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Three-dimensional analysis of the seismic response of guyed masts
Engineering Structures 29 (2007) 2254–2261 www.elsevier.com/locate/engstruct
Three-dimensional analysis of the seismic response of guyed masts Gregory M. Hensley, Raymond H. Plaut ∗ Charles E. Via Jr. Department of Civil and Environmental Engineering, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA Received 17 November 2005; received in revised form 29 September 2006; accepted 6 November 2006 Available online 28 December 2006
Abstract The response of a 120 m-tall guyed mast to three-dimensional seismic excitation is analyzed using the finite-element program ABAQUS. The Northridge earthquake and an amplified El Centro earthquake are applied. The mast is pinned at its base, and guy cables are attached at four equally spaced points along its height. The dynamic tension in the guys is modeled by a nonlinear function based on tests, and periods of slackness in the guys during motion of the system lead to snap tension loads and affect the response. The mast is represented by three-dimensional beam elements. Displacements, bending moments, and base shears are computed for the mast, along with guy tensions. The effects of guy stiffness and pre-tension, mast weight, and directionality of the ground motion are investigated. c 2006 Elsevier Ltd. All rights reserved.
Keywords: Finite element modeling; Guyed mast; Seismic response
1. Introduction A steel mast that is pinned at its base and stabilized by guy cables is considered. The mast has four guy levels (the quarter points and the top), with three cables at each level oriented in vertical planes separated by 120◦ . The mast is subjected to excitations of the Northridge and amplified El Centro earthquakes, including the vertical component and two orthogonal horizontal components of the ground motion. The finite-element program ABAQUS [1] is utilized to obtain time histories of the response. During the motion, some cables become slack and then taut again, leading to snap loads. The constitutive law for the tension in the cables is nonlinear and is based on vibration tests. The self-weight of the mast is one of the factors that affects the response. Madugula [2] presented an overview of work on the dynamic response of guyed masts. Free vibrations have been analyzed in [3] and other studies. Much attention has been focused on the motions of guyed masts caused by wind (e.g., [4–22]). Controlling the vibrations of a guyed mast in wind was the topic of [23,24]. Failures of guyed towers due to ice and/or wind are listed in [25,26]. Buchholdt et al. [27] and Ben Kahla [28,29] examined the response of a guyed mast following the sudden ∗ Corresponding author. Tel.: +1 540 231 6072; fax: +1 540 231 7532.
E-mail address: [email protected] (R.H. Plaut). c 2006 Elsevier Ltd. All rights reserved. 0141-0296/$ - see front matter doi:10.1016/j.engstruct.2006.11.019
rupture of one of the cables. In an example in [30], a guyed mast was subjected to horizontal, uni-directional, simple-harmonic motion at its base. A few researchers have investigated the seismic response of guyed masts. Most of these studies applied the finiteelement method to obtain numerical solutions. Guevara and McClure [31] considered a 24 m-tall mast with two guy levels, and a 107 m-tall mast with six guy levels. A scaled horizontal component from the El Centro earthquake and one from the Parkfield earthquake were applied. The time delay between the excitations at the different anchorage points of the guys and the mast was included in part of the study. Time histories of guy tensions, shear forces, vertical forces, and displacements were presented. Nejad [32] considered a 327 m-tall mast with five guy levels. All three components of seismic ground motion were included. Time histories of displacements and frequency spectra of two displacements and one guy tension were shown, along with lists of maximum displacement, shear force, axial force, and bending moment of the mast. Amiri [33] applied three-dimensional components of the El Centro, Parkfield, and Taft earthquakes to eight guyed telecommunication masts with heights varying from 150 m to 607 m. The base shear, vertical force in the mast, and guy tensions were examined. Finally, Amiri et al. [34] analyzed a 342 m-tall mast with seven guy levels, and a 607 m-tall
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mast with nine levels. A single horizontal component from the 1978 Tabas earthquake in Iran was applied. The study included the spatial effect of the distances between the anchorage points (‘multiple-support excitation’), as previously considered in [31]. Time histories of lateral tip displacement, axial force in the mast, cable and mast base shear, and cable tension were presented. The system to be analyzed here is described in Section 2. Free vibrations are considered in Section 3, and results for seismic response in a ‘standard case’ are presented in Section 4. Section 5 describes the effects of guy pre-tension, mast density, and earthquake direction, and Section 6 contains concluding remarks. 2. Description of mast, guys, and earthquake excitation The structure considered is similar to a mast that was analyzed by Irvine [3] and Desai and Punde [14]. The profile is shown in Fig. 1(a) and the plan view in Fig. 1(b). The mast is 120 m tall and has four guy levels separated by 30 m, with three guys per level, and two sets of guy anchors in each of the three planes. The mast is pinned at the base and is modeled as a beam-column using 12 cubic B33 elements in ABAQUS [1]. The mast’s weight per unit length is 588 N/m, its modulus of elasticity is 209 GPa, and its moment of inertia for bending in any direction is 0.0018 m4 . The constitutive law for the guy cables is based on dynamic tests of synthetic fiber ropes. Drop tests [35] and cyclic vibration tests [36] were conducted. The ropes stiffened nonlinearly as they extended, and the force was approximately proportional to the elongation raised to the power 1.3 (under conditions of no pre-tension). Here the ‘equivalent’ linear value E A = 3 × 104 kN is used in the ‘standard case’. The ropes are under pre-tension to stabilize the mast. In the standard case, the pre-tension at equilibrium is 25 kN, and the axial force (when positive) is assumed to be 1.2(E A/L)(δ + δ0 )1.3 , where δ is the elongation from the equilibrium length L. At guy levels 1, 2, 3, and 4, respectively, δ0 = 0.066, 0.095, 0.137, and 0.161 m. In other words, the ropes are modeled as nonlinear springs which do not resist compression. The results could be applicable to guy cables constructed of other materials if their inertia and transverse vibrations do not greatly affect the response. The anchors of the guys are assumed to be rigid. The ground motion is assumed to be the same at the cable anchorage points and the base of the mast. The height of this structure is at the lower end of the spectrum for guyed masts, and thus it should not be particularly sensitive to multiple-support excitation. The components of the Northridge earthquake taken at the Sylmar County Medical Center (SYL) are applied here in the standard case. This ground motion record exhibits some nearfield effects that are not common in many other records. The strongest direction of recorded ground motion is SYL360, with a peak ground acceleration (PGA) of 0.843g. The motion in the orthogonal horizontal direction, SYL090, has a PGA of 0.604g, and the vertical record, SYL-UP, has a PGA of 0.535g. Data are taken from [37], which lists accelerations with time increments of 0.02 s, and 20 s of the earthquake record are used.
Fig. 1. (a) Profile view of guyed mast; (b) plan view of direction of strong horizontal component.
In addition to excitation by the Northridge earthquake, an amplified version of the 1940 El Centro earthquake will be considered in Section 5. The strongest recorded direction is ELC180, with a PGA of 0.313g. The other horizontal direction, ELC270, has a PGA of 0.215g, and the vertical record ELC-UP has a PGA of 0.205g. Again, data are taken from [37], now in time increments of 0.01 s over a time of 40 s. This event has more high-frequency content than the Northridge event, but less intensity. The El Centro accelerations are amplified by a factor of 1.5 and then applied to the guyed mast. For each earthquake, the recorded horizontal component with the higher range between maximum and minimum accelerations is denoted the ‘strong direction’. If the acceleration records in the two measured horizontal directions are considered and their locus is plotted in a plane, the direction with the largest range turns out to be close to this direction for both earthquakes [38]. The arrows in Fig. 1(b) indicate the strong direction of the earthquake, and direction A will be used in the standard case. The ground displacements during the first 15 s of motion of the Northridge earthquake are plotted in Fig. 2, beginning at the origin. Fig. 2(a) depicts the displacements in the horizontal plane, with SYL360 as the abscissa. Displacements in the vertical plane including this component are shown in Fig. 2(b), and those in the orthogonal vertical plane are presented in Fig. 2(c). In the horizontal plane, the direction with the largest
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29.3 rad/s, respectively, which closely match values obtained from linear beam theory using equivalent transverse springs for the guys [38]. The first three modes are sketched in Fig. 3, with numbers showing the locations of the guy levels. They are similar to the modes shown in [3] and [14]. For the seismic analysis, the first and fourth modes were used to determine the parameters for Rayleigh damping that would approach 2% of critical damping for the most relevant modes [39]. This yielded coefficients of 0.237 and 0.00140 of the mass and stiffness matrices, respectively, and the linear combination of these two scaled matrices formed the assumed damping matrix. 4. Standard case
Fig. 2. Ground displacements for Northridge earthquake: (a) horizontal plane; (b) vertical plane including SYL360; vertical plane including SYL090.
range of displacements is oriented approximately 20◦ from the strong direction for the Northridge earthquake (Fig. 2(a)), and approximately 90◦ from the strong direction for the El Centro earthquake [38]. 3. Free vibrations Small-amplitude free vibrations of the guyed mast were analyzed first. The lowest five natural frequencies for vibrations in the vertical plane of direction A (neglecting axial force in the mast) were computed to be 8.37, 10.5, 16.1, 20.3, and
Results for the standard case are presented first. As mentioned prevously, this involves the Northridge earthquake with its strong motion (SYL360) in direction A in Fig. 1(b), a pre-tension of 25 kN in the guys, an ‘equivalent’ linear value of E A = 3 × 104 kN for the guys, and a weight per unit length of 588 N/m for the mast. Horizontal displacements were determined at the base of the mast and at the guy levels (labeled 1–4 in Fig. 1(a)). The plots of displacements at the guy levels will present displacements relative to the ground motion (rather than absolute displacements), whereas the plots of base displacement will show the ground motion (since the relative displacement there is zero). Fig. 4(a) depicts the ground motion along direction A, and Fig. 4(b) and (c) depict the time histories of the relative displacements along that direction at levels 2 and 4, respectively. Fig. 5(a), (b), and (c) show the corresponding displacements along the orthogonal SYL090 direction. The displacements at the top of the mast (level 4) tend to be larger than those at the center of the mast (level 2). Of particular interest is the fact that after the first five seconds, the peaks of the level-2 and level-4 displacements along the strong-motion direction (Fig. 4(b) and (c)) are smaller than those along the orthogonal horizontal direction (Fig. 5(b) and (c)). Also, the motion in the orthogonal direction tends to be more regular and indicates that the first vibration mode dominates the response in that direction. In the SYL360 direction, the maximum peak displacement for all the guy levels occurs at level 4 (the top of the mast), but it occurs at level 3 in the orthogonal direction [38]. The bending moments are zero at the base and at level 4. At levels 1, 2, and 3, they demonstrate significant variability during the response, with the largest peak moment in direction A being about 150 kN m and occurring at level 2. In the orthogonal direction, the largest peak moment is approximately 100 kN m and occurs at level 1 [38]. Fig. 6 presents the time history of the tension in a guy attached at level 3 in a vertical plane at an angle of 120◦ with direction A. The plot covers the time from 3 s to 10 s after initiation of the earthquake record. Periods of slackness occur starting at around a time of 6.5 s, and the guy exhibits snap loads, i.e., a sudden rise in tension and then a return to slackness shortly afterward [40]. Following a time of 6.5 s, the
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Fig. 3. Lowest modes of free vibration: (a) first; (b) second; (c) third.
Fig. 4. Displacements in strong direction for standard case: (a) base; (b) level 2; (c) level 4.
guy attached at level 3 at an angle of −120◦ with direction A tends to be slack when the guy in Fig. 6 is taut, and
Fig. 5. Displacements in SYL090 direction for standard case: (a) base; (b) level 2; (c) level 4.
vice versa [38]. These snap loads, associated with slack–taut transitions, appear to cause a significant increase in some of the
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Fig. 6. Tension in cable at guy level 3 for standard case.
Fig. 7. Base shear for standard case.
Fig. 9. Sequence of mast shapes in strong earthquake direction for standard case between (a) 3.5 s and 6.0 s, (b) 7.0 s and 8.0 s.
accelerations), and in Fig. 9(b) they correspond to shapes in 0.1 s increments from 7.0 s to 8.0 s. The form of the mast changes significantly during the motion, sometimes being almost linear and often having one local relative minimum and one local relative maximum in the transverse displacement. The highest magnitude of the total displacement (ground motion plus deformation) in Fig. 9 is 0.5 m at a height of 100 m (curve 4 in Fig. 9(a)). The corresponding plots for projections of the mast in the orthogonal plane exhibit a maximum total displacement of 0.4 m [38]. Fig. 8. Vertical force in mast at level 2 for standard case.
guy-level displacements and bending moments in the SYL090 direction. The time history of the base shear of the mast in direction A is plotted in Fig. 7. The mast weighs 70.6 kN, and the maximum base shear in Fig. 7 is about 15 kN, i.e., a little over 20% of the weight. Fig. 8 shows the time history of the vertical compressive force in the mast at a location just below level 2. The static axial force at that point, due to the self-weight of the top half of the mast and the vertical components of the guys at levels 2, 3, and 4, is 149 kN. During motion, this force decreases to almost 100 kN and increases to almost 200 kN, i.e., fluctuations of approximately ±30% about the static value. It is interesting to observe the changing shape of the mast during the dynamic response. The projections of the shapes along direction A are plotted in Fig. 9. In Fig. 9(a) the numbers 1–11 at the tops of the shapes correspond, respectively, to the shapes in 0.25 s increments at times from 3.5 s to 6.0 s in the record (during the period with largest ground
5. Effects of pre-tension, mast density, and earthquake direction The effect of the pre-tension force in the guys is investigated first. In the figures, the strong earthquake components, SYL360 and ELC180, are oriented along direction A in Fig. 1(b). The equivalent linear value of E A for the guys was 3 × 104 kN in the standard case. Additional values of 2 × 104 and 4 × 104 kN are included now. The designations N-2, N-3, and N-4 denote the Northridge earthquake and pre-tensions 2 × 104 , 3 × 104 , and 4 × 104 kN, respectively, and E-2, E-3, and E-4 denote the corresponding cases for the amplified El Centro earthquake. Results were computed for guy pre-tension forces of 15, 20, 25 (the standard case), 30, and 35 kN. Fig. 10 shows absolute values of maximum relative displacement (i.e., deformation with respect to the ground motion) at the top of the mast (level 4). Fig. 10(a) presents displacements in direction A, and Fig. 10(b) depicts displacements in the orthogonal horizontal direction. Further results can be found in [38]. Maximum displacements tend to
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Fig. 10. Variation of maximum relative displacements at top as function of pretension: (a) strong direction; (b) orthogonal direction.
decrease as the pre-tension in the guys increases and as the stiffness of the guys increases. This trend is more noticeable at levels 3 and 4 than at levels 1 and 2. Maximum absolute values of bending moments at level 1 (the lowest guy level) are plotted in Fig. 11. Bending in direction A is considered in Fig. 11(a), and bending in the orthogonal direction in Fig. 11(b). There are no definite trends as the pre-tension or stiffness increases. Sometimes the bending moment decreases, and sometimes it increases. Fig. 12 considers the N-3 case and shows how the maximum tension during the response of each of the 12 guys depends on the value of the pre-tension. The notation i– j refers to level i and guy j, where guy 3 is in direction A (strong motion), guy 1 is oriented 120◦ clockwise from direction A (viewed from above), and guy 2 is 120◦ counter-clockwise from direction A (Fig. 1(b)). As the pre-tension increases, the maximum tensions in the guys tend to increase, as expected, but the dynamic portion of the tension (i.e., relative to the pre-tension) tends to decrease. Maximum guy tensions reach 400% of the pretension in some cases. Base shears are plotted as functions of guy pre-tension in Fig. 13. Parts (a) and (b) show the shears in direction A and the orthogonal direction, respectively, for both earthquakes and the three values of E A. No general trend seems to be exhibited regarding the manner in which the base shear changes as the guys’ pre-tension or stiffness increases. Some simulations were conducted for different crosssectional properties of the mast [38]. In the results presented in Figs. 4–13, the mast’s weight per unit length is µ = 588 N/m and its moment of inertia for bending in any direction is
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Fig. 11. Maximum bending moments at level 1 as function of pre-tension: (a) strong direction; (b) orthogonal direction.
Fig. 12. Maximum guy tensions as function of pre-tension: solid lines, level 1; dashed lines, level 2; dotted lines, level 3; dash-dot lines: level 4.
I = 0.0018 m4 . Two other cases were treated: (i) µ = 523 N/m and I = 0.0016 m4 , and (ii) µ = 653 N/m and I = 0.0020 m4 (i.e., µ and I were reduced by 11% in case (i), and increased by 11% in case (ii)). Other than these changes, the standard case was considered. For the Northridge earthquake, as µ and I were increased, the displacements in direction A at guy levels 1, 3, and 4 increased, while those at level 2 remained about the same. For the amplified El Centro earthquake, the displacements at all four guy levels increased from case (i) to the standard case and
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Fig. 13. Maximum base shears as function of pre-tension: (a) strong direction; (b) orthogonal direction.
then decreased for case (ii). In general, as the weight of the mast increases, the base shears and the bending moments at levels 1, 2, and 3 increase. Finally, the orientation of the earthquake was varied [38]. Directions A, B, and C are defined in Fig. 1(b). Maximum base shears, displacements, and bending moments at the guy levels were computed. The results were quite similar for the three orientations. Therefore the relative orientation of the strongest ground motion does not have a significant effect on the response of this mast, for which the guys are oriented in vertical planes separated by 120◦ . 6. Concluding remarks Seismic forces are sometimes an important factor in the design of guyed structures. A finite-element analysis of a 120 m-tall guyed mast, pinned at its base, has been conducted in this study. Three-dimensional ground excitation was applied, using the vertical component and two horizontal components of (a) the Northridge earthquake and (b) an amplification of the El Centro earthquake. Large responses were computed, and the guy wires, attached at four equally spaced levels of the mast, exhibited periods of slackness followed by snap loads with a sharp rise in tension. When in tension, the guy force was assumed to be a nonlinear function of the elongation, based on vibration tests conducted on synthetic fiber ropes. The assumed stiffening behavior also should be relevant for some guy cables made of other materials. Free vibrations were examined first. Most of the results that are presented were for the standard case. In that case, the stronger of the two recorded horizontal components of the
ground acceleration was directed parallel to one of the three vertical planes containing the guys. Horizontal and vertical displacements, bending moments, base shears, and vertical forces were computed for the mast, along with tensions in the guys [38]. Maximum horizontal displacements did not always occur in the strong direction of the earthquake. They corresponded either to the top of the mast or points within the top half. The shape of the mast changed during motion, and tended to be dominated by the first three or four vibration modes. Snap loads in the guys tended to induce increased bending moments in the mast. The maximum base shear was about 20% of the mast weight. The dynamic component of the vertical force in the mast, caused largely by the vertical component of ground motion, was sometimes significant compared to the mast weight. Similarly, the dynamic component of tension in the guys was sometimes large compared to the pre-tension. As the pre-tension or stiffness of the guys was increased, maximum horizontal displacements often were reduced and maximum vertical displacements of the mast tended to increase. As the weight of the mast was increased, bending moments and base shears tended to decrease, and horizontal displacements often increased. Variations in the orientation of the ground motion relative to the guys did not cause a significant change in the response. Seismic ground motions are three-dimensional. It is important to include all three components of the ground motion when analyzing the response of a guyed mast. It is also important to include three-dimensional motions of the mast, which may become quite complex during the response. Acknowledgement This material is based upon work supported by the US National Science Foundation under grant no. 0114709. References [1] ABAQUS. ABAQUS/Standard user’s manual, version 6.4. Providence (RI): ABAQUS, Inc; 2005. [2] Madugula MKS, editor. Dynamic response of lattice towers and guyed masts. Reston (VA): ASCE; 2002. [3] Irvine HM. Cable structures. Cambridge (MA): The MIT Press; 1981. [4] Augusti G, Borri C, Marradi L, Spinelli P. On the time-domain analysis of wind response of structures. J Wind Eng Ind Aerodyn 1986;23:449–63. [5] Iannuzzi A. Aerodynamic response of guyed masts: A deterministic approach. Bull IASS 1986;26:47–59. [6] Iannuzzi A, Spinelli P. Response of a guyed mast to real and simulated wind. Bull IASS 1989;30:38–45. [7] Augusti G, Borri C, Gusella V. Simulation of wind loading and response of geometrically non-linear structures with particular reference to large antennas. Struct Safety 1990;8:161–79. [8] Borri C, Zahlten W. Fully simulated nonlinear analysis of large structures subjected to turbulent artificial wind. Mech Struct Machines 1991;19: 213–50. [9] Augusti G, Bartoli G, Borri C, Gusella V, Spinelli P. Wind load and response of broadcasting antennas: Three years of research work in cooperation with RAI. J Wind Eng Ind Aerodyn 1992;41–44:2077–88. [10] Peil U, Nolle H. Guyed masts under wind load. J Wind Eng Ind Aerodyn 1992;43:2129–40.
G.M. Hensley, R.H. Plaut / Engineering Structures 29 (2007) 2254–2261 [11] Peil U, Nolle H, Wang ZH. Dynamic behaviour of guys under turbulent wind load. J Wind Eng Ind Aerodyn 1996;65:43–54. [12] Sparling BF, Davenport AG. Three-dimensional dynamic response of guyed towers to wind turbulence. Can J Civ Eng 1998;25:512–25. [13] Millar MA, Barghian M. Snap-through behaviour of cables in flexible structures. Comput Struct 2000;77:361–9. [14] Desai YM, Punde S. Simple model for dynamic analysis of cable supported structures. Eng Struct 2001;23:271–9. [15] He YL, Ma X, Wang ZM, Dong SL. Wind-induced response of guyed masts. J Int Assoc Shell Spatial Struct 2001;42:129–38. [16] Sparling BF, Davenport AG. Evaluation of simplified guy models in the dynamic analysis of guyed masts. J Int Assoc Shell Spatial Struct 2002; 43:41–55. [17] Harikrishna P, Annadurai A, Gomathinayagam S, Lakshmanan N. Full scale measurements of the structural response of a 50 m guyed mast under wind loading. Eng Struct 2003;25:859–67. [18] He YL, Xing M, Wang ZM. Nonlinear discrete analysis method for random vibration of guyed masts under wind load. J Wind Eng Ind Aerodyn 2003;91:513–25. [19] He YL, Chen WJ, Dong SL, Wang ZM. The dynamic stability analysis of guyed masts under random wind loads. Wind Struct 2003;6:151–64. [20] Xing M, Zhonggang W, Hongzhou D, Zhaomin W. Theoretical and experimental research on dynamic behavior of guyed masts under wind load. Acta Mech Solida Sinica 2004;17:166–71. [21] Law SS, Bu JQ, Zhu XQ, Chan SL. Response prediction of a 50 m guyed mast under typhoon conditions. Wind Struct 2006;9:397–412. [22] Sparling BF, Wegner LD. Comparison of frequency- and time-domain analyses for guyed masts in turbulent winds. Can J Civ Eng 2006;33: 169–82. [23] Blachowski BD. Active vibration control of guyed masts. In: Proceedings of the AMAS/ECCOMAS/STC workshop on smart materials and structures, 2005. pp 123–9. [24] Blachowski BD, Gutkowski W. Optimal vibration control of guyed masts. In: Gutkowski W, Kowalewski TA, editors. Mechanics for the 21st century: Proceedings of the 21st international congress of theoretical and applied mechanics. Berlin: Springer; 2005. [25] Magued MH, Bruneau M, Dryburgh RB. Evolution of design standards and recorded failures of guyed towers in Canada. Can J Civ Eng 1989;16: 725–32.
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[26] Mulherin ND. Atmospheric icing and communication tower failure in the United States. Cold Regions Sci Tech 1998;27:91–104. [27] Buchholdt HA, Moossavinejad S, Iannuzzi A. Non-linear dynamic analysis of guyed masts subjected to wind and guy ruptures. Proc Inst Civ Engrs Pt 2: Research Theory 1986;81:353–95. [28] Ben Kahla N. Nonlinear dynamic response of a guyed tower to a sudden guy rupture. Eng Struct 1997;19:879–90. [29] Ben Kahla N. Response of a guyed tower to a guy rupture under no wind pressure. Eng Struct 2000;22:699–706. [30] Argyris J, Mlejnek HP. Dynamics of structures. Amsterdam: NorthHolland; 1991. [31] Guevara E, McClure G. Nonlinear seismic response of antenna-supporting structures. Comput Struct 1993;47:711–24. [32] Moossavi Nejad SE. Dynamic response of guyed masts to strong motion earthquake. In: Proceedings of the 11th world conference on earthquake engineering, Paper No. 289. 1996. [33] Amiri GG. Seismic sensitivity indicators for tall guyed telecommunication towers. Comput Struct 2002;80:349–64. [34] Amiri GG, Zahedi M, Jalali RS. Multiple-support seismic excitation of tall guyed telecommunication towers. In: Proceedings of the 13th world conference on earthquake engineering, Paper No. 212. 2004. [35] Hennessey CM, Pearson NJ, Plaut RH. Experimental snap loading of synthetic ropes. Shock Vib 2005;12:163–75. [36] Ryan JC. Analytical and experimental investigation of improvement to seismic performance of steel moment frame structures using synthetic rope devices. PhD dissertation, Blacksburg (VA): Virginia Polytechnic Institute and State University; 2006. [37] PEER. Strong motion dababase. Pacific Earthquake Engineering Research Center, Berkeley, CA (http://peer.berkeley.edu/smcat/index.html). 2005. [38] Hensley GM. Finite element analysis of the seismic behavior of guyed masts. MS thesis, Blacksburg, VA: Virginia Polytechnic Institute and State University (http://scholar.lib.vt.edu/theses/available/etd-07062005161028). 2005. [39] Chopra AK. Dynamics of structures: Theory and applications to earthquake engineering. 2nd ed. Upper Saddle River (NJ): Prentice Hall; 2001. [40] Plaut RH, Archilla JC, Mays TW. Snap loads in mooring lines during large three-dimensional motions of a cylinder. Nonlinear Dynam 2000; 23:271–84.
Three-dimensional analysis of the seismic response of guyed masts Gregory M. Hensley, Raymond H. Plaut ∗ Charles E. Via Jr. Department of Civil and Environmental Engineering, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA Received 17 November 2005; received in revised form 29 September 2006; accepted 6 November 2006 Available online 28 December 2006
Abstract The response of a 120 m-tall guyed mast to three-dimensional seismic excitation is analyzed using the finite-element program ABAQUS. The Northridge earthquake and an amplified El Centro earthquake are applied. The mast is pinned at its base, and guy cables are attached at four equally spaced points along its height. The dynamic tension in the guys is modeled by a nonlinear function based on tests, and periods of slackness in the guys during motion of the system lead to snap tension loads and affect the response. The mast is represented by three-dimensional beam elements. Displacements, bending moments, and base shears are computed for the mast, along with guy tensions. The effects of guy stiffness and pre-tension, mast weight, and directionality of the ground motion are investigated. c 2006 Elsevier Ltd. All rights reserved.
Keywords: Finite element modeling; Guyed mast; Seismic response
1. Introduction A steel mast that is pinned at its base and stabilized by guy cables is considered. The mast has four guy levels (the quarter points and the top), with three cables at each level oriented in vertical planes separated by 120◦ . The mast is subjected to excitations of the Northridge and amplified El Centro earthquakes, including the vertical component and two orthogonal horizontal components of the ground motion. The finite-element program ABAQUS [1] is utilized to obtain time histories of the response. During the motion, some cables become slack and then taut again, leading to snap loads. The constitutive law for the tension in the cables is nonlinear and is based on vibration tests. The self-weight of the mast is one of the factors that affects the response. Madugula [2] presented an overview of work on the dynamic response of guyed masts. Free vibrations have been analyzed in [3] and other studies. Much attention has been focused on the motions of guyed masts caused by wind (e.g., [4–22]). Controlling the vibrations of a guyed mast in wind was the topic of [23,24]. Failures of guyed towers due to ice and/or wind are listed in [25,26]. Buchholdt et al. [27] and Ben Kahla [28,29] examined the response of a guyed mast following the sudden ∗ Corresponding author. Tel.: +1 540 231 6072; fax: +1 540 231 7532.
E-mail address: [email protected] (R.H. Plaut). c 2006 Elsevier Ltd. All rights reserved. 0141-0296/$ - see front matter doi:10.1016/j.engstruct.2006.11.019
rupture of one of the cables. In an example in [30], a guyed mast was subjected to horizontal, uni-directional, simple-harmonic motion at its base. A few researchers have investigated the seismic response of guyed masts. Most of these studies applied the finiteelement method to obtain numerical solutions. Guevara and McClure [31] considered a 24 m-tall mast with two guy levels, and a 107 m-tall mast with six guy levels. A scaled horizontal component from the El Centro earthquake and one from the Parkfield earthquake were applied. The time delay between the excitations at the different anchorage points of the guys and the mast was included in part of the study. Time histories of guy tensions, shear forces, vertical forces, and displacements were presented. Nejad [32] considered a 327 m-tall mast with five guy levels. All three components of seismic ground motion were included. Time histories of displacements and frequency spectra of two displacements and one guy tension were shown, along with lists of maximum displacement, shear force, axial force, and bending moment of the mast. Amiri [33] applied three-dimensional components of the El Centro, Parkfield, and Taft earthquakes to eight guyed telecommunication masts with heights varying from 150 m to 607 m. The base shear, vertical force in the mast, and guy tensions were examined. Finally, Amiri et al. [34] analyzed a 342 m-tall mast with seven guy levels, and a 607 m-tall
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mast with nine levels. A single horizontal component from the 1978 Tabas earthquake in Iran was applied. The study included the spatial effect of the distances between the anchorage points (‘multiple-support excitation’), as previously considered in [31]. Time histories of lateral tip displacement, axial force in the mast, cable and mast base shear, and cable tension were presented. The system to be analyzed here is described in Section 2. Free vibrations are considered in Section 3, and results for seismic response in a ‘standard case’ are presented in Section 4. Section 5 describes the effects of guy pre-tension, mast density, and earthquake direction, and Section 6 contains concluding remarks. 2. Description of mast, guys, and earthquake excitation The structure considered is similar to a mast that was analyzed by Irvine [3] and Desai and Punde [14]. The profile is shown in Fig. 1(a) and the plan view in Fig. 1(b). The mast is 120 m tall and has four guy levels separated by 30 m, with three guys per level, and two sets of guy anchors in each of the three planes. The mast is pinned at the base and is modeled as a beam-column using 12 cubic B33 elements in ABAQUS [1]. The mast’s weight per unit length is 588 N/m, its modulus of elasticity is 209 GPa, and its moment of inertia for bending in any direction is 0.0018 m4 . The constitutive law for the guy cables is based on dynamic tests of synthetic fiber ropes. Drop tests [35] and cyclic vibration tests [36] were conducted. The ropes stiffened nonlinearly as they extended, and the force was approximately proportional to the elongation raised to the power 1.3 (under conditions of no pre-tension). Here the ‘equivalent’ linear value E A = 3 × 104 kN is used in the ‘standard case’. The ropes are under pre-tension to stabilize the mast. In the standard case, the pre-tension at equilibrium is 25 kN, and the axial force (when positive) is assumed to be 1.2(E A/L)(δ + δ0 )1.3 , where δ is the elongation from the equilibrium length L. At guy levels 1, 2, 3, and 4, respectively, δ0 = 0.066, 0.095, 0.137, and 0.161 m. In other words, the ropes are modeled as nonlinear springs which do not resist compression. The results could be applicable to guy cables constructed of other materials if their inertia and transverse vibrations do not greatly affect the response. The anchors of the guys are assumed to be rigid. The ground motion is assumed to be the same at the cable anchorage points and the base of the mast. The height of this structure is at the lower end of the spectrum for guyed masts, and thus it should not be particularly sensitive to multiple-support excitation. The components of the Northridge earthquake taken at the Sylmar County Medical Center (SYL) are applied here in the standard case. This ground motion record exhibits some nearfield effects that are not common in many other records. The strongest direction of recorded ground motion is SYL360, with a peak ground acceleration (PGA) of 0.843g. The motion in the orthogonal horizontal direction, SYL090, has a PGA of 0.604g, and the vertical record, SYL-UP, has a PGA of 0.535g. Data are taken from [37], which lists accelerations with time increments of 0.02 s, and 20 s of the earthquake record are used.
Fig. 1. (a) Profile view of guyed mast; (b) plan view of direction of strong horizontal component.
In addition to excitation by the Northridge earthquake, an amplified version of the 1940 El Centro earthquake will be considered in Section 5. The strongest recorded direction is ELC180, with a PGA of 0.313g. The other horizontal direction, ELC270, has a PGA of 0.215g, and the vertical record ELC-UP has a PGA of 0.205g. Again, data are taken from [37], now in time increments of 0.01 s over a time of 40 s. This event has more high-frequency content than the Northridge event, but less intensity. The El Centro accelerations are amplified by a factor of 1.5 and then applied to the guyed mast. For each earthquake, the recorded horizontal component with the higher range between maximum and minimum accelerations is denoted the ‘strong direction’. If the acceleration records in the two measured horizontal directions are considered and their locus is plotted in a plane, the direction with the largest range turns out to be close to this direction for both earthquakes [38]. The arrows in Fig. 1(b) indicate the strong direction of the earthquake, and direction A will be used in the standard case. The ground displacements during the first 15 s of motion of the Northridge earthquake are plotted in Fig. 2, beginning at the origin. Fig. 2(a) depicts the displacements in the horizontal plane, with SYL360 as the abscissa. Displacements in the vertical plane including this component are shown in Fig. 2(b), and those in the orthogonal vertical plane are presented in Fig. 2(c). In the horizontal plane, the direction with the largest
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29.3 rad/s, respectively, which closely match values obtained from linear beam theory using equivalent transverse springs for the guys [38]. The first three modes are sketched in Fig. 3, with numbers showing the locations of the guy levels. They are similar to the modes shown in [3] and [14]. For the seismic analysis, the first and fourth modes were used to determine the parameters for Rayleigh damping that would approach 2% of critical damping for the most relevant modes [39]. This yielded coefficients of 0.237 and 0.00140 of the mass and stiffness matrices, respectively, and the linear combination of these two scaled matrices formed the assumed damping matrix. 4. Standard case
Fig. 2. Ground displacements for Northridge earthquake: (a) horizontal plane; (b) vertical plane including SYL360; vertical plane including SYL090.
range of displacements is oriented approximately 20◦ from the strong direction for the Northridge earthquake (Fig. 2(a)), and approximately 90◦ from the strong direction for the El Centro earthquake [38]. 3. Free vibrations Small-amplitude free vibrations of the guyed mast were analyzed first. The lowest five natural frequencies for vibrations in the vertical plane of direction A (neglecting axial force in the mast) were computed to be 8.37, 10.5, 16.1, 20.3, and
Results for the standard case are presented first. As mentioned prevously, this involves the Northridge earthquake with its strong motion (SYL360) in direction A in Fig. 1(b), a pre-tension of 25 kN in the guys, an ‘equivalent’ linear value of E A = 3 × 104 kN for the guys, and a weight per unit length of 588 N/m for the mast. Horizontal displacements were determined at the base of the mast and at the guy levels (labeled 1–4 in Fig. 1(a)). The plots of displacements at the guy levels will present displacements relative to the ground motion (rather than absolute displacements), whereas the plots of base displacement will show the ground motion (since the relative displacement there is zero). Fig. 4(a) depicts the ground motion along direction A, and Fig. 4(b) and (c) depict the time histories of the relative displacements along that direction at levels 2 and 4, respectively. Fig. 5(a), (b), and (c) show the corresponding displacements along the orthogonal SYL090 direction. The displacements at the top of the mast (level 4) tend to be larger than those at the center of the mast (level 2). Of particular interest is the fact that after the first five seconds, the peaks of the level-2 and level-4 displacements along the strong-motion direction (Fig. 4(b) and (c)) are smaller than those along the orthogonal horizontal direction (Fig. 5(b) and (c)). Also, the motion in the orthogonal direction tends to be more regular and indicates that the first vibration mode dominates the response in that direction. In the SYL360 direction, the maximum peak displacement for all the guy levels occurs at level 4 (the top of the mast), but it occurs at level 3 in the orthogonal direction [38]. The bending moments are zero at the base and at level 4. At levels 1, 2, and 3, they demonstrate significant variability during the response, with the largest peak moment in direction A being about 150 kN m and occurring at level 2. In the orthogonal direction, the largest peak moment is approximately 100 kN m and occurs at level 1 [38]. Fig. 6 presents the time history of the tension in a guy attached at level 3 in a vertical plane at an angle of 120◦ with direction A. The plot covers the time from 3 s to 10 s after initiation of the earthquake record. Periods of slackness occur starting at around a time of 6.5 s, and the guy exhibits snap loads, i.e., a sudden rise in tension and then a return to slackness shortly afterward [40]. Following a time of 6.5 s, the
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Fig. 3. Lowest modes of free vibration: (a) first; (b) second; (c) third.
Fig. 4. Displacements in strong direction for standard case: (a) base; (b) level 2; (c) level 4.
guy attached at level 3 at an angle of −120◦ with direction A tends to be slack when the guy in Fig. 6 is taut, and
Fig. 5. Displacements in SYL090 direction for standard case: (a) base; (b) level 2; (c) level 4.
vice versa [38]. These snap loads, associated with slack–taut transitions, appear to cause a significant increase in some of the
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Fig. 6. Tension in cable at guy level 3 for standard case.
Fig. 7. Base shear for standard case.
Fig. 9. Sequence of mast shapes in strong earthquake direction for standard case between (a) 3.5 s and 6.0 s, (b) 7.0 s and 8.0 s.
accelerations), and in Fig. 9(b) they correspond to shapes in 0.1 s increments from 7.0 s to 8.0 s. The form of the mast changes significantly during the motion, sometimes being almost linear and often having one local relative minimum and one local relative maximum in the transverse displacement. The highest magnitude of the total displacement (ground motion plus deformation) in Fig. 9 is 0.5 m at a height of 100 m (curve 4 in Fig. 9(a)). The corresponding plots for projections of the mast in the orthogonal plane exhibit a maximum total displacement of 0.4 m [38]. Fig. 8. Vertical force in mast at level 2 for standard case.
guy-level displacements and bending moments in the SYL090 direction. The time history of the base shear of the mast in direction A is plotted in Fig. 7. The mast weighs 70.6 kN, and the maximum base shear in Fig. 7 is about 15 kN, i.e., a little over 20% of the weight. Fig. 8 shows the time history of the vertical compressive force in the mast at a location just below level 2. The static axial force at that point, due to the self-weight of the top half of the mast and the vertical components of the guys at levels 2, 3, and 4, is 149 kN. During motion, this force decreases to almost 100 kN and increases to almost 200 kN, i.e., fluctuations of approximately ±30% about the static value. It is interesting to observe the changing shape of the mast during the dynamic response. The projections of the shapes along direction A are plotted in Fig. 9. In Fig. 9(a) the numbers 1–11 at the tops of the shapes correspond, respectively, to the shapes in 0.25 s increments at times from 3.5 s to 6.0 s in the record (during the period with largest ground
5. Effects of pre-tension, mast density, and earthquake direction The effect of the pre-tension force in the guys is investigated first. In the figures, the strong earthquake components, SYL360 and ELC180, are oriented along direction A in Fig. 1(b). The equivalent linear value of E A for the guys was 3 × 104 kN in the standard case. Additional values of 2 × 104 and 4 × 104 kN are included now. The designations N-2, N-3, and N-4 denote the Northridge earthquake and pre-tensions 2 × 104 , 3 × 104 , and 4 × 104 kN, respectively, and E-2, E-3, and E-4 denote the corresponding cases for the amplified El Centro earthquake. Results were computed for guy pre-tension forces of 15, 20, 25 (the standard case), 30, and 35 kN. Fig. 10 shows absolute values of maximum relative displacement (i.e., deformation with respect to the ground motion) at the top of the mast (level 4). Fig. 10(a) presents displacements in direction A, and Fig. 10(b) depicts displacements in the orthogonal horizontal direction. Further results can be found in [38]. Maximum displacements tend to
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Fig. 10. Variation of maximum relative displacements at top as function of pretension: (a) strong direction; (b) orthogonal direction.
decrease as the pre-tension in the guys increases and as the stiffness of the guys increases. This trend is more noticeable at levels 3 and 4 than at levels 1 and 2. Maximum absolute values of bending moments at level 1 (the lowest guy level) are plotted in Fig. 11. Bending in direction A is considered in Fig. 11(a), and bending in the orthogonal direction in Fig. 11(b). There are no definite trends as the pre-tension or stiffness increases. Sometimes the bending moment decreases, and sometimes it increases. Fig. 12 considers the N-3 case and shows how the maximum tension during the response of each of the 12 guys depends on the value of the pre-tension. The notation i– j refers to level i and guy j, where guy 3 is in direction A (strong motion), guy 1 is oriented 120◦ clockwise from direction A (viewed from above), and guy 2 is 120◦ counter-clockwise from direction A (Fig. 1(b)). As the pre-tension increases, the maximum tensions in the guys tend to increase, as expected, but the dynamic portion of the tension (i.e., relative to the pre-tension) tends to decrease. Maximum guy tensions reach 400% of the pretension in some cases. Base shears are plotted as functions of guy pre-tension in Fig. 13. Parts (a) and (b) show the shears in direction A and the orthogonal direction, respectively, for both earthquakes and the three values of E A. No general trend seems to be exhibited regarding the manner in which the base shear changes as the guys’ pre-tension or stiffness increases. Some simulations were conducted for different crosssectional properties of the mast [38]. In the results presented in Figs. 4–13, the mast’s weight per unit length is µ = 588 N/m and its moment of inertia for bending in any direction is
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Fig. 11. Maximum bending moments at level 1 as function of pre-tension: (a) strong direction; (b) orthogonal direction.
Fig. 12. Maximum guy tensions as function of pre-tension: solid lines, level 1; dashed lines, level 2; dotted lines, level 3; dash-dot lines: level 4.
I = 0.0018 m4 . Two other cases were treated: (i) µ = 523 N/m and I = 0.0016 m4 , and (ii) µ = 653 N/m and I = 0.0020 m4 (i.e., µ and I were reduced by 11% in case (i), and increased by 11% in case (ii)). Other than these changes, the standard case was considered. For the Northridge earthquake, as µ and I were increased, the displacements in direction A at guy levels 1, 3, and 4 increased, while those at level 2 remained about the same. For the amplified El Centro earthquake, the displacements at all four guy levels increased from case (i) to the standard case and
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Fig. 13. Maximum base shears as function of pre-tension: (a) strong direction; (b) orthogonal direction.
then decreased for case (ii). In general, as the weight of the mast increases, the base shears and the bending moments at levels 1, 2, and 3 increase. Finally, the orientation of the earthquake was varied [38]. Directions A, B, and C are defined in Fig. 1(b). Maximum base shears, displacements, and bending moments at the guy levels were computed. The results were quite similar for the three orientations. Therefore the relative orientation of the strongest ground motion does not have a significant effect on the response of this mast, for which the guys are oriented in vertical planes separated by 120◦ . 6. Concluding remarks Seismic forces are sometimes an important factor in the design of guyed structures. A finite-element analysis of a 120 m-tall guyed mast, pinned at its base, has been conducted in this study. Three-dimensional ground excitation was applied, using the vertical component and two horizontal components of (a) the Northridge earthquake and (b) an amplification of the El Centro earthquake. Large responses were computed, and the guy wires, attached at four equally spaced levels of the mast, exhibited periods of slackness followed by snap loads with a sharp rise in tension. When in tension, the guy force was assumed to be a nonlinear function of the elongation, based on vibration tests conducted on synthetic fiber ropes. The assumed stiffening behavior also should be relevant for some guy cables made of other materials. Free vibrations were examined first. Most of the results that are presented were for the standard case. In that case, the stronger of the two recorded horizontal components of the
ground acceleration was directed parallel to one of the three vertical planes containing the guys. Horizontal and vertical displacements, bending moments, base shears, and vertical forces were computed for the mast, along with tensions in the guys [38]. Maximum horizontal displacements did not always occur in the strong direction of the earthquake. They corresponded either to the top of the mast or points within the top half. The shape of the mast changed during motion, and tended to be dominated by the first three or four vibration modes. Snap loads in the guys tended to induce increased bending moments in the mast. The maximum base shear was about 20% of the mast weight. The dynamic component of the vertical force in the mast, caused largely by the vertical component of ground motion, was sometimes significant compared to the mast weight. Similarly, the dynamic component of tension in the guys was sometimes large compared to the pre-tension. As the pre-tension or stiffness of the guys was increased, maximum horizontal displacements often were reduced and maximum vertical displacements of the mast tended to increase. As the weight of the mast was increased, bending moments and base shears tended to decrease, and horizontal displacements often increased. Variations in the orientation of the ground motion relative to the guys did not cause a significant change in the response. Seismic ground motions are three-dimensional. It is important to include all three components of the ground motion when analyzing the response of a guyed mast. It is also important to include three-dimensional motions of the mast, which may become quite complex during the response. Acknowledgement This material is based upon work supported by the US National Science Foundation under grant no. 0114709. References [1] ABAQUS. ABAQUS/Standard user’s manual, version 6.4. Providence (RI): ABAQUS, Inc; 2005. [2] Madugula MKS, editor. Dynamic response of lattice towers and guyed masts. Reston (VA): ASCE; 2002. [3] Irvine HM. Cable structures. Cambridge (MA): The MIT Press; 1981. [4] Augusti G, Borri C, Marradi L, Spinelli P. On the time-domain analysis of wind response of structures. J Wind Eng Ind Aerodyn 1986;23:449–63. [5] Iannuzzi A. Aerodynamic response of guyed masts: A deterministic approach. Bull IASS 1986;26:47–59. [6] Iannuzzi A, Spinelli P. Response of a guyed mast to real and simulated wind. Bull IASS 1989;30:38–45. [7] Augusti G, Borri C, Gusella V. Simulation of wind loading and response of geometrically non-linear structures with particular reference to large antennas. Struct Safety 1990;8:161–79. [8] Borri C, Zahlten W. Fully simulated nonlinear analysis of large structures subjected to turbulent artificial wind. Mech Struct Machines 1991;19: 213–50. [9] Augusti G, Bartoli G, Borri C, Gusella V, Spinelli P. Wind load and response of broadcasting antennas: Three years of research work in cooperation with RAI. J Wind Eng Ind Aerodyn 1992;41–44:2077–88. [10] Peil U, Nolle H. Guyed masts under wind load. J Wind Eng Ind Aerodyn 1992;43:2129–40.
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