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NOTE / NOTE A novel method for determining effective length factors for solid r

NOTE / NOTE A novel method for determining effective length factors for solid round steel leg members of guyed lattice towers with cross-bracing Yongcong Ding, Faouzi Ghrib, Sudip Bhattacharjee, and Murty K.S. Madugula Abstract: A novel method is presented for determining the effective length factors (K factors) for solid round steel leg (vertical) members in all-welded guyed communication towers with cross-bracing. Deflections due to an arbitrary load were used to determine the restraining moments and rotational stiffness of the legs. Using the closed-form solution for buckling of a column with rotational restraint at both ends, K factors were computed. From the study it is found that the K factors varied from 0.61 to 1.00, depending on the relative sizes of the leg and bracing members. An expression for estimating the effective length factors for solid round steel leg members of all-welded guyed lattice towers with cross-bracing is developed. Key words: cross-bracing, effective length factor (K factor), finite element analysis, guyed lattice tower, solid round steel leg member, welded tower. Résumé : Cet article présente une nouvelle méthode pour déterminer les facteurs de longueur efficace (coefficients K) pour les pieds (verticaux) ronds en acier plein des tours de transmission haubanées soudées comportant des contreven- tements. Les fléchissements causés par une charge arbitraire ont été utilisés pour déterminer les moments de retenue et la rigidité rotationnelle des pieds. Les coefficients K ont été calculés en utilisant la solution analytique pour le flambe- ment d’une colonne munie d’une entrave à la rotation à chaque extrémité. L’étude a démontré que les coefficients K variaient entre 0,61 et 1,00, selon les dimensions relatives du pied et des membrures de contreventement. Une expres- sion mathématique est développée pour estimer les facteurs de longueur efficace pour les pieds ronds en acier plein des tours de transmission haubanées soudées comportant des contreventements. Mots clés : contreventement, facteur de longueur efficace (coefficient K), analyse par éléments finis, tour haubanée, membrure ronde en acier plein, tour soudée. [Traduit par la Rédaction] Ding et al. 786 1. Introduction Radio and television broadcasting and telecommunications require elevated antennas, which are usually supported on ei- ther self-supporting or guyed lattice towers. When welded, guyed lattice towers usually consist of solid round steel rods for legs, diagonals, and sometimes girts (horizontal members). The compressive resistance of the leg member depends on its effective length. In the American Standard ANSI/TIA/ EIA-222-G (Draft) Structural standard for antenna support- ing structures and antennas (TIA 2002), the Australian Standard AS 3995-1994 Design of steel lattice towers and masts (Standards Australia 1994), and the Canadian Stan- dard CSA-S37-01 Antennas, towers and antenna-supporting structures (CSA 2001), the effective length factor K for solid round steel leg members is specified as 1.0. Eurocode 3, Design of steel structures, Part 3.1: Towers and masts (CEN 1996), also gives an effective length factor for symmetrically braced solid round leg members as 1.0, with a note that a reduced K value may be justified by analysis. Most practicing engineers use in their design of guyed lat- tice towers the default values given in the standards. How- ever, some designers feel that the bracing members, when sufficiently stiff, provide rotational rigidity to the leg mem- bers and assume a K factor as low as 0.8 for leg members. Therefore, there is a need to derive rational values of K de- pending on relative sizes of the leg and bracing members. More accurate effective length factors also help in increasing the compressive resistance of leg members of existing tow- ers designed with an effective length factor of 1.0. This will be beneficial if new antennas are to be added to the tower because the need for expensive strengthening of the leg members in the field is eliminated. This paper presents a novel method of determining the K factors for solid round steel leg members of all-welded guyed lattice towers with cross-bracing, based on the relative Can. J. Civ. Eng. 30: 780–786 (2003) doi: 10.1139/L03-027 © 2003 NRC Canada 780 Received 29 July 2002. Revision accepted 19 March 2003. Published on the NRC Research Press Web site at http://cjce.nrc.ca on 12 August 2003. Y. Ding, F. Ghrib, S. Bhattacharjee, and M.K.S. Madugula.1 Department of Civil and Environmental Engineering, University of Windsor, Windsor, ON N9B 3P4, Canada. Written discussion of this note is welcomed and will be received by the Editor until 31 December 2003. 1Corresponding author (e-mail: madugul@uwindsor.ca). flexural stiffness (I/L, where I is the moment of inertia of the leg member and L is the length of the leg member) of leg and bracing members meeting at the joint. The present study is limited to determining the effect of bracing members on the effective length factors of legs. Therefore, other factors such as bar defects and fabrication sag, which certainly have an important effect on the strength of the leg members, are not included in the investigation. 2. Methodology The following procedure is used to determine the K factor of leg members. (1) Since adjacent leg members buckle at approximately the same time, the leg member gets restraint only from bracing members and not from the adjacent leg mem- bers. In case the adjacent leg members do not fail at the same time, this assumption leads to conservative results. Therefore, the two leg members in the panels adjacent to the leg member under investigation were removed from the tower section before analyzing the remaining section. The leg member under investigation was sup- ported at its ends, and an arbitrary central transverse concentrated load was applied in the vertical direction (the direction in which the restraining effect of the brac- ing members is the least). The deflection at the center was then computed by finite element analysis. (The ar- bitrary load need not be a concentrated load: it could have been a distributed load or any other load.) (2) From the midpanel deflection, the rotational stiffness at each end of the leg member was determined. (3) Using the classical results for the buckling load of an elastically restrained column, the effective length factor of the leg member was determined. 3. Details of the method 3.1. Finite element analysis The commercial computer software package ABAQUS (Hibbitt, Karlson & Sorenson, Inc. 2001) was used to com- pute the deflection of the leg member under transverse load. Two different configurations (six and eight panels) of tower sections with four leg sizes (38.1, 50.8, 57.2, and 69.9 mm) and five cross-bracing member sizes (12.7, 14.3, 15.9, 19.1, and 22.2 mm) are included in the investigation. These con- figurations and member sizes are considered typical of cur- rent industry practice. Details of tower sections included in the investigation are given in Figs. 1a and 1b and Tables 1 and 2. A finite element model of a typical six-panel tower section, with the two leg members adjacent to the leg mem- ber under investigation removed, is shown in Fig. 2. In the analysis, the cross-bracing members are assumed to be rig- idly connected to the leg members. To determine whether in practice the welds between the bracing members and the legs are sufficiently “rigid” to prevent any rotation between them, tests on 15 six-panel and 16 eight-panel tower sections were carried out at the University of Windsor (Sun 1999). The results clearly showed that the connection between the leg and bracing members can be assumed to be rigid and that the bracing members are indeed capable of providing the rotational restraint to leg members. The leg member under investigation was simply supported at both ends and an arbitrary transverse load of 10 kN was applied in the vertical direction at midspan. The resulting vertical deflection of the leg member was computed by lin- ear elastic finite element analysis. 3.2. Rotational stiffness at ends of leg member The free-body diagram of the leg member, the bending moment diagram due to the restraining moment M, and the © 2003 NRC Canada Ding et al. 781 Fig. 1. Dimensions of (a) six-panel and (b) eight-panel tower sections. simply-supported bending moment diagram under transverse loading W are shown in Figs. 3a–3c. From the second mo- ment-area theorem, the deflection at the middle of the leg member, δ, is given by © 2003 NRC Canada 782 Can. J. Civ. Eng. Vol. 30, 2003 Finite element model No. Diameter of leg member (mm) Moment of inertia of leg member, I (×103 mm4) Diameter of bracing member (mm) Moment of inertia of bracing member, IB (×103 mm4) Transverse deflection by finite element analysis, δ (mm) Rotational stiffness, M/θ (kN·m/rad) 1 38.1 103 12.7 1.28 4.142 5.63 2 38.1 103 14.3 2.05 3.884 11.2 3 38.1 103 15.9 3.14 3.689 16.2 4 38.1 103 19.1 6.53 3.404 24.9 5 38.1 103 22.2 11.9 2.950 44.5 6 50.8 327 12.7 1.28 1.372 6.37 7 50.8 327 14.3 2.05 1.356 9.21 8 50.8 327 15.9 3.14 1.334 13.3 9 50.8 327 19.1 6.53 1.272 25.7 10 50.8 327 22.2 11.9 1.189 45.3 11 57.2 525 12.7 1.28 0.862 6.43 12 57.2 525 14.3 uploads/Litterature/ k-factor-rpviewdoc 1 .pdf