Analisis Miembros Antena

8/2/2019 Analisis Miembros Antena

1/118

EFFECTIVE LENGTH FACTORSFOR SOLID ROUND CHORD MEMBERS OFGUYEDTOWERS

A ThesisSubmined to the Coilege o f Graduate Studies and ResearchthroughCivil & Environmental Engineering

in Partial Fulfillment of the Requirements forthe Degree o f Master ofApplid Scienceat theUniversityofWindsor

Windsor, Ontario,Canada1999


2/118

National Library1*1 of Canada Bibliothque nationaledu CanadaAcquisitions and Acquisitions etBibliographie Services services bibliographiques395 Wellington Street 395. rueWeilingtOIIOttawaON K l A W OltawaON K 1 A Wcanada canada

The axthor has granted a non-exclusive licence aliowing theNational Library of Canada toreproduce, loan,distribute or sellcopies of this thesis in microform,paper or electronic formats.

L'auteur a accord une licence nonexclusive permettant a laBibliothque nationale du Canada dereproduire, prter, distribuer ouvendre des copies de cette thse sousla forme de microfiche/film, dereproduction sur papier ou sur formatlectronique.

The author retains ownership of the L'auteur conserve la proprit ducopyright in this thesis. Neither the droit d'auteur qui protge cette thse.thesis nor substantial extracts fiom it Ni la thse ni des extraits substantielsmay be printed or otherwise de celle-ci ne doivent tre imprimsreproduced without the author's ou autrement reproduits sans sonpermission. autorisation.


3/118


4/118

Chord mernbes of twenty-five dl-welded guyed-latticed communication steel towersections were tested in the Structurai Engineering Laboratory of the University ofWindsor to detemine the effective length factors of the chord memben. Two diffaentmanufacturers, viz., Pirod Inc., Plymouth, Indiana, and ERI Inc., Chandler, Indiana,provided the test specimens. Ali tower sections were fabricated fiom solid roundmembers and were trimgular in cross section. Tower sections provided by ER1 were 4.57m (1 5.0 fi.) long with continuous diagonal bracings welded to the chord members,whilethose provided by Pirod were 6-09 m (20.0 A.) long with the diagonal bracings cut andwelded to the chord members.The diameters of th e chord members varied fiom 38.1 mm(1.5 in.) to 69.85 mm (2.75 in.) while the diametersof the diagonal bracings vared fiom12.7 mm (0.5 in.) to 22.3 mm 0.875 in.).

The tower sections were tested in a horizontal position. One chord member of the towerwas cut and tested by applying a load at its center while the diagonal bracings remainedattached to the chord members. Deflections were recorded manually while the appliedIoad was recorded using a data acquisition system. The load-deflection data were used tocalculate the effective length factors for the chord members which were found to bevarying from 0.95 to 0.99. Good agreement was observed between the expenmentaldeflections at th e center of the chord member and those obtained fiom the cornputeranalysis software package ABAQUS.


5/118

The stiffness contributionof diagonal bracings to the ends of the chordmembers was alsocomputed numerically by structural analysis package STAADmro and the effkctivelength factors were calculated using the equilibrium equation given by the structuralstability research council (SSRC, 976). The effective length factor dculated by thisrnethod varkd h m .93 to 0.99. Based on the experimental and numerical results, itappears reasonable to assume an effective length factor of 1 .O in actual design practice.


6/118

DEDICATEDTO MYLOVING PARENTS

DR.ABDUL W I M QURESHIANDDR. SAKINAQURESHI


7/118

First of dl, the author expresses his most sincere gratitude to Aimighty G d withoutWhose help nothing would have been possible.

The author wishes to express his deep appreciation and gratitude to his Principal AdvisorDr. Murty K. S. Maduguk Professor, Department of Civil and EnvironmentalEngineering, for his constant guidance, c ~ p e r a t i o nnd encouragement dunng the entireprocess of this research. To him I say, "Thank you very much".

Special hanks are also reserved for the Co-Advisor Dr. Sudip Bhattacharjee, AssistantProfessor, Deparmient of Civil and Enviro~lmentalEngineering, for his valuablesuggestions and supe~s ionuring the whole process of this work.

The author is also thankful to Pirod Inc. and ER1 Inc. for providing the test specimensused in this investigation.

Thanks are also reserved for the technicians Richard Clark, Lucian Pop and PatrickSeguin for their vital help during the experimental part of this project.

The author also acknowledges the help provided by his fiiends and colleagues Messrs.Zonghua Chen, Yongcong Ding, Robert Rea and Yean Sun. The support provided by the

vi


8/118

Information Technology services of the University of Windsor including the services oftheCADICAM laboratory are also greatly appreciated.

The financial support provided by the Department of Civil and EnvironmentalEngineering, University of Windsor, and the Natural Sciences and Engineering ResearchCouncil of Canada is grateflly acknowiedged.

In th e end the author also feels very much obliged and indebted to al1 of his familymembers who provided continued mord and financial support during this whole periodof study and research at theUniversityof Windsor.

vii


9/118

TABLEOF CONTENTS

ABSTRACTDEDICATIONACKNOWLEDGEMENTSLIST OF TABLESLIST OF FIGURESNOTATION

ivvi

vi ixi

xi iXV

CaAPTERONE- NTRODUCTION. .1 GENERAL 1. 2 CLASSIFICATION OF COMMUNICATION TOWER S 1. .3 STRUCTURALCONFIGURATION 4..4 NEED FOR INVESTIGATION 5. .5 OBJECTIVE OF THE PRESENT RESEARCH 6. .6 OUTLINEOFTHETHESIS 6

CEIAPTER 'IWO-BACKGROUNDOF RESEARCHAND REVIEW O FLITERATWRE2.1 GENERAL 82.2 JOINTEFFECT IN TRANSMISSION TOWERS 92.3 EFFECTIVE LENGTH FACTOR IN BRACED FRAMES IO2.4 THE G-FACTOR AND EFFECTIVE LENGTHSOFCOLUMNS 132.5 CODES, TANDARDSAND SPECIFICATIONS 16

2.5.1 CSA S37-94 "ANTENNAS,TOWERS,ANDANTENNA-SUPPORTING STRUCTUREStt 16

2.5.2 CAN/CSA-S16.1-94 "LIMIT STATES DESIGN OFSTEEL STRUCTURES" 162.5.3 AISC - LRFD "LOADANDRESISTANCE FACTORDESIGN SPECIFICATION FOR STRUCTURALSTEEL BUILDINGS" 172.5.4 EUROCODE 3: PART .1 : 1997 "DESIGNOF STEEL

STRUCTURES OWERS, MASTS AND CKIMNEYS-TOWERS ANDMASTS" 20


10/118

CHAPTERTHREE EXPERIMENTAL INVESTIGATION3.1 GEBERAL3.2 DETAILS OF SPECIMENS

3.2.1 E N PECIMENS3.2.2 PIROD SPECIMENS

3.3 TESTSETUP3.3.1 SUPPORT ASSEMBLY3.3.2 LOAD APPLICATION3.3.3 DATA ACQUISITION SYSTEM3.3.4 TESTINGOF CHORD MEMBERCHAPTER FOUR-ANALYSIS OF LOAD-DEFLEC'MONDATA

4.1 GENERAL 334.2 CALCULATION OF CHORDROTATION ATTHE ENDS 334.3 CALCULATION OF EFFECTTVE LENGTH FACTOR 364.4 COMPARISON OF DEFLECTIONS WITH ABAQUSRESULTS 424.5 DISCUSSION OF RESULTS 4 3

CHAPTER FlVE - ANALYTICAL INVESTIGATION5.1 GENERAL 485.2 EQUILIBIUUM EQUATION 485.3 DETERMINATION OF ROTATIONAL STIFFNESS USINGSTAADPro 5 15.4 CALCULATION OF EFFECTTVE LENGW FACTOR 565.5 DISCUSSION OF RESULTS 56

CHAPTER SIX- ONCLUSIONS6.1 GENERAL6.2 CONCLUSIONS6.3 RECOMMENDATION

REFERENCES 60APPENDTX-A FIGURES 64APPENDIX- ABAQUS INPUT FILES 90VITA AUCTORIS 103


11/118

LIST O F TABLES

TABLE 3.1 Details of tower sectionsTABLE 4.1 Rotational stiffiness of hord membersTABLE 4.2 Effective Iength factors (experimental)TABLE 5.1 Vaiue ofG calculated using STAADProTABLE 5.2 Effective length factors (STAADPro)


12/118

LIST OF FIGURES

Self-supporting towerGuyed TowerEfiective Length Factors for six idealuedconditionsNomographERI Tower-Weld anangementPirod Tower-Weld arrangementP i r d Towers lying in the laboratoryChord member cut and loadedTest setupTest setup showing the loading coiumn and the suppon assemblyData acquisition systemChord member fixed at both endsFree body diagram of a coium.Mode1 generated byABAQUSGraphical comparison for deflections of specimen-S-1 (A)Graphical comparison for deflections of specimen- -3 (A)Graphical comparison for deflections of spefimen- -5 (A)Framed structureBuckling of a colunin in a h e d tructureTowermode1generatedusing STAAD/ProSection of tower talcen out for analysis

xi


13/118

Moment applied on tower jointRotation of tower jointLoad vs. Deflection&ta for chordmember- SpecimenS-1 (A)Load vs. Deflectiondaia for chord member- pecimenS-1 (B)Load vs. Deflection&ta for chord member- pecimenS-2 (A)Load vs. Deflectiondata for chord member- p e c bS-2 (B)Load vs. Deflection&ta for chordmember.- pecimen 5-2 (C)Load vs. Deflectiondata for chord m e m k- pecimenS-3 (A)Load vs. Deflectiondata for chord member- pecimenS-3 (B)Load vs. Deflection&ta for chord member- pecimenS-4 (B)Load vs. Deflectiondata for chord mernber- pechen S 4 C)Load vs. Deflection data for chordmember- pecimen S-5 (A)Load vs. Deflectiondata for chord mernber- pecimen S-5 (B)Load vs. Deflection data for chordmember-Specimen S-5 (C)Load vs. Deflectiondata for cho d memberSpecimenS-6 (A)Load vs. Deflection data for chord member- pecimen S-6 (B)Load vs. Deflection data for chord member- pecimen S-6 (C)Load vs. Deflectiondata for chordmember- pecimenP l (A)Load vs. Deflectiondata for chord member- Specirnen P l (B)Load vs. Deflection data for chord rnember- Specimen P2 (B )Load vs. Deflectiondata for chordmember- Specimen P3 (A)Load vs. Defiection grapii for chord rnember- pecimen P3 (B)


14/118

A.2 1 Load vs. Defection&ta for cbord member- pecimenP4 (A)A.22 Load vs. Deflection&ta for chord mernber- Spechen P6 (A)A.23 Load vs. Deflection&ta for chord member- SpecimenP6 (B)A.24 Load vs. Deflection data for chordmember- SpecimenP8 A)A.25 Load vs. Deflection data for chord member4pecimen P8 B)

xiv


15/118

NOTATION

Cross-sectional areaUniformly distributed loaded lengthDiameter of cross sectionYoung's modulus of elasticityRelative joint stiffiiess ratio, lowerRelative joint stifniess ratio, upperMoment of inertiaMoment of inertia of column/chordmemberMoment of inertia of girderldiagonal bracingEffective length factorPanel length ofcolumnLength ofgirderMoment at ends of chord memberFixed end momentRestraining MomentConcentrated load appliedintensity of oad appliedDifierence in DeflectionExperimental deflectionCalculated DeflectionAngle of Rotation


16/118

CIiAPTER ONEINTRODUCTION

1.1 GENERAL

With the world entering into the 21" century, the demand for an effective and usefultelecommunication system is pwing. This need has resulted in an increase in theproduction and developmcnt of telecommunication tools. Some of thesetelecommunication tools are mobile phones and pagen. To transmit and receive thesignals properly, antennas are use& which are either parabolic or pole type. Theseantennas are usuallymounted on steel communication towers.

1.2 CLASSIFICATIONO F COMMUNICATION TOWERS

From the structural point of view, communication towers can be classified into threetypes:

(a) Monopoles, which are cantileveredtubeswith heights up to 70 m,

@) Self-supporting towers (Figure 1.1), commoniy used for heights up to 120 m, and

(c) Guyed towers ( F i p e 1.2)which have been utilized for taller structures up to 620 m.


17/118

Figure 1.1 Self-supportingtower(Source: Pirod inc.)


18/118

Figure 1-2 GuyedTower(Source:Pirod inc.)


19/118

Guyed towers require more land area than the self-supporting ones. This is because of theguys that provide lateral support to the structure. These guys are steel cabla madeespecially for this purpose. They are hooked up fimily with guy anchors that areanchored in the nearby land, thus occupying a huge surmunding area. Because of thisdisadvantage, guyed towers are generally not recommended for installation in denselypopulated residential areas or such places where availability of adequate space is aproblem. Usually the nurnber of guy levels holding a tower ranges h m hree to five(Wahba, 1999).

1.3 STRUCTURALCONFIGURATION

A guyed tower consists of a mast usually of a constant iangular cross section. Thesetowers have a latticed structure with al1 the joints welded or bolted. The triangular mastof a guyed tower has three vertical members (chords). These chord members are weldedor bolted with horizontals and diagonals (bracing) to form a latticed structure. Al1 themembers are usually solid rounds with varying diameters. Tubes and angles are alsosometimes used for these chords and bracings.

A guyed toww is fabricated in sections and is erected by mounting these sections one ontop of the other in the field by a special method used to build ta11 communicationstructures with antennas fixed on it. Helicopters are also occasionally used to carry outthis installation operation.


20/118

1.4 NEED FOR INVESTIGATION

The effective length factor of the chord members of triangular guyed-steel-communication towers is the main topic of this study. To the best of author's knowledge,most of the e x m e n t a l investigation c e e d out so far on guyed-latticedcommunication steel towers, with reference to finding out the effective length factors, ison cross braced diagonais. So far, no investigation has ken carried out on solid roundchord members of these towers. Th, here is a strong need to experimentally determineth e effective length factors of chord members hat are primary structural components of asteel communication tower.

Section 6.2 of the Canadian Standards Association S37-94, "Antennas, Towers andAntenna-SupportngStructures," (CSA 1994), deals with members under compression. 1gives the effective length factor for the leg membm of towers as unity. Most practicingengineers fiequently use this value. However, there are some designers who feel that thebracing members ptovide rotational rigidity ?O the chord members and so rake theeffective length factor as 0.8.

Thus, there is a need to cany out the experimental investigation of the chord membersand find out the value of the effective length factor. The research is based on thehypothesis that the diagonal bracings provide rotational ratraint to the chord memberwith which they are attacha thus, resulting in an effective length factor which is lessthan unity.


21/118

1.5 OBJECTIVEOF THE PRESENTRESEARCH

The objective of tbe research is to detexmine, experimentally and theoreticaily, theeffective length factors for solid round chord members of guyed-latticed steelcommunication towers.

1.6 OUTILiNEOF THETHESIS

Chapter Two, BACKGROUND OF THE RESEARCH AND REVIEW OFL I T E R A m , includes bnef review of available research material on communicationtowers, effective length factors of members under compression and the recommendationsgiven by different codes, standardsand specificationsin this regard.

Chapter Three, EXPERJMENTAL INVESTIGATION, describes in detail theexperimental investigation canied out on chord members of tower specimens in theStructural Engineering Laboratory of the University of Windsor. h e procedure isexplained step-by-step and is accompanied by photographs taken at the t h e ofexperirnentatioa.

Chapter Four,ANALYSISOF LOAD - DEFLECTION DATA, deals with the analysis ofthe &ta obtained through experiments. The rotational stifThess at the ends of the chordmembers has been calculated h m he loaddeflectiondata and effective length factor ofthe chord member is determined. The defiection data of the chord member observed


22/118

experimenrally is also compared with the results obtained from the computer softwareABAQUS (Version 5.8, 1998).

Chapter Five, NUMERICAL INVESTIGATION, describes the procas for thedetermination of the relative joint stifniess ratio lmown as G-factor, by using structuralanalysis software STAADPro (1998). This factor is used subsequently to obtain theeffective length factor from the equilibrium equation (Equationof the Nomograpb).

Finally, Chapter Six, CONCLUSIONS, gives the s ~ l ~ l l l l a r yf the work done, and theconclusions reached.


23/118

CIiAPTERTWOBACKGROUNDOF RESEARCH AND REVIEW OF LITERATURE

2.1 GENERAL

Guyed transmission towers are highly indeterminate three-dimensional structures with areiatively complex structural composition. These structuresare among some of the tallestin the world. They stand as high as 15 f e t or 500 rn above ground level. Due to thepeculiar structurai composition of guyed towers, much work has been done in this ana bymany researchers.

One general assumption tbat is usually made in d y s i s of guyed towers is that thedifferent structural members meet at a single point and he joints have a pinned effect.However, this assumption is not satisfied in usual fabrication. When the joint is no tpinned, there are chances hat secondary stresses will arise in the member that may have asignificant influence on the critical load. This discrepancy between design assumptionand actual fabrication may undemine the ultimate load carrying capacity of the tower.

In the case of steel communication towers, the panel joints (end conditions) are notsimple pin connections. The diagonal mernbers of each panel provide some resistance toa moment. This restraining moment s h i h the location of the points of inflection fiom the


24/118

end points to inside, thus reducing the effective length of the member. The amount ofreduction is proportional to the resistance provided by the restrainingmembers.

In triangulated fkame structures (tnisses), the loads are usually applied at the joints. Thusif the joints are truly pinne, then the members are axially loaded. Defiections of thejoints and the tniss as a whole are caused by the axial deformation of the members. Theangles between members meeting at a joint also change because of these deformations. Ifthe members are connected together at the joints by welds or bolts, the angle changescause secondary bending stresses. Tbese have Iittle effect on the buckling strength of thetniss members. Because of the local yielding of extreme fibers of the membm near thejoints, as the truss is loaded to ultimate, the seconciary moments gradually dissipate.(Galarnbos, 1988).

2.2 JOINT EFFECT LN TRANSMISSIONTOWERS

Knight and Santhakumar (1993) studied the joint effects on behavior of transmissiontowers. A full-scalequadrant of the lowermost panel of a transmission tower designed asa pin-jointed truss was tested according to ICP (1978). The behavior of the tower wasobserved under normal loading conditions. The lowermost panel was chosen forexperimental observationas the vertical, transverse and longitudinal loads were assumedto be the maximum on that panel. It was concluded that the failure of the chord membersdepends on the axial forces as well as the moments generated because of forces in thesecondary members. Thus, it was concluded that the consideration of moments


25/118

introduced by the secondary members is a necessity in order to arrive at a realisticestimate of the failure laad of the tower andhence the consideration of joint effect is veryimportant as it may result in a premature failure or inappropriate analysis of the wholetower.

2 3 EFFECTIVE LENGTH FACTOR IN BRACED FRAMES

With reference to the design of steel fiames, the effective length concept is extensivelyused for finding out the effects of the interaction of other members (beams, columns orother) on the member under compression in a frame. Much work has been done withreference to the analysis of columns in partially and Mly restrained fiames.

Kishi, Goto and Komuro (1995) observeci the effective length factor for columns inbraced and flexibly jointed fiames. Their discussion was with reference to AISC-LRFDspecification (AISC 1993) which States that in order to design a partially or fiillyrestrained fiame, th e bending moments of members are to be obtained estimating thesecond-order effects @-A and P fkcts) and the effective length factor-K of colitmns,considering the nonlinear moment-rotation characteristics of semi-rigid connections.They used the alignment chart approach to denve the gov&g equation for detenniningthe columns K-factor in braced and flexibly jointed fkames. They showed that alignmentchart can be used to tind the K-factor for a column by modimg the relative sti!Tnessfactors.


26/118

Duan and Chen (1988) studied the effective length factor for columns in braced fiames.They derived the general effective length factor equations for columns in braced h e scomesponding to five different boundary conditions of top and bottom columns. Theyconcluded that the far end conditions of the columns above and below the column havesignificant effect on the effective length factor K of the column king investigated- Adirect use of the alignment chart, without modifications, results in an effective lengthfactor that can be either tao conservative or even unsafe depending on the boundaryconditions of the columns. They gave a modified improved alignment chart procedurewhich included the usual rigid far ends of the top and bottom colurnn as a special caseand also considerd the casesof fixed orhinged far endsof the top and bottom columns.

Fraser (1983) gave a method for evaluating the effective length factor in braced h e s . Itcomprises of an iterative procedure and is helpfbl in the anaiysis of structures in whichthe flexural sti fies s of the restrainingmembers is significantlyreduced by the presenceof axial forces. The whole procedure is much simplified by the use of linearized stabilityfnctionsexpressed in terms of effective length factors.

Cheong-Siat-Moy (1 997) analyzed the possibility of the design charts used by practicingdesign engheers giving unconservative values for effective length factors for bracedM e s because they assume that the lateral restraints are infinite. T he basis for thisinfinite value is the assumption hat there is no sway in the member or in other words it issway-prevented. It was concluded that the use of the assumed-sway-preventdK-ac or inpractical braced m e s ould sometima be tw unconservative.


27/118

Some researchers, taking into consideration the complex and indefinite procedure ofdeterminhg the K-factor, went as far as pmposing the designof steel fiames without theconsideration of effective length. White and Hajjar (1997)were among those researchers.Limits have been suggested in their paper for use of K=l . Also AISC-LRFDspecifications are discussed with reference to neglecting P-Delta moments in design.Generai equations are presented in the paper that give the emr in the ABC-LRFD beam-colurnn interaction equations associateci with the use of K=l . The infiuence of keyvariables on the error is analyzed. Suggestionsare given when the design of steel -esby AISC LRFD may be based on K=l. The paper ends by cornparhg design strengthswith and without effective length to the result fiom elastic-plastic hinge and rigorousplastic zone andysis for several 'maximum error' examples. The whole discussionprovides an assessrnent of the accuracy of upper-bound error estimates, and of theimplications of using K=l relative to the theoretical inelastic fiame behavior. Therecommendationsand discussions are applicable for any type of steel frame that includesM e s with fully or partially r e s h e d connections, and unbraced or partially bracedfiames.

Wood (1974) studied the effective length of columns in multi-story buildings. He gavecomprehensive design charts for ef fs tive length of columnswith any local degree of endrestraint,both for sway and non-sway conditions.


28/118

2.4 THEGFACTOR ANDEFFECTIVELENGTHS OF COLUMNS

Framed members under compression interact with th e resaaining beams at member ends.They also corne in contact with other members that are under compression above andbelow the member under consideration. This interaction is due to the c o ~ e c t i n ghorizontal bearns that are comrnonly in contact with both compression members. Theseinteractions are complex and peculiar in behavior. It is, however, a cornmon practice thatto avoid more complexity in such analysis the compression rnembers are snidied asisolated, with the end restraints defned in terms of simple sti fies s ratios, known as G-factors. The issue of the accuracy of the G-factor has been addresseci by many researcherssince it plays an important part in the overall design and assurnptions made for as nicture.

Bridge and Fraser (1987) gave an improved G-factor method for evaluating effetivelength of columns with reference to the Nomograph that is widely used by practicingengineers to find out the effective length factor. It requires the engineer to evaluate G ateach end of the column. The value of G should always be positive. In the case of swayprevented structures, however, the effective length factors are always less than unity. It ispractically possible that columns may have values of K greater than unity thatcorresponds to a value of G that is negative. The paper concludes that the failure toinclude the effects of axial load in adjacent members restraining a critical bucklingcolumn could result in an error in determining the elastic-buckling load. Such acalculatioo can be dealt with if the axial forces in restraining beams and in adjacent


29/118

columns are taken care of properly. Sucb a procedure can be achieved by incorporatingthe negative G-factors. Hence, an nproved G-factor method was given which takes intoaccount al1 these factors.

Hellesland and Bjorhovde (1 996) discussed the methods for determining effective lengthsof members under compression in a fiame n terms of exact results and general principlesof mechanics of buckling. The conventional G-term (relative joint stiffiiess d o ) hasbeen shown to be inaccurate and conceptuaily flawed. A novel concept of a restraintdemand factor is introduced. This is done in order to allow for improved development ofvertical interaction, which also includes effective length predictions

Hajjar and White (1995) swnmarized the objectives and contents of ASCE cornmitteereport entitled "Effective Length and equivalent imperfection approaches for assessingframe stability: Implications for Arnerican Steel Design". The discussion is withreference to the procedure used commody for colurnn design in th e United States that isbased primarily on the Nomograph effective length approach. The report discussed theprocedure for the calculation of effective lengthwithin partially restrained (or semi-rigid)framing. Among other key issues, onewas the issue of the validity of the effective lengthconcept for naming in which the restraining elements, that includes beams and theuconnections, are relatively flexible and show significant non-lindty. For use indeteminhg effective length, the selection of proper connection stifiess is alsodiscussed, dongwith concepts for cdculation of effective lm& in fiameci structures.


30/118

Hellesland and Bjorhovde (1996) also proposed a rnethod which involved ps t -processing of effective lengths fiom isolated column analysis to arrive at improved,weighted mean values. As such, the metho is tenned as the "method of means". Thismethod involves pst-processing of effective lengths fiom isolated columns analyses toarrive at improved, weighted mean values. The approach is applicable to braced and to arange of unbraced b e s .

Duan and Chen (1989) analyzed the effective length factor for columns in unbracedm e s with reference to the specifications of AISC (1986) which makes use of thealignment charts to determine effective length factor for colurnns in both types of *es,braced and unbraced. It was fond that the efktive length factor of a h e d olumn isnot only dependent on the relative bending stiffaess ratio of the jointed columns andgirders, i.e., the G-factors at the ends of its unbraced length in an unbraced conthuousframe, but is aiso dependent on the boundary conditions of far ends of restrainingcolumns.

Kishi, Chen and Goto (1997) analyzed the effective length factor of colurnns in semi-rigid and unbraced frames. The paper is with reference to the engineering practice ofevaluating the columns in h e s with rigid and semi-rigid connections whereby theestimation of the effective length factor (K-factor) is necessary. This estimation is doneconsidering the effects of the nonlinear moment-rotation characteristics of beam-to-column connections. The paper States that with a proper evaluation of the tangentconnection stifnms for semi-rigid beam-to-column connections at buckling state and


31/118

with the intniduction of the modified relative stifniess factors, the alignment chart in thepresent American Institute of Steel construction - Load and Resistance Factor Designspecification can be used to find the corresponding K-factor for columns in semi-rigidh e s

CODES, STANDARDSAND SPECIFICATIONS

CSA S37-94 uANTIENNAS,TOWERS,AND ANTENNA-SUPPORTINGSTRUCTURES"

Section 6.2.1.1 of the Canadian Standards Association S37-94 "Antennas, Towers andAntenna-Supporthg Structures", 1994, gives the effective length factor for leg mernbersof towers as unity.

2.5.2 CAWCSA-S16.1-94 "LLMIT STATES DESIGN OF STEELSTRUCTURES*

The tandard States about the effective length of colw~ i~lss:

"The effective length,KL, may be thought of as the actual unbraced length, L, multipliexiby a factor, K, such tbat the product, KL, s equal to the length of a pin ended column ofequal capacity to the actual member. The effective length factor, K, of a column of finiteunbraced length therefore depends on the conditions of restraintafforded to the column atits braced location. A variation in K between 0.65 and 2.0 would apply to the majority of


32/118

cases likely to be encountered in a c t d structures. Figure 2.1 illustrates six idealizedcases in which joint rotation and translation are either klly realized or non-existent".

The standard also gives a Nomograph (Figure 2.2) which is based on the assumption thatal1 the column in the portion of the fiamework considered reach their individual criticalload simultaneously. This may be used to determine the effective length factors for in-plane behavior of compression members of tnissa designad as axially loaded memberseven though the joints are rigid. In this case, there should be no in-plane eccentricitiesand al1 the members of the uss meeting at the joint must not reach their ultimate loadsimultaneously. Further, the standard says that if it cannot be shown that al1 members atthe joint do not reach their ultimate load simulbneously, then the effective length factorof the compression members shall be taken as 1 O.

2.53 AISC-LRFD "LOAD AND RESISTANCE FACTOR DESIGNSPECIFICATIONFOR STRUCTURAL STEEL BUILDINGS"

Chapter E of the LRFD specification deals with columns and other compression-prismatic members hat are subjected to axial compression througb their centroidal axis.Section E l deal with th e Effective Length Factor - K and States that for structuralmembers under compression it should not be taken as l e s than unity.


33/118

End condition code

Figure 2.1 Effective Length Factors for six idealized conditions(Source: SSRC, 1976)


34/118

Figure 2.2 Nomograph(Source: SSRC, 976)


35/118

2.5.4 EUROCODE 3 PART 3.1: 1997 uDESIGN OF STEEL STRUCTURESTOWERS, MASTSAND 'ilIMNEYS -TOWRS AND MASTS,"

Sub-clause5.6.2.2 of the code refers to single members and reads as:

"The following rules shouid be used for single angles, tubular sections or solid roundswhere used for chord sections. For chords or chords with axial compression bracedsymmetrically in two normal planes, or planes 60' apart in the case of triangularstructures, the slendemess should bedetennined from the system length between nodes.

"Where bracing is staggered in two normal planes or planes 60' apart in the case oftriangular stnictures, the system length should be taken as the length between nodes onone face. For angle sections the radiusof gyration about the minor axis should be used tocalculate the slendemess".

Clause 5.7 refers to effective slenderness factor K and States that if the chord membersare solid rounds then the value of K hould be taken as 1 O. This value is for both types oftowers having symmetricalor asymmetrical bracings.


36/118

Investigations were carried out on fifieen specimens provided by Electroaics Research,Inc., Chandler, Indiana, USA, nd ten specimens provided by P i r d Inc., Plymouth,Indiana, USA. The specimens were actual tower segments fabricated by these twocompanies. The specirnens provided by ERI hc . were 4.57 m (15.0 feet) long while thoseprovided by Pirod were 6.09 m (20.0 feet) Long. Most of the specimens wererepresentative samples of acnial tower segments but some were specially fabricated forthese investigations.

3.2 DETAILS OF SPECIMEN

3.2.1 ER1 SPECIMENS:

Fifieen specimens were provided by Electronic Research Institute, Inc. Three differentchord sizes, 69.9, 50.8 and 38.1 mm, were usai with three different diagonal sizes, 15.9,14.3 and 12.7 mm, respectively. The diagonal bracings were continuous and bent at thepoint of welding with the chord memben.Refer to Figure 3.1.


37/118

Figure 3.1 ER1 Tower-Weld arrangement

Figure 3.2 Pirod Tower -Weld arrangement


38/118

None of the specimens were galvanized. A11 the chord members were made fiomstandard solid rounds with different diameters. The total length of each tower sectionprovided by ERI hc. was 4.57 m. Each tower section had six panels. AI1 sections were oftriangular cross section with chord, and diagonal mernbers comected together bywelding.

Ten specimens were provided by Pirod Inc. Three different chord sizes, 57.2, 50.8 and38.1 mm, were provided with four different diagonal bracing sizes, 22.2, 19.1, 15.9 and22.2 mm. The diagonal bracings were cut and welded to the chord member as sh ow inFigure 3.2. The total length of each tower section was 6.09 m. Figure 3.3 shows threePirod towers section lying in the laboratory. Each section had eight panels. Refer to Table3.1 for the details of the tower sections.

3.3 TEST SETUP

Al1 the tower specimens were tested in a similar fashion at the Structural EngineeringLaboratory of the University of Windsor. They were first placed in a horizontal positionresting on two roller supports, 1219mm long and 152.4 mm in diameter. One panel fkomthe bottom chord was chosen for the test and was loaded by the hydraulic jack that wasattached to the loading frame in the laboratory


39/118

TABLE .1 DETAILS OF TOWER SECTIONSSpecimen Total No. ofS.NO. Supplier 1.D Lenflb Panels

ERi

S-1 (A), s-1 (BIS-2 (A)

S-2 (B)S-2 (C)S-3 (A)

, s-3(B) .S-4 (B)

PanelLengtbmm

Face Dia. of Dia.ofWidth chords diagonds


40/118

Figure 3.3 Pirod Towers lying in the Laboratory


41/118

Figure 3.4 Chord member cut and Ioaded

The chord member was cut beyond its pane1 points/joints on either side of the test sectionas shown in Figure 3 -4. The diagonal bracings, however, remained attached :O the chordmember. The cutting was done either with the help of a welding torch or an electricpower saw. The panel, king isolated h m oth sides , was then simply supported oneither of its sides on two srnail rollers, 127 mm long and 38.1 mm in diameter. Load wasapplied on that panel at the mid-point through a loading block as shown n the test setupin Figure 3.5

33.1 SUPPORTASSEMBLY

A support assembly that consisted of several built up steel sections supported the tower atthe ends. These steel sections were assembled together to furnish enough height to beable to test the specimen ushg the testing fiame available in the laboratory.


42/118

Figure 3.5 Test setup


43/118

3.3.2 LOADAPPLICATION

The load was applied through a hydraulic jack attached to the testing h e . A 448 kN(100 kip) load cell was screwed to the bottom of the hydradic jack. An extension wasprovided to the load ce11 in the shape of a long solid round cylindrical column, 76.2 mmin diameter and 1220 mm long. This extension aras needed to fumish the ciifferencebetween the heights of th e load ceil attached to the roof barn and the chord member ofthe tower specimen. Refer to Figure 3.6.

The load was applied through loading blocks that were specially machined and grwvedto sit properiy on the chord members having varying diameters. The blocks measured101 6 ~ 7 6 . 2 ~92.08 mm, 120.65~76.2xi92mm nd 139.7~76.2~92 mm to sit properly onthe chords having diameters 38.1, 50.8 and 69.9 mm respectively. In between the loadapplication column and the load block, a ball-and-socket joint, as shown in Figure 3.5,was introuced in order to take care of any eccentricity of the applied loading. The loadwas applied using a mechanical purnp connected to the hydraulic jack. To measwe theapplied load, a data acquisition system was used which gave readings in kilo-newtons.

3 3 3 DATAACQUISITION SYSTEM

An automatic &ta acquisition system was used to record the load applied by thehydraulic jack to the chordmember. This was a Datascan 7000 series of Data AcquisitionModules. A Type 732 1measurementprocessor was used that provided directly 8 anaiog


44/118

Figure 3.6 Test setup showing the loading column and thesupport assembly


45/118

inputs, including channel excitation and full local channel expansion capability.A type7021 analog expansion scanner module was used which provided 8 analog input channelsper scanner, with excitation for transducers. In addition to that, a type 7036 digitalexpansion scanner module was used which provided 8 digital input and 8 digital outputchannel capability. One arnong the 40 channels was used to know the applied load on thechord member. The system was hooked up with a 486 DX2 host cornputer (Figure 3.7).Proper settings were made and the load ce11 was configured to the system before the startof the actual test.

3.3.4 TESTINGOF CHORD MEMBER

A small load was first applied and released to make sure that the whole assembly isperfectly seated and well placed. In order to measure the deflection due to the appliedloading, a dia1 indicator was fixed exactly below the point where the load was kingapplied to the chord member. The indicator was firmly fixed to the adjacent steel section,which was lying on the floor. No movement was permitteci to enswe undisturbedreadings of the dial indicator once the actual testing procedure was started.

The load was applied in srnaiier incrernents. To make sure that the material does notyield, the maximum load appliedwas kept within the elastic range of the chord member.The maximum load applied varied for each chord depending upon its diameter. For eachload increment, the dia1 gauge reading and the correspondhg applied load were recorded.


46/118

Load deflection data were thus obtained for al1 these specimens. Graphs for al1 twenty-five specimensare given inAppendix A.


47/118

Figure 3.7 Data acquisition system


48/118

CEAPTERFOURANALYSIS OF LOAbDEF'LECI'IONDATA

The load deflection data gathered through th e experimental procedure were analysed todetermine the effective length factor K. The deflections of the chord memberof the towerwere dso compared with the values obtained numerically by using the commercialsoftware package ABAQUS.

4.2 CALCULATIONOFCHORDROTATIONAT TEE ENDS

The deflections, A,, of the chord member of the tower were first calculateciassumingbothends o be compietely fixed as shown in the Figure 4.1. The downward deflection at mid-span of the beam, fixed at both ends and loaded with a unifonnly dismbuted load oflength 'c' at its center is given by:

where q is the intensity of loading, 1 is the panel length of the tower, c is the uniforrnlydistrbuted loaded length, E is Young's modulus of elasticity, 1 is the moment of nertiaa . n d y = c / l .


49/118

Figure 4.1 Chord member fixeci at both ends

Equation 4.1 gives the deflection of a beam that is completely fixed on both ends and isacted upon by a partial u n i f o d y distributed load. This equation was chosen to take intoaccount the setup in the laboratory where a rectangular block was placed on the chord atits center (refer to Figure 3.4). It may be aoted that the parameter 'c' in Equation 4.1 istaken as the width of the rectangular block (76.2 mm) whereas '1' is the panel length(varying for both types of towers).

The load versus deflection data was calculated for al1 the towers with the help ofEquation 4.1. The graphs are ploned by the side of the experimental deflections and aregiven in Appendix A. The difference in the deflection in both the cases is aven by:

AE is the experimental and Ax is the calculated deflection of the chord member. Theclifference occurs because the ends of the beamare not totally restrained in real situation.


50/118

For a beam partially restrained at both ends, he siope at each end, 8, is givenby:

where, M is the moment at both ends of the chordmember.Deflectioq 6, at mid span isgiven as:

(4.5)

From Equation 4.4 and 4.5, slope8, t the ends of the bearn is given by:

where 6 is the differential displacement as calculated in Equation 4.2. The end slopecalculated through Equation 4.6 is given in Table 4.1. Load of 6.0 kN was chosenarbitrarily for al1 the tower sections provided by both the fabricating companies. At thisload the chord member remained elastic and did not yield.


51/118

4 3 CALCULATIONOF EFFECTIVE LENGTH FACTOR

R e f e h g to Figure 4.1, the equation for fixecl-end moment is given as:

The restmiiingmomentMR.which is causedby the diagonal bracings at the point of weldwith the chord member is equal to the fxed end moment less the moment released due toa rotation 0 of the member. Mathematically, fiom Equations 4.4 and 4.7, MR can bedefined as:

The rotational stiflhess provided by the diagonal bracings to the ends of chord membercm be caiculated by dividiag the restraining moment with the end slope as given inEquation 4.9 as:

Rotationai Stifiess - MR 9Equation 4.9 is solved by taking values from Equations 4.6 and 4.8. The results are givenin Table 4.1. The table also gives the value of the restraining moment calculated withEquation 4.8.


52/118

..

S.NO.

1234567

, 89I O1 i12

, 1314, 1516, 1 718, 1920, 2122, 2 32425

SpecimenID

S-1 (A), S-1 (B )S-2 (A)S-2(B)S-2(C)S-3 A)S-3 (B )S-4(U)S-4 (C )S-5 (A )S-5(B)S-5 (C )S-6(A )S-6 (B)S-6(C)Pl (A )PI(R)P2 (B)P3 A)P3 (B)P4 (A )P6(A)P6(B)PB (A)P8 (B)

Dia. ofchordmm38.138.138.138.138.150.850.850.850.869.969.969.969.969.969.938.138.138.150.850.850.857.257.238.138.1

STIFFNESS OFDiffcrence

mm1.73Et001.64Et001.94Et001.98Et001.97Et005.71E-O16.01E-O 15.91E-016.21E-011.72E-011.67E-O11.62E-O1.52E-O I1.52E-011.42E-O11.18Et001.16E+00I.SIE+OO4.378-014.228-017.608-022.91E-O12.74E-018.1 3E-017.43E-01

TABLE 4.1Expt. Deflection

(for 6 kN)m m2.392.302.602.642 -630.780.8 10.800.830.230.230.220.210.210.201.721.702.O50.6 10.590.250.400.381,351.28

ROTATIONALFixed beam def.

(for 6 kN)mm6.62E-016,62E-016.628-016.628-06.62E-012.09E-O12.09E-012.09E-012.09E-O5.84E-025.84E-O25.84E-02

5 ME-025.84E-025.84E-025.37E-015.37E-015.37E-01.70E-011.70E-01.70E-011.ME-01.06E-015,37E-015.37E-01

Rot. Stiffness(Momenl/Slopc)

(Eq. 4.9)N-mrnhdian -8.50E+061.19Et07 ,1,69E+065.6 1Et058.40E+051.86Et079.08Et061.21Et073.26Et061.73Et073.63Et075.64E+071 .O1Et081 .O I Et08 ,1.SIE+O82.17Et072.30Et074.24Et063.2 1E+073.99E+071.06Et092.91 E+074.94Et075.80Et07

- 6.89Et07

CHORDEnd Slope(Eq. 4.6)Radians9.07E-038.60E-031.02E-021.ME-021.03E-022.998-033.1 SE-033.1 OE-033.26E-039.01E-O48.74E-048.488-047.96E-047.968-047.43E-046.65E-036.548-038.51E-032.468-032.37E-034.288-041.64E-031.54E-034.57E-034,18E-03

MEMBERSResirainingMoment(Eq.4.8)N-mm7.7 1Et041.03Et051,72E+045.83Et038.68Et035.57E+042.868+043.76E+041.06Et04

1.56E+043.1 7E+M4.79Et048.02Et048.02Et041. 12E+051.44Et051.51E+O53.61Et047.898+049.48Et044.53EtOS4.76Et047.61Et042.658+052.88Et05


53/118

At buckling, the forces that act on a columnMth elastically restrainedends, are shown inthe fiee body diagram Figure 4.2.

Figure 4.2 Free body diagram ofa column- - - - - - - - - - - - - - - - -

The differential equation of the column acted upon by a load P as shown in Figure 4.2can be written as:


54/118

where M is the end moment induced as a result of buckling of the column due to theapplied loading. The solution to Equation 4.10, with consideration of deflectionboundaryconditions, is given by:

and

dy M a aL- -(-tan - cosax+ sin ax)dx P 2

From equation 4.13

The lefbhand side of Equation 4.14 is the rotational stiffhess as calculated throughequation 4.9, where


55/118

Euler's equation forelastic bucklingof columns is given as:

Equations 4.14 wa s solved for P, which was substituted subsequently in Equation 4.16 tofind out the effective length factor, K. Young's rnodulus ofelasticity was taken as200 000 MPa while , the moment of inertia, and 1, the panel length, varied depending onthe type of tower under consideration- The values of the effective length factor(Experirnental)so obtained are given in Table 4.2.


56/118

TABLE 4.2 EFFECTIVE LENGTH FACTORS (EXPERIMENTAL)

1 1 S-1 (A) 1 38.1 1 12.7 1

S.No

3 1 S-2 (A) 1 38.1 1 12.7 1

6 1 S-3 (A ) 1 50.8 1 14.3 1

SpecirnenID

7 1 S-3 (B) 1 50.8 1 14.3 19 1 S-4 (C) 1 50.8 1 14.3 1

Dia. ofchordmm

1 1 1 S-5 (B) 1 69.9 1 15.9 112 1 S-5 (C) 1 69.9 1 15.9 1

Dia. ofDiagonalmm

15 ( S-6 (C) 1 69.9 1 15.9 1

PanelLengthmm

1314

M of 1ChordMembermm4

S-6(A)S-6 (B)

16171819202122232425

M o f IDiagonalBracingmm4

69.969.9

K

15.915.9

Pl(A)pl (BIp2 (BIp3 (A)p3 (BIP4 (A)P6 (A)p6 (B)P8 (A)p8 (BI

103 000103000103000327 O00327 000327 O 0 0525 000525000103 000

1

IO3 000

31403 14031406530653065301190011900119001 1900

711.2

.38.1 1 15.9 30.96

I0.960.970.970.970.980.980.980.95

I

0.95

38-138.150.850.850.857.257.238.138.1

A15.915.919.119.119.1

122.2122.2

22.222.2


57/118

4.4 COMPARISONOF DEFLECTIONSWITHABAQUS RESULTS

Numerical investigation was carried out with the commerciai cornputer software packageABAQUS in order to compare theoretical results with the deflections of the chordmembers obtained experimentally. A model of a typicai six-panel tower section, ERIspecirnen, was generated with 40 nodes as s h o w in the Figure 4.3. The boundaryconditions were defined in the same manner as the tower was placed in the laboratory fortesting.

The eiement library in ABAQUS contains several types of beam elements. Differentbeam elements in ABAQUS use different assumptions. The beam elements chosen forthe model definition of the tower are the Timoshenko (shear flexible) beams. Thesebearns allow for transverse shear deformation. They are used since they provide usefilresults for such beams that are made fiom unifom material. ABAQUS assumes that thetransverse shear behaviour of Timoshenko bearns is linear elastic with a fked modulusand, thus, independent of the response of the beam sections to axial stretch and bending.Refer to Appendix B for INPUTFlLES FOR ABAQUS.

It may be noted that in the model, Figure 4.3, with reference to the chord member onwhich the load is applied the adjacent chord mernbers are not provided. This is toresemble the situation in the laboratory whereby the chord member was cut beyond thepanel junction. In order to apply a central concentrated load at the middle of the chordmember an extra node was defined at that point.


58/118

Figure 4.3 Mode1 generatedby ABAQUS

Analysis was camed out keeping the magnitude of the applied load qua1 to 6 kN, forwhich the experimental deflections were also availabie. The results are compared withthose obtained experimenally and are shown in Figures 4.4.4.5 and 4.6. The cornparisonshows the experimental and numencal deflections of tower sections S-1 (A), S-3 A) andS-5 (A), having diameters o f the chord member 38.1,50.8 and 69.9 respectively.

4.5 DISCUSSION O F RESULTS

As expected, the effective length factors for both types of towers came out to be less thanunity. This proved that the diagonal bracings were providing resistance (rotationairigidity) to the chord memben, though, in most cases not very significant.The effectivelength factors varied from 0.95 to 0.99. It was also observed that the thicker the chord


59/118

member the more closer the effective length factor was to unity. The difference in theweid arrangementof two towen did not have a significant effect on the resuits. Thus, itcan aiso be said that the diagonals do not provide significant rotational resnaint to thechord member, though they do provide lateral restraint. The results obtained throughcommercial software package ABAQUS gave deflections that were a little less than theexperimental ones. This may be due to the reason that the mode1 generated in thecornputer was stiffer than the one tested experimentally and had ideal loading andsupporting conditions. The error between the numerical and experimental results wasobserved to be less than 5% .


60/118


61/118

Chord Diagonal Dia. LOADTower IDDEFLECTION BY

APPLIED DEFLECTION(Average) ABAQUSmm mm kN mm mm

COMPARISON OF RESULTS

DEFLECTION BYABAQUS

EXPERIMENTALDEFLECTION

(Average)

DEFLECTION (mm)

FIGURE 4.5 CRAPHICAL COMPARISONS FOR DEFLECTIONS - SPECIMEN S-3 (A)


62/118


63/118

CH A P ' r E R r nAIYALYTICAL INVESTIGATION

The effective length factors of the chord members of guyed towers were also calculatedanalytically using the equation of equilibrium. The relative stifiess ratio, the G-factor,was calculated using the structural anal ysis cornputer s o h a r e STAAD/Pro.

5.2 EQUILIBRIUM EQUATION

Ln this case the chord member of the tower, which was attached with diagonal bracings,is considered a case similar to an axially loaded coiumn in a h e d tructure. Wherecontinuous construction is used, the beam-to-column connections are rigid. In thissituation, the column-end rotation, that take place during the bending motion of thecolumn will induce corresponding rotations in the &am-ends. Bending moments will bedeveloped at the connections, and these will restrain the motion of the column andttiereby increase its strength.

To understand the analpis procedure, the buckling strength of a column in a typicalframe s considered as shown in the Figure 5.1. Buckling would occur on the application


64/118

of an axial force. During the buckling motion, the ends of the columns rotate th rougi^angles Ou Figure5.2).

Ln order to consider the buckling strength of he column IJL, he portion o f the fiame hasbeen isolated in Figure 5.2. The dotted line shows the position of the frame before load isbeing applied to it. At buckling the additional deformation of the column and adjacentmembers are also shown in the Figure. Assurnhg that during the buckling motion, theends of columns rotate through an angle 8~ r 8 ~ ;he bar n ends are forced through thissame amount of rotation. A resisting moment, MU=, ill be created at the beam-to-column connection given by he followingequation:


65/118

where L, is the beam length and 1 is the Moment of Inertia of he beam about its axis ofbending.

Figure 5.2 Buckiingofcolumn in a Framed structure

The other beams at the joint will develop similar moments and mist the rotation of thecolumn ends during buckling. The beams at a paiticular joint will provide mistance to


66/118

columns both above and below that joint. Assuming the mistance to buckling is inproportion to the stifniess, &, of the column considered, the net resisting moment, Mu,acting on the columnwill be given by:

The symbol I: indicates a summation for al1 memben rigidly ~ 0 ~ e C t e do the joint andlying in the plane in which buckling is king considered. is the column moment ofinertia and L, is the length of the column (Kulak, Adams andGilmor, 1985).

5.3 DETERMINATIONOF ROTATIONAL STWFNESS USING STAAD/Pro

The differential equation of equilibriurn is no longer that for a pin ended column, butmust account for the presence of the end moments and shears. The solution to thedifferential equation can be expressed by the following equation given by the structuralstability research council (SSRC, 976).


67/118

where Gc and Gi. are defined in equation 5.4 for joints U and L. respectively. K is theeffective length factor of the colurnn under consideration and is defined as the ratio of theeffective iength to the actual length. Equation 5.4 h a s been plotted as a Nomopph. asshown previously in Figure 2 . 2 . The chord member of the tower. having similarconditions at both ends reduces Equation 5.4 into Equation 5.5:

The va lue of relative stiffness ration. G. was deterrnined analytically by using th estructural analysis software STAADPro. The rnodel of the whole tower was generatedusing the modellinp facility given in STAADIPro. which is an interactive menu-dnvengraphics oriented procedure for creating a rnodel. The model had 39 nodes. (Refer toFigure 5.3). The value of E was taken as 100 000MPa and Poisson's ratio as 0.3.

Figure 5-3 Tower model generated using STAADPro


68/118

The dimensions of the chord members (panel length and diameter) were changed fordifferent tower specimens as they vared for both the fabricating companies.

Figure 5.4 Section of tower taken out for analysis

One joint region of the tower, which included two chord members with four diagonalbracings attached to it. as shown highlighted in Figure 5.1. was taken out for analysis.This was to find out the rotationai stiffness of the chord member for use in theequilibrium Equation 5.5.

1''Figure 5.5 Moment applied on tower joint


69/118

The far ends of the chord mernbers and the diagonal bncings were defined as fixed andpinned respectively. Moment was than applied. as shown in Figure 5.5. The applicationof the moment caused the joint to rotate (Figure 5.6) through an angle01.

Figure 5.6 Rotation of tower joint

The same amount of moment was again applied on the sarne tower joint but without thediagonal bncings attached to it . The application of the moment caused the chord to rotatethrough an angle 0 7 . From equation 5.3, the value of G can now be given as:

Equation 5.6 can also be written as:

Refer to Table 5.1 for the value of G cdcdated this way for different tower specimens.5 4


70/118

Table 5.1 VALUE OF G CALCULATEDUSINCSTAADPro

Specimen Chord Panel Diagonal SfAAD/ProS.No G/ 1 ID 1 Dia 1 Length 1 Dia 1

4 S-2 B) 38.1 762 12.7 4.553 4.62 66.65 S - 2 ( C ) 38.1 762 12.7 4.553 4.62 66.66 S-3(A) 50.8 762 14.3 1.449 1.466 86.07 S-3 B) 50.8 762 14.3 1.449 1A66 86.08 S-4 B) 50.8 762 14.3 1 -449 1-466 86.0

12

10 S-5(A) 69.85 762 15.9 0.410 0.4 13 147.411 S-5 (B) 69.85 762 15.9 0.4 0 0.4 13 147.4

S-l(A)S-1 (B)

131415161718

mm38.138.

2122232425

S-6 (A)S-6 (B)S-6(C)Pl(A)Pl@)P2 (B)

mm762762

P4(A)P6 (A)P6 (B)PS(A)P8@)

69.8569.8569.8538.138.138.1

mm12.712.7

50.857.1557.1538.138.1

76276276271 1.271 1.271 1.2

71 1.271 1.2711.2711.271 1.2

66.666.6

06 MEMBERSradians4.5534.553

15.915.915.915.915.915.9

02MEMBERSradians4.624.62

19-0522.22522.22522.22522.225

0.4 O0.4100.4104.1334.1334.133

1.3310.8290.8293.6903.690

0.4130.4 130.4134.6004.6004 - 6 0

147.4147.4147.48.88.88.8

1.4600.9 30.9134.6004.600

10.39.99.94.14.1


71/118

5.4 CALCULATION OF EFFECTIVELENGTH FACTOR

By substituthg the vaiue of G in Equation 5.5 the effective length factor of the chordmember was calculated. Equation 5.5 was solved such that the sum of al1 the values onthe left-hand side equals to unity. Solver option of theMicrosoft Excel package was usedfor t h i s purpose.

Microsoft Excel Solver uses the Generalized Reduced Gradient. With Solver, an optimalvalue for a formula in one cell, called the target cell on a worksheet, was calculated.Solver worked with a group of cells that were relate. either directly or indirectly, to theformula in the target cell. Solver adjusted the values in the changing cells which werespecified. called th e adjustable cells, to produce the resuit which were specified from thetarget ce11 foxmula. Constraints were applied to restrict the values solver can use in themodel. Refer to Table 5.2 for the values ofK alculated through his method.

5.5 DISCUSSION OF RESULTSThe effective length factors found through this procedure matched with the resuitsobtained through the analysis of the load deflection data and varied fkom 0.93 to 0.99.The rotational stifbess values of chord and diagonal members were calculated fiom thestructural analysis cornputer soffware STAADRro. The relative stifnless factor, G, variedh m 47 to 4. The effective length factors so determinecl were found to be in accordancewith the Nomograph provided in CANKSA S16.1, which is based on the G factor.


72/118

TABLE 5.2 EFFECTIVE LEGTH F ACTORS (STAADIPro)Y

-"i-II-

- "II-4-

--II

-.c. -I I

I II-.W C-.- "

-.-.W .

m.

-

Diagona M.1 of chord Diagonal' 1 g ! 6I Length memberBracin

Specimen Dia. of Dia. ofID 1 chord 1 Diagonal PanelLength

S-3 (A) 1 50.8 1 14.3S-3 (B) 50.8 14.3S-4 (B) 50.8 14.3S-4 (C) 50.8 14.3S-5 (A) 69.9 15.9S-5 (B) 69.9 15.9S-S(C) 69.9 15.9S-6 (A) 69.9 15.9

P i (A) 1 38.1 1 15.9 1

P8 B) 1 38.1 1 22.2 1


73/118

CHAPTER SIXCONCLUSIONSAND RECOMMENDATION

Chord members of twenty-five specimens, actuai tower sections provided by twocompanies, were tested by applying a load at the center. Al1 the tower sections had atriangular cross section and were made up of solid rounds. The joints were welded andnone of th e tower sectionswas galvanized. The data gathered through experiments wasused to find out the effkctive length factors. Effective length factors were also calculatedby using he equilibriumequation.

6.2 CONCLUSIONS

Based on the analyticai and numencai analysis, the following conclusions can be drawn:

1. The effective length factor of the tower sections calculated through the analysis ofthe experimentaldata, was found out to be less than unity.The value ranges fiom0.95 to 0.99. Thus it was determined that the rotational restraint provided bydiagonal bracing to the chord member is not very significant, specially for thecase where a relatively thicker chord member is fabricated wih a relativelythimer diagonal bracing.


74/118

2. The values of the effative length factors obtained by making use of theequilibrium equation given by structural stability research council (SSRC, 976)were found to be close to the values obtained through the analysis of the loaddeflection data. Thus, the equilibrium equation or more conveniently theNomograph (Figure 2.2), c m be used to detennine the efiective length factor ofchordmembem of guyed towers.

3. Good agreementwas observai betweetl the results obtained through commercialsoftwarepackage ABAQUS and the experimental data.

Since the effective length factors were found to be very close to 1O so it is recommendedthat the effective length factor for solid round chord members of guyed towers should betaken as unity as recomrnended by Canadian Standards Association S37-94"ANTENNAS, TOWERS, AND ANTENNA-SUPPORTINGSTRUCTURES".


75/118

REFERENCES

ABAQUS, 1998, Version 5.8, Hibbitt, Karlsson & Sorenson, Inc.RI, USA.

American Institute of Steel Construction, 1993, "Load and Resistant Factor DesignSpecificstion for Structurai Steel Building," American Institute for Steel Construction,Chicago, IL, November.

Amencan Society of Civil Engineers, 1998, "Design of Latticed Steel TransmissionStructures".

Aydm, R. nd Gonen, H., 994, "Lateral Stinnes of Frames and Effcetive Length ofFramed Columns," J o u d of Structural Engineering, ASCE, Vol. 120, No. 5, May, pp.1455-1470.

Bridge, R.Q. and Fraser, D.J., 1987, 'Improve GFactor Method For EvaluatingEffective Lengths of Columns," Journal of Structural Engineering, ASCE, Vol. 1 13,No. 6, June,pp. 1341-1356.

Canadian Standards Association, 1994, "Antennas, Towers, and Antenna-SupportingStructures,"CSA S37-94, Etobicoke,Ontario.

Canadian Standards Association, 1994, " L M t States Design of Steel Structures,"CANICSA-S16.1-M94,Etobicoke, Ontario.


76/118

CEN, 1997, Eurocode 3 Part 3.1: 1997, "Design of steel structures Towers, masts andChimneys - Towers and Masts", CEN, European Cornmittee for Standardisation,Brussels, Belgium.

Chen, W.F., 1987, "Structural Stability Tbeory and Implementation," ElsevierScience Publishing Co., Inc., New York.

Cheong-Siat-Moy, F., 1997, 'K-fpctors fo r braced h m e s , " Engineering Stnictures,No. 9, Elsevier Science Ltd, pp. 760-763.

Duan, L. and Chen, W., 1988, '


77/118

Hajar, J. F. and White, D.W., 1995, uEffktve Length and Equivalent ImperfectionApproaches for Assessing Frime Stabiiity," Restnicniring: America and Beyond;Proceedings of StructuresCongres XIII held in Boston, Massachusetts, April, pp. 1785-1788.

Hellesland, J. and Bjorhovde, R, 1996%YImproved Frime Stability Analysis withEffective Lengths," Journal of Structural Engineering, ASCE, Vol. 122, No. 11,November, pp.1275-1283.

Hellesland, J. and Bjorhovde, R., 1996b, "Ratraint Demand Factoton and EffectiveLengths of Bnce Columns," Journal of Strucurai Engineering, ASCE, Vo1.122,No.10,October, pp. 121601224.

Indian Standard code of Practice, 1.S 802-1978, Part 3 testing, Governent of India, NewDehli, India.

Kishi, N., Chen, W.F., Goto, Y., 1997, "Effective Length Factor of Columns inSemirigid and UnbracedFrimes," oumal of Smcturai Engineering, ASCE, Vo1.123,No.3, pp. 313-320.

Kishi, N., Goto, Y., Komuro, M., 1995, "Effective Length Factor for Columns inBraced and Flexibly Jointed Frames," Engineering Mechanics: Proceedings of the IO&Conference, University of Colorado at Boulder, Boulder, Colorado, May.


78/118

Knight, G.M.S. and Santhakumar, A-R, 1993, "Joint effects on behavior oftransmission towers," Journal ofSmcurai Engineering,ASCE, Vol. 19,No.3,March,pp.668- 12 .

Kulak, Adams and Gilmor, 1985, "Limit States Design in Stmcturai Steel," CanadianInstitute of Steel Constniction.

SSRC, 1976, Stnicniral Stability Research Council 'GiMe to sabiiity criterion formetal structures", 3d Edition, John Wiley & Sons, NewYork,USA.

STAAD/Pro, 1997-1998, Release 3.0, Research Engineers, Research Engineers Corp.Headquarters, CA, USA

Wahba, Y.M.F., 1999, "Static and Dynamic Analyses of Guyed Antenna Towers,"Ph.D. Thesis, University ofWindsor, Windsor, Ontario, Canada.

White, D.W. and Hajjar, J.F., 1997, 'Design of steel trames without consideration ofef'fective length," EngineeringStructures,Vol.19,No. O,pp. 797-8 0.

Wood, R.H., 1974, Effective Lengtb of columns in multi-story buiidings," TheS mctural Engineer, Volume 52, No.7, July.


79/118

APPENDIX-AFIGURES


80/118


81/118

I TheoreticalLord Experimental deflec,ionApplicd defiection (Fixed Ends)

1-Theoretlcal (Fixe Ends)+ xperimentalFigure A.2 Load vs. Deflection data for chord member - Specimen S I B)


82/118

ThcareticalLoad Experimental deflec ionApplied deflections (Fixed Ends)

0.0 0.5 1O 1.5 2.0 2.5 3.0Deflectlon (mm)

1- heoretlcalF i x d Ends) +- xperlmental 1Figure A.3 Load vs. Deflection Data for Chord Member - Speeimen S-2 (A)


83/118


84/118

I TheoreticalLoad ExperimentrlApplied defiedion deiiection(Fired Ends) 6.054 .0P8 3.0cil 2.0

1Oo e o

0.00 0.50 1O0 1O 2.00 2.50 3.00Deflection (mm)

1- h e a r e t k i lFlxed Ends) -O- Esperrnentd 1Figure A.5 Load vs. Deflection data for chord member - Specimen 5-2 (C)


85/118


86/118

TheoreicalLoad Experimental deflectionA ~ ~ l i e denedion (FixedEnds)

0.00 0.20 0.40 0.60 0.80 1O0Dofection(mm)

1-~heoretical (Fixcd Ends)+ xperirnental /

Figure A.7 Load Vs. Deflection data for chord member - Specimen S-3 (B)


87/118


88/118


89/118

O 0.05 0.1 0.15 0.2 0.25Deflection (mm)1 -~heoreticil (Fixcd Ends) -+ xpcrimcntil

Figure A.10 Load vs. Deflection data for chord member - Specimen S-5 (A)


90/118

Load Experimental TheoreticalApplied deflection deflectionIFixedEnds)

O 0.05 0.1 0.15 0.2 0.2sDeflection(mm)

Figure A.1 I Load vs. Deflection data for chord member - Specimen S5 (B)


91/118

Load Experimental TheoreticalApplied deflection deflection(Fixed End81

0.05 0.10 0.15 0.20Deflecion (mm)

1- heoretiralFixcd Ends)+ xperimcntrlFigure A.12 Load vs. Deflection data for chord member - Specimen S-5 (C)


92/118

TheoreticalLoad Experimental deflectioi

0.00 0.05 0.10 0.15 0.20 0.2sDenection (mm)

Figure A.13 Load vs. Deflection data for chord member - Specimen S-6 (A)


93/118

I TheoreticalLoad Experimental deflectionApplied deflection (Fixed Endsl

O 0.05 0.1 0.15 0.2Deflection(mm)

I1 -~heoretical (Fixed Ends)+ xperimcntrl

Figure A.14 Load vs. Deflection data for chord member - Specimen S-6 (B)


94/118


95/118

I TheoreticalLoad Enpetimental deiiectionApplied deflection (Fined Ends)

0.00 0.50 1O0 1.50 2.00Deflcction (mm)

1-~hearcticd (Fired Ends) +- nperimentil

Figure A.16 Load vs. Deflectlondata for chord member - Specimen P I (A)


96/118


97/118

I TheoreticalLoad ExpcrimentalApplied deflection lFixed Ends)

0.00 0.50 1O0 1SO 2.00 2.50Deflection (mm)

1- heoreticiilFixcd Ends)+ xperimeitdFigure A.18 Load vs. Deflection data for chord member - Specimen P2(B)


98/118


99/118


100/118

I TheoreticalLoad Experimental defiectionA ~ ~ l i e denec''' {PiscdEnds)

0.00 0.05 0.10 O. 15 0.20 0.25 0.30Defiec on (mm)1 -theoretical (Fixcd Ends)+-Experimcntal

Figure A.21 Load vs. Deflection data for chord member - Specimen P4 (A)


101/118


102/118

TheoreticalLoad Experimental deflectionAppUed deflection

0.00 0.10 0.20 0.30 0.401-~heoretiral (Fixed Ends)+ xperimental 1

Figure A.23 Load Vs. Deflection data for chord member - Specimen P6 (B)


103/118

TheoreticalLoad Experimental deflection

0.50 1.00 1 50Deflection (mm)

1- hcaretialFired Ends)+ xperimental\Figure A.24 Load vs. Deflection data for chord member - Specimen P8 A)


104/118


105/118

APPENDIX-BABAQUS INPUT FILES


106/118

B. 1 ABAQUS INPUT S - 1 (A) x t*HEADINGER1 TOWERSI Units (mm, N)*RESTART WRITE**Mode1 Definition* **NODE1, o . , o . , o .2 , O . , O . , 914 .43, O . , 791 .718 , 457 .2

7 6 2 . f O . , O . ,5, 762. , O . , 914.4,6, 762. , 791 .718 , 457.2 ,7 , 1 5 2 4 . , O . , 914 .48, 1 5 2 4 . , 791.718, 457.2 ,9 , 1 5 2 4 . , O , , O . ,

2286 . , O . , O . ,2 2 8 6 . , O . , 914.4 ,2286 . , 791 .718 , 457 .2 ,3048 . , O . , O . ,3O48., O . , 914.4 ,3048 . , 791 .718 , 457 .2 ,3810 . , O . , O . ,3810 , , 791 .718 , 457 .2 ,4572 . , O . , O . ,0572 . , O . , 914.4 ,4572 . , 791 .718 , 457 .2 ,381 . , 395 .732 , 228 .6 ,1143 . , 395 .732 , 228 .6 ,1 9 0 5 . , 3 9 5 . 7 32 , 2 2 8 . 6 ,2 6 6 7 . , 3 9 5 . 7 3 2 , 228.6,4191., 395 .732 , 228 .6 ,381 . , 395 .732 , 685 .8 ,1143 , , 395 .732 , 685 .8 ,1905 . , 395 .732 , 685 .8 ,381., O . , 457.2 ,1 1 4 3 . , O . , 457.2,1 9 0 5 . , o . , 457.2,2667 . , O . , 457.2 ,3429 . , O . , 457.2 ,4 1 9 1 . , O . , 457.2,2667. , 39 5 .732, 685.8 ,3429 . , 395 .732 , 685 .8 ,4191., 395 .732 , 685 .8 ,3810 . , O . , 914.4 ,3429 . , 395 .732 , 228.6,2667 . , O . , O . ,


107/118

B. 1 ABAQUS INPUT S 1 (A) x t4, 11, 145, 14, 386, 38, 197 , 3, 68, 6 , 99, 9, 1210, 12, 1511, 15, 1712, 17, 2013, 1, 414, 4, 718, 16, 18103, 13, 40104, 40, 10108, 4, 9*BEAM SECTION, ELSET=Ml 8 MATERIAL=STEEL# SECTION=CIRC19.05


108/118

B.1 ABAQUS INPUT S - 1 (A) t x t

*BEAM SECTION, ELSET=M2, MATERIAL=STEEL, SECTION=CIRC6.35


109/118

B.1 ABAQUS INPUT S-1 (A) . t x t21, 2, 322, 20, 1824, 19, 20*BEAM SECTION, ELSET=M3, MATERIAL=STEEL, SECTION=CIRC9 . 5 2 5

*ELEMENT, TYPE=B31, ELSET=M420,1,2,323,18,19,20*BEAM SECTION, ELSET=M4, MATERIAL=STEEL, SECTION=CIRC9.525

*STEP PERTURBATION6000N CONCENTRATED LOAD*STATIC*BOUNDARY4, 25, 210, 213, 218, 1, 319, 1, 3*CLOAD40, 2, -6000*ELPRINT*NODE PRINT'il*END STEP


110/118

B.2 ABAQUS INPUT S - 3 (A ) . t x t*HEADINGER1 TOWERSI Units (mm, N)*RESTART, WRITE**Mode1 Definition* **NODE1, o . , o . , o .2, O . , O . , 914.43, O . , 791 .718 , 457 .24 , 762. , O . , O . ,5, 762. , O , , 914.4,6 , 762. , 791.718, 457.2,7 , 1524 . , O . , 914.48, 1524 . , 791 .718 , 457 .2 ,9 , 1524 . , O . , O . ,o., o.,

O . , 914.4,791 .718 , 457 .2 ,o . , o . ,O . , 914.4,791 .718 , 457 .2 ,o . , O * ,791 .718 , 457 .2 ,o . , o . ,O . , 914 .4 ,791.718, 457.2,381,, 395.732, 228.6,

1143. , 395.732, 228.6,1905., 395.732, 228.6,2667 . , 395 .732 , 228 .6 ,4191 . , 395 .732 , 228 .6 ,3 8 l . , 395 .732 , 685 .8 ,1143 . , 395 .732 , 685 .8 ,1905 . , 395 .732 , 685 .8 ,381 . , O . , 457.2,

O * , 457 .2 ,o . , 457.2,o . , 457 .2 ,o . , 457.2,o . , 457.2,395 .732 , 685 .8 ,395 .732 , 6 8 5 . 8 ,395 .732 , 685 .8 ,O . , 914.4,395 .732 , 228 .6 ,o . , o . ,


111/118

4, 11, 145, 14, 386, 38, 197 , 3 , 681 6 , 99, 9, 1210, 12, 1511, 15, 1712, 17, 2013, 1, 414, 4, 718, 16, 18103, 13, 40104, 40, 10108, 4 , 9*BEAM SECTION, ELSET=Ml, MATERIAL=STEEL, SECTION=CIRC25 - 4


112/118

B . 2 ABAQUS I N P U T S - 3 (A) . t x t

100, 38, 37101, 3 7 , 20105, 2 , 5106, 5, 7107, 7, 11109, 30, 9110, 31, 9111, 21, 1112, 6, 8113, 8, 12*BEAM SECTION, ELSET=M2, MATERIAL=STEEL, SECTION=CIRC7.15


113/118

B. 2 ABAQUS INPUT S 3 (A) . x t21, 2, 322, 20, 1824, 19, 20*BEAM SECTION, ELSET=M39.525*ELEMENT, TYPE=B31, ELSET=M420,1,2,323,18,19,20*BEAM SECTION, ELSET=M4, MATERIAL=STEEL,9.525

*STEP, PERTURBATION6000N CONCENTRATED LOAD*STATIC*BOUNDARY4, 25, 210, 213, 218, 1, 319, 1, 3*CLOAD40, 2, -6000*ELPRINT*NODE PRINTU*END STEP


114/118

B.3 ABAQUS INPUT S - 5 (A ) . t x t* H W I N GER1 TOWERSI Units (mm, N)*RESTART, WRITE**Mode1 Definition* **NODE1, O,, o. , O.2, O., O., 914.43, O., 791.718, 457.24, 762., O., O.,5, 762., O., 914.4,6, 762., 791.718, 457.2,7, 1524., O., 914.48, 1524., 791.718, 457.2,9, 1524., O., O.,10, 2286., O., O.,11, 2286., O., 914.4,12, 2286., 791.718, 457.2,13, 3048., O., O.,14, 3048., O., 914.4,15, 3048., 791.718, 457.2,16, 3810,, O., O.,17, 3810., 791.718, 457.2,18, 4572., O., O, ,19, 4572., O., 914.4,20, 4572., 791.718, 457.2,21, 381., 395.732, 228.6,22, 1143., 395.732, 228.6,23, 1905., 395.732, 228.6,24, 2667., 395.732, 228.6,25, 4191., 395.732, 228.6,26, 381,, 395.732, 685.8,27, 1143., 395.732, 685.8,28, 1905., 395.732, 685.8,29, 381., O., 457.2,30, 1143., O., 457.2,31, 1905., O., 457.2,32, 2667 . , O., 457.2,33, 3429., O., 457.2,34, 4191., O., 457.2,35, 2667., 395.732, 685.8,36, 3429., 395.732, 685.8,37, 4191., 395.732, 685.8,38, 3810., O., 914.4,39, 3429., 395.732, 228.6,40, 2667., O., O.,*ELEMENT, TYPE=B31, ELSET=Ml1, 2, 32, 5 , 83, 8, 11


115/118

B.3 ABAQUS INPUT S - 5 (A) t x t4, 11, 145, 14, 386, 38, 1971 3, 68, 6 , 99 , 9, 1 210, 12, 1511, 15, 1712, 17, 2013, 1, 414, 4, 718, 16, 18103, 13, 40104, 40, 10108, 4, 9*BEAM SECTION, ELSET=Ml, MATERIAL=STEEL, SECTION=CIRC3 4 . 9 5


116/118

B.3 ABAQUS INPUT S-5 (A) t x t

100, 38, 37101, 37 , 20105, 2, 5106, 5 , 7107, 7, 11109, 30, 9110, 31, 9111, 21, 1112, 6 , 8113, 8, 1 2"BEAM SECTION, ELSET=MS, MATERIAL=STEEL, SECTION=CIRC7.95


117/118

B. 3 ABAQUS INPUT S - 5 (A) x t21, 2, 322, 20, 1824, 19, 20*BEAM SECTION, ELSET=M3, MATERIAL=STEEL, SECTION=CIRC9 . 5 2 5*ELEMENT, TYPE=B31, ELSETzM420,1,2,323,18,19,20*BEAM SECTION, ELSET=M4, MATERIAL=STEEL, SECTION=CIRC9 . 5 2 5

*STEP, PERTURBATION6000N CONCENTRATED LOAD*STATIC*BOUNDARY4, 251 210, 213, 218, 1, 319, 1, 3*CLOAD40, 2, -6000*ELPRINT*NODE PRINTu*END STEP


118/118

VITA AUCTORIS

Adnan Karim Qureshi was born in 1971 in Hala, a city of Province o f Sind, Pakistan. Hegraduated from Public School Hyderabad in 1988. From there he went to MehranUniversity of Engineering & Technology, Jamshoro, Pakistan, where he obtained h i sBachelor's with Honors in Civil Engineering in 1995. Mer his Bachelor's he worked forabout three years in the field o f Civil Engineeringin Pakistan and in Norway.

He is currently a candidate for the Master's degree in Structural Engineering at theUniversity of Windsor and hopes to graduate in Fall 1999.

Analisis Miembros Antena

Documents

Transcript of Analisis Miembros Antena