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Seismic sensitivity indicators for tall guyed telecommunication towers
Computers and Structures 80 (2002) 349–364 www.elsevier.com/locate/compstruc
Seismic sensitivity indicators for tall guyed telecommunication towers G. Ghodrati Amiri * Department of Civil Engineering, Iran University of Science and Technology, P.O. Box 16765-163, Narmak, Tehran 16844, Iran Received 10 January 2001; accepted 30 October 2001
Abstract Very tall towers are a fundamental component of post-disaster communication systems and their protection during a severe earthquake is of high priority. In North America, there are no simple rules to evaluate the seismic sensitivity of guyed masts, and very limited attention has been paid to the seismic behaviour of such structures to date. Since guyed towers may exhibit significant geometric nonlinearities, their detailed nonlinear seismic analysis is complex and timeconsuming. In addition, climatic loads such as wind and ice are likely to govern their design in most cases. As a result, earthquake effects are often ignored or improperly evaluated. The objective of this paper is to propose some seismic sensitivity indicators for tall guyed masts, which would help tower designers decide whether seismic effects are important and whether detailed dynamic analysis of the structure is required. The indicators proposed relate to the maximum base shear and the dynamic component of the axial force in the mast and guy cable tensions. A conceptual model is also presented to explain the distribution of earthquake effects along tower elevation. The study is based on detailed nonlinear seismic analyses of eight existing guyed telecommunication towers with heights varying from 150 to 607 m. 2002 Elsevier Science Ltd. All rights reserved. Keywords: Earthquake; Dynamic analysis; Telecommunication towers; Guyed towers; Sensitivity indicators; Geometric nonlinearity
1. Introduction In the wireless, microwave, and satellite communications industry, tall guyed towers are one of the important structural subsystems. They support a variety of antenna––broadcasting systems at great heights, or are themselves radiators in order to transmit radio, television, and telephone signals over long distances. Guyed towers normally provide an economical and efficient solution for tall towers of 150 m and above, compared to self-supporting ones. As shown in Fig. 1, the main component of these structures is a slender lattice steel mast, usually of triangular cross-section, which is pinned at its base. Sets of inclined pretensioned guy cables support laterally the mast at several levels along its
*
Tel.: +98-21-7391-3124; fax: +98-21-745-4053. E-mail address: [email protected] (G. Ghodrati Amiri).
height. In some towers, the guy cables are connected to a stabilizer or outrigger at some stay levels (see cable Set 4 in Fig. 1), in order to improve the torsional stiffness of the structure. The structural behaviour of tall guyed towers is complex. This complexity arises from significant geometric nonlinear behaviour due to, in first order, the sagging tendency of the guy cables and the interaction between the cables and the mast, and in second order, the slenderness of the mast (beam-column effects). As a result, earthquake-resistant design of these structures cannot simply be extrapolated from simple rules available for buildings. The purpose of this paper is to propose some seismic sensitivity indicators for tall guyed masts, which would help tower designers decide whether seismic effects are important and whether detailed dynamic analysis of the structure is required. The indicators proposed relate to the maximum base shear and the dynamic component of
0045-7949/02/$ - see front matter 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 4 5 - 7 9 4 9 ( 0 1 ) 0 0 1 7 5 - 4
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Nomenclature A BA BS E g H h IT k k0 L
cross-sectional area of a guy cable dynamic component of the axial force at the base of the mast maximum base shear modulus of elasticity gravity acceleration tower height sectional elevation of the mast initial tension in a guy cable lateral stiffness of guy cluster lateral stiffness of guy cluster without sag effects cable length
Pdyn PGA PGV r T UTS W a b c
maximum dynamic component of the axial force in the mast at a section of elevation h peak horizontal ground acceleration peak horizontal ground velocity coefficient of correlation natural period ultimate tensile strength of a cable total weight of guyed tower angle of guy cable to horizon second parameter of Newmark-b direct integration operator first parameter of Newmark-b direct integration operator
Fig. 1. Typical geometry of tall guyed tower.
the axial force in the mast and guy cable tensions. A conceptual model is also presented to explain the distribution of earthquake effects along tower elevation.
The study is based on detailed nonlinear seismic analyses of eight existing guyed telecommunication towers with heights varying from 150 to 607 m.
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2. Background The structural behaviour of guyed towers has been studied by many researchers, but mostly from a static perspective and in relation to wind and ice loads. Only a few previous studies have focused on seismic response, and it is difficult to draw general conclusions from them because they each address very specific applications. Moossavi Nejad [3] carried out numerical simulations of the nonlinear seismic response of a 327-m guyed radio tower with triangular cross-section and five stay levels. Three ground accelerograms (two orthogonal horizontal components and one vertical) were artificially generated, which had peak ground intensities of 0.3 and 0.2 g in the horizontal and vertical directions, respectively. The mast was modelled as an equivalent beamcolumn, and a mesh of five to nine two-node elements was used for guy cable modelling. The results have indicated that the strong ground motions simulated produce large dynamic tensions in the guy cables, in the range of 55–250% of initial tensions. A comparison of the seismic analysis results with those of static wind analysis has shown that the maximum base axial force and the maximum bending moment in the mast due to earthquake loads were much larger than the corresponding results of the wind load analysis, by 30% and 170%, respectively. However, the maximum tensions in the guy cables due to earthquake loads were found smaller (but in the same range) than those due to wind. Similar numerical simulations had been reported earlier by Guevara and McClure (cf. [5–8]) on three guyed towers, using natural accelerograms (El Centro and Parkfield). A 342-m mast with seven stay levels was modelled as a three-dimensional truss and two smaller ones (24-m with two stay levels and 107-m with six stay levels) with equivalent Timoshenko beam-column elements. The two main conclusions of these numerical studies were that: (1) vertical input accelerations may amplify the dynamic interactions between the mast and the cables, these amplifications being more significant in the axial force in the mast and in the guy wire tensions of the top and bottom clusters; and (2) only correct modelling of inertia properties of both the mast and the guy wires can fully simulate these potential dynamic interactions. Another numerical study by Argyris and Mlejnek [11] on a 152.5-m transmitter tower with two stay levels, used an idealized sinusoidal horizontal accelerogram (amplitude of 0.5 g and period of 1 s) as a rough simulation of an earthquake. Large tower displacements (1.5 m) were calculated, probably an indication of nearresonance of the input with the fundamental flexural mode of the structure. The International Association for Shell and Spatial Structures (IASS [13]) presents general recommendations for seismic analysis of guyed masts, in a special
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report published by its Working Group 4 on masts and towers. This report suggests that an equivalent static lateral load proportional to the tower weight may be used to model earthquake effects, as it is done in most building design codes. Designers are then advised to use their national standards for more specific guidelines on dynamic amplification factors and force distribution: the problem is that such guidelines simply do not exist in North American tower standards. The report also recommends several simplifying assumptions in order to linearize the analysis and use modal superposition. However, it acknowledges that very tall guyed masts or unusual towers may exhibit significant geometric nonlinearities. A comprehensive study of the behaviour of tall guyed towers under seismic excitation is required in order to develop more complete seismic analysis and design guidelines than summarized above. Reliable nonlinear dynamic analysis software is commercially available, and the focus can now shift from the discussion of modelling considerations to the structural response itself.
3. Methodology This study relies entirely on ‘‘numerical experiments’’, i.e. detailed full-scale simulations using the finite element method. Why? first of all, experimental results relating to the overall seismic behaviour of guyed telecommunication towers are very rare (not to mention that they are inexistent). Also, since tall guyed towers exhibit significant geometric nonlinearities, experimental studies of scaled-down physical models are very complex and of limited usefulness. Furthermore, experimental work on full-scale tall guyed towers is not feasible at present. A numerical modelling study of eight existing tall guyed telecommunication towers subjected to three different seismic excitations has been carried out. The tower models are very detailed to include geometric nonlinearities and to allow for potential dynamic interaction between the mast and guy wires. The selection of the towers was made to cover a wide range of tower heights, from 150 to 607 m, in order to identify some common trends in behaviour.
4. Description of towers 4.1. Tower geometry In practice, guyed towers taller than 150 m usually provide economical solutions comparing to self-supporting towers. Therefore, the lower height limitation of 150 m is a common criterion to classify towers with
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Table 1 Geometry of guyed towers analysed Tower height (m)
No. of guying stay levels
No. of anchor groups
No. of outriggers
Panel width (m)
Panel height (m)
Location (source)
607.1
9
3
0
3
2.25
342.2 313.9 213.4 200
7 5 7 8
2 2 2 3
1 0 0 0
2 2.14 1.52 1.8
1.52 1.52 1.52 1
198.1
6
2
1
2.13
1.52
152.4
8
2
2
0.84
0.61
150
7
2
3
1.3
1
U.S.A., California, Sacramento (LeBLANC & Royle Telcom Inc.) Canada (Wahba et al. [10]) Canada (Wahba et al. [10]) Canada (Wahba et al. [10]) Argentina, Buenos Aires (Estudio Ing. M. Oberlander) Canada, Prince Edward Island, Charlottetown (Trylon Manufacturing Co. Ltd.) Canada, Alberta, Elk River (AGT, LeBLANC & Royle Telcom Inc.) Canada, Alberta, Little Buffalo (LeBLANC & Royle Telcom Inc., AGT)
Note: All towers have a mast of triangular cross-section, except that of the 200-m tower which is square.
respect to their heights. In this regard, those available data for guyed towers taller than 150 m were selected for the simulations. These towers are listed in Table 1. It should be mentioned that the 607-m tower, located in Sacramento, California, is one of the tallest guyed telecommunication towers in North America. The 200-m tower is located in Buenos Aires, Argentina, and the other six towers are located in Canada. All masts have a triangular cross-section, except for the 200-m tower which is square. 4.2. Weights of mast and cables The detailed weights of mast and cables for the eight guyed towers studied are listed in Table 2. It is observed that the mast accounts for 69–77% of the total tower weight, leaving 23–31% to the guy cables. It should be noted that tower attachments such as antennae and accessories (e.g. ladders, platforms, transmission lines, lights for aircraft warning, etc.) are not included in the total weights. The mass distribution along the mast for the eight towers is almost uniform with average values ranging from 60 kg/m (152-m tower) to 600 kg/m (607-m tower). Since the antennae and ancillary components are not modelled, the only discontinuities in the mass profile are due to the presence of torsional resistors (outriggers) in the 150-m and 152-m towers.
Table 2 Weights of masts and cables Tower height (m)
Structural component
Weight (kN)
% of total weight
607
Cable Mast Total
1595 3578 5173
31 69 100
342
Cable Mast Total
287 754 1041
28 72 100
313
Cable Mast Total
306 1033 1339
23 77 100
213
Cable Mast Total
91 205 296
31 69 100
200
Cable Mast Total
95 249 344
28 72 100
198
Cable Mast Total
180 535 715
25 75 100
152
Cable Mast Total
36 91 127
28 72 100
150
Cable Mast Total
105 229 334
31 69 100
4.3. Initial tensions of guy clusters The initial average cable tensile stress prescribed by design in each guy cluster is listed in Table 3 for all eight
towers. It represents the cable tension in the equilibrium configuration of the tower under its self weight and the
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Table 3 Initial tension of guy clusters Tower height (m)
Set no. 1
2
3
4
5
6
7
8
9
607
I.T. (I.T./UTS)%
130 12
108 10
108 10
108 10
108 10
108 10
108 10
108 10
97 9.0
Anchor I.T. (I.T./UTS)%
Inner 159 15
91 8.4
100 9.3
Intermediate 100 103 9.3 9.5
97 9.0
Outer 97 9.0
Anchor
Inner
I.T. (I.T./UTS)%
129 12
Anchor
Inner
I.T. (I.T./UTS)%
129 12
127 12
127 12
129 12
Anchor
Inner
I.T. (I.T./UTS)%
62 5.7
43 4.0
44 4.1
47 4.4
35 3.2
Anchor
Inner
I.T. (I.T./UTS)%
105 9.7
Anchor
Inner
I.T. (I.T./UTS)%
87 8.1
91 8.4
90 8.3
Anchor
Inner
I.T. (I.T./UTS)%
108 10
Anchor
Inner
342
313
213
200
198
152
150
Outer 108 10
108 10
85 7.9
86 7.9
Outer 129 12
129 12
129 12 Outer
51 4.7
51 4.7
43 4.0
Intermediate 107 9.9
107 9.9
Outer
107 9.9
108 10
108 10
91 8.4
91 8.4
Outer 87 8.1
92 8.5
92 8.5
Outer 105 9.7
106 9.8
105 9.7
108 10
108 10
108 10
Outer
I.T. ¼ initial cable tensile stress of guy cluster (MPa); UTS ¼ ultimate tensile strength ¼ 1080 MPa.
effect of the prestressing forces in the guy wires. This tension is usually expressed as a percentage of the cable ultimate tensile strength (UTS), and in most cases the initial tension varies from 8% to 12% of the UTS. The only exception is the 200-m tower with initial cable tensions as low as 3–6% of the UTS: this will be found to have an important influence on its seismic response.
4.4. Equivalent lateral stiffness of guy clusters Data on initial cable tensions is used to calculate the equivalent lateral stiffness of each guy cluster. This calculation is not necessary to perform dynamic analyses, as the stiffness contributions of each element are as-
sembled automatically by the analysis software, but it will prove useful when discussing the results. The equivalent lateral stiffness of each guy cluster, k, expressed in kN/m, is listed in Table 4. It was obtained from nonlinear static analysis in the displaced tower configuration. Since the variation of k was expected to be nonlinear with respect to the amplitude of the horizontal displacement of the anchor point on the tower, the lateral stiffness was evaluated for a range of lateral displacements of the mast varying from 10% to 100% of the maximum displacement observed in seismic analyses. However, the calculations have shown no significant variations (in general, variations are in the range of 10– 15%) and the average value is taken to represent the lateral stiffness of each guy cluster.
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Table 4 Lateral stiffness of guy clusters (kN/m) Tower height (m)
Set no. 1
2
3
4
5
6
7
8
9
607
k0 k (k=k0 )%
1886 1054 56
2006 855 43
1166 509 44
610 79 13
590 81 14
664 98 15
338 23 7
558 40 7
290 17 6
Anchor k0 k (k=k0 )%
Inner 482 386 80
620 115a 19
350 72 21
Intermediate 348 320 86 56 25 18
290 44 15
Outer 218 36 17
Anchor
Inner
k0 k (k=k0 )%
1026 639 62
Anchor
Inner
k0 k (k=k0 )%
810 517 64
256 114 45
266 84 32
266 117 44
Anchor
Inner
k0 k (k=k0 )%
804 441 55
244 24 10
304 17 6
226 16 7
288 10 3
Anchor
Inner
k0 k (k=k0 )%
1438 821 57
Anchor
Inner
k0 k (k=k0 )%
272 188 69
244 97a 40
142 54 38
Anchor
Inner
k0 k (k=k0 )%
988 907 92
Anchor
Inner
342
313
213
200
198
152
150
a
Outer 906 410 45
470 216 46
810 125 15
356 54 15
Outer 492 330 67
312 209 67
278 126 45 Outer
602 302 50
374 49 13
348 31 9
Intermediate 1226 747 61
650 400 62
1096 499a 46
Outer 514 239 46
510 219 43
182 92 51
156 64 41
Outer 230 117 51
490 230a 47
188 94 50
Outer 1264 887a 70
700 480 69
584 399 68
560 284a 51
478 238 50
1030 406a 39
Outer
Outrigger location.
For comparison, the lateral stiffness of perfectly taut guy clusters (without sag), k0 , is also tabulated, which represents the stiffness of a straight rods. Since the guy wires are prestressed, all the cables in a cluster are assumed to contribute to the lateral stiffness according to k0 ¼ ð2AE cos2 aÞ=L
ð1Þ
where k0 is the lateral stiffness of taut guy cluster, A the cross-sectional area of cable, E the modulus of elasticity
of cable, a the angle of inclination of cable with horizon, L the length of cable. Eq. (1) applies only to guy clusters with three or four cables symmetrically arranged. Table 4 also includes the percentage ratio of k to k0 . This ratio indicates the relative degree of lateral support provided to the mast when sagging effects are accounted for, compared to the straight rod assumption. In general, the lateral stiffness provided by guy clusters tends to decrease with tower elevation, as longer cables tend to
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have more sag. The adjustment of the initial tension therefore plays an important role. Values presented in the table indicate that relatively large stiffnesses are achieved for the lower levels (k1 varies from 55% to 92% of k0 ) compared to the top levels, in particular for the 200-m and the 607-m towers for which the k=k0 ratio of the top cluster is only 3% and 6%, respectively. In the first case, this small lateral stiffness is due to a low initial tension (only 3.2% of the UTS––see Table 3), whereas in the tallest tower, this is mainly due to the large cable length (over 750 m) since the initial tension is of 9% UTS.
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The nonlinear dynamic analysis is done by direct step-by-step integration in the time domain, using the Newmark-b operator with parameters c ¼ 0:5 and b ¼ 0:25 (constant average acceleration method or trapezoidal rule). The Broyden–Fletcher–Goldfarb– Shanno (BFGS) stiffness update is used and stiffness matrix updates are performed at every time step. An energy-based convergence criterion is used to bound the iteration process. The subspace iteration procedure is used for the frequency analysis. More details about the finite element modelling aspects of the problem can be found in Amiri [2] and in the ADINA Users Manual (ADINA R&D, Inc. [9]).
5. Numerical modelling considerations 5.2. Earthquake accelerograms 5.1. Tower modelling assumptions The nonlinear finite element software automatic dynamic incremental nonlinear analysis (ADINA) is used in this research (ADINA R&D, Inc. [9]). The mast is a spatial structure with response in all three dimensions. The members making up the mast are rolled steel sections, and a detailed three-dimensional truss model is used where all elements are pin-ended. Previous work by Guevara and McClure [7] has shown that, for tall towers, there is little saving in computing effort when using the equivalent beam model for the mast. Furthermore, torsional behaviour is very difficult to model with equivalent properties as warping effects become important in long masts with nonsymmetrical bracing patterns. All members are assumed to remain in the linear elastic range of material behaviour. As the displacements and rotations of the mast may be large, a large kinematics formulation (with small strains) is necessary in order to account for potential geometric nonlinearities. Foundations are assumed perfectly rigid. Each guy cable is modelled as a linkage of ten prestressed truss elements. Three-node isoparametric elements are used with two Gauss integration points each, for a total of 20 sampling points along each cable. A large kinematics formulation is also used to account for full geometric nonlinearities. A tension-only linear elastic stress–strain law is defined to model cable slackening effects. A lumped mass formulation is used for both the cable elements and the individual truss members. Added mass due to tower attachments is not included in this study, but it could easily be inputted in the form of concentrated masses at tributary nodes. Structural damping is modelled by using an equivalent viscous damper with a value of 2% of critical viscous damping in parallel with each element of the mast and cables. Earthquake loads are assumed to occur under still air conditions (IASS [13]) so aerodynamic damping has not been modelled.
In this study, three classical horizontal earthquake accelerograms have been selected, representing different types of earthquake loading. The first one is the S00E 1940 El Centro earthquake containing a wide range of frequencies and several episodes of strong ground motion; the second one is the N65E 1966 Parkfield earthquake representing a single pulse loading with dominant lower frequencies; and the third one is the S69E 1952 Taft earthquake with high frequency content and strong shaking with long duration. These earthquakes are selected to reflect realistic frequency contents as exhibited by real ground motions. The earthquake direction, indicated in Fig. 1, was selected to coincide with a principal direction of the mast cross-section so as to create maximum lateral effects without any torsional response. The earthquake records were scaled to fit as much as possible the elastic design spectrum of the 1995 National Building Code of Canada [4] for the region of Victoria, British Columbia. This region has one of the highest seismicity levels in Canada with a peak horizontal ground acceleration (PGA) of 0.34 g and a peak horizontal ground velocity (PGV) of 0.29 m/s. It is understood that none of the towers studied is actually located in Victoria, but the scaling allows the comparison of the response of the towers for different accelerograms. Schiff’s scaling procedure (Schiff [12]) is used and values of the scaling factors are given in Amiri [2] along with the plot of each earthquake record. Unfortunately, no realistic data was available on the vertical accelerograms corresponding to the horizontal ground motions selected. Referring to the National Building Code of Canada 1995––Commentary J (NRC [4]), and considering that the ratio of vertical-to-horizontal accelerations depends on site conditions and varies widely, a value of 3/4 is used. This means that the vertical accelerograms are assumed perfectly synchronous with the horizontal ones, and that the intensity of the acceleration is uniformly scaled down by a factor of 3/4.
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6. Results of dynamic analysis 6.1. Frequency analysis A frequency analysis in the deformed configuration under self weight and cable prestressing forces has been carried out for each tower. The period calculated for the fundamental flexural mode is plotted in Fig. 2 as a function of tower height. The graph shows an almost linear increase of the first flexural period of the towers with height, the only exception being the 200-m tower, which is very flexible with respect to the other comparable towers due to its slacker guy cables (much less than 10% of their UTS). Fundamental periods vary from 0.6 to 4.3 s (or 0.2 to 1.7 Hz in terms of frequency), which already suggests a wide range of seismic response. Most towers have their lowest four lateral modes in the frequency range of 0.5–3 Hz. The following formula (essentially a curve fit) is suggested to get an estimate of the fundamental flexural natural period of guyed towers with cable initial tensions in the range of 10% of their UTS: T ¼ 0:0083H 0:74
ð2Þ
where T is the fundamental period in s and H is the tower height in m. This formula is restricted to steel lattice guyed masts of triangular cross-sections with heights in the range of 150–607 m, but due to the lack of data for towers above 350 m, it is cautious to limit its use to the 150–350 m range.
Another important observation obtained from the frequency analysis is the shape of the fundamental flexural mode. Since the mass of the masts is more or less uniformly distributed along the height, it is the location of the stay levels and the relative lateral stiffness of the guy clusters that will affect most the lateral mode shapes. Table 4 indicates that for most of the towers (150-m, 152-m, 198-m, 213-m and 342-m towers), one of the guy clusters close to the top region is stiffer than the other intermediate ones, probably as a result of design serviceability limitations in the vicinity of antennae. For these masts, the top part behaves more or less like a pinned support, whereas the large stiffness of the bottom clusters mimics a fixed support. Therefore, the lowest flexural mode of the mast is similar to the second lowest frequency mode of a cantilever structure. However, the other three towers (200-m, 313-m and 607-m towers) have guy clusters with lateral flexibility increasing with tower height: as a result, their fundamental flexural mode is similar to the lowest frequency mode of a cantilever structure. Two examples of these dominant mode shape configurations will be discussed later in conjunction with a simple conceptual model to explain lateral earthquake effects, in conjunction with Figs. 6 and 7. 6.2. Sesimic sensitivity indicators Results from the detailed time history analyses were studied in order to detect common trends in behaviour among the various towers and under the various inputs.
Fig. 2. Natural period of fundamental flexural mode versus tower height.
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Although frequency content and time at peak response were also considered, the emphasis was put on the envelope curves of the response, i.e. on the maximum amplitude of seismic effects. The following nine response indicators were selected for a refined analysis of the results: (1) the resultant cable reaction force on the mast at each stay level; (2) the dynamic component of cable tension in the most critical cable of each guy cluster; (3) the horizontal shear force along the mast; (4) the dynamic component of the axial force along the mast; (5) the bending moment along the mast; (6) the dynamic component of the cable oscillation; (7) the lateral displacement along the mast; (8) the dynamic component of the axial displacement of the mast; and (9) the tilting rotation along the mast. All of them are important in design: the first five relate to strength and stability whereas the last four relate to serviceability considerations. Some seismic sensitivity indicators are now proposed in terms of base shear, axial reaction in the mast, and amplification of cable tension. 6.2.1. Base shear Fig. 3 shows the maximum base shear calculated for the eight towers subjected to the three earthquake accelerograms. The base shears were obtained by the summation of all horizontal reactions at the ground supports of cables and mast. Contributions to the base shear are presented separately for the cables and the
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total tower, and they are expressed as a percentage of tower weight. For towers shorter than 200 m, the total base shear varies in the range of 40–80% of their total weight, while this range is narrowed to 15–30% of the total weight for towers taller than 300 m. The relative contribution of the inertia effects of the cables to the total base shear is almost constant, in the range of 3–5% of the total weight of tower. It is interesting to note that for towers shorter than 200 m, although the cables make up 25–30% of the total mass (see Table 2) only 5– 10% of the total base shear is contributed by inertia effects in the cables. For the towers taller than 300 m, the relative contribution of the cables compared to that of the mast increases slightly with height. It is observed that increased cable-mast interactions are generated for the 607-m tower, especially under the El Centro earthquake. The total base shear being one of the most important response indicators for earthquake-resistant design purposes, the maximum base shear of the towers for the three earthquake excitations is investigated more precisely. Fig. 4 shows the percentage ratio of the maximum base shear to the total tower weight versus the tower height. A curve fit is suggested to predict the maximum base shear for different tower heights in the range of 150–607 m. The following equation is suggested which has a correlation factor of 0.95 for the eight results available: BS ¼ 28 300H 1:17 ð% of W Þ
Fig. 3. Base shear versus tower height.
ð3Þ
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Fig. 4. Curve fit for maximum base shear versus tower height.
where BS is the maximum base shear as a percentage of total tower weight, W , and H is the tower height in m. It is important to recall that this formula is restricted to sensitivity levels comparable to the Victoria region in Canada. It is also cautious to limit its use to the 150–350 m range due to the lack of data for taller towers. In conclusion, from the viewpoint of total base shear, guyed towers in the 150–350 m range may be sensitive to seismic effects, but since the amount of base shear can be predicted using Eq. (3) (for the seismicity level studied here), a detailed modelling study is not necessary. At the other end of the spectrum, towers taller than 350 m will have relatively low natural frequencies in their lowest modes, which will likely be matched by the dominant frequency content of the ground accelerograms. As a result, important dynamic amplifications will occur. At this stage, considering the lack of data for towers in that range of heights, a detailed dynamic analysis is recommended. 6.2.2. Mast axial force at the base The maximum dynamic component of the axial force at the base of the mast is reported in Fig. 5 as a percentage of the total weight of each tower, and the relative contributions of the inertia effects of the cables are plotted separately. Since the axial effects are more important when the vertical component of the earthquake is also considered, these results correspond to the case of combined vertical and horizontal accelerations.
As shown in Fig. 5, the relative contribution of the cable mass to the total dynamic component of the axial force is small, in the range of 4–10% of the total weights: its maximum envelope can be taken as 10% of total weight for all the towers except for the 200-m which is 15%. The dynamic component of the axial force at the base of the mast varies widely with the three accelerograms studied, from 25% to 125% of the tower weight. Typically, more response is obtained whenever there is frequency coincidence between the lowest axial modes of the tower and the dominant frequencies of the accelerogram. The Taft accelerogram is richer in high frequencies (above 4 Hz) than El Centro and Parkfield, which amplifies the response in the axial modes. The behaviour of the 607-m tower appears to contradict that analysis, but in this case, the lowest axial modes have lower frequencies (below 4 Hz) than for the other towers, and the El Centro accelerogram causes the maximum amplification. Fig. 5 also shows that the 200-m tower is very sensitive axially (especially under Taft) compared to other towers of similar heights. As already discussed in terms of flexural behaviour, this tower is exceptional because of the slackness of its guying system, and its dynamic sensitivity was confirmed in almost all of the response indicators studied. As an upper bond estimate, one could use an axial force at the base of the mast in the order of 80% of the tower weight (with the exception of the 200-m tower). However, realistic vertical accelerograms were not used,
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Fig. 5. Dynamic axial force at the base of the mast versus tower height (under combined horizontal and vertical accelerations).
which would normally show higher dominant frequencies than the horizontal accelerograms and would likely shift the sensitive frequency range in the vicinity of the lowest axial modes of many towers (4–10 Hz for the 150–350 m range). For this reason, we believe that the numerical values obtained in the analysis cannot lead to general quantitative conclusions. 6.2.3. Seismic amplification factor of cable tension Dynamic cable tensions can also be an important indicator of the severity of the dynamic effects in a guyed mast. In this study, the results were calculated for two load cases: (1) horizontal earthquake (H) and (2) combined horizontal and vertical earthquake (H þ V) accelerograms. The maximum dynamic component of cable tension in each guy cluster was recorded for all eight towers and three earthquake accelerograms. Table 5 summarizes the results for the clusters with largest dynamic effects among those connected to the outer ground anchors (i.e. the upper clusters) and to the inner ground anchors (i.e. the lower clusters). For convenience in discussing the results, a seismic amplification factor is defined as the percentage ratio of the dynamic component of cable tension to the initial tension. Overall, this seismic amplificator factor varies between 30% (upper clusters of the 607-m tower) to 290% (upper clusters of the 200-m tower). More typical values for the 150–350 m range are between 50% (upper clusters of the 150-m tower) and 200% (lower clusters of the 152-m tower). It is interesting to note that the 607-m
tower does not experience much dynamic effects in its cables compared to the other ones, whereas, not surprisingly, the 200-m tower shows the most sensitivity because of its slacker guying system. It is also observed that for all towers the combination of horizontal and vertical accelerograms (H þ V loading) does not significantly increase the dynamic cable tensions, compared to the horizontal loading case. As a matter of fact, this combined loading even tends to reduce the amplifications in some towers. In general, for all towers, the lower clusters are subjected to larger amplifications than the upper ones. 6.3. Distribution of earthquake effects with tower elevation 6.3.1. Lateral earthquake force The lateral earthquake force is defined here as the resultant horizontal force generated by an earthquake on the mast at the cable attachment points. As discussed earlier, the contribution of the mass of the cables to the total base shear (Fig. 3) and to the seismic component of the mast axial force at the base (Fig. 5) is almost constant and is small compared to that of the mast. Therefore, only the inertia of the mast has a major effect on the lateral earthquake forces. A simplified conceptual model is proposed to explain the lateral force distribution along tower height, which makes use of three important tower characteristics: (1) the predominant mode shape of mast; (2) its mass
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Table 5 Dynamic amplification of cable tensions Tower height (m)
Loading case
607
H HþV H HþV H HþV H HþV H HþV H HþV H HþV H HþV
342 313 213 200 198 152 150
Amplification factor ¼ (maximum tension/initial tension) 100 Outer ground anchor
Inner ground anchor
El Centro
Parkfield
Taft
El Centro
Parkfield
Taft
66 51 64 78 49 44 58 55 161 195 87 91 120 119 90 93
84 46 66 70 48 46 83 89 154 289 87 91 108 96 105 100
32 35 70 71 62 77 90 98 267 159 127 121 54 62 50 57
86 92 148 151 83 83 110 111 171 174 137 138 177 178 71 73
78 78 80 86 77 83 113 114 172 170 138 142 188 184 78 80
96 102 96 104 108 106 91 87 171 175 101 94 195 194 74 73
distribution; and (3) the presence of discontinuities in lateral stiffness. The predominant mode shape of the mast can be used to represent the horizontal acceleration profile along the tower height. This acceleration profile combined to the mass profile can represent the distribution of the horizontal inertia forces developed in the mast. These distributed inertia forces will result in lateral forces at the cable attachment points on the mast, in accordance with the relative lateral stiffness of the guy clusters. Two examples are selected to illustrate the concept, the first one is the 198-m tower (Fig. 6) and the second one is the 313-m tower (Fig. 7). Each figure is divided into three graphs showing (a) the mass profile, (b) the
fundamental lateral mode shape, and finally in (c) the percentage ratio of the lateral earthquake force to the total base shear. The position of the stay levels with reference to their ground anchor point is clearly identified on the ordinates of figures (a) and (c). The first example is representative of towers with a predominant mode shape similar to that of a propped cantilever (Fig. 6(b)) and the other is representative of towers with slacker upper clusters with a fundamental mode shape closer to that of a cantilever (Fig. 7(b)). It is noted that the lateral earthquake forces along the mast are an envelope of maximum amplitudes, and therefore their summation exceeds the total base shear. For most towers, the maximum lateral force varies in the
Fig. 6. Distribution of lateral earthquake forces in the 198-m tower.
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Fig. 7. Distribution of lateral earthquake forces in the 313-m tower.
range of 25–35% of the total base shear. Although the detailed results are not shown here, it is interesting to report that the 200-m tower, with its slack guying system, has a more or less uniform distribution of lateral forces at cable attachment points, of magnitude around 10% of the base shear. The distribution of the lateral earthquake forces generated by the cables on the mast cannot be explained only by the mass profile of the mast and its fundamental mode shape, as is the case for lateral effects in common lightly damped buildings. The nonuniform distribution of the lateral stiffness provided by the guying system (already discussed in conjunction with Table 4) also plays an important role. The largest variations in lateral
stiffness occur when there is a change of ground anchor point: this is due to a change in cable length and sometimes also a change in cable properties and/or initial tension. As a result, the portion of the mast in between or adjacent to these transition points becomes particularly sensitive to dynamic lateral effects, as will be discussed later. 6.3.2. Axial forces The distribution of the maximum dynamic component of the axial forces in the mast along the height for all eight towers is summarized in Fig. 8, for combined horizontal and vertical earthquake motions. The horizontal axis represents the percentage ratio of the
Fig. 8. Distribution of dynamic axial force in the mast along tower height (under combined horizontal and vertical accelerations).
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maximum dynamic component of mast axial forces to the maximum dynamic component of mast base axial force, and the vertical axis is the ratio of the sectional elevation of mast to the total tower height, h=H . It can be seen that the results of all the towers are relatively close to each other and form a narrow band. They can conservatively be represented by a parabolic curve fit (bold line in Fig. 8) with the following equation: ðPdyn =BAÞ ¼ 100–95ðh=H Þ2
ðin %Þ
ð4Þ
where ðPdyn =BAÞ is the percentage ratio of the maximum dynamic component of axial force in the mast at a section of elevation h to the maximum dynamic component of the axial force at the base of the mast. 6.3.3. Maximum shear and bending moment Table 6 summarizes results obtained for the maximum mast shear and bending moment for the eight towers studied. For comparison purposes, these values have been normalized as follows. The maximum mast shear is divided by the total weight of the towers, and the maximum mast bending moment is divided by the product of the panel width and the total weight. We find it reasonable to use the panel width to normalize the moment since the bending moment in the mast is in direct relation with the lever arm provided by the
Table 6 Maximum shear and bending moment of mast Tower height (m)
Maximum mast shear % (Total weight)
Maximum mast moment % (Panel width Total weight)
607 342 313 213 200 198 152 150
2.5 3.7 4.7 6.3 5.8 6.7 6.6 6.8
14 25 35 42 36 39 48 40
spacing between the tower leg members. Results indicate that in the lower range of tower heights (150–213 m), the maximum shear varies around 6–7% of tower weight and the ratio of maximum bending moment to the product of panel width and total weight varies from 36% to 48%. In the upper range of tower heights (313–607 m), the effects are slightly smaller in percentage (but not in absolute values) with the maximum shear in the range of 2.5–5% of tower weight and the corresponding bending moment ratio in the range of 14–35%. Although the envelope curves of shear and bending moments are not shown here, it was confirmed that the maximum values of mast shear occur directly at stay levels and the minimum shear occurs at midspan between two stay levels, and vice versa for the mast bending moment. 6.3.4. Serviceability considerations Antenna-supporting towers must meet strict serviceability criteria that are established by their owners in view of the particular use of the tower. Seismic amplifications of displacements and rotations may affect the top part of the tower where the antennas are attached, but they should not result in any local permanent deformation after the earthquake. Such deformations may yield to a loss of serviceability resulting in unacceptable signal attenuation. Table 7 summarizes the results obtained for maximum tower displacements and rotations. The maximum horizontal displacements are given both in absolute value and as a percentage ratio of the tower height, in columns (2) and (3) respectively. The lateral displacements in the earthquake direction are small, in the range of 0.05–0.12% of the tower height, which confirms that in spite of being slender, the towers are not very flexible. The maximum flexural rotation (tilting) of the top of the mast, listed in column (4), is below 0.4 for each tower, which should ensure the serviceability of most reflector antennae. The maximum dynamic component of cable oscillation, i.e. the amplitude of the transverse cable vibrations, is listed in column (5). Amplitudes between 0.3 and 1.3 m are calculated, which are small compared to the corresponding tower heights.
Table 7 Serviceability response indicators Tower height (m)
Maximum horizontal displacement (m)
Maximum horizontal displacement % (Tower height)
Maximum rotation (deg)
Maximum dynamic component of cable oscillation (m)
607 342 313 213 200 198 152 150
0.31 0.26 0.23 0.21 0.22 0.23 0.14 0.16
0.05 0.08 0.07 0.10 0.11 0.12 0.09 0.11
0.27 0.28 0.29 0.34 0.31 0.37 0.36 0.33
1.01 0.80 1.31 0.35 0.48 0.43 0.31 0.31
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Finally, the dynamic component of the maximum axial displacement of the mast was found negligible in most towers, with values inferior to a few centimetres. In general, then, for the seismicity level considered in the study, all the towers appear to behave within reasonable serviceability limits.
7. Conclusions 7.1. Summary of important parameters of the study Before stating any conclusions, it is important to recall some of the important parameters of the study which have a direct influence on the numerical results. • The seismicity level considered is constant with PGA ¼ 0.34 g and PGV ¼ 0.29 m/s. This corresponds to Victoria, British Columbia, which is one of the most severe seismic regions in Canada. • The vertical accelerograms are identical to the horizontal ones, but multiplied by 3/4. They are not representative of realistic vertical accelerograms which typically exhibit higher frequency content than horizontal ground motions. • The heights of the guyed towers studied in this research vary in the range of 150–607 m. However, since only one tower is modelled with a height above 350 m, it is cautious not to extrapolate results beyond that limit. • Most of these towers are located in Canada and were designed for both wind and ice loads, which necessarily influences their mass and stiffness characteristics. 7.2. Summary of the proposed seismic sensitivity indicators The following seismic sensitivity indicators are proposed to assess whether a particular tower is sensitive to earthquake effects, and if so, whether a detailed nonlinear dynamic analysis is necessary: Fundamental natural period of the tower: Eq. (2) is proposed in the 150–350 m range of tower heights. A simple check can be made with the earthquake design spectrum to assess whether or not lateral effects will be important. Values corresponding to 0.5–3 Hz are considered in the sensitive range for horizontal response. Base shear: For towers with heights ranging from 150 to 350 m, important base shears may develop, in the order of 40–80% of tower weight. The magnitude of the base shear can be predicted using Eq. (3) and a detailed nonlinear dynamic analysis could be avoided. For taller towers, the relative contributions of the inertia effects in the cables and the mast are unpredictable, and it is not possible to suggest a simple estimator. Therefore, until
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more data and knowledge is available, a detailed dynamic analysis is recommended for towers higher than 350 m. Axial forces in the mast: When seismic vertical effects are considered, guyed towers with slack cables, i.e. with initial cable tension below 5% UTS, are sensitive to the combination of vertical and horizontal earthquake motions. Since their behaviour is unpredictable, a detailed nonlinear dynamic analysis is recommended. For guyed towers with usual initial cable tension (i.e. around 10% UTS or more), the maximum dynamic component of the axial force at the base of the mast due to combined vertical and horizontal earthquake motions is about 80% of their total weight. A detailed numerical study is not necessary unless there is some detailed information available for the vertical accelerograms which suggests that their frequency content is much higher than the horizontal accelerograms. The variation of the maximum axial force in the mast can be predicted using Eq. (4). Cable tensions: The dynamic component of cable tensions can vary by an order of magnitude, from 30% to 300% of the initial tension. Typical values for the 150–350 m range of tower heights are between 50% and 200%. 7.3. Summary of trends in tower behaviour • A simplified conceptual model for the distribution of earthquake forces along the height of the towers has been proposed which accounts for the fundamental mode shape of the mast, its mass distribution, and the relative lateral stiffness provided by the guy clusters. • Discontinuities in behaviour are expected in portions of the mast between or close to adjacent guy cable clusters with significantly different lateral stiffnesses. Most response indicators were found maximum in these sensitive sections of the mast. • Inertia effects in the guy cables contribute little to lateral seismic effects in the mast, and can be neglected. However, when vertical accelerations are studied, dynamic interactions occur between the cables and the mast: these interactions are difficult to predict and detailed analysis is recommended for very tall towers (above 350 m) or towers with relatively slack guy clusters. • All the towers studied behaved within serviceability limits.
Acknowledgements The following engineers have provided detailed tower data for this study, and their contribution is greatly
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appreciated: Mr. Donald G. Marshall, Special Consultant for LeBlanc & Royle Telcom Inc., Oakville, Ontario; Mr. M. Oberlander, Estudio Ing. M. Oberlander, Buenos Aires, Argentina; Mr. K. Penfold of Trylon Manufacturing Co. Ltd., Elmira, Ontario; and K.R. Jawanda of AGT Limited, Edmonton, Alberta. Financial supports from the Ministry of Science, Research & Technology of Iran and the Natural Sciences and Engineering Research Council of Canada are acknowledged.
[6]
[7]
[8]
[9]
References [10] [2] Amiri GG. Seismic sensitivity of tall guyed telecommunication towers. PhD thesis. Department of Civil Engineering and Applied Mechanics, McGill University; Canada: 1997. [3] Moossavi Nejad SE. Dynamic response of guyed masts to strong motion earthquake. Proc 11th World Conf on Earthquake Eng. Acapulco: Mexico; 23–28 June 1996, Paper #289. [4] NRC. National building code of Canada, National Research Council of Canada, 11th ed. Canada, Ottawa: 1995. [5] McClure G, Guevara EI. Seismic behaviour of tall guyed telecommunication towers. Proc IASS-ASCE International
[11] [12]
[13]
Symposium 1994 on Spatial, Lattice and Tension Structures. Georgia, Atlanta: 24–28 April 1994. p. 259–268. Guevara EI. Nonlinear seismic analysis of antenna-supporting structures. M. Eng Project Report no. G93-14. Department of Civil Engineering and Applied Mechanics, McGill University; Canada: 1993. Guevara EI, McClure G. Nonlinear seismic response of antenna-supporting structures. Comput Struct 1993;47(4/ 5):711–24. McClure G, Guevara EI, Lin N. Dynamic analysis of antenna-supporting structures. Proc CSCE Annual Conf New Brunswick, Canada 1993;II:335–44. ADINA R&D, Inc. ADINA Reference Manuals; Reports ARD 4, 7, 8, 10. 71 Elton Avenue, Watertown, MA 02172, ADINA R&D, Inc., USA. Wahba YMF, Madugula MKS, Monforton GR. Limit states design of antenna towers. Final report submitted to the CSA technical committee S37 on antenna towers. Department of Civil and Environmental Engineering. University of Windsor; Ontario, Canada: 1992. Argyris J, Mlejnek HP. Dynamics of structures. New York: Texts on Computational Mechanics 1991;5:505–10. Schiff SD. Seismic design studies of low-rise steel frames. PhD thesis. Department of Civil Engineering, University of Illinois at Urbana-Champaign, 1988. IASS. Recommendations for guyed masts. Madrid, Spain: International Association for Shell and Spatial Structures; Working Group no. 4, 1981.
Seismic sensitivity indicators for tall guyed telecommunication towers G. Ghodrati Amiri * Department of Civil Engineering, Iran University of Science and Technology, P.O. Box 16765-163, Narmak, Tehran 16844, Iran Received 10 January 2001; accepted 30 October 2001
Abstract Very tall towers are a fundamental component of post-disaster communication systems and their protection during a severe earthquake is of high priority. In North America, there are no simple rules to evaluate the seismic sensitivity of guyed masts, and very limited attention has been paid to the seismic behaviour of such structures to date. Since guyed towers may exhibit significant geometric nonlinearities, their detailed nonlinear seismic analysis is complex and timeconsuming. In addition, climatic loads such as wind and ice are likely to govern their design in most cases. As a result, earthquake effects are often ignored or improperly evaluated. The objective of this paper is to propose some seismic sensitivity indicators for tall guyed masts, which would help tower designers decide whether seismic effects are important and whether detailed dynamic analysis of the structure is required. The indicators proposed relate to the maximum base shear and the dynamic component of the axial force in the mast and guy cable tensions. A conceptual model is also presented to explain the distribution of earthquake effects along tower elevation. The study is based on detailed nonlinear seismic analyses of eight existing guyed telecommunication towers with heights varying from 150 to 607 m. 2002 Elsevier Science Ltd. All rights reserved. Keywords: Earthquake; Dynamic analysis; Telecommunication towers; Guyed towers; Sensitivity indicators; Geometric nonlinearity
1. Introduction In the wireless, microwave, and satellite communications industry, tall guyed towers are one of the important structural subsystems. They support a variety of antenna––broadcasting systems at great heights, or are themselves radiators in order to transmit radio, television, and telephone signals over long distances. Guyed towers normally provide an economical and efficient solution for tall towers of 150 m and above, compared to self-supporting ones. As shown in Fig. 1, the main component of these structures is a slender lattice steel mast, usually of triangular cross-section, which is pinned at its base. Sets of inclined pretensioned guy cables support laterally the mast at several levels along its
*
Tel.: +98-21-7391-3124; fax: +98-21-745-4053. E-mail address: [email protected] (G. Ghodrati Amiri).
height. In some towers, the guy cables are connected to a stabilizer or outrigger at some stay levels (see cable Set 4 in Fig. 1), in order to improve the torsional stiffness of the structure. The structural behaviour of tall guyed towers is complex. This complexity arises from significant geometric nonlinear behaviour due to, in first order, the sagging tendency of the guy cables and the interaction between the cables and the mast, and in second order, the slenderness of the mast (beam-column effects). As a result, earthquake-resistant design of these structures cannot simply be extrapolated from simple rules available for buildings. The purpose of this paper is to propose some seismic sensitivity indicators for tall guyed masts, which would help tower designers decide whether seismic effects are important and whether detailed dynamic analysis of the structure is required. The indicators proposed relate to the maximum base shear and the dynamic component of
0045-7949/02/$ - see front matter 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 4 5 - 7 9 4 9 ( 0 1 ) 0 0 1 7 5 - 4
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Nomenclature A BA BS E g H h IT k k0 L
cross-sectional area of a guy cable dynamic component of the axial force at the base of the mast maximum base shear modulus of elasticity gravity acceleration tower height sectional elevation of the mast initial tension in a guy cable lateral stiffness of guy cluster lateral stiffness of guy cluster without sag effects cable length
Pdyn PGA PGV r T UTS W a b c
maximum dynamic component of the axial force in the mast at a section of elevation h peak horizontal ground acceleration peak horizontal ground velocity coefficient of correlation natural period ultimate tensile strength of a cable total weight of guyed tower angle of guy cable to horizon second parameter of Newmark-b direct integration operator first parameter of Newmark-b direct integration operator
Fig. 1. Typical geometry of tall guyed tower.
the axial force in the mast and guy cable tensions. A conceptual model is also presented to explain the distribution of earthquake effects along tower elevation.
The study is based on detailed nonlinear seismic analyses of eight existing guyed telecommunication towers with heights varying from 150 to 607 m.
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2. Background The structural behaviour of guyed towers has been studied by many researchers, but mostly from a static perspective and in relation to wind and ice loads. Only a few previous studies have focused on seismic response, and it is difficult to draw general conclusions from them because they each address very specific applications. Moossavi Nejad [3] carried out numerical simulations of the nonlinear seismic response of a 327-m guyed radio tower with triangular cross-section and five stay levels. Three ground accelerograms (two orthogonal horizontal components and one vertical) were artificially generated, which had peak ground intensities of 0.3 and 0.2 g in the horizontal and vertical directions, respectively. The mast was modelled as an equivalent beamcolumn, and a mesh of five to nine two-node elements was used for guy cable modelling. The results have indicated that the strong ground motions simulated produce large dynamic tensions in the guy cables, in the range of 55–250% of initial tensions. A comparison of the seismic analysis results with those of static wind analysis has shown that the maximum base axial force and the maximum bending moment in the mast due to earthquake loads were much larger than the corresponding results of the wind load analysis, by 30% and 170%, respectively. However, the maximum tensions in the guy cables due to earthquake loads were found smaller (but in the same range) than those due to wind. Similar numerical simulations had been reported earlier by Guevara and McClure (cf. [5–8]) on three guyed towers, using natural accelerograms (El Centro and Parkfield). A 342-m mast with seven stay levels was modelled as a three-dimensional truss and two smaller ones (24-m with two stay levels and 107-m with six stay levels) with equivalent Timoshenko beam-column elements. The two main conclusions of these numerical studies were that: (1) vertical input accelerations may amplify the dynamic interactions between the mast and the cables, these amplifications being more significant in the axial force in the mast and in the guy wire tensions of the top and bottom clusters; and (2) only correct modelling of inertia properties of both the mast and the guy wires can fully simulate these potential dynamic interactions. Another numerical study by Argyris and Mlejnek [11] on a 152.5-m transmitter tower with two stay levels, used an idealized sinusoidal horizontal accelerogram (amplitude of 0.5 g and period of 1 s) as a rough simulation of an earthquake. Large tower displacements (1.5 m) were calculated, probably an indication of nearresonance of the input with the fundamental flexural mode of the structure. The International Association for Shell and Spatial Structures (IASS [13]) presents general recommendations for seismic analysis of guyed masts, in a special
351
report published by its Working Group 4 on masts and towers. This report suggests that an equivalent static lateral load proportional to the tower weight may be used to model earthquake effects, as it is done in most building design codes. Designers are then advised to use their national standards for more specific guidelines on dynamic amplification factors and force distribution: the problem is that such guidelines simply do not exist in North American tower standards. The report also recommends several simplifying assumptions in order to linearize the analysis and use modal superposition. However, it acknowledges that very tall guyed masts or unusual towers may exhibit significant geometric nonlinearities. A comprehensive study of the behaviour of tall guyed towers under seismic excitation is required in order to develop more complete seismic analysis and design guidelines than summarized above. Reliable nonlinear dynamic analysis software is commercially available, and the focus can now shift from the discussion of modelling considerations to the structural response itself.
3. Methodology This study relies entirely on ‘‘numerical experiments’’, i.e. detailed full-scale simulations using the finite element method. Why? first of all, experimental results relating to the overall seismic behaviour of guyed telecommunication towers are very rare (not to mention that they are inexistent). Also, since tall guyed towers exhibit significant geometric nonlinearities, experimental studies of scaled-down physical models are very complex and of limited usefulness. Furthermore, experimental work on full-scale tall guyed towers is not feasible at present. A numerical modelling study of eight existing tall guyed telecommunication towers subjected to three different seismic excitations has been carried out. The tower models are very detailed to include geometric nonlinearities and to allow for potential dynamic interaction between the mast and guy wires. The selection of the towers was made to cover a wide range of tower heights, from 150 to 607 m, in order to identify some common trends in behaviour.
4. Description of towers 4.1. Tower geometry In practice, guyed towers taller than 150 m usually provide economical solutions comparing to self-supporting towers. Therefore, the lower height limitation of 150 m is a common criterion to classify towers with
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Table 1 Geometry of guyed towers analysed Tower height (m)
No. of guying stay levels
No. of anchor groups
No. of outriggers
Panel width (m)
Panel height (m)
Location (source)
607.1
9
3
0
3
2.25
342.2 313.9 213.4 200
7 5 7 8
2 2 2 3
1 0 0 0
2 2.14 1.52 1.8
1.52 1.52 1.52 1
198.1
6
2
1
2.13
1.52
152.4
8
2
2
0.84
0.61
150
7
2
3
1.3
1
U.S.A., California, Sacramento (LeBLANC & Royle Telcom Inc.) Canada (Wahba et al. [10]) Canada (Wahba et al. [10]) Canada (Wahba et al. [10]) Argentina, Buenos Aires (Estudio Ing. M. Oberlander) Canada, Prince Edward Island, Charlottetown (Trylon Manufacturing Co. Ltd.) Canada, Alberta, Elk River (AGT, LeBLANC & Royle Telcom Inc.) Canada, Alberta, Little Buffalo (LeBLANC & Royle Telcom Inc., AGT)
Note: All towers have a mast of triangular cross-section, except that of the 200-m tower which is square.
respect to their heights. In this regard, those available data for guyed towers taller than 150 m were selected for the simulations. These towers are listed in Table 1. It should be mentioned that the 607-m tower, located in Sacramento, California, is one of the tallest guyed telecommunication towers in North America. The 200-m tower is located in Buenos Aires, Argentina, and the other six towers are located in Canada. All masts have a triangular cross-section, except for the 200-m tower which is square. 4.2. Weights of mast and cables The detailed weights of mast and cables for the eight guyed towers studied are listed in Table 2. It is observed that the mast accounts for 69–77% of the total tower weight, leaving 23–31% to the guy cables. It should be noted that tower attachments such as antennae and accessories (e.g. ladders, platforms, transmission lines, lights for aircraft warning, etc.) are not included in the total weights. The mass distribution along the mast for the eight towers is almost uniform with average values ranging from 60 kg/m (152-m tower) to 600 kg/m (607-m tower). Since the antennae and ancillary components are not modelled, the only discontinuities in the mass profile are due to the presence of torsional resistors (outriggers) in the 150-m and 152-m towers.
Table 2 Weights of masts and cables Tower height (m)
Structural component
Weight (kN)
% of total weight
607
Cable Mast Total
1595 3578 5173
31 69 100
342
Cable Mast Total
287 754 1041
28 72 100
313
Cable Mast Total
306 1033 1339
23 77 100
213
Cable Mast Total
91 205 296
31 69 100
200
Cable Mast Total
95 249 344
28 72 100
198
Cable Mast Total
180 535 715
25 75 100
152
Cable Mast Total
36 91 127
28 72 100
150
Cable Mast Total
105 229 334
31 69 100
4.3. Initial tensions of guy clusters The initial average cable tensile stress prescribed by design in each guy cluster is listed in Table 3 for all eight
towers. It represents the cable tension in the equilibrium configuration of the tower under its self weight and the
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Table 3 Initial tension of guy clusters Tower height (m)
Set no. 1
2
3
4
5
6
7
8
9
607
I.T. (I.T./UTS)%
130 12
108 10
108 10
108 10
108 10
108 10
108 10
108 10
97 9.0
Anchor I.T. (I.T./UTS)%
Inner 159 15
91 8.4
100 9.3
Intermediate 100 103 9.3 9.5
97 9.0
Outer 97 9.0
Anchor
Inner
I.T. (I.T./UTS)%
129 12
Anchor
Inner
I.T. (I.T./UTS)%
129 12
127 12
127 12
129 12
Anchor
Inner
I.T. (I.T./UTS)%
62 5.7
43 4.0
44 4.1
47 4.4
35 3.2
Anchor
Inner
I.T. (I.T./UTS)%
105 9.7
Anchor
Inner
I.T. (I.T./UTS)%
87 8.1
91 8.4
90 8.3
Anchor
Inner
I.T. (I.T./UTS)%
108 10
Anchor
Inner
342
313
213
200
198
152
150
Outer 108 10
108 10
85 7.9
86 7.9
Outer 129 12
129 12
129 12 Outer
51 4.7
51 4.7
43 4.0
Intermediate 107 9.9
107 9.9
Outer
107 9.9
108 10
108 10
91 8.4
91 8.4
Outer 87 8.1
92 8.5
92 8.5
Outer 105 9.7
106 9.8
105 9.7
108 10
108 10
108 10
Outer
I.T. ¼ initial cable tensile stress of guy cluster (MPa); UTS ¼ ultimate tensile strength ¼ 1080 MPa.
effect of the prestressing forces in the guy wires. This tension is usually expressed as a percentage of the cable ultimate tensile strength (UTS), and in most cases the initial tension varies from 8% to 12% of the UTS. The only exception is the 200-m tower with initial cable tensions as low as 3–6% of the UTS: this will be found to have an important influence on its seismic response.
4.4. Equivalent lateral stiffness of guy clusters Data on initial cable tensions is used to calculate the equivalent lateral stiffness of each guy cluster. This calculation is not necessary to perform dynamic analyses, as the stiffness contributions of each element are as-
sembled automatically by the analysis software, but it will prove useful when discussing the results. The equivalent lateral stiffness of each guy cluster, k, expressed in kN/m, is listed in Table 4. It was obtained from nonlinear static analysis in the displaced tower configuration. Since the variation of k was expected to be nonlinear with respect to the amplitude of the horizontal displacement of the anchor point on the tower, the lateral stiffness was evaluated for a range of lateral displacements of the mast varying from 10% to 100% of the maximum displacement observed in seismic analyses. However, the calculations have shown no significant variations (in general, variations are in the range of 10– 15%) and the average value is taken to represent the lateral stiffness of each guy cluster.
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Table 4 Lateral stiffness of guy clusters (kN/m) Tower height (m)
Set no. 1
2
3
4
5
6
7
8
9
607
k0 k (k=k0 )%
1886 1054 56
2006 855 43
1166 509 44
610 79 13
590 81 14
664 98 15
338 23 7
558 40 7
290 17 6
Anchor k0 k (k=k0 )%
Inner 482 386 80
620 115a 19
350 72 21
Intermediate 348 320 86 56 25 18
290 44 15
Outer 218 36 17
Anchor
Inner
k0 k (k=k0 )%
1026 639 62
Anchor
Inner
k0 k (k=k0 )%
810 517 64
256 114 45
266 84 32
266 117 44
Anchor
Inner
k0 k (k=k0 )%
804 441 55
244 24 10
304 17 6
226 16 7
288 10 3
Anchor
Inner
k0 k (k=k0 )%
1438 821 57
Anchor
Inner
k0 k (k=k0 )%
272 188 69
244 97a 40
142 54 38
Anchor
Inner
k0 k (k=k0 )%
988 907 92
Anchor
Inner
342
313
213
200
198
152
150
a
Outer 906 410 45
470 216 46
810 125 15
356 54 15
Outer 492 330 67
312 209 67
278 126 45 Outer
602 302 50
374 49 13
348 31 9
Intermediate 1226 747 61
650 400 62
1096 499a 46
Outer 514 239 46
510 219 43
182 92 51
156 64 41
Outer 230 117 51
490 230a 47
188 94 50
Outer 1264 887a 70
700 480 69
584 399 68
560 284a 51
478 238 50
1030 406a 39
Outer
Outrigger location.
For comparison, the lateral stiffness of perfectly taut guy clusters (without sag), k0 , is also tabulated, which represents the stiffness of a straight rods. Since the guy wires are prestressed, all the cables in a cluster are assumed to contribute to the lateral stiffness according to k0 ¼ ð2AE cos2 aÞ=L
ð1Þ
where k0 is the lateral stiffness of taut guy cluster, A the cross-sectional area of cable, E the modulus of elasticity
of cable, a the angle of inclination of cable with horizon, L the length of cable. Eq. (1) applies only to guy clusters with three or four cables symmetrically arranged. Table 4 also includes the percentage ratio of k to k0 . This ratio indicates the relative degree of lateral support provided to the mast when sagging effects are accounted for, compared to the straight rod assumption. In general, the lateral stiffness provided by guy clusters tends to decrease with tower elevation, as longer cables tend to
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have more sag. The adjustment of the initial tension therefore plays an important role. Values presented in the table indicate that relatively large stiffnesses are achieved for the lower levels (k1 varies from 55% to 92% of k0 ) compared to the top levels, in particular for the 200-m and the 607-m towers for which the k=k0 ratio of the top cluster is only 3% and 6%, respectively. In the first case, this small lateral stiffness is due to a low initial tension (only 3.2% of the UTS––see Table 3), whereas in the tallest tower, this is mainly due to the large cable length (over 750 m) since the initial tension is of 9% UTS.
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The nonlinear dynamic analysis is done by direct step-by-step integration in the time domain, using the Newmark-b operator with parameters c ¼ 0:5 and b ¼ 0:25 (constant average acceleration method or trapezoidal rule). The Broyden–Fletcher–Goldfarb– Shanno (BFGS) stiffness update is used and stiffness matrix updates are performed at every time step. An energy-based convergence criterion is used to bound the iteration process. The subspace iteration procedure is used for the frequency analysis. More details about the finite element modelling aspects of the problem can be found in Amiri [2] and in the ADINA Users Manual (ADINA R&D, Inc. [9]).
5. Numerical modelling considerations 5.2. Earthquake accelerograms 5.1. Tower modelling assumptions The nonlinear finite element software automatic dynamic incremental nonlinear analysis (ADINA) is used in this research (ADINA R&D, Inc. [9]). The mast is a spatial structure with response in all three dimensions. The members making up the mast are rolled steel sections, and a detailed three-dimensional truss model is used where all elements are pin-ended. Previous work by Guevara and McClure [7] has shown that, for tall towers, there is little saving in computing effort when using the equivalent beam model for the mast. Furthermore, torsional behaviour is very difficult to model with equivalent properties as warping effects become important in long masts with nonsymmetrical bracing patterns. All members are assumed to remain in the linear elastic range of material behaviour. As the displacements and rotations of the mast may be large, a large kinematics formulation (with small strains) is necessary in order to account for potential geometric nonlinearities. Foundations are assumed perfectly rigid. Each guy cable is modelled as a linkage of ten prestressed truss elements. Three-node isoparametric elements are used with two Gauss integration points each, for a total of 20 sampling points along each cable. A large kinematics formulation is also used to account for full geometric nonlinearities. A tension-only linear elastic stress–strain law is defined to model cable slackening effects. A lumped mass formulation is used for both the cable elements and the individual truss members. Added mass due to tower attachments is not included in this study, but it could easily be inputted in the form of concentrated masses at tributary nodes. Structural damping is modelled by using an equivalent viscous damper with a value of 2% of critical viscous damping in parallel with each element of the mast and cables. Earthquake loads are assumed to occur under still air conditions (IASS [13]) so aerodynamic damping has not been modelled.
In this study, three classical horizontal earthquake accelerograms have been selected, representing different types of earthquake loading. The first one is the S00E 1940 El Centro earthquake containing a wide range of frequencies and several episodes of strong ground motion; the second one is the N65E 1966 Parkfield earthquake representing a single pulse loading with dominant lower frequencies; and the third one is the S69E 1952 Taft earthquake with high frequency content and strong shaking with long duration. These earthquakes are selected to reflect realistic frequency contents as exhibited by real ground motions. The earthquake direction, indicated in Fig. 1, was selected to coincide with a principal direction of the mast cross-section so as to create maximum lateral effects without any torsional response. The earthquake records were scaled to fit as much as possible the elastic design spectrum of the 1995 National Building Code of Canada [4] for the region of Victoria, British Columbia. This region has one of the highest seismicity levels in Canada with a peak horizontal ground acceleration (PGA) of 0.34 g and a peak horizontal ground velocity (PGV) of 0.29 m/s. It is understood that none of the towers studied is actually located in Victoria, but the scaling allows the comparison of the response of the towers for different accelerograms. Schiff’s scaling procedure (Schiff [12]) is used and values of the scaling factors are given in Amiri [2] along with the plot of each earthquake record. Unfortunately, no realistic data was available on the vertical accelerograms corresponding to the horizontal ground motions selected. Referring to the National Building Code of Canada 1995––Commentary J (NRC [4]), and considering that the ratio of vertical-to-horizontal accelerations depends on site conditions and varies widely, a value of 3/4 is used. This means that the vertical accelerograms are assumed perfectly synchronous with the horizontal ones, and that the intensity of the acceleration is uniformly scaled down by a factor of 3/4.
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6. Results of dynamic analysis 6.1. Frequency analysis A frequency analysis in the deformed configuration under self weight and cable prestressing forces has been carried out for each tower. The period calculated for the fundamental flexural mode is plotted in Fig. 2 as a function of tower height. The graph shows an almost linear increase of the first flexural period of the towers with height, the only exception being the 200-m tower, which is very flexible with respect to the other comparable towers due to its slacker guy cables (much less than 10% of their UTS). Fundamental periods vary from 0.6 to 4.3 s (or 0.2 to 1.7 Hz in terms of frequency), which already suggests a wide range of seismic response. Most towers have their lowest four lateral modes in the frequency range of 0.5–3 Hz. The following formula (essentially a curve fit) is suggested to get an estimate of the fundamental flexural natural period of guyed towers with cable initial tensions in the range of 10% of their UTS: T ¼ 0:0083H 0:74
ð2Þ
where T is the fundamental period in s and H is the tower height in m. This formula is restricted to steel lattice guyed masts of triangular cross-sections with heights in the range of 150–607 m, but due to the lack of data for towers above 350 m, it is cautious to limit its use to the 150–350 m range.
Another important observation obtained from the frequency analysis is the shape of the fundamental flexural mode. Since the mass of the masts is more or less uniformly distributed along the height, it is the location of the stay levels and the relative lateral stiffness of the guy clusters that will affect most the lateral mode shapes. Table 4 indicates that for most of the towers (150-m, 152-m, 198-m, 213-m and 342-m towers), one of the guy clusters close to the top region is stiffer than the other intermediate ones, probably as a result of design serviceability limitations in the vicinity of antennae. For these masts, the top part behaves more or less like a pinned support, whereas the large stiffness of the bottom clusters mimics a fixed support. Therefore, the lowest flexural mode of the mast is similar to the second lowest frequency mode of a cantilever structure. However, the other three towers (200-m, 313-m and 607-m towers) have guy clusters with lateral flexibility increasing with tower height: as a result, their fundamental flexural mode is similar to the lowest frequency mode of a cantilever structure. Two examples of these dominant mode shape configurations will be discussed later in conjunction with a simple conceptual model to explain lateral earthquake effects, in conjunction with Figs. 6 and 7. 6.2. Sesimic sensitivity indicators Results from the detailed time history analyses were studied in order to detect common trends in behaviour among the various towers and under the various inputs.
Fig. 2. Natural period of fundamental flexural mode versus tower height.
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Although frequency content and time at peak response were also considered, the emphasis was put on the envelope curves of the response, i.e. on the maximum amplitude of seismic effects. The following nine response indicators were selected for a refined analysis of the results: (1) the resultant cable reaction force on the mast at each stay level; (2) the dynamic component of cable tension in the most critical cable of each guy cluster; (3) the horizontal shear force along the mast; (4) the dynamic component of the axial force along the mast; (5) the bending moment along the mast; (6) the dynamic component of the cable oscillation; (7) the lateral displacement along the mast; (8) the dynamic component of the axial displacement of the mast; and (9) the tilting rotation along the mast. All of them are important in design: the first five relate to strength and stability whereas the last four relate to serviceability considerations. Some seismic sensitivity indicators are now proposed in terms of base shear, axial reaction in the mast, and amplification of cable tension. 6.2.1. Base shear Fig. 3 shows the maximum base shear calculated for the eight towers subjected to the three earthquake accelerograms. The base shears were obtained by the summation of all horizontal reactions at the ground supports of cables and mast. Contributions to the base shear are presented separately for the cables and the
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total tower, and they are expressed as a percentage of tower weight. For towers shorter than 200 m, the total base shear varies in the range of 40–80% of their total weight, while this range is narrowed to 15–30% of the total weight for towers taller than 300 m. The relative contribution of the inertia effects of the cables to the total base shear is almost constant, in the range of 3–5% of the total weight of tower. It is interesting to note that for towers shorter than 200 m, although the cables make up 25–30% of the total mass (see Table 2) only 5– 10% of the total base shear is contributed by inertia effects in the cables. For the towers taller than 300 m, the relative contribution of the cables compared to that of the mast increases slightly with height. It is observed that increased cable-mast interactions are generated for the 607-m tower, especially under the El Centro earthquake. The total base shear being one of the most important response indicators for earthquake-resistant design purposes, the maximum base shear of the towers for the three earthquake excitations is investigated more precisely. Fig. 4 shows the percentage ratio of the maximum base shear to the total tower weight versus the tower height. A curve fit is suggested to predict the maximum base shear for different tower heights in the range of 150–607 m. The following equation is suggested which has a correlation factor of 0.95 for the eight results available: BS ¼ 28 300H 1:17 ð% of W Þ
Fig. 3. Base shear versus tower height.
ð3Þ
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Fig. 4. Curve fit for maximum base shear versus tower height.
where BS is the maximum base shear as a percentage of total tower weight, W , and H is the tower height in m. It is important to recall that this formula is restricted to sensitivity levels comparable to the Victoria region in Canada. It is also cautious to limit its use to the 150–350 m range due to the lack of data for taller towers. In conclusion, from the viewpoint of total base shear, guyed towers in the 150–350 m range may be sensitive to seismic effects, but since the amount of base shear can be predicted using Eq. (3) (for the seismicity level studied here), a detailed modelling study is not necessary. At the other end of the spectrum, towers taller than 350 m will have relatively low natural frequencies in their lowest modes, which will likely be matched by the dominant frequency content of the ground accelerograms. As a result, important dynamic amplifications will occur. At this stage, considering the lack of data for towers in that range of heights, a detailed dynamic analysis is recommended. 6.2.2. Mast axial force at the base The maximum dynamic component of the axial force at the base of the mast is reported in Fig. 5 as a percentage of the total weight of each tower, and the relative contributions of the inertia effects of the cables are plotted separately. Since the axial effects are more important when the vertical component of the earthquake is also considered, these results correspond to the case of combined vertical and horizontal accelerations.
As shown in Fig. 5, the relative contribution of the cable mass to the total dynamic component of the axial force is small, in the range of 4–10% of the total weights: its maximum envelope can be taken as 10% of total weight for all the towers except for the 200-m which is 15%. The dynamic component of the axial force at the base of the mast varies widely with the three accelerograms studied, from 25% to 125% of the tower weight. Typically, more response is obtained whenever there is frequency coincidence between the lowest axial modes of the tower and the dominant frequencies of the accelerogram. The Taft accelerogram is richer in high frequencies (above 4 Hz) than El Centro and Parkfield, which amplifies the response in the axial modes. The behaviour of the 607-m tower appears to contradict that analysis, but in this case, the lowest axial modes have lower frequencies (below 4 Hz) than for the other towers, and the El Centro accelerogram causes the maximum amplification. Fig. 5 also shows that the 200-m tower is very sensitive axially (especially under Taft) compared to other towers of similar heights. As already discussed in terms of flexural behaviour, this tower is exceptional because of the slackness of its guying system, and its dynamic sensitivity was confirmed in almost all of the response indicators studied. As an upper bond estimate, one could use an axial force at the base of the mast in the order of 80% of the tower weight (with the exception of the 200-m tower). However, realistic vertical accelerograms were not used,
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Fig. 5. Dynamic axial force at the base of the mast versus tower height (under combined horizontal and vertical accelerations).
which would normally show higher dominant frequencies than the horizontal accelerograms and would likely shift the sensitive frequency range in the vicinity of the lowest axial modes of many towers (4–10 Hz for the 150–350 m range). For this reason, we believe that the numerical values obtained in the analysis cannot lead to general quantitative conclusions. 6.2.3. Seismic amplification factor of cable tension Dynamic cable tensions can also be an important indicator of the severity of the dynamic effects in a guyed mast. In this study, the results were calculated for two load cases: (1) horizontal earthquake (H) and (2) combined horizontal and vertical earthquake (H þ V) accelerograms. The maximum dynamic component of cable tension in each guy cluster was recorded for all eight towers and three earthquake accelerograms. Table 5 summarizes the results for the clusters with largest dynamic effects among those connected to the outer ground anchors (i.e. the upper clusters) and to the inner ground anchors (i.e. the lower clusters). For convenience in discussing the results, a seismic amplification factor is defined as the percentage ratio of the dynamic component of cable tension to the initial tension. Overall, this seismic amplificator factor varies between 30% (upper clusters of the 607-m tower) to 290% (upper clusters of the 200-m tower). More typical values for the 150–350 m range are between 50% (upper clusters of the 150-m tower) and 200% (lower clusters of the 152-m tower). It is interesting to note that the 607-m
tower does not experience much dynamic effects in its cables compared to the other ones, whereas, not surprisingly, the 200-m tower shows the most sensitivity because of its slacker guying system. It is also observed that for all towers the combination of horizontal and vertical accelerograms (H þ V loading) does not significantly increase the dynamic cable tensions, compared to the horizontal loading case. As a matter of fact, this combined loading even tends to reduce the amplifications in some towers. In general, for all towers, the lower clusters are subjected to larger amplifications than the upper ones. 6.3. Distribution of earthquake effects with tower elevation 6.3.1. Lateral earthquake force The lateral earthquake force is defined here as the resultant horizontal force generated by an earthquake on the mast at the cable attachment points. As discussed earlier, the contribution of the mass of the cables to the total base shear (Fig. 3) and to the seismic component of the mast axial force at the base (Fig. 5) is almost constant and is small compared to that of the mast. Therefore, only the inertia of the mast has a major effect on the lateral earthquake forces. A simplified conceptual model is proposed to explain the lateral force distribution along tower height, which makes use of three important tower characteristics: (1) the predominant mode shape of mast; (2) its mass
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Table 5 Dynamic amplification of cable tensions Tower height (m)
Loading case
607
H HþV H HþV H HþV H HþV H HþV H HþV H HþV H HþV
342 313 213 200 198 152 150
Amplification factor ¼ (maximum tension/initial tension) 100 Outer ground anchor
Inner ground anchor
El Centro
Parkfield
Taft
El Centro
Parkfield
Taft
66 51 64 78 49 44 58 55 161 195 87 91 120 119 90 93
84 46 66 70 48 46 83 89 154 289 87 91 108 96 105 100
32 35 70 71 62 77 90 98 267 159 127 121 54 62 50 57
86 92 148 151 83 83 110 111 171 174 137 138 177 178 71 73
78 78 80 86 77 83 113 114 172 170 138 142 188 184 78 80
96 102 96 104 108 106 91 87 171 175 101 94 195 194 74 73
distribution; and (3) the presence of discontinuities in lateral stiffness. The predominant mode shape of the mast can be used to represent the horizontal acceleration profile along the tower height. This acceleration profile combined to the mass profile can represent the distribution of the horizontal inertia forces developed in the mast. These distributed inertia forces will result in lateral forces at the cable attachment points on the mast, in accordance with the relative lateral stiffness of the guy clusters. Two examples are selected to illustrate the concept, the first one is the 198-m tower (Fig. 6) and the second one is the 313-m tower (Fig. 7). Each figure is divided into three graphs showing (a) the mass profile, (b) the
fundamental lateral mode shape, and finally in (c) the percentage ratio of the lateral earthquake force to the total base shear. The position of the stay levels with reference to their ground anchor point is clearly identified on the ordinates of figures (a) and (c). The first example is representative of towers with a predominant mode shape similar to that of a propped cantilever (Fig. 6(b)) and the other is representative of towers with slacker upper clusters with a fundamental mode shape closer to that of a cantilever (Fig. 7(b)). It is noted that the lateral earthquake forces along the mast are an envelope of maximum amplitudes, and therefore their summation exceeds the total base shear. For most towers, the maximum lateral force varies in the
Fig. 6. Distribution of lateral earthquake forces in the 198-m tower.
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Fig. 7. Distribution of lateral earthquake forces in the 313-m tower.
range of 25–35% of the total base shear. Although the detailed results are not shown here, it is interesting to report that the 200-m tower, with its slack guying system, has a more or less uniform distribution of lateral forces at cable attachment points, of magnitude around 10% of the base shear. The distribution of the lateral earthquake forces generated by the cables on the mast cannot be explained only by the mass profile of the mast and its fundamental mode shape, as is the case for lateral effects in common lightly damped buildings. The nonuniform distribution of the lateral stiffness provided by the guying system (already discussed in conjunction with Table 4) also plays an important role. The largest variations in lateral
stiffness occur when there is a change of ground anchor point: this is due to a change in cable length and sometimes also a change in cable properties and/or initial tension. As a result, the portion of the mast in between or adjacent to these transition points becomes particularly sensitive to dynamic lateral effects, as will be discussed later. 6.3.2. Axial forces The distribution of the maximum dynamic component of the axial forces in the mast along the height for all eight towers is summarized in Fig. 8, for combined horizontal and vertical earthquake motions. The horizontal axis represents the percentage ratio of the
Fig. 8. Distribution of dynamic axial force in the mast along tower height (under combined horizontal and vertical accelerations).
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maximum dynamic component of mast axial forces to the maximum dynamic component of mast base axial force, and the vertical axis is the ratio of the sectional elevation of mast to the total tower height, h=H . It can be seen that the results of all the towers are relatively close to each other and form a narrow band. They can conservatively be represented by a parabolic curve fit (bold line in Fig. 8) with the following equation: ðPdyn =BAÞ ¼ 100–95ðh=H Þ2
ðin %Þ
ð4Þ
where ðPdyn =BAÞ is the percentage ratio of the maximum dynamic component of axial force in the mast at a section of elevation h to the maximum dynamic component of the axial force at the base of the mast. 6.3.3. Maximum shear and bending moment Table 6 summarizes results obtained for the maximum mast shear and bending moment for the eight towers studied. For comparison purposes, these values have been normalized as follows. The maximum mast shear is divided by the total weight of the towers, and the maximum mast bending moment is divided by the product of the panel width and the total weight. We find it reasonable to use the panel width to normalize the moment since the bending moment in the mast is in direct relation with the lever arm provided by the
Table 6 Maximum shear and bending moment of mast Tower height (m)
Maximum mast shear % (Total weight)
Maximum mast moment % (Panel width Total weight)
607 342 313 213 200 198 152 150
2.5 3.7 4.7 6.3 5.8 6.7 6.6 6.8
14 25 35 42 36 39 48 40
spacing between the tower leg members. Results indicate that in the lower range of tower heights (150–213 m), the maximum shear varies around 6–7% of tower weight and the ratio of maximum bending moment to the product of panel width and total weight varies from 36% to 48%. In the upper range of tower heights (313–607 m), the effects are slightly smaller in percentage (but not in absolute values) with the maximum shear in the range of 2.5–5% of tower weight and the corresponding bending moment ratio in the range of 14–35%. Although the envelope curves of shear and bending moments are not shown here, it was confirmed that the maximum values of mast shear occur directly at stay levels and the minimum shear occurs at midspan between two stay levels, and vice versa for the mast bending moment. 6.3.4. Serviceability considerations Antenna-supporting towers must meet strict serviceability criteria that are established by their owners in view of the particular use of the tower. Seismic amplifications of displacements and rotations may affect the top part of the tower where the antennas are attached, but they should not result in any local permanent deformation after the earthquake. Such deformations may yield to a loss of serviceability resulting in unacceptable signal attenuation. Table 7 summarizes the results obtained for maximum tower displacements and rotations. The maximum horizontal displacements are given both in absolute value and as a percentage ratio of the tower height, in columns (2) and (3) respectively. The lateral displacements in the earthquake direction are small, in the range of 0.05–0.12% of the tower height, which confirms that in spite of being slender, the towers are not very flexible. The maximum flexural rotation (tilting) of the top of the mast, listed in column (4), is below 0.4 for each tower, which should ensure the serviceability of most reflector antennae. The maximum dynamic component of cable oscillation, i.e. the amplitude of the transverse cable vibrations, is listed in column (5). Amplitudes between 0.3 and 1.3 m are calculated, which are small compared to the corresponding tower heights.
Table 7 Serviceability response indicators Tower height (m)
Maximum horizontal displacement (m)
Maximum horizontal displacement % (Tower height)
Maximum rotation (deg)
Maximum dynamic component of cable oscillation (m)
607 342 313 213 200 198 152 150
0.31 0.26 0.23 0.21 0.22 0.23 0.14 0.16
0.05 0.08 0.07 0.10 0.11 0.12 0.09 0.11
0.27 0.28 0.29 0.34 0.31 0.37 0.36 0.33
1.01 0.80 1.31 0.35 0.48 0.43 0.31 0.31
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Finally, the dynamic component of the maximum axial displacement of the mast was found negligible in most towers, with values inferior to a few centimetres. In general, then, for the seismicity level considered in the study, all the towers appear to behave within reasonable serviceability limits.
7. Conclusions 7.1. Summary of important parameters of the study Before stating any conclusions, it is important to recall some of the important parameters of the study which have a direct influence on the numerical results. • The seismicity level considered is constant with PGA ¼ 0.34 g and PGV ¼ 0.29 m/s. This corresponds to Victoria, British Columbia, which is one of the most severe seismic regions in Canada. • The vertical accelerograms are identical to the horizontal ones, but multiplied by 3/4. They are not representative of realistic vertical accelerograms which typically exhibit higher frequency content than horizontal ground motions. • The heights of the guyed towers studied in this research vary in the range of 150–607 m. However, since only one tower is modelled with a height above 350 m, it is cautious not to extrapolate results beyond that limit. • Most of these towers are located in Canada and were designed for both wind and ice loads, which necessarily influences their mass and stiffness characteristics. 7.2. Summary of the proposed seismic sensitivity indicators The following seismic sensitivity indicators are proposed to assess whether a particular tower is sensitive to earthquake effects, and if so, whether a detailed nonlinear dynamic analysis is necessary: Fundamental natural period of the tower: Eq. (2) is proposed in the 150–350 m range of tower heights. A simple check can be made with the earthquake design spectrum to assess whether or not lateral effects will be important. Values corresponding to 0.5–3 Hz are considered in the sensitive range for horizontal response. Base shear: For towers with heights ranging from 150 to 350 m, important base shears may develop, in the order of 40–80% of tower weight. The magnitude of the base shear can be predicted using Eq. (3) and a detailed nonlinear dynamic analysis could be avoided. For taller towers, the relative contributions of the inertia effects in the cables and the mast are unpredictable, and it is not possible to suggest a simple estimator. Therefore, until
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more data and knowledge is available, a detailed dynamic analysis is recommended for towers higher than 350 m. Axial forces in the mast: When seismic vertical effects are considered, guyed towers with slack cables, i.e. with initial cable tension below 5% UTS, are sensitive to the combination of vertical and horizontal earthquake motions. Since their behaviour is unpredictable, a detailed nonlinear dynamic analysis is recommended. For guyed towers with usual initial cable tension (i.e. around 10% UTS or more), the maximum dynamic component of the axial force at the base of the mast due to combined vertical and horizontal earthquake motions is about 80% of their total weight. A detailed numerical study is not necessary unless there is some detailed information available for the vertical accelerograms which suggests that their frequency content is much higher than the horizontal accelerograms. The variation of the maximum axial force in the mast can be predicted using Eq. (4). Cable tensions: The dynamic component of cable tensions can vary by an order of magnitude, from 30% to 300% of the initial tension. Typical values for the 150–350 m range of tower heights are between 50% and 200%. 7.3. Summary of trends in tower behaviour • A simplified conceptual model for the distribution of earthquake forces along the height of the towers has been proposed which accounts for the fundamental mode shape of the mast, its mass distribution, and the relative lateral stiffness provided by the guy clusters. • Discontinuities in behaviour are expected in portions of the mast between or close to adjacent guy cable clusters with significantly different lateral stiffnesses. Most response indicators were found maximum in these sensitive sections of the mast. • Inertia effects in the guy cables contribute little to lateral seismic effects in the mast, and can be neglected. However, when vertical accelerations are studied, dynamic interactions occur between the cables and the mast: these interactions are difficult to predict and detailed analysis is recommended for very tall towers (above 350 m) or towers with relatively slack guy clusters. • All the towers studied behaved within serviceability limits.
Acknowledgements The following engineers have provided detailed tower data for this study, and their contribution is greatly
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appreciated: Mr. Donald G. Marshall, Special Consultant for LeBlanc & Royle Telcom Inc., Oakville, Ontario; Mr. M. Oberlander, Estudio Ing. M. Oberlander, Buenos Aires, Argentina; Mr. K. Penfold of Trylon Manufacturing Co. Ltd., Elmira, Ontario; and K.R. Jawanda of AGT Limited, Edmonton, Alberta. Financial supports from the Ministry of Science, Research & Technology of Iran and the Natural Sciences and Engineering Research Council of Canada are acknowledged.
[6]
[7]
[8]
[9]
References [10] [2] Amiri GG. Seismic sensitivity of tall guyed telecommunication towers. PhD thesis. Department of Civil Engineering and Applied Mechanics, McGill University; Canada: 1997. [3] Moossavi Nejad SE. Dynamic response of guyed masts to strong motion earthquake. Proc 11th World Conf on Earthquake Eng. Acapulco: Mexico; 23–28 June 1996, Paper #289. [4] NRC. National building code of Canada, National Research Council of Canada, 11th ed. Canada, Ottawa: 1995. [5] McClure G, Guevara EI. Seismic behaviour of tall guyed telecommunication towers. Proc IASS-ASCE International
[11] [12]
[13]
Symposium 1994 on Spatial, Lattice and Tension Structures. Georgia, Atlanta: 24–28 April 1994. p. 259–268. Guevara EI. Nonlinear seismic analysis of antenna-supporting structures. M. Eng Project Report no. G93-14. Department of Civil Engineering and Applied Mechanics, McGill University; Canada: 1993. Guevara EI, McClure G. Nonlinear seismic response of antenna-supporting structures. Comput Struct 1993;47(4/ 5):711–24. McClure G, Guevara EI, Lin N. Dynamic analysis of antenna-supporting structures. Proc CSCE Annual Conf New Brunswick, Canada 1993;II:335–44. ADINA R&D, Inc. ADINA Reference Manuals; Reports ARD 4, 7, 8, 10. 71 Elton Avenue, Watertown, MA 02172, ADINA R&D, Inc., USA. Wahba YMF, Madugula MKS, Monforton GR. Limit states design of antenna towers. Final report submitted to the CSA technical committee S37 on antenna towers. Department of Civil and Environmental Engineering. University of Windsor; Ontario, Canada: 1992. Argyris J, Mlejnek HP. Dynamics of structures. New York: Texts on Computational Mechanics 1991;5:505–10. Schiff SD. Seismic design studies of low-rise steel frames. PhD thesis. Department of Civil Engineering, University of Illinois at Urbana-Champaign, 1988. IASS. Recommendations for guyed masts. Madrid, Spain: International Association for Shell and Spatial Structures; Working Group no. 4, 1981.