EFFECTIVE LENGTH FACTORS FOR SOLID ROUND CHORD MEMBERS OF

GUYEDTOWERS

A Thesis Submined to the Coilege of Graduate Studies and Research

through Civil & Environmental Engineering in Partial Fulfillment of the Requirements for

the Degree of Master of Applid Science at the University of Windsor

Windsor, Ontario, Canada

1999

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Chord mernbes of twenty-five dl-welded guyed-latticed communication steel tower

sections were tested in the Structurai Engineering Laboratory of the University of

Windsor to detemine the effective length factors of the chord memben. Two diffaent

manufacturers, viz., Pirod Inc., Plymouth, Indiana, and ERI Inc., Chandler, Indiana,

provided the test specimens. Ali tower sections were fabricated fiom solid round

members and were trimgular in cross section. Tower sections provided by ER1 were 4.57

m (1 5.0 fi.) long with continuous diagonal bracings welded to the chord members, while

those provided by Pirod were 6-09 m (20.0 A.) long with the diagonal bracings cut and

welded to the chord members. The diameters of the chord members varied fiom 38.1 mm

(1.5 in.) to 69.85 mm (2.75 in.) while the diameters of the diagonal bracings varïed fiom

12.7 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 tower

was cut and tested by applying a load at its center while the diagonal bracings remained

attached to the chord members. Deflections were recorded manually while the applied

Ioad was recorded using a data acquisition system. The load-deflection data were used to

calculate the effective length factors for the chord members which were found to be

varying from 0.95 to 0.99. Good agreement was observed between the expenmental

deflections at the center of the chord member and those obtained fiom the cornputer

anal ysis software package ABAQUS.

The stiffness contribution of diagonal bracings to the ends of the chord members was also

computed numerically by structural analysis package STAADmro and the effkctive

length factors were calculated using the equilibrium equation given by the structural

stability research council (SSRC, 1976). The effective length factor dculated by this

rnethod varkd h m 0.93 to 0.99. Based on the experimental and numerical results, it

appears reasonable to assume an effective length factor of 1 .O in actual design practice.

DEDICATED TO MY

LOVING PARENTS DR. ABDUL W I M QURESHI

AND DR. SAKINA QURESHI

First of dl , the author expresses his most sincere gratitude to Aimighty G d without

Whose help nothing would have been possible.

The author wishes to express his deep appreciation and gratitude to his Principal Advisor

Dr. Murty K. S. Maduguk Professor, Department of Civil and Environmental

Engineering, for his constant guidance, c~pe ra t i on and encouragement dunng the entire

process of this research. To him I say, "Thank you very much".

Special hanks are also reserved for the Co-Advisor Dr. Sudip Bhattacharjee, Assistant

Professor, Deparmient of Civil and Enviro~lmental Engineering, for his valuable

suggestions and supe~s ion during the whole process of this work.

The author is also thankful to Pirod Inc. and ER1 Inc. for providing the test specimens

used in this investigation.

Thanks are also reserved for the technicians Richard Clark, Lucian Pop and Patrick

Seguin 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

vii

Information Technology services of the University of Windsor including the services of

the CADICAM laboratory are also greatly appreciated.

The financial support provided by the Department of Civil and Environmental

Engineering, University of Windsor, and the Natural Sciences and Engineering Research

Council of Canada is gratefûlly acknowiedged.

In the end the author also feels very much obliged and indebted to al1 of his family

members who provided continued mord and financial support during this whole period

of study and research at the University of Windsor.

viii

TABLE OF CONTENTS

ABSTRACT

DEDICATION

ACKNOWLEDGEMENTS

LIST OF TABLES

LIST OF FIGURES

NOTATION

iv

vi

vii

xi

xii

XV

CaAPTER ONE - INTRODUCTION

..1 GENERAL 1

. .2 CLASSIFICATION OF COMMUNICATION TOWERS 1

. .3 STRUCTURAL CONFIGURATION 4

. .4 NEED FOR INVESTIGATION 5

. .5 OBJECTIVE OF THE PRESENT RESEARCH 6

..6 OUTLINEOFTHETHESIS 6

CEIAPTER 'IWO - BACKGROUND OF RESEARCH AND REVIEW OF LITERATWRE

2.1 GENERAL 8 2.2 JOINT EFFECT IN TRANSMISSION TOWERS 9 2.3 EFFECTIVE LENGTH FACTOR IN BRACED FRAMES IO 2.4 THE G-FACTOR AND EFFECTIVE LENGTHS OF

COLUMNS 13 2.5 CODES, STANDARDS AND SPECIFICATIONS 16

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

2.5.2 CAN/CSA-S16.1-94 "LIMIT STATES DESIGN OF STEEL STRUCTURES" 16

2.5.3 AISC - LRFD "LOAD AND RESISTANCE FACTOR DESIGN SPECIFICATION FOR STRUCTURAL STEEL BUILDINGS" 17

2.5.4 EUROCODE 3: PART 3.1 : 1997 "DESIGN OF STEEL STRUCTURES TOWERS, MASTS AND CKIMNEYS - TOWERS AND MASTS" 20

CHAPTER THREE - EXPERIMENTAL INVESTIGATION

3.1 GEBERAL 3.2 DETAILS OF SPECIMENS

3.2.1 E N SPECIMENS 3.2.2 PIROD SPECIMENS

3.3 TESTSETUP 3.3.1 SUPPORT ASSEMBLY 3.3.2 LOAD APPLICATION 3.3.3 DATA ACQUISITION SYSTEM 3.3.4 TESTING OF CHORD MEMBER

CHAPTER FOUR - ANALYSIS OF LOAD-DEFLEC'MON DATA

4.1 GENERAL 33 4.2 CALCULATION OF CHORD ROTATION AT THE ENDS 33 4.3 CALCULATION OF EFFECTTVE LENGTH FACTOR 36 4.4 COMPARISON OF DEFLECTIONS WITH ABAQUS

RESULTS 42 4.5 DISCUSSION OF RESULTS 43

CHAPTER FlVE - ANALYTICAL INVESTIGATION 5.1 GENERAL 48 5.2 EQUILIBIUUM EQUATION 48 5.3 DETERMINATION OF ROTATIONAL STIFFNESS USING

STAADPro 5 1 5.4 CALCULATION OF EFFECTTVE LENGW FACTOR 56 5.5 DISCUSSION OF RESULTS 56

CHAPTER SIX - CONCLUSIONS

6.1 GENERAL 6.2 CONCLUSIONS 6.3 RECOMMENDATION

REFERENCES 60

APPENDTX - A FIGURES 64

APPENDIX - B ABAQUS INPUT FILES 90

VITA AUCTORIS 1 03

LIST O F TABLES

TABLE 3.1 Details of tower sections

TABLE 4.1 Rotational stiffiness of chord members

TABLE 4.2 Effective Iength factors (experimental)

TABLE 5.1 Vaiue of G calculated using STAADPro

TABLE 5.2 Effective length factors (STAADPro)

LIST OF FIGURES

Self-supporting tower

Guyed Tower

Efiective Length Factors for six idealued conditions

Nomograph

ERI Tower - Weld anangement

Pirod Tower - Weld arrangement

P i r d Towers lying in the laboratory

Chord member cut and loaded

Test setup

Test setup showing the loading coiumn and the suppon assembly

Data acquisition system

Chord member fixed at both ends

Free body diagram of a coium.

Mode1 generated by ABAQUS

Graphical comparison for deflections of specimen - S- 1 (A)

Graphical comparison for deflections of specimen - S-3 (A)

Graphical comparison for deflections of spefimen - S-5 (A)

Framed structure

Buckling of a colunin in a h e d Structure

Tower mode1 generated using STAAD/Pro

Section of tower talcen out for analysis

xii

Moment applied on tower joint

Rotation of tower joint

Load vs. Deflection &ta for chord member - Specimen S-1 (A)

Load vs. Deflection daia for chord member - Specimen S-1 (B)

Load vs. Deflection &ta for chord member - Specimen S-2 (A)

Load vs. Deflection data for chord member - S p e c b S-2 (B)

Load vs. Deflection &ta for chord member. - Specimen 5-2 (C)

Load vs. Deflection data for chord m e m k - Specimen S-3 (A)

Load vs. Deflection data for chord member - Specimen S-3 (B)

Load vs. Deflection &ta for chord member - Specimen S-4 (B)

Load vs. Deflection data for chord mernber - Spechen S 4 (C)

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

Load vs. Deflection data for chord mernber - Specimen S-5 (B)

Load vs. Deflection data for chord member - Specimen S-5 (C)

Load vs. Deflection data for chod member Specimen S-6 (A)

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

Load vs. Deflection data for chord member - Specimen S-6 (C)

Load vs. Deflection data for chord member - Specimen Pl (A)

Load vs. Deflection data for chord member - Specirnen P l (B)

Load vs. Deflection data for chord rnember - Specimen P2 (B)

Load vs. Deflection data for chord member - Specimen P3 (A)

Load vs. Defiection grapii for chord rnember - Specimen P3 (B)

A.2 1 Load vs. Defîection &ta for cbord member - Specimen P4 (A)

A.22 Load vs. Deflection &ta for chord mernber - Spechen P6 (A)

A.23 Load vs. Deflection &ta for chord member - Specimen P6 (B)

A.24 Load vs. Deflection data for chord member - Specimen P8 (A)

A.25 Load vs. Deflection data for chord member 4pecimen P8 (B)

xiv

NOTATION

Cross-sectional area

Uniformly distributed loaded length

Diameter of cross section

Young's modulus of elasticity

Relative joint stiffiiess ratio, lower

Relative joint stifniess ratio, upper

Moment of inertia

Moment of inertia of column/chord member

Moment of inertia of girderldiagonal bracing

Effective length factor

Panel length of column

Length of girder

Moment at ends of chord member

Fixed end moment

Restraining Moment

Concentrated load applied

intensity of load applied

Difierence in Deflection

Experimental deflection

Calculated Deflection

Angle of Rotation

CIiAPTER ONE

INTRODUCTION

1.1 GENERAL

With the world entering into the 21" century, the demand for an effective and useful

telecommunication system is pwing. This need has resulted in an increase in the

production and developmcnt of telecommunication tools. Some of these

telecommunication tools are mobile phones and pagen. To transmit and receive the

signals properly, antennas are use& which are either parabolic or pole type. These

antennas are usually mounted on steel communication towers.

1.2 CLASSIFICATION OF COMMUNICATION TOWERS

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

types :

(a) Monopoles, which are cantilevered tubes with 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.

Figure 1.1 Self-supporting tower

(Source: Pirod inc.)

Figure 1 -2 Guyed Tower

(Source: Pirod inc.)

Guyed towers require more land area than the self-supporting ones. This is because of the

guys that provide lateral support to the structure. These guys are steel cabla made

especially for this purpose. They are hooked up fimily with guy anchors that are

anchored in the nearby land, thus occupying a huge surmunding area. Because of this

disadvantage, guyed towers are generally not recommended for installation in densely

populated residential areas or such places where availability of adequate space is a

problem. Usually the nurnber of guy levels holding a tower ranges h m three to five

(Wahba, 1999).

1.3 STRUCTURAL CONFIGURATION

A guyed tower consists of a mast usually of a constant ûiangular cross section. These

towers have a latticed structure with al1 the joints welded or bolted. The triangular mast

of a guyed tower has three vertical members (chords). These chord members are welded

or bolted with horizontals and diagonals (bracing) to form a latticed structure. Al1 the

members are usually solid rounds with varying diameters. Tubes and angles are also

sometimes used for these chords and bracings.

A guyed toww is fabricated in sections and is erected by mounting these sections one on

top of the other in the field by a special method used to build ta11 communication

structures with antennas fixed on it. Helicopters are also occasionally used to carry out

this installation operation.

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-latticed

communication steel towers, with reference to finding out the effective length factors, is

on cross braced diagonais. So far, no investigation has k e n carried out on solid round

chord members of these towers. Th, there is a strong need to experimentally determine

the effective length factors of chord members that are primary structural components of a

steel communication tower.

Section 6.2 of the Canadian Standards Association S37-94, "Antennas, Towers and

Antenna-Supportïng Structures," (CS A 1 994), deals with members under compression. 1 t

gives the effective length factor for the leg membm of towers as unity. Most practicing

engineers fiequently use this value. However, there are some designers who feel that the

bracing members ptovide rotational rigidity ?O the chord members and so rake the

effective length factor as 0.8.

Thus, there is a need to cany out the experimental investigation of the chord members

and find out the value of the effective length factor. The research is based on the

hypothesis that the diagonal bracings provide rotational ratraint to the chord member

with which they are attacha thus, resulting in an effective length factor which is less

than unity.

1.5 OBJECTIVE OF THE PRESENT RESEARCH

The objective of tbe research is to detexmine, experimentally and theoreticaily, the

effective length factors for solid round chord members of guyed-latticed steel

communication towers.

1.6 OUTILiNE OF THE THESIS

Chapter Two, BACKGROUND OF THE RESEARCH AND REVIEW OF

L I T E R A m , includes bnef review of available research material on communication

towers, effective length factors of members under compression and the recommendations

given by different codes, standards and specifications in this regard.

Chapter Three, EXPERJMENTAL INVESTIGATION, describes in detail the

experimental investigation canied out on chord members of tower specimens in the

Structural Engineering Laboratory of the University of Windsor. î h e procedure is

explained step-by-step and is accompanied by photographs taken at the t h e of

experirnentatioa.

Chapter Four, ANALYSIS OF LOAD - DEFLECTION DATA, deals with the analysis of

the &ta obtained through experiments. The rotational stifThess at the ends of the chord

members has been calculated h m the loaddeflection data and effective length factor of

the chord member is determined. The defiection data of the chord member observed

experimenrally is also compared with the results obtained from the computer software

ABAQUS (Version 5.8, 1 998).

Chapter Five, NUMERICAL INVESTIGATION, describes the procas for the

determination of the relative joint stifniess ratio lmown as G-factor, by using structural

analysis software STAADPro (1998). This factor is used subsequently to obtain the

effective length factor from the equilibrium equation (Equation of the Nomograpb).

Finally, Chapter Six, CONCLUSIONS, gives the s ~ l ~ l l l l a r y of the work done, and the

conclusions reached.

CIiAPTERTWO

BACKGROUND OF RESEARCH AND REVIEW OF LITERATURE

2.1 GENERAL

Guyed transmission towers are highly indeterminate three-dimensional structures with a

reiatively complex structural composition. These structures are among some of the tallest

in the world. They stand as high as 1 5ûû f e t or 500 rn above ground level. Due to the

peculiar structurai composition of guyed towers, much work has been done in this ana by

many researchers.

One general assumption tbat is usually made in d y s i s of guyed towers is that the

different structural members meet at a single point and îhe joints have a pinned effect.

However, this assumption is not satisfied in usual fabrication. When the joint is not

pinned, there are chances that secondary stresses will arise in the member that may have a

significant influence on the critical load. This discrepancy between design assumption

and 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 not

simple pin connections. The diagonal mernbers of each panel provide some resistance to

a moment. This restraining moment sh ih the location of the points of inflection fiom the

end points to inside, thus reducing the effective length of the member. The amount of

reduction is proportional to the resistance provided by the restraining members.

In triangulated fkame structures (tnisses), the loads are usually applied at the joints. Thus

if the joints are truly pinneà, then the members are axially loaded. Defiections of the

joints and the tniss as a whole are caused by the axial deformation of the members. The

angles between members meeting at a joint also change because of these deformations. If

the members are connected together at the joints by welds or bolts, the angle changes

cause secondary bending stresses. Tbese have Iittle effect on the buckling strength of the

tniss members. Because of the local yielding of extreme fibers of the membm near the

joints, as the truss is loaded to ultimate, the seconciary moments gradually dissipate.

(Galarnbos, 1988).

2.2 JOINT EFFECT LN TRANSMISSION TOWERS

Knight and Santhakumar (1993) studied the joint effects on behavior of transmission

towers. A full-scale quadrant of the lowermost panel of a transmission tower designed as

a pin-jointed truss was tested according to ICP (1978). The behavior of the tower was

observed under normal loading conditions. The lowermost panel was chosen for

experimental observation as the vertical, transverse and longitudinal loads were assumed

to be the maximum on that panel. It was concluded that the failure of the chord members

depends on the axial forces as well as the moments generated because of forces in the

secondary members. Thus, it was concluded that the consideration of moments

introduced by the secondary members is a necessity in order to arrive at a realistic

estimate of the failure laad of the tower and hence the consideration of joint effect is very

important as it may result in a premature failure or inappropriate analysis of the whole

tower.

23 EFFECTIVE LENGTH FACTOR IN BRACED FRAMES

With reference to the design of steel fiames, the effective length concept is extensively

used for finding out the effects of the interaction of other members (beams, columns or

other) on the member under compression in a frame. Much work has been done with

reference to the analysis of columns in partially and Mly restrained fiames.

Kishi, Goto and Komuro (1995) observeci the effective length factor for columns in

braced and flexibly jointed fiames. Their discussion was with reference to AISC-LRFD

specification (AISC 1993) which States that in order to design a partially or fiilly

restrained fiame, the bending moments of members are to be obtained estimating the

second-order effects @-A and Pâ efkcts) 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 detennining

the columns K-factor in braced and flexibly jointed fkames. They showed that alignment

chart can be used to tind the K-factor for a column by modimg the relative sti!Tness

factors.

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 s

comesponding to five different boundary conditions of top and bottom columns. They

concluded that the far end conditions of the columns above and below the column have

significant effect on the effective length factor K of the column king investigated- A

direct use of the alignment chart, without modifications, results in an effective length

factor that can be either tao conservative or even unsafe depending on the boundary

conditions of the columns. They gave a modified improved alignment chart procedure

which included the usual rigid far ends of the top and bottom colurnn as a special case

and also considerd the cases of fixed or hinged far ends of the top and bottom columns.

Fraser (1 983) gave a method for evaluating the effective length factor in braced h e s . It

comprises of an iterative procedure and is helpfbl in the anaiysis of structures in which

the flexural stifiess of the restraining members is significantly reduced by the presence

of axial forces. The whole procedure is much simplified by the use of linearized stability

fùnctions expressed in terms of effective length factors.

Cheong-Siat-Moy (1 997) analyzed the possibility of the design charts used by practicing

design engheers giving unconservative values for effective length factors for braced

M e s because they assume that the lateral restraints are infinite. The basis for this

infinite value is the assumption îhat there is no sway in the member or in other words it is

swa y-prevented. It was concluded that the use of the assumed-sway-preventd K- fac tor in

prac tical braced m e s could sometima be tw unconservative.

Some researchers, taking into consideration the complex and indefinite procedure of

determinhg the K-factor, went as far as pmposing the design of steel fiames without the

consideration of effective length. White and Hajjar ( 1 997) were among those researchers.

Limits have been suggested in their paper for use of K=l. Also AISC-LRFD

specifications 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 key

variables on the error is analyzed. Suggestions are given when the design of steel -es

by AISC LRFD may be based on K=l. The paper ends by cornparhg design strengths

with and without effective length to the result fiom elastic-plastic hinge and rigorous

plastic zone andysis for several 'maximum error' examples. The whole discussion

provides an assessrnent of the accuracy of upper-bound error estimates, and of the

implications of using K=l relative to the theoretical inelastic fiame behavior. The

recommendations and discussions are applicable for any type of steel frame that includes

M e s with fully or partially r e s h e d connections, and unbraced or partially braced

fiames.

Wood (1974) studied the effective length of columns in multi-story buildings. He gave

comprehensive design charts for effstive length of columns with any local degree of end

restraint, both for sway and non-sway conditions.

2.4 THE GFACTOR AND EFFECTIVE LENGTHS OF COLUMNS

Framed members under compression interact with the resaaining beams at member ends.

They also corne in contact with other members that are under compression above and

below the member under consideration. This interaction is due to the c o ~ e c t i n g

horizontal bearns that are comrnonly in contact with both compression members. These

interactions are complex and peculiar in behavior. It is, however, a cornmon practice that

to avoid more complexity in such analysis the compression rnembers are snidied as

isolated, with the end restraints defïned in terms of simple stifiess ratios, known as G-

factors. The issue of the accuracy of the G-factor has been addresseci by many researchers

since it plays an important part in the overall design and assurnptions made for a

s tnicture.

Bridge and Fraser (1987) gave an improved G-factor method for evaluating effeçtive

length of columns with reference to the Nomograph that is widely used by practicing

engineers to find out the effective length factor. It requires the engineer to evaluate G at

each end of the column. The value of G should always be positive. In the case of sway

prevented structures, however, the effective length factors are always less than unity. It is

practically possible that columns may have values of K greater than unity that

corresponds to a value of G that is negative. The paper concludes that the failure to

include the effects of axial load in adjacent members restraining a critical buckling

column could result in an error in determining the elastic-buckling load. Such a

calculatioo can be dealt with if the axial forces in restraining beams and in adjacent

columns are taken care of properly. Sucb a procedure can be achieved by incorporating

the negative G-factors. Hence, an ùnproved G-factor method was given which takes into

account al1 these factors.

Hellesland and Bjorhovde (1 996) discussed the methods for determining effective lengths

of members under compression in a fiame in terms of exact results and general principles

of mechanics of buckling. The conventional G-term (relative joint stiffiiess d o ) has

been shown to be inaccurate and conceptuaily flawed. A novel concept of a restraint

demand factor is introduced. This is done in order to allow for improved development of

vertical interaction, which also includes effective length predictions

Hajjar and White (1995) swnmarized the objectives and contents of ASCE cornmittee

report entitled "Effective Length and equivalent imperfection approaches for assessing

frame stability: Implications for Arnerican Steel Design". The discussion is with

reference to the procedure used commody for colurnn design in the United States that is

based primarily on the Nomograph effective length approach. The report discussed the

procedure for the calculation of effective length within partially restrained (or semi-rigid)

framing. Among other key issues, one was the issue of the validity of the effective length

concept for naming in which the restraining elements, that includes beams and theu

connections, are relatively flexible and show significant non-lindty. For use in

deteminhg effective length, the selection of proper connection stifiess is also

discussed, dong with concepts for cdculation of effective lm& in fiameci structures.

Hellesland and Bjorhovde (1996) also proposed a rnethod which involved pst -

processing of effective lengths fiom isolated column analysis to arrive at improved,

weighted mean values. As such, the methoà is tenned as the "method of means". This

method involves pst-processing of effective lengths fiom isolated columns analyses to

arrive at improved, weighted mean values. The approach is applicable to braced and to a

range of unbraced b e s .

Duan and Chen (1989) analyzed the effective length factor for columns in unbraced

m e s with reference to the specifications of AISC (1986) which makes use of the

alignment 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 column is

not only dependent on the relative bending stiffaess ratio of the jointed columns and

girders, i.e., the G-factors at the ends of its unbraced length in an unbraced conthuous

frame, but is aiso dependent on the boundary conditions of far ends of restraining

columns.

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 of

evaluating the columns in h e s with rigid and semi-rigid connections whereby the

estimation of the effective length factor (K-factor) is necessary. This estimation is done

considering the effects of the nonlinear moment-rotation characteristics of beam-to-

column connections. The paper States that with a proper evaluation of the tangent

connection sti fnms for semi-rigid beam-to-column connections at buckling state and

with the intniduction of the modified relative stifniess factors, the alignment chart in the

present American Institute of Steel construction - Load and Resistance Factor Design

specification can be used to find the corresponding K-factor for columns in semi-rigid

h e s

CODES, STANDARDS AND SPECIFICATIONS

CSA S37-94 uANTIENNAS, TOWERS, AND ANTENNA-SUPPORTING

STRUCTURES"

Section 6.2.1.1 of the Canadian Standards Association S37-94 "Antennas, Towers and

Antenna-Supporthg Structures", 1994, gives the effective length factor for leg mernbers

of towers as unity.

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

The standard States about the effective length of colw~i~ls as:

"The effective length, KL, may be thought of as the actual unbraced length, L, multipliexi

by a factor, K, such tbat the product, KL, is equal to the length of a pin ended column of

equal capacity to the actual member. The effective length factor, K, of a column of finite

unbraced length therefore depends on the conditions of restraint afforded to the column at

its braced location. A variation in K between 0.65 and 2.0 would apply to the majority of

cases likely to be encountered in a c t d structures. Figure 2.1 illustrates six idealized

cases 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 that

al1 the columnç in the portion of the fiamework considered reach their individual critical

load simultaneously. This may be used to determine the effective length factors for in-

plane behavior of compression members of tnissa designad as axially loaded members

even though the joints are rigid. In this case, there should be no in-plane eccentricities

and al1 the members of the ûuss meeting at the joint must not reach their ultimate load

simultaneously. Further, the standard says that if it cannot be shown that al1 members at

the joint do not reach their ultimate load simulbneously, then the effective length factor

of the compression members shall be taken as 1 .O.

2.53 AISC-LRFD "LOAD AND RESISTANCE FACTOR DESIGN

SPECIFICATION FOR STRUCTURAL STEEL BUILDINGS"

Chapter E of the LRFD specification deals with columns and other compression-

prismatic members that are subjected to axial compression througb their centroidal axis.

Section E l deal with the Effective Length Factor - K and States that for structural

members under compression it should not be taken as l e s than unity.

End condition code

Figure 2.1 Effective Length Factors for six idealized conditions

(Source: SSRC, 1976)

Figure 2.2 Nomograph

(Source: SSRC, 1976)

2.5.4 EUROCODE 3 PART 3.1: 1997 uDESIGN OF STEEL STRUCTURES

TOWERS, MASTS AND t'ilIMNEYS -TOWRS AND MASTS,"

Sub-clause 5.6.2.2 of the code refers to single members and reads as:

"The following rules shouid be used for single angles, tubular sections or solid rounds

where used for chord sections. For chords or chords with axial compression braced

symmetrically in two normal planes, or planes 60' apart in the case of triangular

structures, the slendemess should be detennined from the system length between nodes.

"Where bracing is staggered in two normal planes or planes 60' apart in the case of

triangular stnictures, the system length should be taken as the length between nodes on

one face. For angle sections the radius of gyration about the minor axis should be used to

calculate the slendemess".

Clause 5.7 refers to effective slenderness factor K and States that if the chord members

are solid rounds then the value of K should be taken as 1 .O. This value is for both types of

towers having symmetrical or asymmetrical bracings.

Investigations were carried out on fifieen specimens provided by Electroaics Research,

Inc., Chandler, Indiana, USA, and ten specimens provided by P i r d Inc., Plymouth,

Indiana, USA. The specimens were actual tower segments fabricated by these two

companies. The specirnens provided by ERI hc . were 4.57 m (1 5.0 feet) long while those

provided by Pirod were 6.09 m (20.0 feet) Long. Most of the specimens were

representative samples of acnial tower segments but some were specially fabricated for

these investigations.

3.2 DETAILS OF SPECIMEN

3.2.1 ER1 SPECIMENS:

Fifieen specimens were provided by Electronic Research Institute, Inc. Three different

chord 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 the

point of welding with the chord memben. Refer to Figure 3.1.

Figure 3.1 ER1 Tower - Weld arrangement

Figure 3.2 Pirod Tower - Weld arrangement

None of the specimens were galvanized. A11 the chord members were made fiom

standard solid rounds with different diameters. The total length of each tower section

provided by ERI hc. was 4.57 m. Each tower section had six panels. AI1 sections were of

triangular cross section with chord, and diagonal mernbers comected together by

welding.

Ten specimens were provided by Pirod Inc. Three different chord sizes, 57.2, 50.8 and

3 8.1 mm, were provided with four different diagonal bracing sizes, 22.2, 1 9.1, 1 5.9 and

22.2 mm. The diagonal bracings were cut and welded to the chord member as s h o w in

Figure 3.2. The total length of each tower section was 6.09 m. Figure 3.3 shows three

Pirod towers section lying in the laboratory. Each section had eight panels. Refer to Table

3.1 for the details of the tower sections.

3.3 TEST SETUP

Al1 the tower specimens were tested in a similar fashion at the Structural Engineering

Laboratory of the University of Windsor. They were first placed in a horizontal position

resting on two roller supports, 1219 mm long and 152.4 mm in diameter. One panel fkom

the bottom chord was chosen for the test and was loaded by the hydraulic jack that was

attached to the loading frame in the laboratory

TABLE 3.1 DETAILS OF TOWER SECTIONS

Specimen Total No. of S.NO. Supplier 1.D Lenflb Panels

ERi

S-1 (A) , s-1 (BI

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

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

Panel Lengtb

mm

Face Dia. of Dia.of Width chords diagonds

Figure 3.3 Pirod Towers lying in the Laboratory

Figure 3.4 Chord member cut and Ioaded

The chord member was cut beyond its pane1 points/joints on either side of the test section

as shown in Figure 3 -4. The diagonal bracings, however, remained attached :O the chord

member. The cutting was done either with the help of a welding torch or an electric

power saw. The panel, king isolated h m both sides , was then simply supported on

either of its sides on two srnail rollers, 127 mm long and 38.1 mm in diameter. Load was

applied on that panel at the mid-point through a loading block as shown in the test setup

in Figure 3.5

33.1 SUPPORT ASSEMBLY

A support assembly that consisted of several built up steel sections supported the tower at

the ends. These steel sections were assembled together to furnish enough height to be

able to test the specimen ushg the testing fiame available in the laboratory.

Figure 3.5 Test setup

3.3.2 LOAD APPLICATION

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 was

provided to the load ce11 in the shape of a long solid round cylindrical column, 76.2 mm

in diameter and 1220 mm long. This extension aras needed to fumish the ciifference

between the heights of the load ceil attached to the roof barn and the chord member of

the tower specimen. Refer to Figure 3.6.

The load was applied through loading blocks that were specially machined and grwved

to sit properiy on the chord members having varying diameters. The blocks measured

10 1 - 6 ~ 7 6 . 2 ~ 192.08 mm, 120.65~76.2xi92mm and 139.7~76.2~ 192 mm to sit properly on

the chords having diameters 38.1, 50.8 and 69.9 mm respectively. In between the load

application column and the load block, a ball-and-socket joint, as shown in Figure 3.5,

was introâuced in order to take care of any eccentricity of the applied loading. The load

was applied using a mechanical purnp connected to the hydraulic jack. To measwe the

applied load, a data acquisition system was used which gave readings in kilo-newtons.

3 3 3 DATA ACQUISITION SYSTEM

An automatic &ta acquisition system was used to record the load applied by the

hydraulic jack to the chord member. This was a Datascan 7000 series of Data Acquisition

Modules. A Type 732 1 measurement processor was used that provided directly 8 anaiog

Figure 3.6 Test setup showing the loading column and the support assembly

inputs, including channel excitation and full local channel expansion capability . A type

702 1 analog expansion scanner module was used which provided 8 analog input channels

per scanner, with excitation for transducers. In addition to that, a type 7036 digital

expansion scanner module was used which provided 8 digital input and 8 digital output

channel capability. One arnong the 40 channels was used to know the applied load on the

chord 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 start

of the actual test.

3.3.4 TESTING OF CHORD MEMBER

A small load was first applied and released to make sure that the whole assembly is

perfectly seated and well placed. In order to measure the deflection due to the applied

loading, a dia1 indicator was fixed exactly below the point where the load was king

applied 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 undisturbed

readings 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 not

yield, the maximum load applied was kept within the elastic range of the chord member.

The maximum load applied varied for each chord depending upon its diameter. For each

load increment, the dia1 gauge reading and the correspondhg applied load were recorded.

Load deflection data were thus obtained for al1 these specimens. Graphs for al1 twenty-

five specimens are given in Appendix A.

Figure 3.7 Data acquisition system

CEAPTER FOUR

ANALYSIS OF LOAbDEF'LECI'ION DATA

The load deflection data gathered through the experimental procedure were analysed to

determine the effective length factor K. The deflections of the chord member of the tower

were dso compared with the values obtained numerically by using the commercial

software package ABAQUS.

4.2 CALCULATION OF CHORD ROTATION AT TEE ENDS

The deflections, A,, of the chord member of the tower were first calculateci assuming both

ends to 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 of

length '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 uniforrnly

distrïbuted loaded length, E is Young's modulus of elasticity, 1 is the moment of inertia

a . n d y = c / l .

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 is

acted upon by a partial u n i f o d y distributed load. This equation was chosen to take into

account the setup in the laboratory where a rectangular block was placed on the chord at

its center (refer to Figure 3.4). It may be aoted that the parameter 'c' in Equation 4.1 is

taken 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 of

Equation 4.1 . The graphs are ploned b y the side of the experimental deflections and are

given 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. The

clifference occurs because the ends of the beam are not totally restrained in real situation.

For a beam partially restrained at both ends, the siope at each end, 8, is given by:

where, M is the moment at both ends of the chord member. Deflectioq 6, at mid span is

given as:

(4.5)

From Equation 4.4 and 4.5, slope 8, at the ends of the bearn is given by:

where 6 is the differential displacement as calculated in Equation 4.2. The end slope

calculated through Equation 4.6 is given in Table 4.1. Load of 6.0 kN was chosen

arbitrarily for al1 the tower sections provided by both the fabricating companies. At this

load the chord member remained elastic and did not yield.

4 3 CALCULATION OF EFFECTIVE LENGTH FACTOR

Refehg to Figure 4.1, the equation for fixecl-end moment is given as:

The restmiiing moment MR. which is caused by the diagonal bracings at the point of weld

with the chord member is equal to the fïxed end moment less the moment released due to

a rotation 0 of the member. Mathematically, fiom Equations 4.4 and 4.7, MR can be

defined as:

The rotational stiflhess provided by the diagonal bracings to the ends of chord member

cm be caiculated by dividiag the restraining moment with the end slope as given in

Equation 4.9 as:

Rotationai Stifiess - MR 1 9

Equation 4.9 is solved by taking values from Equations 4.6 and 4.8. The results are given

in Table 4.1. The table also gives the value of the restraining moment calculated with

Equation 4.8.

..

S.NO.

1 2 3 4 5 6 7

, 8 9 I O 1 i 12

, 13 14

, 15 16

, 1 7 18

, 19 20

, 21 22

, 2 3 24 25

Specimen ID

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. of c hord

mm 38.1 38.1 38.1 38.1 38.1 50.8 50.8 50.8 50.8 69.9 69.9 69.9 69.9 69.9 69.9 38.1 38.1 38.1 50.8 50.8 50.8 57.2 57.2 38.1 38.1

STIFFNESS OF

Di ffcrence

mm 1.73Et00 1.64Et00 1.94Et00 1.98Et00 1.97Et00 5.7 1 E-O 1 6.01 E-O 1 5.91 E-01 6.2 1 E-01 1.72E-01 1.67E-O1 1.62E-O 1 1.52E-O I 1.52E-01 1.42E-O 1 1.18Et00 1.16E+00 I.SIE+OO 4.378-0 1 4.228-01 7.608-02 2.9 1 E-O1 2.74E-01 8.1 3E-0 1 7.43E-01

TABLE 4.1

Expt. Deflection (for 6 kN)

mm 2.39 2.30 2.60 2.64 2 -63 0.78 0.8 1 0.80 0.83 0.23 0.23 0.22 0.2 1 0.2 1 0.20 1.72 1.70 2 .O5 0.6 1 0.59 0.25 0.40 0.38 1,35 1.28

ROTATIONAL

Fixed beam def. (for 6 kN)

mm 6.62E-01 6,62E-01 6.628-01 6.628-0 1 6.62E-01 2.09E-O1 2.09E-01 2.09E-01 2.09E-O I 5.84E-02 5.84E-O2 5.84E-02 5 .ME-02 5.84E-02 5.84E-02 5.37E-01 5.37E-01 5.37E-0 1 1.70E-0 1 1.70E-0 1 1.70E-01 1 .ME-0 1 1.06E-01 5,37E-01 5.37E-01

Rot. Sti ffness (Momenl/Slopc)

(Eq. 4.9) N-mrnhdian -

8.50E+06 1.19Et07 ,

1,69E+06 5.6 1 Et05 8.40E+05 1.86Et07 9.08Et06 1.2 1 Et07 3.26Et06 1.73Et07 3.63Et07 5.64E+07 1 .O 1 Et08 1 .O I Et08 ,

1 .SIE+O8 2.17Et07 2.30Et07 4.24Et06 3.2 1 E+07 3.99E+07 1.06Et09 2.91 E+07 4.94Et07 5.80Et07

- 6.89Et07

CHORD

End Slope (Eq. 4.6)

Radians 9.07E-03 8.60E-03 1.02E-02 1 .ME-02 1.03E-02 2.998-03 3.1 SE-03 3.1 OE-03 3.26E-03 9.01 E-O4 8.74E-04 8.488-04 7.96E-04 7.968-04 7.43E-04 6.65E-03 6.548-03 8.51E-03 2.468-03 2.37E-03 4.288-04 1.64E-03 1.54E-03 4.57E-03 4,18E-03

MEMBERS

Resiraining Moment (Eq.4.8) N-mm

7.7 1 Et04 1.03Et05 1,72E+04 5.83Et03 8.68Et03 5.57E+04 2.868+04 3.76E+04 1.06Et04 1 .56E+04 3.1 7E+M 4.79Et04 8.02Et04 8.02Et04 1.1 2E+05 1.44Et05 1.5 1 E+O5 3.61Et04 7.898+04 9.48Et04 4.53EtOS 4.76Et04 7.6 1 Et04 2.658+05 2.88Et05

At buckling, the forces that act on a column Mth elasticall y restrained ends, are shown in

the fiee body diagram Figure 4.2.

Figure 4.2 Free body diagram of a column - - - - - - - - - - - -

- - - - -

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

can be written as:

where M is the end moment induced as a result of buckling of the column due to the

applied loading. The solution to Equation 4.10, with consideration of deflection boundary

conditions, 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 through

equation 4.9, where

Euler's equation for elastic buckling of columns is given as:

Equations 4.14 was solved for P, which was substituted subsequently in Equation 4.16 to

find out the effective length factor, K. Young's rnodulus of elasticity was taken as

200 000 MPa while ï, the moment of inertia, and 1, the panel length, varied depending on

the type of tower under consideration- The values of the effective length factor

(Experirnental) so obtained are given in Table 4.2.

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

Specirnen ID

7 1 S-3 (B) 1 50.8 1 14.3 1

9 1 S-4 (C) 1 50.8 1 14.3 1

Dia. of chord

mm

1 1 1 S-5 (B) 1 69.9 1 15.9 1 12 1 S-5 (C) 1 69.9 1 15.9 1

Dia. of Diagonal

mm

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

Panel Length

mm

13

14

M of 1 Chord Member

mm4

S-6(A)

S-6 (B)

16

17

18

19

20

21

22

23

24

25

MofI Diagonal Bracing

mm4

69.9

69.9

K

15.9

15.9

Pl(A)

pl (BI

p2 (BI

p3 (A)

p3 (BI

P4 (A)

P6 (A)

p6 (B)

P8 (A)

p8 (BI

103 000

103 000

103 000

327 O00

327 000

327 O00

525 000

525 000

103 000 1

IO3 000

3 140

3 140

3140

6530

6530

6530

11900

1 1900

11900

1 1900

71 1.2

. 38.1 1 15.9

3

0.96 I

0.96

0.97

0.97

0.97

0.98

0.98

0.98

0.95 I

0.95

38- 1

38.1

50.8

50.8

50.8

57.2

57.2

38.1

38.1

A

15.9

15.9

19.1

19.1

19.1 1

22.2 1

22.2

22.2

22.2

4.4 COMPARISON OF DEFLECTIONS WITH ABAQUS RESULTS

Numerical investigation was carried out with the commerciai cornputer software package

ABAQUS in order to compare theoretical results with the deflections of the chord

members obtained experimentally. A model of a typicai six-panel tower section, ERI

specirnen, was generated with 40 nodes as s h o w in the Figure 4.3. The boundary

conditions were defined in the same manner as the tower was placed in the laboratory for

testing.

The eiement library in ABAQUS contains several types of beam elements. Different

beam elements in ABAQUS use different assumptions. The beam elements chosen for

the model definition of the tower are the Timoshenko (shear flexible) beams. These

bearns allow for transverse shear deformation. They are used since they provide usefil

results for such beams that are made fiom unifom material. ABAQUS assumes that the

transverse shear behaviour of Timoshenko bearns is linear elastic with a fked modulus

and, thus, independent of the response of the beam sections to axial stretch and bending.

Refer to Appendix B for INPUT FlLES FOR ABAQUS.

It may be noted that in the model, Figure 4.3, with reference to the chord member on

which the load is applied the adjacent chord mernbers are not provided. This is to

resemble the situation in the laboratory whereby the chord member was cut beyond the

panel junction. In order to apply a central concentrated load at the middle of the chord

member an extra node was defined at that point.

Figure 4.3 Mode1 generated by ABAQUS

Analysis was camed out keeping the magnitude of the applied load qua1 to 6 kN, for

which the experimental deflections were also availabie. The results are compared with

those obtained experimenîally and are shown in Figures 4.4.4.5 and 4.6. The cornparison

shows the experimental and numencal deflections of tower sections S-1 (A), S-3 (A) and

S-5 (A), having diameters of the chord member 38.1,50.8 and 69.9 respectively.

4.5 DISCUSSION OF RESULTS

As expected, the effective length factors for both types of towers came out to be less than

unity. This proved that the diagonal bracings were providing resistance (rotationai

rigidity) to the chord memben, though, in most cases not very significant. The effective

length factors varied from 0.95 to 0.99. It was also observed that the thicker the chord

member the more closer the effective length factor was to unity. The difference in the

weid arrangement of two towen did not have a significant effect on the resuits. Thus, it

can aiso be said that the diagonals do not provide significant rotational resnaint to the

chord member, though they do provide lateral restraint. The results obtained through

commercial software package ABAQUS gave deflections that were a little less than the

experimental ones. This may be due to the reason that the mode1 generated in the

cornputer was stiffer than the one tested experimentally and had ideal loading and

supporting conditions. The error between the numerical and experimental results was

observed to be less than 5%.

Chord Diagonal Dia.

LOAD Tower ID

DEFLECTION BY APPLIED

DEFLECTION (Average)

ABAQUS

mm mm kN mm mm

COMPARISON OF RESULTS

DEFLECTION BY ABAQUS

EXPERIMENTAL DEFLECTION

(Average)

DEFLECTION (mm)

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

C H A P ' r E R r n

AIYALYTICAL INVESTIGATION

The effective length factors of the chord members of guyed towers were also calculated

analytically 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 structure. Where

continuous construction is used, the beam-to-column connections are rigid. In this

situation, the column-end rotation, that take place during the bending motion of the

column will induce corresponding rotations in the &am-ends. Bending moments will be

developed at the connections, and these will restrain the motion of the column and

ttiereby increase its strength.

To understand the analpis procedure, the buckling strength of a column in a typical

frame is considered as shown in the Figure 5.1. Buckling would occur on the application

of an axial force. During the buckling motion, the ends of the columns rotate th rougi^

angles Ou (Figure 5.2).

Ln order to consider the buckling strength of the column IJL, the portion of the fiame has

been isolated in Figure 5.2. The dotted line shows the position of the frame before load is

being applied to it. At buckling the additional deformation of the column and adjacent

members are also shown in the Figure. Assurnhg that during the buckling motion, the

ends of columns rotate through an angle 8~ or 8 ~ ; the barn ends are forced through this

same amount of rotation. A resisting moment, MU=, will be created at the beam-to-

column connection given by the following equation:

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

bending.

Figure 5.2 Buckiing of column in a Framed structure

The other beams at the joint will develop similar moments and mist the rotation of the

column ends during buckling. The beams at a paiticular joint will provide mistance to

columns both above and below that joint. Assuming the mistance to buckling is in

proportion to the stifniess, ï&, of the column considered, the net resisting moment, Mu,

acting on the column will be given by:

The symbol I: indicates a summation for al1 memben rigidly ~ 0 ~ e C t e d to the joint and

lying in the plane in which buckling is king considered. is the column moment of

inertia and L, is the length of the column (Kulak, Adams and Gilmor, 1985).

5.3 DETERMINATION OF ROTATIONAL STWFNESS USING STAAD/Pro

The differential equation of equilibriurn is no longer that for a pin ended column, but

must account for the presence of the end moments and shears. The solution to the

differential equation can be expressed by the following equation given by the structural

stability research council (SSRC, 1976).

where Gc and Gi. are defined in equation 5.4 for joints U and L. respectively. K is the

effective length factor of the colurnn under consideration and is defined as the ratio of the

effective iength to the actual length. Equation 5.4 has been plotted as a Nomopph. as

shown previously in Figure 2.2. The chord member of the tower. having similar

conditions at both ends reduces Equation 5.4 into Equation 5.5:

The value of relative stiffness ration. G. was deterrnined analytically by using the

structural analysis software STAADPro. The rnodel of the whole tower was generated

using the modellinp facility given in STAADIPro. which is an interactive menu-dnven

graphics oriented procedure for creating a rnodel. The model had 39 nodes. (Refer to

Figure 5.3). The value of E was taken as 100 000 MPa and Poisson's ratio as 0.3.

Figure 5 -3 Tower model generated using STAADPro

The dimensions of the chord members (panel length and diameter) were changed for

different tower specimens as they varïed 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 diagonal

bracings 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 the

equilibrium Equation 5.5.

1''

Figure 5.5 Moment applied on tower joint

The far ends of the chord mernbers and the diagonal bncings were defined as fixed and

pinned respectively. Moment was than applied. as shown in Figure 5.5. The application

of the moment caused the joint to rotate (Figure 5.6) through an angle 01.

Figure 5.6 Rotation of tower joint

The same amount of moment was again applied on the sarne tower joint but without the

diagonal bncings attached to it. The application of the moment caused the chord to rotate

through 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.

54

Table 5.1 VALUE OF G CALCULATED USINC STAADPro

Specimen Chord Panel Diagonal SfAAD/Pro S.No

G / 1 ID 1 Dia 1 Length 1 Dia 1

4 S-2 (B) 38.1 762 12.7 4.553 4.62 1 66.6 5 S-2(C) 38.1 762 12.7 4.553 4.62 1 66.6 6 S-3 (A) 50.8 762 14.3 1.449 1.466 86.0 7 S-3 (B) 50.8 762 14.3 1.449 1 A66 86.0 8 S-4 (B) 50.8 762 14.3 1 -449 1 -466 86.0

1

2

10 S-5 (A) 69.85 762 15.9 0.41 0 0.4 13 147.4

11 S-5 (B) 69.85 762 15.9 0.4 10 0.4 13 147.4

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

13

14

15 16 17 18

mm 38.1 38. t

21

22 23 24 25

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

Pl(A) Pl@) P2 (B)

mm 762 762

P4 (A) P6 (A) P6 (B)

PS(A) P8@)

69.85 69.85

69.85

38.1 38.1 38.1

mm 12.7 12.7

50.8 57.15 57.15 38.1 38.1

762 762

762 71 1.2 71 1.2 71 1.2

71 1.2 71 1.2 71 1.2 711.2 71 1.2

66.6 66.6

06 MEMBERS radians 4.553 4.553

15.9 15.9 15.9 15.9 15.9 15.9

02 MEMBERS radians 4.62 1 4.62 1

19-05 22.225 22.225 22.225 22.225

0.4 1 O 0.410 0.410 4.133 4.133 4.133

1.331 0.829 0.829 3.690 3.690

0.4 13 0.4 13 0.413 4.600 4.600 4-60

147.4 147.4 147.4 8.8 8.8 8.8

1.460 0.9 13 0.9 13 4.600 4.600

10.3

9.9 9.9 4.1 4.1

5.4 CALCULATION OF EFFECTIVE LENGTH FACTOR

By substituthg the vaiue of G in Equation 5.5 the effective length factor of the chord

member was calculated. Equation 5.5 was solved such that the sum of al1 the values on

the left-hand side equals to unity. Solver option of the Microsoft Excel package was used

for t h i s purpose.

Microsoft Excel Solver uses the Generalized Reduced Gradient. With Solver, an optimal

value 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 the

formula in the target cell. Solver adjusted the values in the changing cells which were

specified. called the adjustable cells, to produce the resuit which were specified from the

target ce11 foxmula. Constraints were applied to restrict the values solver can use in the

model. Refer to Table 5.2 for the values of K calculated through this method.

5.5 DISCUSSION OF RESULTS

The effective length factors found through this procedure matched with the resuits

obtained 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 the

structural analysis cornputer soffware STAADRro. The relative stifnless factor, G, varied

h m 147 to 4. The effective length factors so determinecl were found to be in accordance

with the Nomograph provided in CANKSA S 16.1, which is based on the G factor.

TABLE 5.2 EFFECTIVE LEGTH FACTORS (STAADIPro)

Y

- - "i

- II-

-" I i

I - - - 4-

- - -

I I

- I

.c. - I I

I I

I - -. W C

-.

-"

-. -. W .

m.

-

Diagona M.1 of chord Diagonal M ' 1 g ! 6

I Length member Bracin

Specimen Dia. of Dia. of ID 1 chord 1 Diagonal

Panel Length

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

Pi (A) 1 38.1 1 15.9 1

P8 (B) 1 38.1 1 22.2 1

CHAPTER SIX

CONCLUSIONS AND RECOMMENDATION

Chord members of twenty-five specimens, actuai tower sections provided by two

companies, were tested by applying a load at the center. Al1 the tower sections had a

triangular cross section and were made up of solid rounds. The joints were welded and

none of the tower sections was galvanized. The data gathered through experiments was

used to find out the effkctive length factors. Effective length factors were also calculated

by using the equilibrium equation.

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 of

the experimental data, was found out to be less than unity. The value ranges fiom

0.95 to 0.99. Thus it was determined that the rotational restraint provided by

diagonal bracing to the chord member is not very significant, specially for the

case where a relatively thicker chord member is fabricated wiîh a relatively

thimer diagonal bracing.

2. The values of the effative length factors obtained by making use of the

equilibrium equation given by structural stability research council (SSRC, 1976)

were found to be close to the values obtained through the analysis of the load

deflection data. Thus, the equilibrium equation or more conveniently the

Nomograph (Figure 2.2), c m be used to detennine the efiective length factor of

chord membem of guyed towers.

3. Good agreement was observai betweetl the results obtained through commercial

software package ABAQUS and the experimental data.

Since the effective length factors were found to be very close to 1 .O so it is recommended

that the effective length factor for solid round chord members of guyed towers should be

taken as unity as recomrnended by Canadian Standards Association S37-94

"ANTENNAS, TOWERS, AND ANTENNA-SUPPORTING STRUCTURES".

REFERENCES

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

American Institute of Steel Construction, 1993, "Load and Resistant Factor Design

Specificstion for Structurai Steel Building," American Institute for Steel Construction,

Chicago, IL, November.

Amencan Society of Civil Engineers, 1998, "Design of Latticed Steel Transmission

Structures".

Aydm, R. and Gonen, H., 1994, "Lateral Stinnes of Frames and Effcetive Length of

Framed 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, 'Improveà GFactor Method For Evaluating

Effective 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-Supporting

Structures," CSA S37-94, Etobicoke, Ontario.

Canadian Standards Association, 1994, "LMt States Design of Steel Structures,"

CANICSA-S 16.1-M94, Etobicoke, Ontario.

CEN, 1997, Eurocode 3 Part 3.1: 1997, "Design of steel structures Towers, masts and

C himneys - Towers and Masts", CEN, European Cornmittee for Standardisation,

Brussels, Belgium.

Chen, W.F., 1987, "Structural Stability Tbeory and Implementation," Elsevier

Science Publishing Co., Inc., New York.

Cheong-Siat-Moy, F., 1997, 'K-fpctors for braced hmes," Engineering Stnictures,

No. 9, Elsevier Science Ltd, pp. 760-763.

Duan, L. and Chen, W., 1988, '<Effective Length Factor for Columis in Braced

Frames," Journal of Stnicnual Engineering, ASCE, Vo1.114, No.10, October, pp. 2357-

2370.

Duan, L. and Chen, W., 1989, Tffcctive length Factor for Columns in Unbraced

Frames," Journal of Structural Engineering, ASCE, Vol. 115, No. 1 , January, pp. 149-

165.

Fraser, D.J., 1983, uEvaiuation of effective lengtb factors in braced frimes,"

Canadian Joumal of Civil Engineering, 10, pp. 1 8-26.

Galarnbos, T.V., 1998, 'CGuide to Stabüity Design Criteria for Metal Structums,~

Fourth Edition, John Wiley & Sons, Inc.

Hajar, J. F. and White, D.W., 1995, uEffktïve Length and Equivalent Imperfection

Approaches for Assessing Frime Stabiiity," Restnicniring: America and Beyond;

Proceedings of Structures Congres XIII held in Boston, Massachusetts, April, pp. 1785-

1788.

Hellesland, J. and Bjorhovde, R, 1996% YImproved Frime Stability Analysis with

Effective Lengths," Journal of Structural Engineering, ASCE, Vol. 1 22, No. 1 1,

November, pp. 1275-1 283.

Hellesland, J. and Bjorhovde, R., 1996b, "Ratraint Demand Factoton and Effective

Lengths of Bnceà Columns," Journal of Strucîurai Engineering, ASCE, Vo1.122,

No.10, October, pp. 121601224.

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

Dehli, India.

Kishi, N., Chen, W.F., Goto, Y., 1997, "Effective Length Factor of Columns in

Semirigid and Unbraced Frimes," Joumal of Smcturai Engineering, ASCE, Vo1.123,

No.3, pp. 3 13-320.

Kishi, N., Goto, Y., Komuro, M., 1995, "Effective Length Factor for Columns in

Braced and Flexibly Jointed Frames," Engineering Mechanics: Proceedings of the IO&

Conference, University of Colorado at Boulder, Boulder, Colorado, May.

Knight, G.M.S. and Santhakumar, A-R, 1993, "Joint effects on behavior of

transmission towers," Journal of Smcîurai Engineering, ASCE, Vol. 11 9, No.3, March,

pp. 668- 712.

Kulak, Adams and Gilmor, 1985, "Limit States Design in Stmcturai Steel," Canadian

Institute of Steel Constniction.

SSRC, 1976, Stnicniral Stability Research Council 'GiMe to sîabiiity criterion for

metal 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 of Windsor, Windsor, Ontario, Canada.

White, D.W. and Hajjar, J.F., 1997, 'Design of steel trames without consideration of

ef'fective length," Engineering Structures, Vol. 19, No. 1 O, pp. 797-8 10.

Wood, R.H., 1974, 'Effective Lengtb of columns in multi-story buiidings," The

S tmctural Engineer, Volume 5 2, No.7, Jul y.

APPENDIX - A FIGURES

I Theoretical Lord Experimental deflec,ion

Applicd de fiect ion (Fixed Ends)

1 -Theoretlcal (Fixeâ Ends) + Experimental 1

Figure A.2 Load vs. Deflection data for chord member - Specimen S I (B)

Thcaretical Load Experimental

deflec tion Applied deflections

(Fixed Ends)

0.0 0.5 1 .O 1.5 2.0 2.5 3.0

Deflectlon (mm)

1 - ~heoretlcal (Fixd Ends) +- Experlmental 1

Figure A.3 Load vs. Deflection Data for Chord Member - Speeimen S-2 (A)

I Theoretical Load Experimentrl

Applied defiedion deiiection

(Fired Ends) 6.0

5 4 4.0 P 8 3.0 cil

2.0

1 .O

o e o

0.00 0.50 1 .O0 1 .!!O 2.00 2.50 3.00

Deflection (mm)

1 - ~hearetkil (Flxed Ends) -O- Esperîrnentd 1

Figure A.5 Load vs. Deflection data for chord member - Specimen 5-2 (C)

T heoreîical Load Experimental

deflection A~~lied denedion (Fixed Ends)

0.00 0.20 0.40 0.60 0.80 1 .O0

Dofîection (mm)

1-~heoretical (Fixcd Ends) + Experirnental /

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

O 0.05 0. 1 0.15 0.2 0.25

Deflection (mm)

1 -~heoreticil (Fixcd Ends) -+ Expcrimcntil 1

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

Load Experimental Theoretical

Applied deflection deflection IFixed Ends)

O 0.05 0. 1 0.15 0.2 0.2s

Deflect ion (mm)

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

Load Experimental T heoretical

Applied deflection deflection (Fixed End81

0.05 0.10 0.15 0.20

Deflecîion (mm)

1 - ~heoretiral (Fixcd Ends) + Experimcntrl

Figure A.12 Load vs. Deflection data for chord member - Specimen S-5 (C)

Theoretical Load Experimental deflectioi

0.00 0.05 0.10 0.15 0.20 0.2s

Denection (mm)

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

I Theoretical Load Experimental

deflection Applied deflection

(Fixed Endsl

O 0.05 0. 1 0.15 0.2

Deflection (mm) I 1 -~heoretical (Fixed Ends) + Experimcntrl 1

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

I Theoretical Load Enpetimental

deiiection Applied deflection (Fined Ends)

0.00 0.50 1 .O0 1.50 2.00

Deflcction (mm)

1-~hearcticd (Fired Ends) +- Enperimentil 1

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

I Theoretical Load Expcrimental

Applied deflection lFixed Ends)

0.00 0.50 1 .O0 1 .SO 2.00 2.50

Deflection (mm)

1 - ~heoreticiil (Fixcd Ends) + Experimeitd )

Figure A.18 Load vs. Deflection data for chord member - Specimen P2 (B)

I Theoretical Load Experimental

defiection A~~lied denec''' { Piscd Ends)

0.00 0.05 0.10 O. 15 0.20 0.25 0.30

Defiec t ion (mm)

1 -theoretical (Fixcd Ends) +- Experimcntal 1

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

Theoretical Load Ex perimental deflection

AppUed deflection

0.00 0.10 0.20 0.30 0.40

1-~heoretiral (Fixed Ends) + Experimental 1

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

Theoretical Load Experimental

deflection

0.50 1.00 1 -50

Deflection (mm)

1 - ~hcaretiçal (Fired Ends) + Experimental\

Figure A.24 Load vs. Deflection data for chord member - Specimen P8 (A)

APPENDIX - B ABAQUS INPUT FILES

B. 1 ABAQUS INPUT S - 1 (A) . t x t *HEADING E R 1 TOWER SI Units (mm, N) *RESTART , WRITE **Mode1 Definition * * *NODE 1, o., o . , o . 2 , O . , O . , 914 .4 3, 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 .4 8, 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, 4 5 7 . 2 , 3048 . , O . , O . , 3O48., O . , 914 .4 , 3048 . , 791 .718, 4 5 7 . 2 , 3810 . , O . , O . , 3810 , , 791 .718, 4 5 7 . 2 , 4572 . , O . , O . , 0572 . , O . , 914.4, 4 5 7 2 . , 791 .718, 457 .2 , 3 8 1 . , 395 .732, 228 .6 , 1 1 4 3 . , 395.732, 228 .6 , 1 9 0 5 . , 395 .732, 228 .6 , 2 6 6 7 . , 395 .732, 228.6, 4191., 395.732, 228 .6 , 3 8 1 . , 395 .732, 685.8, 1 1 4 3 , , 395 .732, 685.8, 1 9 0 5 . , 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. , 395 .732, 685 .8 , 3 4 2 9 . , 395 .732, 685 .8 , 4191., 395.732, 6 8 5 . 8 , 3810 . , O . , 914.4, 3429 . , 395 .732, 228.6, 2667 . , O. , O. ,

B. 1 ABAQUS INPUT S - 1 (A) . tx t 4, 11, 14 5, 14, 38 6, 38, 19 7 , 3, 6 8, 6 , 9 9, 9, 12 10, 12, 15 11, 15, 17 12, 17, 20 13, 1, 4 14, 4, 7 18, 16, 18 103, 13, 40 104, 40, 10 108, 4, 9 *BEAM SECTION, ELSET=Ml8 MATERIAL=STEEL# SECTION=CIRC 19.05

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

*BEAM SECTION, ELSET=M2, MATERIAL=STEEL, SECTION=CIRC 6.35

B.1 ABAQUS INPUT S-1 (A) . t x t 21, 2, 3 22, 20, 18 24, 19, 20 *BEAM SECTION, ELSET=M3, MATERIAL=STEEL, SECTION=CIRC 9 . 5 2 5

*ELEMENT, TYPE=B31, ELSET=M4 20,1,2,3 23,18,19,20 *BEAM SECTION, ELSET=M4, MATERIAL=STEEL, SECTION=CIRC 9.525

* STEP , PERTURBATION 6000N CONCENTRATED LOAD *STATIC *BOUNDARY 4, 2 5, 2 10, 2 13, 2 18, 1, 3 19, 1, 3 * CLOAD 40, 2, -6000 *ELPRINT *NODE PRINT 'il *END STEP

B.2 ABAQUS INPUT S - 3 (A) . t x t *HEADING E R 1 TOWER SI Units (mm, N) *RESTART, WRITE **Mode1 Definition * * *NODE 1, o., o. , o . 2, O . , O . , 914.4 3, O . , 791.718, 457.2 4 , 762. , O . , O. , 5, 762. , O , , 914.4, 6, 762. , 791.718, 457.2, 7 , 1524. , O. , 914.4 8, 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 . ,

4, 11, 14 5, 14, 38 6, 38, 19 7 , 3, 6 81 6 , 9 9, 9, 12 10, 12, 15 11, 15, 17 12, 17, 20 13, 1, 4 14, 4, 7 18, 16, 18 103, 13, 40 104, 40, 10 108, 4 , 9 *BEAM SECTION, ELSET=Ml, MATERIAL=STEEL, SECTION=CIRC 25 - 4

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

100, 38, 37 101, 3 7 , 20 105, 2 , 5 106, 5, 7 107, 7, 11 109, 30, 9 110, 31, 9 111, 21, 1 112, 6, 8 113, 8, 12 *BEAM SECTION, ELSET=M2, MATERIAL=STEEL, SECTION=CIRC 7.15

B. 2 ABAQUS INPUT S - 3 (A) . t x t 21, 2, 3 22, 20, 18 24, 19, 20 *BEAM SECTION, ELSET=M3 9.525

*ELEMENT, TYPE=B31, ELSET=M4 20,1,2,3 23,18,19,20 *BEAM SECTION, ELSET=M4, MATERIAL=STEEL, 9.525

*STEP, PERTURBATION 6000N CONCENTRATED LOAD *STATIC * BOUNDARY 4, 2 5, 2 10, 2 13, 2 18, 1, 3 19, 1, 3 *CLOAD 40, 2, -6000 *ELPRINT *NODE PRINT U *END STEP

B.3 ABAQUS INPUT S - 5 (A) . t x t * H W I N G ER1 TOWER SI Units (mm, N) *RESTART, WRITE **Mode1 Definition * * *NODE 1, O,, o. , O. 2, O., O., 914.4 3, O., 791.718, 457.2 4, 762., O., O., 5, 762., O., 914.4, 6, 762., 791.718, 457.2, 7, 1524., O., 914.4 8, 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=Ml 1, 2, 3 2, 5 , 8 3, 8, 11

B.3 ABAQUS INPUT S-5 (A) . t x t 4, 11, 14 5, 14, 38 6, 38, 19 71 3, 6 8, 6 , 9 9 , 9, 1 2 10, 12, 15 11, 15, 17 12, 17, 20 13, 1, 4 14, 4, 7 18, 16, 18 103, 13, 40 104, 40, 10 108, 4, 9 *BEAM SECTION, ELSET=Ml, MATERIAL=STEEL, SECTION=CIRC 3 4 . 9 5

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

100, 38, 37 101, 3 7 , 20 105, 2, 5 106, 5 , 7 107, 7, 11 109, 30, 9 110, 31, 9 111, 21, 1 112, 6, 8 113, 8, 1 2 "BEAM SECTION, ELSET=MS, MATERIAL=STEEL, SECTION=CIRC 7.95

B. 3 ABAQUS INPUT S - 5 (A) . t x t 21, 2, 3 22, 20, 18 24, 19, 20 *BEAM SECTION, ELSET=M3, MATERIAL=STEEL, SECTION=CIRC 9 . 5 2 5

*ELEMENT, TYPE=B31, ELSETzM4 20,1,2,3 23,18,19,20 *BEAM SECTION, ELSET=M4, MATERIAL=STEEL, SECTION=CIRC 9 . 5 2 5

*STEP, PERTURBATION 6000N CONCENTRATED LOAD *STATIC * BOUNDARY 4, 2 51 2 10, 2 13, 2 18, 1, 3 19, 1, 3 *CLOAD 40, 2, -6000 *ELPRINT *NODE PRINT u *END STEP

VITA AUCTORIS

Adnan Karim Qureshi was born in 1971 in Hala, a city of Province of Sind, Pakistan. He

graduated from Public School Hyderabad in 1988. From there he went to Mehran

University of Engineering & Technology, Jamshoro, Pakistan, where he obtained h i s

Bachelor's with Honors in Civil Engineering in 1 995. Mer his Bachelor's he worked for

about three years in the field of Civil Engineering in Pakistan and in Norway.

He is currently a candidate for the Master's degree in Structural Engineering at the

University of Windsor and hopes to graduate in Fall 1999.