behaviour of cold formed lipped angles in transmission line towers 2006

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Thin-Walled Structures 44 (2006) 1017–1030

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Behaviour of cold formed lipped angles in transmission line towers

S.J. Mohana,�, S. Rahima Shabeenb, G.M. Samuel Knightc

aStructural Engineering Research Centre, Chennai 600 113, IndiabDepartment of Civil Engineering, Indian Institute of Technology, Chennai 600 036, India

cDepartment of Civil Engineering, Anna University, Chennai 600 025, India

Received 2 March 2006; accepted 26 July 2006

.

l

Abstract

The importance and use of equivalent radius of gyration method is discussed and necessary expressions are derived in this paper. The

limiting values of slenderness ratio for the equivalent radius of gyration with the least radius of gyration are discussed to establish the

buckling behaviour of lipped angles. Finite Element Analysis on the buckling behaviour of the mathematical models of individual lipped

angle members and a full scale X-panel was carried out to compare the values predicted by equivalent radius of gyration. A series of

compression tests were carried out on lipped angle sections and their behaviour is studied in the elastic and in the inelastic ranges of

loading. These tests were broadly classified under two categories; concentrically loaded members and eccentrically loaded members

Experimental investigations on full scale tower panels with conventional patterns of leg and diagonals were also carried out. The results

of the experiments were compared with analytical predictions using torsional flexural buckling equations, Finite Element Analysis and

the equivalent radius of gyration approach.

r 2006 Published by Elsevier Ltd.

Keywords: Lipped angle; Torsional–flexural buckling; Equivalent radius of gyration; Transmission line tower; Tower panel; Leg member; Diagona

bracing member

1. Introduction

The torsional–flexural buckling equations for singlysymmetric cold formed sections are used to predict themember capacity in most of the International Standardslike AISI [1], AS/NZS:4600 [2], BS 5950-Part 5 [3] and IS-801 [4]. However, these codes of practices do notadequately cover the design of transmission line towerssince they are formulated for the use in general buildingconstructions. The American Standard ASCE 10-97 [5]meant for the design of transmission line towers suggests amethod for designing the cold formed sections which isbased on equivalent radius of gyration approach. Althoughcodes of practice on cold formed steel sections areavailable, their accuracy in predicting the actual behaviourhas not been fully established. Studies have been carriedout to quantify this accuracy for members of latticed masts.

ee front matter r 2006 Published by Elsevier Ltd.

ws.2006.07.006

ding author. Tel./fax: +91 044 22641734.

ress: [email protected] (S.J. Mohan).

Latticed mast consists of a continuous leg member bracedon both flanges and assumed concentrically loaded. Thelattices or the bracings are connected on only one flangeand hence they are always assumed as eccentrically loadedmembers. While bracings are connected with leg membersframing eccentricity may also bound to occur. Normalframing eccentricity implies that the centroid of the boltpattern is located between the heel of the angle and thecentre line of the connected leg. When this is not the case,due consideration should be given to the additional stressesinduced in the member. ASCE Standard 10-97 [5] accountfor these considerations in terms of effective length byusing equivalent radius of gyration instead of least radiusof gyration. Though the theory of torsional–flexuralbuckling of mono-symmetric thin walled open sectionsunder concentric loading is well developed, determinationof the torsional–flexural buckling load with normalframing eccentricities is extremely tedious. Equivalentradius of gyration approach is an attempt to understandthe theory of torsional flexural buckling by taking into

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istteales.eddds.desal

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x

y

x0

C.Gx0

y

x

v

i

Shear Centre

Shear Centre

C.G

S.J. Mohan et al. / Thin-Walled Structures 44 (2006) 1017–10301018

account all the basic parameters that contribute to thbuckling. Since literature was very scant on equivalenradius of gyration, its derivation is included as a separasection. Finite Element Analysis on cold formed individumembers and as a component in full scale structure wercarried out to asses the buckling capacities of lipped angleStructure level FEM analysis was used to compare thbuckling mode and load carrying capacity of axially loadeleg members. Experiments were carried out on axially aneccentrically loaded individual compression members analso on full scale panels with normal framing eccentricitieThe load versus axial shortening, deflection and strain fielacross the cross section were compared between thindividual element test and structural test. Buckling loadpredicted from analytical methods, FEM and experimentresults are analysed and concluded.

fsy-issafnssnsyselydalr-dsed-he

isaleofe

)

d

)

)

)

ggerertis

–ryenl–

)

)

u

Fig. 1. Torsional–flexural buckling deformations.

2. Equivalent radius of gyration concept

Theoretical studies on torsional flexural buckling omono-symmetric sections carried out during recent timeare briefly mentioned. Szymczak [6] presented sensitivitanalysis of thin walled members with open monosymmetric or bisymmetric cross sections. The analyswas based upon the classical theory of thin walled beamwith non-deformable cross section. Liu et al. [7] presentedglobal optimization approach suited for optimization ocold formed cross sections. An expert based approach othe knowledge gained in the design optimization procewas considered as a basis for the design optimizatioproblems. The design problem of angle sections whose legare slender, subjected to local buckling was addressed bRasmussen [8]. It was stated that for slender angle sectionthe local buckling mode is identical to the torsional modand traditional design procedure becomes excessiveconservative. Young [9] presented the behaviour of colformed steel lipped equal angle columns. The initial locimperfections, residual stresses and corner material propeties of the cold formed steel angles were measured anreported. Further an extensive parametric study wacarried out using the finite element method to study theffect of cross section geometries on the strength anbehaviour of lipped angle columns. However, the prediction of buckling strength of mono-symmetric sections witnormal framing eccentricities were not addressed in threcent research works.

A typical torsional flexural buckling deformationshown in Fig. 1. When a member buckles in the torsionmode the equivalent torsional radius of gyration may bobtained by equating the torsional buckling stress tflexural buckling stress assuming that the radius ogyration used is the torsional radius of gyration [10].Thfollowing expression is used:

p2E

ðl=rtÞ2¼

1

Ar20GJ þ

p2ECw

l2

� �. (1

The term Ar20 is the polar moment of inertia and denoteby IP, then

r2t ¼l2

p2EIPGJ þ

p2ECw

l2

� �. (2

Assuming G ¼ 0:4E and p2 ¼ 9:87

r2t ¼0:04Jl2 þ Cw

IP

� �, (3

rt ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0:04Jl2 þ Cw

IP

s, (4

where A is the gross area of cross section, Cw the warpinconstant, E the modulus of elasticity, G the shearinmodulus of elasticity, IP the polar moment of inertia, J thSt. Venant torsional constant, i the rotation about sheacentre w.r.t x-axis, l the unbraced length of column, r thleast radius of gyration, r0 the polar radius of gyration,the torsional radius of gyration, u and v the X- and Y-axtranslations.Similarly, when the member buckles in the torsional

flexural mode the torsional–flexural radius of gyration othe equivalent radius of gyration may be obtained bequating the torsional–flexural buckling stress to thflexural buckling stress assuming the radius of gyratioused is the equivalent radius of gyration. The torsionaflexural buckling stress is given by

stf ¼1

2kðsx þ stÞ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðsx þ stÞ

2� 4ksxst

q� �, (5

where

k ¼ 1�x0

r0

� �2

. (6

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0

1

0 150 250 300

Rat

io o

f rtf/

r min

75x75x30x3.15100x100x30x4150x150x50x6

Slenderness ratio (l/rmin)

50 100 200

1.4

1.2

0.8

0.6

0.4

0.2

Fig. 2. Behaviour of equivalent radius of gyration for compact sections.

0

1

0 100 150 200 250 300

75x75x30x1.5100x100x40x3.15150x150x50x4

r tf /

r min

1.4

1.2

0.8

0.6

0.4

0.2

Slenderness ratio (l/rmin)

50

Fig. 3. Behaviour of equivalent radius of gyration for non-compact

sections.

S.J. Mohan et al. / Thin-Walled Structures 44 (2006) 1017–1030 1019

x0 and y0 are the distances from shear centre to centroidalong the x- and y-axis, respectively.

st is the torsional buckling stress, stf the torsional–flex-ural buckling stress, sx the flexural buckling stress.

The torsional and torsional–flexural buckling stressescan then be written in terms of flexural buckling stresses tofind an expression for equivalent radius of gyration. i.e.,

stf ¼p2Er2tf

l2, (7)

st ¼p2Er2t

l2, (8)

sx ¼p2Er2x

l2, (9)

where rtf is the equivalent radius of gyration, rx the radiusof gyration about x-axis, ry radius of gyration about y-axis.

Substituting the above values in the Eq. (5) andcanceling the common factor (p2E/l2) we obtain

r2tf ¼1

2kðr2x þ r2t Þ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðr2x þ r2t Þ

2� 4kr2xr2t

q� �. (10)

Considering the reciprocal term:

1

r2tf¼ 2k

1

ðr2x þ r2t Þ �


2� 4kr2xr2t

q� � , (11)

multiplying and dividing by the complementary terms

1

r2tf¼ 2k

ðr2x þ r2t Þ þ


2� 4kr2xr2t

qðr2x þ r2t Þ

2� ðr2x þ r2t Þ

2� 4kr2xr2t

� 8<:

9=;, (12)

simplifying the above expression

2

r2tf¼

1

r2tþ

1

r2xþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

r4tþ

1

r4xþ

2

r2xr2t�

4k

r2xr2t

s. (13)

Adding and subtracting the term 2=r2xr2t within thesquare root term and simplifying

2

r2tf¼

1

r2tþ

1

r2xþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

r2t�

1

r2x

� �2

þ4ð1� kÞ

r2xr2t

s. (14)

Substituting the value of k from Eq. (6)

2

r2tf¼

1

r2tþ

1

r2xþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

r2t�

1

r2x

� �2

þ 4x0

rxrrr0

� �2s

. (15)

The equivalent radius of gyration (rtf) is a function of (rt)the torsional radius of gyration which is a function ofeffective length of the member. Though the equivalentradius of gyration is valid for all mono-symmetric sectionsits characteristics behaviour relevant to cold formed lippedangle sections alone exclusively studied here. Various sizesof lipped angles were considered under a broad classifica-tion of compact and non-compact sections. The variationsof equivalent radius of gyration on a non-dimensional form

of graphs are shown in Figs. 2 and 3 for compact and non-compact sections, respectively.For compact sections (Fig. 2) the value rtf/rmin ¼ 1 refer

to change of radius of gyration from rtf to rmin. In otherwords when rtf/rmino1.0 the least radius of gyration is rtfand when rtf/rmin41.0 the least radius of gyration is rmin

rtf/rmin41.0 correspond to a slenderness ratio of 160 andmore, hence column will buckle in flexure. Then theequivalent radius of gyration may not necessarily be arepresentative parameter to characterize the buckling oflong columns. However, for intermediate columns the rtfvalue is lower than the least radius of gyration and resultsin lowest buckling value. For non-compact sections thescatter of rtf/rmin is very wide and also rtf is a minimumeven for long columns when slenderness ratio is more than160. In general at these ranges of slenderness ratios, mostof the long columns will buckle in flexure withoutinteraction of torsional–flexural. Hence, the significanceof equivalent radius of gyration for non-compact sectionsmay not be a true representative parameter for buckling.Further for non-compact sections the local plate bucklingcoefficient and distortional buckling plays a significant rolewhich was not accounted in the derivation of equivalentradius of gyration.

tr

e

s

tt-

t

ss

le.sss

eer-s

t-

efe

e

-

.t

es

re

se

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0

50

100

150

200

250

300

350

400

0 4000

Allo

wab

le S

tres

s (M

Pa)

Section - 75x75x30x3.15 mm

Effective Length (mm)

2000 30001000

Yield = 350 Mpa

Fa (ry)Fa (rtf)

Fig. 4. Variation of stress with respect to equivalent radius of gyration.

Table 1

Analytical models

S. no. Details Section Dimension

1 Long column 75� 75� 30� 3.15 2500

2 Stub column 75� 75� 30� 3.15 720

3 Bracing member 75� 75� 30� 3.15 1512

(Unsupported)

4 3D full scale panel

transverse and

vertical loads

75� 75� 30� 3.15 2500 (Leg

member)

5 3D panel for

torsional loads

75� 75� 30� 3.15 2500 (Leg

member)

Cross-overjoint

Fig. 5. Cross-bracing.


The member strength can be worked out using the leasradius of gyration and equivalent radius of gyration. Fothe lipped angle 75� 75� 30� 3.15mm the bucklingvalues for various lengths were worked as per the formulagiven in ASCE Standard 10-97 [5] and plotted as shown inFig. 4. When the least radius of gyration ry is used, thbuckling stress calculated correspond to flexural bucklingstress: whereas when rtf is used, the buckling representtorsional–flexural buckling stress which is lower thanflexural buckling stress. Hence, it was suggested in ASCEManual 52 [11] and ASCE Standard 10-97 [5] thaequivalent radius of gyration should be taken into accounwhen cold formed lipped angles are designed for transmission line towers.

3. Finite element analysis

Finite element analysis on lipped angles were carried ouat the individual member levels as well as a members inthree-dimensional analytical models of full scale panels. Ageneral purpose finite element program NE-NASTRAN[12] was used to analyse the buckling behaviour and also topredict the ultimate capacity of cold formed sections. Thiprogram facilitates to analyse any field problem, and iintegrated with a pre and post processor called FEMAP[13]. FEMAP [13] has the capability to develop the physicamodels for sophisticated analyses of stress, temperaturand dynamic performance of any engineering problemNE-NASTRAN [12] is basically a solver, which receivethe FEMAP [13] codes, processes the analysis, monitorthe analysis and allows to change the settings and optionrelated to application of loads, boundary conditions andany other superimposed restraints in the model. FEMAP[13] can then import the results from the solver and provida wide variety of tools for visualizing and reporting thresults. This program has a special capability to solve lineabuckling analysis, nonlinear buckling analysis and nonlinear transient response analysis. The analytical modelgenerated for studying the buckling behaviour of cold

formed lipped angles are given in Table 1. The simplesformulation of four noded linear quadrilateral isoparametric plate elements was used for discretizing the coldformed sections. This element typically resists in-planmembrane shear and bending. The rotational degrees ofreedom at the connecting nodal points and normal to thelements are active in the element formulation and requiredto be constrained only at fixed end boundaries. Thelement has six degrees of freedom per node.Members which are loaded concentrically are classified

as leg member and further classified as long column andstub column depending on its slenderness ratios. Bracingmembers are eccentrically loaded with uni axial eccentricity. In a latticed towers the cross bracings are subjected tocompression and tension alternatively as shown in Fig. 5The behaviour of the bracing system in a panel is such thawithin each set of cross bracing, the strut is restrained bythe tie member at the cross-over joint (Kemp and Behnek[14]). The behaviour of cross bracing can be idealised ashown in Fig. 6. Flexible bracing system can bend alongwith the tie as a whole as shown in Fig. 6a. However, fobracings which are commonly used in a latticed mast, thstrut will be restrained by the tie and buckling mechanismis as shown in Fig. 6b. In some occasions if plan bracingat cross-over joints and other secondary bracings ar

ARTICLE IN PRESS

(a)

(b)

(c)

Tie

Strut

Member eccentricity

Fig. 6. Behaviour of cross-bracing.

Fig. 7. Displacement contours of members.

Fig. 8. Deflected contours of the 3D panel.


introduced then the bracing system will behave as shown inFig. 6c. For the purposes of this study the commonly usedbracing system (Fig. 6b) was considered. In the FiniteElement model the mid-segment bracing bolt hole transla-tion degrees of freedom perpendicular to the member axiswas arrested and all other degrees of freedoms wereallowed.

A linear elastic buckling analysis was preformed on allthese mathematical models followed by a nonlinearultimate strength analysis to predict the ultimate loadcapacity. The displacement contours for all these threesample members are shown in Fig. 7.

Full scale three-dimensional panel made of cold formedlipped angle of dimension 2.0� 2.0m at base 1.5� 1.5m at

top and 2.5m in height was also analysed. The mid surfacemodelling technique was adopted for generating the modelof the 3-D panel. The panel was analysed for hinged basecondition and hence the bottom most nodes were arrestedfor translation in all the direction letting the rotationaldegrees of freedom as free. The mathematical model wasanalysed for two load cases, one to simulate the normalservice condition of transmission line towers and the otherfor the broken conductor load case. The details of paneldimensions and loadings are explained with relevantsketches in Appendix A. The deflected shapes of the panelsare shown in Fig. 8. Analytically predicted buckling valuesof the individual members and as a member in the panelsare compared in Table 2.

4. Experimental investigation

Concentrically loaded column tests on cold formedmembers have indicated that the accuracy of the codes ofpractices vary with wide margin of difference and also varywith respect to various cross sections and shapes [15]. Itwas stated that when lipped channels subjected toconcentric loads, the effective area causes a shift in thecentroidal axis. Experimental results have indicated thatthe shift predicted by codes are more than the actual shiftleading to conservative buckling stresses [16]. Experimentalvalues on compression members connected by only one legof angle sections were found to be less than the predictionby AISC [17] and ASCE Standard 10-97 [5] for hot rolledsections [18].

4.1. Test on cold formed members

A series of tests was carried out on lipped angles of size75� 75� 30� 3.15mm as compression members in theelastic and in the inelastic ranges. The members werecategorised under two basic types: members concentricallyloaded corresponding to leg members of a latticed mastand members eccentrically loaded simulating a diagonalbracing member of a tower. The test setup consists of aMTS displacement controlled machine with spherical joints

f

s

s

e

te

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e

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ARTICLE IN PRESS

Table 2

Comparison of experimental and torsional–flexural capacities

S.

no.

Specimen

code

l/rmin l/rtf Torsional

flexural capacity

Ptf (kN)

Equivalent radius

of gyration Per

(kN)

FEM analysis

(kN)

Experimental

value Pex (kN)

(Ptf)/

(Pex)

(Per)/

(Pex)

(PFE)/

(Pex)

1 EC1 119.51 152.96 51.83 52.59 43.50 47.33 1.09 1.11 0.92

2 EC2 119.51 152.96 51.83 52.59 43.50 40.28 1.28 1.31 1.08

3 EC3 119.51 152.96 51.83 52.59 43.50 39.52 1.31 1.33 1.10

4 IC1 29.56 62.56 193.56 158.84 171.52 143.00 1.35 1.11 1.20

5 IC2 29.56 62.56 193.56 158.84 171.52 148.00 1.31 1.07 1.16

6 IC3 29.56 62.56 193.56 158.84 171.52 146.00 1.33 1.09 1.17

7 IE1 74.48 122.63 119.55 81.83 86.50 82.23 1.45 0.99 1.05

8 IE2 74.48 122.63 119.55 81.83 86.50 83.55 1.43 0.98 1.04

9 IE3 74.48 122.63 119.55 81.83 86.50 80.89 1.48 1.01 1.06

10 ECP1 119.51 152.96 51.83 52.59 49.25 39.85 1.30 1.32 1.24

11 ECP2 119.51 152.96 51.83 52.59 48.95 36.16 1.43 1.45 1.35

Specimen codes: EC—elastic range concentrically loaded; IC—inelastic range concentrically loaded; IE—inelastic range eccentrically loaded; ECP—elastic

range concentrically loaded panel test.

C.G

90x90x6 Anglewelded tobase plate

200x200x20 Base plate

75x75x30x3.15Lipped angle

16 dia Bolt

Fig. 9. End fixtures.

75

75

30

30

10

10 10

10

15

15

Gau

ge 4

Gau

ge 1

Gau

ge 5

Gauge 6

Gauge 2Gauge 3

Fig. 10. Strain gauge locations.

0

10

20

30

40

50

60

70

80

90

0 5 10 15 20 25 30

Load

(kN

)

Ultimate load of leg = 39.6 kNUltimate Load of Bracing = 82.2 kN

2500

1512

1118

Leg Bracing

Axial shortening (x 10-3mm)

Leg

Bracing

Strain Gauges

Strain Gauges

Fig. 11. Load versus axial shortening of leg and bracing members.


and ball bearing headings at both ends. The end sections othe specimens were connected to the joint by means of aspecially made end fixtures as shown in Fig. 9. For stubcolumns tests the same end fixtures were used.

For eccentrically loaded specimens the same setup waused by connecting only one flange of the specimen withthe welded angle. Necessary accessories were added to thisetup to restrain the connected flange at a discrete locationwhich simulate the X-bracing tension member support. Thspecimens were instrumented with strain gauges at sixlocations across the cross section as shown in Fig. 10 tomeasure the strains across the cross section of the lippedangle during testing. Measurements of strains were taken amid height of the leg member and at the mid length of thlonger unsupported span of the bracing member.

The load versus axial shortening behaviour of the legand bracing member is shown in Fig. 11. For the legmember there is a sudden drop in the load after thultimate load. Whereas for the bracing member, thbehaviour is nonlinear right from the onset of loadingand there is a long horizontal plateau showing larg

ductility before the ultimate load is reached. The restraint provided between the length of bracing membeincreases the load carrying capacity significantly and at th

ARTICLE IN PRESS

0

10

20

30

40

-3000 -2000 -1000 0 1000 2000 3000

Load

(kN

)

εy εy

123

4

56

4 3

1

2

56

Ultimate load39.6 kN

Strain (x10-6)

Fig. 12. Load versus strain behaviour for leg member.

0

10

20

30

40

50

60

70

80

-3000 -2000 -1000 0 1000 3000

Unc

onne

cted

fla

nge

1

234

5

6

1

2

3

456

Strain (X10-6)

2000

Ultimate Load = 82.2 kN

Load

(kN

)

Connected flange

Fig. 13. Load versus strain behaviour for bracing member.

Fig. 14. Buckling of specimens.


same time retains the ultimate load for larger axialdeformations.

The load versus strain behaviour of the leg member isshown in Fig. 12. The behaviour is symmetrical and the lipsare in tension throughout the loading ranges. The heelof the section is always in compression indicating thatthe neutral axis passed through the opposite corners of thelipped angle. A portion on the flange near the lip ofthe angle has a small magnitude of compressive stress till95% of ultimate load beyond which there is a change overto tension. The load versus strain variations of the bracingmember is shown in Fig. 13. Initially strains at all thelocations remained compressive and strains near theunconnected lips changed over to tensile at 60% ofultimate load. The strains in the lip closer to the connectedleg remained compressive till the ultimate load andchanged over to tension at the ultimate load. The strainsat the connected leg remained predominantly compressive.An unsymmetrical behaviour is noticed in the case ofbracing members. It can be seen that the buckling mode ofthe leg member is flexure and that of the bracing member istorsional–flexure. The modes of buckling for leg member,stub column and bracing member are shown in Fig. 14.

4.2. Test on cold formed panels

Studies on the cross bracing made of hot rolled angleswith typical bolted connection have been examined byKemp and Behneke [14] and found that the ASCEStandard 10-97 [5] and European Manuals [19] do notadequately allow for the reduction in strength due to endeccentricities. Hence studies on cold formed compressionmembers at the structure level, were carried out on fullscale tower panels.A typical three-dimensional X-braced panel of dimen-

sion 1.5� 1.5m at the top and 2.0� 2.0m at the bottomwith the height of 2.5m was fabricated and tested tofailure. The size were chosen to simulate an actual panelbelow the bottom cross arm level of a transmission linetower. The panels were designed using lipped angle sectionsof size 75� 75� 30� 3.15mm as leg, diagonal and tophorizontal bracing. Typical schematic views of the panel isshown in Fig. 15. Two panels were tested one normalloading condition and other for broken conductor loadcondition. The details of panel testing is also described inAppendix A. The overall lateral deflections of the top ofthe panel were monitored continuously using a thedolite of1 s accuracy with a scale of 1mm least count. Load sliptests were conducted on both the panels by loading themup to 30% of ultimate load for three cycles. Then the actualtest was conducted with load increments of 5% of theultimate load. Figs. 16(a) and (b) show the load versuslateral deflection behaviour of these panels. For normalload condition the panel responded predominantly bydeveloping resistance in two plane frames in the transversedirection only with less interaction of longitudinal bra-cings. The load versus lateral deflection behaviour waslinear up to 70% of the ultimate load which represents thecombined stiffness of both transverse X-bracing systems.Beyond 70% of the ultimate load the stiffness of X-bracingsystem reduced continuously leading to panel collapse.For broken conductor loads the panel response was

different from that of the normal load condition. After asmall horizontal deflection of the panel the rate of increaseof the panel deflection was less since the compression

.l

f

l

-

-f

,

ARTICLE IN PRESS

1118

2400

20001920100 175

1512

503

15001420

90

130

90

75x7

5x30

x3.1

5 75x75x30x3.15

75x7

5x30

x3.1

5

75x7

5x30

x3.1

5

Transverse

Axis

Longitudinal

Axis

Longitudinal

bracing

Transverse

bracing

Leg

member

Elevation Plan

75x75x30x3.15

Fig. 15. Cold-formed tower panel.

0

10

20

30

40

0 5 10 15 20

Load

(kN

)

0

10

20

30

40

50

0 5 10 15 20

Load

(kN

) Transversedeflection

Deflection (mm) Deflection (mm)

Longitudinal deflection

(a) (b)

Fig. 16. Panel deflection: (a) normal loading condition, and (b) broken conductor condition.


bracing started resisting the load through its axial stiffnesswhich is much higher than the full panel structure stiffnessThis reduced panel deflection persisted till the panecollapse, which was triggered by one of the criticallyloaded leg member. The collapsed panels for normal loadcondition and broken conductor load condition are shownin Figs. 17(a) and (b), respectively.

5. Comparison of ultimate loads

The specimens of cold formed sections used in theseexperiments are all made of press break system. It is ageneral conclusion that the specimens made of press brakemethod, the load carrying capacity is 15% less than that orolled cold formed sections [20]. Also it was stated, thevariation between experiments and classical theories can be

up to 30% if the sections are made of press break processes[15,21,22]. Such wide variations were observed during thisexperimental studies also.Table 2 gives the comparison of experimental values with

other theoretical methods and also finite element analyticavalues of the cold formed angles. It can be seen that theexperimental capacities are always lower than the theoretical capacities calculated based on torsional flexural theory{(Ptf)/(Pex)}. Considering the columns in the inelastic range(IC1, IC2, IC3, IE1, IE2, IE3) the variation between theequivalent radius of gyration method and actual experiment is around 10% {(Per)/(Pex)}. The equivalent radius ogyration method can predict the member capacity of theintermediate columns with an accuracy of 10% and hencemay be an acceptable and rational method. The variationin the elastic range of members (EC1, EC2, EC3, ECP1

ARTICLE IN PRESS

Fig. 17. Panel failure: (a) normal loading condition, and (b) broken

conductor condition.


ECP2) the buckling strength is between 10% and 45%which indicates that the equivalent radius of gyrationmethod may not be considered as rationale for longcolumns. However, while comparing the results of theclassical theories with experiment for the elastic andinelastic ranges of columns the variation is 10–48%.

Comparing the finite element results with experimentalresults except the panel buckling load of leg member othervalues are within the margin of 20% {(PFE)/(Pex)}. Forstub columns the (IC1, IC2, IC3) the FEM estimate thebuckling value 15–20% more than the experimental value.For inelastic and elastic columns (EC1, EC2, EC3, IE1,IE2, IE3) the analytical values are within the variation of10%. However, for panel test the deviations were up to35% since it is difficult to model the imperfections inconnections and initial crookedness of the members of thepanel in the finite element analysis.

6. Conclusion

�
It is more appropriate to use the equivalent radius ofgyration to predict the member capacity for intermedi-ate columns since the equivalent radius of gyration islower than least radius of gyration and also supportedby experimental results.The load versus axial shortening behaviour for the leg � and bracing member tested as an element shows adistinctly different behaviour. There is a long horizontalplateau in the behaviour of the bracing memberindicating the effect of restraint provided. � The load versus strain behaviour of the leg member
showed that, the strain variations are linear upto 90% ofultimate load, irrespective of whether it is in tension orin compression. The behaviour also indicated that theneutral axis passed through the opposite corners of thelipped angles.

�
Comparing the overall deflection of the panel with axialshortening of individual elements it is evident that thestructural stiffness was lower than both the leg and thebracing member for both the load cases. � Due to force redistribution caused by joint rotations the
structural stiffness varied. In the normal load case thepanel lost its stiffness before it reaches the failure load,whereas in broken conductor loading condition, thepanel stiffness increased marginally and then decreasedindicating panel collapse.
� Cold formed sections made of press brake processes may
be unconservative to an extent of 45% compared totorsional flexural buckling theories, when they are usedas structural members in latticed mast with normalframing eccentricities.

Acknowledgements

The authors acknowledge the constant support andencouragement given by Dr. N. Lakshmanan, Director,Structural Engineering Research Centre, Chennai andthank him for his permission to submit this paper forpossible publication.

Appendix A

A.1. Three-dimensional panel tests on cold formed members

The behaviour of cold formed steel compressionmembers at the structure level on a full scale tower panelis studied. A typical X-braced panel of dimension1.5� 1.5m at the top and 2.0� 2.0m at the bottom witha height of 2.5m which is normally used in transmissionline tower was investigated. The full panel is fabricatedwith cold formed steel sections. The panel consists of lippedangle sections of size 75� 75� 30� 3.15mm as leg,diagonal and top horizontal bracings and sections of size50� 50� 20� 3.15mm as top plan diagonal bracings.Schematic views of the panel are shown in Fig. A1. Twopanels were tested for two different loading conditions, oneto simulate the normal service condition of transmissionline towers and the other for the broken conductorcondition. These panels were tested at Tower Testing andResearch Station of Structural Engineering ResearchCentre, (SERC) a National Laboratory under Council ofScientific and Industrial Research (CSIR) Chennai, India.Servo-controlled hydraulic actuators are used for applyingthe loads in the resultant direction at the top nodes of thepanel. The panels were instrumented with dial gauges tomeasure the member deformations as shown in Fig. A2.

A.1.1. Normal load condition

Transmission line towers are generally designed forenvironmental and climatic loads. The self weight of theconductor acts in the direction of gravity. The wind loadson conductor is a horizontal load. Transmission line towers

sddi--raln

es.ds,eoelnsn

lteaisredee.

ARTICLE IN PRESS

75x7

5x30

x3.1

5

75x7

5x30

x3.1

575x75x30x3.15

50x5

0x20

x3.15

50x50x20x3.15

Fig. A1. Schematic views of the panel.

Fig. A2. Deflecto-meters in position.


are normally subjected to these two combinationof loads through out the life span of the structure anhence this combination of load is referred as Normal loacondition. Hence the normal load case is a combnation of vertical load and transverse shear. This condition of load is simulated with loads at the top foucorners of the panel in the horizontal and the verticdirections of equal magnitude at each top node as shown iFig. A3.

A.1.2. Broken conductor load condition

Apart from climatic load conditions transmission lintowers may also be subjected to accidental load casePower conductors may some time snap due to ageing anfatigue effect of components like shackles, insulatorstrainer plates or pins. An unbalanced load sets in thtransmission line system due to the intact conductor next tbroken conductor. This unbalanced load is referred as thbroken conductor load which acts in the longitudinadirection of the tower and induces torsional effects itower. To generate the torsional effect two horizontal loadin orthogonal direction were applied on the panel as showin Fig. A4.

A.1.3. Load application system

Fig. A5 shows the schematic diagram of the controsystem arrangement for the application of the resultanload at the required angle at a typical load point on thpanel. The rams are operated through servo valve withvariable voltage from the individual control channel. This achieved by setting the required test load potentiometeto the full load for the particular test. The load to bapplied to a structure under test and the actual loarealized by the load cell at the point of load application arboth fed as equivalent electrical voltages separately to thtwo inputs of the comparator circuit of the control system

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Gravityload

Windload

TransverseAxis

Longitudinal

Axis

Longitudinalbracing

Transversebracing

Legmember

Fig. A3. Normal load condition and plan view of the panel.

Brokenconductorload

Windload

Fig. A4. Broken conductor loads.


The circuit compares these two inputs and producesa differential output signal under the following threeconditions:

�
Demand load4actual load-inward movement of Ram � Demand loadoactual load-outward movement of
Ram
� Demand load ¼ actual load-stationary position of
Ram

The control channels shown in Fig. A6 incorporate aswitching arrangement, which allows the load transducersto be used at 100%, 50%, 25%, and 12.5% of their ratedvalue for a full scale reading of the recorder.

The double acting hydraulic rams are regulated by aremote control from the control room loading channels.This control is automatic by means of a closed loop servosystem, which adjusts the resultant transverse, and alongitudinal component simultaneously at all load points

and ensures that the loads are proportional from zero tomaximum. Limit switches help to see that the load appliednever crosses the ultimate value. The magnitude of loadand angle of application can be seen on the controlchannel.The transverse and vertical load components are

combined into a resultant force, the longitudinal compo-nent being applied in the horizontal plane. The rope, whichtransmits the resultant load, is attached to the tower endwith a load cell to measure the load and an angle sensor,which is attached along the axis of the load sensor tomonitor the angle of load application. The resultant loadand the angle of application are controlled by the hydraulicram at transverse ram station and the vertical ram at thelower level. The resultant load is applied at the four topnodal points. The panel has been loaded in steps within thelinear range and for many cycles of loading. Strainvariations and lateral deflections in the members wererecorded. The panel was loaded further in the inelasticrange and upto failure to obtain the ultimate load carryingcapacity of the panel. The test set up for normal conditionis shown in Fig. A7.The broken conductor load conditions are normally

simulated during the testing of full scale transmission linetowers. In the broken conductor loading condition, twohorizontal loads are applied at the top of the panel inmutually orthogonal directions equal in magnitude. Unlikethe normal load condition where panel deflects only intransverse direction, the panel deflects in both transverseand longitudinal directions under broken wire condition.Two panel deflection measuring stations were establishedalong the axes of the panel and the deflections aremeasured using theodolites. All other arrangements aresimilar to the normal load condition. The test setup forbroken conductor condition is as shown in Fig. A8.

A.2. Material properties

The material properties of the specimens were deter-mined according to the procedure of the standard tension

gde

ntl

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ConductorLoad

WindLoad

(VE) LOAD FEED BACK

HYDRAULIC RAM AT HILL TOP

SERVO VALVE

SERVO VALVE

TYPICAL INDIVIDUALCONTROLPANEL

HYDRAULICPUMP

HVVHANGLE SELECTION

+V (IN)

+V (OUT)

MANUALSERVO

SERVO AMP

SERVO MANUAL

-V (OUT)

+V (IN)

DRIVE CURRENT

SERVO AMP

V1

V1-VE

VE

LOADCONDITIONING

UNIT

ANGLE CONDITIONING

UNIT

ANGLE DEVIATION

METER

RECORDER

LOAD RPTCENTRALDEMAND

CENTRAL CONTROL PANEL

ANGLE RPT

1 2 3 4

DEMAND LOAD

LOCALCENTRAL

REQUIREDLOAD

Wire rope

Wire rape

LOAD CELL

ANGLE SENSOR

HY

DR

AU

LIC

RA

M A

T G

RO

UN

D L

EV

EL

AN

GL

E F

EE

D B

AC

K

Fig. A5. Schematic diagram of the control system for loading.


tests recommended by the American Society for Testinand Materials (ASTM 2003). Three coupons were testefrom two batch of specimens received. The coupons wer

cut from the centre of the flange of the finished sections ithe longitudinal direction and machined to the exacdimensions as stipulated in ASTM Standards. The typica

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Fig. A7. Test setup for normal load condition.

Gauge length=50

50 5012.5 12.575

200

2012.5

Fig. A9. Dimensions of the tension test coupon.

Fig. A6. Control channels.

Fig. A8. Test setup for broken conductor load condition.

0

100

200

300

400

500

0 5 10 15 20 25 30

Stre

ss (

N/m

m2 )

Yield stress Fy = 350 N/mm2

Ultimate stress Fult = 459 N/mm2

Elongation = 19%Young's Modulus, E = 2.198x105 N/mm2

Strain (x 10-3)

Fig. A10. Stress versus strain behaviour of coupon.


dimensions of the coupon is as shown in Fig. A9. Allphysical dimensions of the coupons were measured atsalient locations and the gauge length was marked andpunched along the centre line. The coupons were tested intension testing machine. The reactions and the loadapplications were achieved using friction grip wedges.The stress–strain behaviour of coupon were obtained fromthe in-built facilities of the machine as shown in Fig. A10.

The mechanical properties of the specimens were sum-marised and the values of Young’s Modulus, yield stress,ultimate stress and the percentage elongation are shown inthe inset.

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l

n

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t

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d

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