167191 -35660 P.T. 15923 24612 13301 7615 1848 19957 165960

80 I.F. 907 101 1943 -1430 116 -2849 -71228 -72168 -24002 P.C. -7260 -2108 -13114 -13451 -17217 -6230 -96709 -99078 -49111 P.T. 7276 11977 5599 10862 2788 16662 49616

10 I.F. 934 -1169 3142 -1383 1772 -4509 -85614 -88337 -3375 P.C. -6930 -7978 -13698 -13451 -20144 -14747 -92009 -88337 -140211 P.T. 8556 12817 10619 10688 12239 20073 3906 53422

498 371 lb which is beyond the 346 356 lb tension capacity of the chords. The peak transient compressive force in strut 1-3 of the panel at level 240 ft, computed to be -14 522 lb, also exceeded its compressive capacity The failure of all

those members will eventually cause the collapse of the guyed tower. For the mast panels below level 160 ft, no member failure occurs (see Figure 9), except for strut 1-3 at the bottom of the mast (level l0 ft) for which the peak

886

Figure 9


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Figure 10 Time variations of mast member forces in panel at elevation lo f t after sudden rupture of G1

Nonl inear dynamic response o f guyed tower to sudden guy rupture : N. Ben Kah la

F igure 11

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..... ....... ...... 1 e 6 ....... i ....... i ...... co 4 . . . . . . . i . . . . . . . ! . . . . . .

0 r ..... . - - . - --1 0 20 40 60

Time (s) 0 x104. .

g

~ 4 . . . . . . . ! . . . . . . . :: . . . . . .

00~2'0 40 60 Time (s)

0x! '. .

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r . . . . . . . . ; . . . . . . ; --

% 20 60 4O T ime (s )

10 x 104 / /

8 - i - - . . . . . . . . . . . . . ~ . . . . . . 4

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. . . . ; . . . . . . 1 e- ..... i ....... i . . . . . .

0 / ; ; , 0 20 40 60

Time (s)

10 x 104

~- o2t .............. ! ...... t 0 20 40 60

Time (s) 10 x 104

6L:: ..... :: ....... i ...... 1 2 I ..... i ....... ! ...... 1 % 2'0 4o 6'0

Time (s)

Time variations, of guys' end tensions at mast attachment points after sudden rupture of G2

887

4 x 104

. . . . . . . i . . . . . . . i . . . . . . .

-2 t -4

0 20 40 60 Time (s)

4 x 11) 4

2f - - - i i

, , - . ::

-% 2'o 4'0 60 T ime (s)

4 x 10 s

" - "2 JO r . .O O3 LL-2

20 40 60 Time (s)

4 x 104

oJ o r . . . . . i . . . . . - :~ : ! . . . . . :1

0 20 40 60 Time (s)

4 x 104

~" 0 " : ..=.i-:~.~..4 u_ _ : ~- . - ; .

_ .

0 20 40 60 Time (s)

4 x 10 s

oO 0 ; ; ' : : : : ' i : ' " " ' i " : ' '

-4 ' i i , 0 20 40 60

Time (s)

4 x 104

- 0 20 40 60 Time (s)

X 104 4

- ~--,i -J = T-IIT:::I~IIIZ~I:I 7 O3 LL- 2

-4 20 40 60 Time (s)

x 10 s

f iiiiii t , ~o " -2 [ . . . . . . . . i . . . . . . . . . ' ""'""

-4 ; 0 20 40 60

Time (s)

F igure 12 Time variations of mast member forces in panel at elevation 400 ft after sudden rupture of G2

888

Figure 13


x 104 4

~2 coO (0

u_-2 -4 0 20 40 60

Time (s) x 10 4

.-. 4 . .

o I~:.;':,.~;:,,:;~',, ~ -~I ....... : ....... ; 1 -4 / /

0 20 40 60 Time (s)

4 x105.

. . . . . . . . . . . . . i . . . . . . .

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Time (s)

x 104 4

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U.-2 -4

0 20 40 60 Time (s)

x 104

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Time (s) 4 x 10 5. .

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'~:~I ....... ' . . . . . . . . . . . . . . 1 0 20 40 60

Time (s)

4X104

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Time (s) 4x-1~.~- .

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eD

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Time (s) x 105

. _ .4 r . ' " t

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_4 ~ ! i 0 20 40 60

Time (s) Time variations of mast member forces in panel at elevation 320 ft after sudden rupture of G2

10 x 104 .

68 ifi i i i i i i i i i i i i i i i i l l j ~ ....... i . . . . . . . :. .......

4 . . . . . . . i . . . . . . . i . . . . . .

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Time (s)

x 104 10[ . .

. . . 8 . . . . . . . i . . . . . . . ; . . . . . .

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2 ~o 4; oo Time (s)

_ _ 10 x 104 . 10 x 104

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Time (s) x 104

10~ , . /

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0;~ 2;0 40 60 Time (s)

0 x 104.

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Figure 14 Time variations of guys' end tensions at mast attachment points after sudden rupture of G3


Figure 15

4 x 104

"2

~0 0 cO LL-2

-40 20 40 60 Time (S)

. x2~ .

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0 20 40 60 Time (s)

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0 20 40 60 Time (s)

Time variat ion of mast member forces

x 104 ,-. 4 . .

" -~ I ............. ~ ...... 1 0 20 40 60

Time (s) X 104

4

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~-2 ~ ~ .....

- ~ 6 0 Time (s)

4 x 105 .

~:EC--~:-! -: ..... i ...... 1 0 20 40 60

Time (s)

in panel at elevation 160 ft after sudden rupture of G3

889

x 1134

I -I ............ i ...... 1 -% ~:0 X0 ~0

Time (s)

4 x 10 L

l " 'WWm~

":~I .............. ~ ...... 1 0 20 40 60

Time (s)

4x_1_c~ .

Time (s)

x 104

2 . . . . . . . ~ . . . . . . . ; . . . . . . ~ 2

OL._;_"=..:-:::I:=::.~ "~ 0

" I ...... ~ ....... ! ...... l ~ -= -4 ' ' -4 0 20 40 60 q

Time (s) 4 x104 4

m

. . . . . . ' . . . . . . . . . . . . . .

0 20 40 60 Time (s)

4 x 10 s

_,,.F ~ i ] 0 20 40 60

Time (s)

"0

1.1_-2 -4

x 104

i:i-!i i i 1 ) 20 40 60

Time (s) X 10 4

t . . . . . . . . . . . . . . i . . . . . .

I : :

0 20 4O 60 Time (s)

x 105

' L ~" 2 =':i,;, ..... '_: .... 0F:;-:-i ....... ~ ......

-2 I- ............ ! . . . . . . -~ 2-'0 4'0 60

Time (s)

Figure 16 Time variations of mast member forces in panel at elevation 10 ft after sudden rupture of G3

890 Nonlinear dynamic response of guyed tower to sudden guy rupture: N. Ben Kahla

transient compressive force was -13 698 lb as determined from Figure 10. This force is larger than its compressive capacity.

The time variations of the guys' end tensions at mast attachment points after the rupture of guy G2, are plotted in Figure 11. The largest tensions were obtained for guys G 1 and G3. The peak transient end tensions for G 1 and G2 are 64 344 lb and 51 365 lb, respectively, corresponding to dynamic amplification factors of 1.76 and 2.05. The conse- quences of the rupture of guy G2 are illustrated by the mag- nitudes of the axial forces in the members of the panels in the vicinity of the G2 mast attachment level. Figure 12 shows the time variations of member forces in the panel at level 400 ft. The plot of F99 gives a peak transient axial force of -244 002 lb in chord 3-6. For the panel at level 320 ft, the time variations of the member forces are plotted in Figure13. The graph of F171 shows a peak of -282 341 lb. Both of these peak forces exceed the chord compression capacity. It is to be noted that all the chords of type 3-6 in the panels between levels 400 ft and 240 ft failed in compression, resulting in the collapse of the structure.

The plots of the time variations of the guys' end tensions are given in Figure 14 for the case when a sudden rupture of guy G3 occurred. The peak transient end tension of G 1 was computed to be 44 968 lb for a dynamic amplification factor of 1.23, that of G2 was 50 380 lb corresponding to a dynamic amplification factor of 1.43. At level 160 ft, chord 3-6 fails in compression, its peak transient axial force as can be seen in the plot of F315 in Figure 15 was -164 349 lb. At the bottom of the mast chords 1-4 and 2- 5 failed. Both of their peak transient axial forces (respectively, -194 692 lb and -197 339 lb) exceeded their

compression capacities. This is illustrated in the plots of F448 and F449 in Figure 16.

5. Conclusions

The present study investigated the nonlinear dynamic reponse of a guyed tower, when a sudden rupture of a selected guy occurred. The investigation was carried out for a nominal wind speed, acting in the plane of the ruptured guy, assumed to be a windward one. The time variations of the guys' end tensions and mast members' axial forces were determined for three cases of guy rupture. The analy- ses showed that the largest tensions were obtained for the initially windward guys and that the collapse of the structure occurs after a set of chord members lose their load carrying capacities. Most of the time the results show that the chords fail in compression.

References 1 'Recommendations for guyed masts', Working Group No. 4, Inter-

national Association for Shell and Spatial Structures, Madrid, Spain, 1981

2 EIA Standard 222-D. 'Structural standards for steel antenna towers and antenna supporting structures', Electronics Industries Associ- ation, Washington, DC, 1986

3 Peyrot, A. H. and Goulois, A. M. 'Analysis of cable structures', Corn- put. Struct. 1979, 10, 805-813

4 Thomas, M. B. 'Broken conductor loads on transmission line structures', Ph.D. thesis, University of Wisconsin-Madison, 1981

5 Ben Kahla, N. 'Dynamics of a single guy cable', Comput. Struct. 1995, 54 (6), 1197-1211

6 Ben Kahla, N. 'Dynamic analysis of guyed towers', Engng Struct. 1994, 16 (4), 287-301.

7 Peyrot, A. H. 'Marine cable structures', J. Struct. Div., ASCE 1980, 106 (12), 2391-2404.

Nonlinear Dynamic Response of Guyed Tower to Sudden Guy Rupture

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